FUNCTI NAL IFF E TIAL E UA I S · partial differential equations, nonlinear problems of mathematical physics, geo metric and algebraic methods in theory of differential equations

FUNCTI NAL IFF E TIAL E UA I S

VOLUME 8, 2001 No. 1-2

THE RESEARCH AUTHORITY THE COLLEGE OF JUDEA & SAMARIA

ARIEL, ISRAEL

The College of Judea and Samaria The Research Authority Executive Director: Dr. Y. Eshel

© All Rights Reserved 2001 Printed in Israel ISSN 0793-1786

PREFACE

The International Conference on Differential and Functional Differential Equations has been held in Moscow, Russia on August 16- 21, 1999. The conference was organized by Moscow State Aviation Institute, the Steklov Institute of Mathematics of the Russian Academy of Sciences, and Moscow Mathematical Society.

The present volume (vol. 8, NO 1 - 4, 2001) is based on the talks presented at the conference. It includes 35 papers of participants from 10 countries, which were delivered as 45-minute invited lectures and 30-minute lectures.

The conference objective was to discuss classical and new tendencies of development and interconnection of different fields of differential equations. The main topics ranged over differential equations with meromorphic coefficients, nonlinear partial differential equations, nonlinear problems of mathematical physics, geometric and algebraic methods in theory of differential equations. In particular, the proceedings include a generalization of the famous Riemann-Hilbert problem, results on the strong solvability of the Navier-Stokes equations, analysis of attractors for the nonlinear Shriidinger equations, etc. Along with the above-mentioned topics this volume also contains nonclassical problems for differential equations: nonlocal boundary value problems and functional differential equations with partial derivatives. These new fields have been intensively developing during the two last decades of this century and have important applications to elasticity theory, plasma theory, theory of Feller semigroups arising in biophysics, and to nonlinear optics.

We hope that this volume will be interesting for a wide spectrum of specialists and graduate students in differential equations, functional differential equations, and mathematical physics.

The editors are greatful to Professor Alexander Scubachevskii for the great help provided by him in preparing the volume for publication.

M. Drakhlin,

E. Litsyn.

TABLE OF CONTENTS

H. Amann. Remarks on the Strong Solvability of the Navier-Stokes Equations. 3

W. Balser. Summability of Formal Power Series of Ordinary and Partial Differential Equations. 11

L. A. Beklaryan. About Canonical Types of Differential Equations with Deviating Argument. 25

Y. Beland, L. Veron and B. Helffer. Long Time Vanishing Property of Solutions of Sublinear Parabolic Equations and Semi-Classical Limit of Schrodinger Operators. 35

M. F. Bidaut-Veron and L. Vivier. Some Semilinear Elliptic Equations with Singularities at the Boundary. 49

A. N. Bogolubov, A. N. Delitsyn and A. G. Sveschikov. On the Existence of the Solution of Maxwell Equations in a Waveguide. 57

A. Bruno. An application of Power Geometry to Finding Self-Similar Solutions. 69

V. P. Burskii. On Well-Posedness of Boundary Value Problems for Some Class of General PDEs in a Generalized Setting. 89

G. Chakvetadze. Some Geometric Aspects of Drilling Modelling. 101

G. A. Chechkin and E. I. Doronina. On the Asymptotics of the Spectrum of a Boundary Value Problem with Nonperiodic Rapidly Alternating Boundary Conditions. 111

V. V. Chepyzhov and M. I. Vishik. Averaging of Thrajectory Attractors of Evolution Equations with Rapidly Oscillating Coefficients. 123

1

A. N. Danilin, E. B. Kuznetsov and V. I. Shalashilin. The Best Parametrization and Numerical Solution of the Cauchy Problem for a System of Ordinary Differential Equations of the Second Order. 141

R. Denk and L. Volevich. Newton's Polygon in the Theory of Singular Perturbations of Boundary Value Problems. 147

Yu. V. Egorov and V. A. Kondratiev. On the Asymptotic Behaviour of Solutions to a Semilinear Elliptic Boundary Problem. 163

A. V. Faminskii. On an Initial Boundary Value Problem in a Bounded Domain for the Generalized Korteweg - De Vries Equation. 183

A. Fasano and V. Solonnikov. On One-Dimensional Unsaturated Flow in a Porous Medium with Hydrophile Grains. 195

E. I. Galakhov. Multidimentional Process with Nonlocal Conditions. 225

FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 8 2001, NO 1-2 PP. 3 - 9

REMARKS ON THE STRONG SOLVABILITY OF THE NAVIER-STOKES EQUATIONS

H. AMANN'

Introduction. Throughout this note m ;::: 3 and either n = JR.m, or n is a half-space of JR_m, or n is a smooth domain in JR_m with a compact boundary an. We consider the Navier-Stokes equations

"'il . v =0 inn,

(1) a1v + (v · "'il)v- v~v = - "'ilp inn, v =0 on an,

v(·, 0) = vo inn.

Of course, there is no boundary condition if n = JR.m.

In a recent paper [1] we have investigated the strong solvability of (1) for initial data v0 belonging to certain spaces of distributions (modulo gradients). In this note we explain some of our main results in a very particular and simple setting. As usual, we concentrate on the velocity field v since the pressure field p is determined up to a constant by v.

Function Spaces. We suppose that 1 < q < oo and denote by H; the Sobolev spaces H;(n, JR.m) for k E N. We write Lq,<T for the closure in Lq = Hg of the spaceD" of all smooth solenoidal vector fields with compact supports inn. Moreover,

"'il·u=O}, "'il·u=O, ulan=O},

• Institut fiir Mathematik, Universitat Zurich, Winterthurerstr. 190, CH-8057 Ziirich, Switzerland

3

.,

4 H. AMANN

We write Bg,r for the Besov spaces Bg,r(D, JRm ), s E JR, 1 < r :':: oo, and refer to [13] for precise definitions. If q > m, then we put

and

B l-mjq ,_ Bl-mjq L q',l,u .- q',I n q',u

B-l+mjq := (Bl,-mfq)' q,oo,u q ,l,u

where q' := qf(q- 1) and the dual space is determined by means of the Lq,o- duality pairing

(u,v) >--+ (u,v) := j u·vdx, n

(u, v) E Lq',u x Lq,u.

Finally, we set

n -l+mjq ·= { q,a . Lq,u I fL . B-Hm/q

C OSUre 0 q,a Ill q,oo,O'

if q = m, ifq > m.

If q > m and [l = JRm, then . I I f""' . B-Hmfq or eqmva ent y, o v"' m q,oo .

n"i,~+mfq is simply the closure of Lq,u,

If 80 i' (/), then the situation is more complicated. To explain it, we put

Then it is shown in [1] that n"i,~+mfq is isometrically isomorphic to the clo-f L /L · B-l+mjq/ (Bl-mjq)J c h (Bl-m/q)j_ · h sure 0 q q,n lll q,oo - q' ,l,u - tOr q > m, W ere q' ,l,u IS t e

annihilator of the closed subspace B~,~1~jq of B~,7."1q. Thus, loosely speaking,

v belongs to n"i,~+mfq for q > m iff v is a distribution in Bq,J:;mfq modulo gradients of functions in Lq,loc(D, JR).

In [1] it is also shown that

(2) L ~ n-l+mfr ~ n-l+mfs q,u r,a s,u ' m<r<s<oo,

d where Y denotes "continuous and dense injection".

ON THE STRONG SOLVABILITY OF THE NAVIER-STOKES EQUATIONS 5

Very Weak, Mild, and Strong Solutions. The solvability of (1) has been investigated by many authors under various hypotheses on v0 and using several seemingly distinct concepts of weak solutions. (We refer to [1 J

for extensive discussions and references.) The following theorem shows that all these concepts coincide.

Suppose that 0 < T ::; oo and q 2: m. By a very weak q-solution of (1) on [0, T) we mean a function

(3) v E C([O, T), Lq,a)

satisfying

for all

T

j {((Bt + v.6.)w, v) + (Vw, v 0 v)} dt = (w(O), v0)

0

wE £1((0, T), H;.,o,a) n W{((O, T), Lq•,a)

vanishing near T.

Denoting by P: Lq --+ Lq,a the Helmholtz projector, we recall that the Stokes operator S := Sq in Lq,a is defined by S := -vP .6. I H;,o,a· If (3) is satisfied, then vis said to be a mild solution in Lq,a of (0,1) on [0, T) if

in Lq,a·

t

v(t) = e-18v0 - j e-(t-T)S P(v 0 v)(r) dr,

0

o::;t<T,

Finally, by a strong q-solution of (0.1) on [0, T) we mean a function

v E C([O, T), n;,;+mfq) n C((O, T), H;,o,a) n C1 ((0, T), Lq,a)

satisfying v ( 0) = v0 and

(4) v+Sv=-P(v·V)v,

Note that (2) implies

C([O, T), Lq,a) ~> C([O, T), n;,~+mfq) .

6 H. AMANN

THEOREM 1. Suppose that q 2 m and v0 E Lq,u· Then the following are equivalent:

(i) v is a very weak q-solution on [0, T);

(ii) v is a mild solution in Lq,u on [0, T);

(iii) vis a strong q-solution in C([O, T), Lq,u)·

It should be remarked that the only related result known so far is due to Fabes, Jones, and Riviere [4]. These authors essentially proved the equivalence of (i) and (ii) in the case where n = JR.m.

Uniqueness. Our next theorem guarantees uniqueness of very weak solutions.

THEOREM 2. If q 2m and v0 E Lq,u then, (1) possesses at most one very weak q-solution on [0, T).

If n = JR3 , then it has recently been shown by Monniaux [10] that there exists at most one mild m-solution. If 80 oF 0, then Lions and Masmoudi [9] have sketched a different proof for uniqueness of mild m-solutions. Our proof in [1 J is rather simple, relying on maximal Lq-regularity if q = m.

Existenc.e. Now we turn to existence and present the following general result.

THEOREM 3. Suppose that q 2m and v0 E n;;,~+mfq.

(i) The Navier-Stokes equations possess a unique maximal strong q-solution satisfying

if q > m;

(ii) If v0 E Lq,u, then

where tt is the maximal existence time;

(iii) For each T > 0, there exists R > 0 such that tt > T whenever v0 satisfies

(iv) v E C(O x (0, tt), JR.m ).


This theorem extends and simplifies corresponding existence results due to Kato [6], Giga and Miyakawa [5], Kobayashi and Muramatu [7], and others (see [1] for extensive references and the relation of our work to previous results). In particular, it should be noted that vq is the unique strong q-solution in C([O, tj), Lq,u) if v0 E Lq,u, as follows from Theorems 1 and 2.

Theorem 3 and (2) imply that, given r > q, problem (1) has a unique maximal strong r-solution Vr on the maximal interval of existence [0, tt). In [1] it is shown that t:;: 2: tj and Vr :J Vq. Denote by n;;;,~u the inductive

limit of the spaces n;,;+m/r, r 2: m, and set t+ := sup{ t:;: ; r 2: q }. Then it follows that there is a unique function

v: [0, t+) -t n-;;,~u

such that vI [0, tt) = Vr for q ::::; r < oo. This function is the unique maximal strong solution of ( 1).

Clearly, v satisfies (4) on (0, t+) and v(t) -t v0 in nq,~+mfq as t -t 0. Moreover, vis smooth on n x (0, t+).

Global Existence. Although Theorem 3 guarantees the existence of a unique maximal strong solution on an arbitrarily large interval for small initial data, it does not imply that v is a global solution, that is, t+ = oo. This fact can be established, however, provided n is bounded and v0 is small in n;;;,~u· Here and below ll·llq is the norm in Lq.

THEOREM 4. Suppose that [! is bounded, q 2: m, and v0 E nq,~+m/q. Given r 2: q, there exists R > 0 such that t+ = oo, provided

(5)

Furthermore, there exists w > 0 such that

llv(t)llr :S: ce-wt , t 2: 0 .

Suppose that v 0 E Lr,u· Then from the fact that n;,;_+m/r is isometrically isomorphic to a closed subspace of B;,);;·mfr /(B;,-1mj")j_ it follows that condition (5) is satisfied, provided ' '

This shows that Theorem 4 is the analogue for bounded domains of a recent result which has been proven by Cannone [2] if n = JR3 and by Cannone,

8 H. AMANN

Planchon, and Schonbek [3] if n is a half-space of R.3 . The proofs of these authors rely heavily on the fact that there are rather explicit representations of the Stokes semigroup as well as of the Helmholtz projector in the situations they consider. This is not the case for bounded domains. Thus our approach is rather different and is based on semigroup, inter- and extrapolation theories.

Leray-Hopf Weak Solutions. If v0 E L 2,u, then u is said to be a weak solution on [0, T) of (1), provided

and

T J { -(0, u) + v('V<p, 'Vu) + (<p, (u · 'V)u)} dt = (<p(O), v0)

0

for all 0 and if u satisfies the energy inequality

t

JJu(t)JJ~ + 2v J IJ'Vu(T)ll~dT :S Jjv0 1l~, t > 0.

0

Recall that it is well known that there exists at least one Leray-Hopf weak solution of (1).

The following theorem establishes relations between the maximal strong solution v and Leray-Hopf weak solutions.

THEOREM 5. Suppose that v0 E L2,u n Lq,u for some q;:::: m.

(i) The unique maximal strong solution v of (1) is a weak solution on [0, T) for every T < t+ and belongs to C([O, t+), L2);

(ii) If u is any Leray-Hopf weak solution then u ::l v. In particular, u is smooth and unique on (0, t+).

This theorem guarantees local uniqueness and smoothness of Leray-Hopf weak solutions without further restrictions. In particular, if v exists globally, then there is a unique Leray-Hopf weak solution and it is smooth for t > 0. This is in contrast to known uniqueness theorems ofSerrin [11], Fabes, Jones, and Riviere [4], Sohr and von Wahl [12], Kozono and Sohr [8], and others,


which are conditional in the sense that they require the solutions to belong to more restricted classes.

The proofs of the above theorems are given in [1] together with many additional details. In particular, there are precise descriptions of the function spaces related to N avier-Stokes equations and useful in precise regularity statements. These descriptions are also essential for the derivation of precise mapping properties of the convection term, a result which is basic for establishing the sharp results given above. In addition, we consider more general domains, the case where m/3 < q < rn, and non-vanishing exterior forces.

REFERENCES

[1] H. Amann, On the strong solvability of the Navier-Stokes equations, Preprint. [2] M. Cannone, Ondelettes, Paraproduits et Navier-Stokes, Diderot Editeur, Paris,

1995. [3] M. Cannone, F. Planchon and M. Schonbek, Strong solutions to the incompressible

Navier-Stokes equations in the half-space, Preprint, 1998. [4] E.B. Fabes, B.F. Jones and N.M. Riviere, The initial value problem for the Navier

Stokes equations with data in LP, Arch. Rat. Mech. Anal., 45 (1972), 222-240. [5] Y. Giga and T. Miyakawa, Solutions in Lr of the Navier-Stokes initial value prob

lem, Arch. Rat. Mech. Anal., 89 (1985), 267-281. [6] T. Kato, Strong LP -solutions of the Navier-Stokes equation in IRm, with applications

to weak solutions, Math. Z., 187 (1984), 471-480. [7] T. Kobayashi and T. Muramatu, Abstract Besov space approach to the non

stationary Navier-Stokes equations, Math. Meth. Appl. Sci., 15 (1992), 599-620. [8] H. Kozono and H. Sohr, Remark on uniqueness of weak solutions to the Navier

Stokes equations, Analysis, 16 (1996), 255-271. [9] P.-L. Lions and N. Masmoudi, Unicite des solutions faibles de Navier-Stokes

dans LN (fl), C.R. Acad. Sc. Paris, 327 (1998), 491-496. [10] S. Monniaux, Uniqueness of mild solutions of the Navier-Stokes equation and

maximal £"-regularity, C.R. Acad. Sc. Paris, 328 (1999), 663-668. [11] J. Serrin, The initial value problem for the Navier-Stokes equations, Nonlinear

Problems, Univ. Wisconsin Press, (1963), 69-98. [12] H. Sohr and W. von Wahl, On the singular set and the uniqueness of weak solutions

of the Navier-Stokes equations, Manuscripta Math., 49 (1984), 27-59. [13] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North

Holland, Amsterdam, 1978.


VOLUME 8 2001, NO 1-2 PP. 11-24

SUMMABILITY OF FORMAL POWER SERIES SOLUTIONS OF ORDINARY AND PARTIAL DIFFERENTIAL

EQUATIONS*

W. BALSER t

Abstract. In this article we shall briefly describe the theory of multisummability of formal power series and discuss its application in different areas of analysis, such as ordinary and partial differential equations as well as difference equations.

1. Four Examples. The recently developed theory of multisummability is concerned with formal power series, i.e., power series in one or several variables whose radius of convergence is equal to zero. Such power series occur as solutions of certain functional equations in various branches of analysis. Their coefficients may not be given in explicit form, but usually satisfy recursion formulas which can be obtained by some tedious although elementary computations. For ordinary differential equations (ODE for short), there exists computer algebra software which helps to do these computations; for a listing of some recently developed packages of this kind, compare [10, Section 13.5]. Here we begin by giving some very elementary examples from four different areas of analysis where such formal power series solutions occur naturally:

A 00

1. The formal power series f(z) = I: n! zn formally satisfies the inhoo

mogeneous ODE

(1.1) z2 x' = (1- z) x- 1.

* The author wishes to express his deep gratitude to the organizers of the conference for giving him this splendid opportunity to present recent work in the theory of formal power series. Financial support by INTAS and a travel grant of Deutsche Forschungsgemeinschaft are also gratefully acknowledged

t Abt.eilung Angewandte Analysis, Universitat. Ulm, 89069 Ulm, Germany

11

12 W. BALSER

Of course, everybody knows how to solve such a simple ODE. However, given a slightly more complicated one, we can no longer explicitly compute its solutions in closed form. Nevertheless, we may still be able to compute solutions in the form of power series. In fact, we shall briefly explain below that every linear system whose coefficients are meromorphic near the origin admits a fundamental solution which, aside from elementary functions, involves such formal power series. So the question arises as to what to do with such a formal solution. In a way, the theory of multisummability provides a final answer to this question - at least as far as solutions of meromorphic ODE are concerned.

2. Consider the homogeneous difference equation

(1.2) x(z + 1) = (1-- az-2) x(z), a E <C \ {0}.

After some elementary calculations, one can show that this differ-, 00

ence equation has a unique solution of the form f(z) = 1 +I:; fn z-n, 1

which is a power series in 1/ z. The coefficients can be uniquely com-puted from the recursion obtained from the difference equation, and they grow, roughly speaking, like n! so that, as in the previous case, the radius of convergence of the power series is equal to zero. Again, this difference equation is so simple that one can explicitly compute its solutions in terms of Gamma functions. But only slightly more complicated difference equations cannot be solved in closed form, while they still have solutions in terms of formal power series.

3. The singularly perturbed inhomogeneous ODE

(1.3) dx

e dt = x- f(t)

00

has the formal solution x(t,e) = "L;c:n fCnl(t), for arbitrary f(t) 0

which we assume holomorphic near the origin, say, for ltl < p. Ac-cording to Cauchy's formula, the coefficients fCnl(t) grow roughly liken!, so once more the radius of convergence of this series is equal to zero. However, note that here we have a power series in the perturbation parameter e whose coefficients, instead of being complex numbers as in the previous examples, are holomorphic functions in the "complex time variable" t. Replacing f(nl(t) by its power series expansion, one can also regard x( t, e) as a power series in two variables. In fact, such a formal power series solution in two variables

SUMMABILITY OF FORMAL POWER SERIES SOLUTIONS 13

also exists for the case when f(t) is no longer a holomorphic function but just a formal power series in t.

4. Consider the following Cauchy problem for the heat equation:

Ut = Uxx, u(O, x) = <p(x),

with a function <p that we assume holomorphic in some disc D. This 00

problem has a unique solution u(t, x) =I:: un(x) tn, with coefficients 0

given by

As above, this may either be regarded as a power series in the variable t, whose coefficients are functions of x that are holomorphic in D, or as a formal power series in two variables. A well-known classical result states that this power series is convergent if and only if <p(x) is an entire function of exponential order 2 and finite type. In general, however, the coefficients un(x), for fixed x E D, grow like n! so that the power series has radius of convergence equal to zero.

The coefficients of the power series in all four examples grew like n! = r(l + n), but in more general cases the rate of growth can also be some (rational) power of n!, or equivalently, like f(1 + sn) with a positive rational constant s. Series of this kind are frequently said to be of Gevrey orders. It is worth mentioning that the theory of multisummability, according to the terminology used here, can only handle power series whose coefficients are of finite Gevrey order; to which of these it indeed may be applied is a somewhat delicate question whose answer at best is of a theoretical nature. For details we refer to [10] or other texts on this matter; here we shall just discuss the situation of solutions of functional equations similar to the ones in the above examples.

2. Multisummability. Given a formal power series solution J of a functional equation satisfying some relatively weak assumptions, and given a sector S of sufficiently small opening, classical results show that the functional equation has a solution f, which is holomorphic in S, and has a formal solution as its asymptotic expansion. While this existence result is usually proven by a fixed point argument, it does not allow to compute the function f directly from the formal series f. In a way, the theory of multisummability (whenever applicable) may be viewed as a means for doing this computation.

14 W. BALSER

A general summability method may be viewed abstractly as a linear map S, defined on some vector space X of series, which we here take to be formal power series in several variables, say, z1, ... , Zm. The coefficients of the power series may be complex numbers, or more generally elements of some Banach algebra, say, of functions in other variables. To each such formal power series, this linear map assigns a generalized sum, which for power series in z1, ... , Zm should be a function in these variables. To be useful for applications to differential and other functional equations, the operator S should have a number of properties:

" The space X should contain all convergent power series, and S should map each convergent series to its natural sum.

" The linear space X should be a differential algebra; i.e., for each power series in X the formal (partial) derivatives should again belong to X, and for any two power series in X, their product should also be in X.

• The operator S should be a homomorphism for differential algebras; that is, it should not only be linear, but should map products to products and derivatives to derivatives .

.. For any formal power series j E X, the function sj should be holomorphic in some sectorial region G, and asymptotic to j when the variable(s) tend to the origin.

For more details, see a discussion of such "good" summation processes in [33]. Observe that, in particular, requiring that S "behaves well" with respect to products of power series rules out many of the clasical summability methods. For power series in one variable and coefficients in C, however, all the above requirements are indeed fulfilled by the following variant of Borel summability which was first introduced and studied by J.-P. Ramis [32]:

- 00

DEFINITION. Let k > 0, d E lR and a power series f(z) = I; fn zn 0

be given. We say that j is k-summable in direction d if the following two conditions hold:

1. The series

(2.1)

has positive radius of convergence. 2. There exists some & > 0 such that the function g ( u) defined above

can be holomorphically continued into the sector ld-argul <&,and


for sufficiently large constants c, K satisfies

(2.2) Jg(u)J ::; c exp[K JuJk]

for all u as above. If this is so, then the integral

00(7)

(2.3) f(z) = z-k j g(u)exp[-(ujzjk]duk,

0

with integration along the ray argu = T, for IT- dl < o, defines a holomorphic function for such z with Re(z-k eikT - K) > 0. Varying T results in holomorphic continuation of f; hence altogether f is defined in a sectorial region of opening larger than 1r j k. We shall consider f as the k-sum off in direction d and write f = Sk,dj. D

Obviously, the above definition, for k = 1, applies to the series j in Example 1: The corresponding function g(u) equals (1- u)-1 which clearly has the required properties for every direction d except for the positive real

- 00 axis. So we conclude that the series f(z) = 2:: n! zn is 1-summable in all

0 directions d '# 0 modulo 27r. For its 1-sum in such directions we obtain

oo(7)

f(z) = z_1 j exp[-u/z] du, 1-u

0

which one may verify to be a solution of (1.1). Due to the pole of the integrand at u = 1, the function f(z) has a branch point at the origin, hence should be studied on the Riemann surface of the (natural) logarithm. In fact, this function f is the unique solution of (1.1) which is asymptotic to j in the sector of opening 37r bisected by the negative real axis.

The formal solution of difference equation (1.2) can also be shown to be 1-summable in all but one direction (modulo 21r); the one exceptional direction depends upon the parameter a. To do this, one has to verify that the function g, defined by (2.1), satisfies a certain integral equation. From this integral equation one can then obtain holomorphic continuation plus growth estimate (2.2).

To deal with the series in Example 3, we think of the variable t as fixed and replace the variable z in the above definition by E. Obviously, the

16 W. BALSER

function

oo f(n) (t) g t ( U) = '\' U n ::______,'--'-

L..., nl 0 .

equals f ( t + u), due to Taylor's formula. So we read off from the definition that (for every fixed t of sufficiently small modulus) x(t, s) is 1-summable in a direction d if and only if the function f can be holomorphically continued into a small sector ld - arg t I < o, and satisfies an estimate of the form lf(t)i :<:; c exp[K It I] there. The corresponding sum then is given by

oo(r)

x(t,c)=s-1 j f(t+u)exp[-u/s]du.

0

We wish to emphasize here that the series x(t, c) is 1-summable in direction d if and only if the data of the equation (here, the function f) have a corresponding property. This kind of result is typical also for Example 4 (the heat equation), as we shall see later.

The above process of k-summability indeed possesses all the properties listed above, but is still too weak to handle all formal power series occuring as solutions of meromorphic ODE, as shall be explained in more detail in the next section. For this reason J. Ecalle [17, 18] introduced what he called multisummability. His original definition was more involved than the one we will give below, but in [7] both have been proven to be equivalent. In order to understand how one can alter the above definition of k-summability and obtain a stronger summation process, observe that the first condition can be relaxed by requiring that the power series in (2.1), instead of being convergent, is assumed to be summable in some other sense. This iteration of summation methods was already familiar to earlier mathematicians such as G. H. Hardy [22] and his student I. J. Good [21], but it was Ecalle who realized its great potential for summation of formal solutions of ODE.

In formal terms, the definition of multisummability uses the following notation. Let p ;::;>: 2 be a natural number. Then a vector K- = (K-1 , .•• , K-q) with positive (real) coordinates will be named a multisummability type. A second vector d = (d1, ... , dq) E JR::' will be called an admissible multidirection (with respect to K-) provided that

2K-j ldj- dj-11 :<:; 1f, 2::; j :<:; q.

For the meaning of this (technical) condition upon d, see [10, Chapter 10]. In these terms, we now say as follows:


DEFINITION. Let a multisummability type K = (K1, ... , Kq) and an admissible multidirection d = (d1, ..• , dq) be given, and consider a formal power

' 00 ' series f(z) = "L.fnzn. We say that f is K-summable in the multidirection d

0 if the following two conditions hold:

1. The series

(2.4) oo J n '( ) L n U

g U = O f(l + nj KJ)

is (K2, ... , Kq)-summable in the multidirection (d2, ... , dq), and we shall write g for the corresponding sum (in particular, for q = 2, this is supposed to mean that fj is K2-summable in direction d2 in the sense of Ramis).

2. There exists some <5 > 0 so that the function g(u) defined above can be holomorphically continued into the sector I d1 - arg u I < <5, and for sufficiently large constants c, K satisfies

for all u as above. If this is so, then the function

oo(r)

(2.5) f(z) = z-~, J g(u)exp[-(u/z)~']du~', ld1 - Kl < <5,

0

is called the K-sum of j in the multidirection d, and we write f = S~,df. D

As was said above, this definition of multisummability is different from the one given by Ecalle; in particular, in his definition a vector k = (k~o ... , kq), k1 > ... > kq > 0, is used to characterize the type of summability. His vector k is related to "' by the conditions

(2.6)

It is not obvious but can be shown that the method of multisummability defined above has all the properties which we listed as desirable when dealing with formal solutions of functional equations. As we shall indicate in the following section, multisummability is a nearly perfect tool to sum formal solutions of meromorphic ODE. However, we shall point out later that this approach is not quite general enough when dealing with difference equations.

18 W .. BALSER

For applications to formal solutions of PDE or singular perturbation problems, one can generalize the definition of multisummability to series with coefficients in a Banach algebra lE; compare [10] for details. E.g., our treatment of Example 3 above may be viewed in this setting, with lE being the set of functions which are holomorphic and bounded in some disc about the origin. However, this point of view prefers one of the variables qccuring in the power series over the others, and examples indicate that this is not always appropriate. For more details upon this, see the section on PDE below.

3. Meromorphic Ordinary Differential Equations. For a natural number r, consider a linear system of ODE of the form

00

zx' = z-r A(z)x, A(z) = LAJ zJ, 0

assuming that the series has a positive radius of convergence. It is well known that every such system has a formal fundamental solution of the form X(z) = F(t) G(z), where F(t) is a formal matrix power series in a root t = z1IP, while G(z) is a matrix of elementary functions such as exponential polynomials in t, integer powers of the logarithm, and general powers of z. The matrix G(z) as well as any finite number of coefficients of the matrix power series F(t) can be computed explicitly in terms of the coefficient matrices AJ, e.g., with the help of a computer algebra package. Due to a result of J.-P. Ramis [32] and (independently) W. Balser [2, 3, 4, 5], the matrix F(t) can be factorized as

P(t) = F1 (t) ..... Fq(t), q 2: 1,

such that each of the matrices FJ(t) is k;-summable in all but finitely many directions, for values k1 > ... > kq > 0 which can be explicitly computed from the exponential polynomials in the matrix G(z), or equivalently, from the coefficients AJ. For q > 1, this factorization is neither unique nor constructive, and it follows directly from the theory of k-summability that the matrix F(t) as a whole is not k-summable in Ramis' sense, for whatever value of k > 0. However, defining a multisummability type r;. by (2.6), general results on multisummability imply that F(t) is r;.-summable in every but "finitely many" multidirections d. The so-called singular multidirections where r;.-summability fails can also be computed explicitly from the coefficients AJ. The dependence of the corresponding sums upon the choice of the non-singular multidirection d gives rise to the Stokes phenomenon of the system of ODE; for this, see B. L. J. Braaksma [12].


More generally, we may look at a nonlinear system of meromorphic ODE; i.e., a system of the form

where g(z, x) is holomorphic in the variables z and x = (xi> ... , Xn) in some polydisc about the origin. If such a system has a formal solution x(z) whose entries are formal power series, then all these series are again K-summable in the above sense, with a multisummability type K which is determined by the linear part of the system. This result was conjectured by Ecalle and proven first by B. L. J. Braaksma [13]. Other proofs, using entirely different methods, have been given somewhat later by Ramis and Sibuya [34], resp. W. Balser [8].

4. Meromorphic Difference Equations. In the analytic theory of linear difference equations, one is concerned with solutions of systems of the form

x(z + 1) = A(z) x(z),

where A(z) is a matrix whose entries are holomorphic in a punctured neighbourhood of infinity and have at most a pole there. In addition, one always assumes that det A(z) does not vanish near infinity. Without going into any details, we mention that such systems always have a formal fundamental solution of the form X = F(t) G(z), where G(z) is a matrix of elementary functions (which here include integer powers of Gamma functions), and F(t) is a matrix of formal power series in t = z-l/P, for a positive integer p. If the leading term A0 of the Laurent expansion of A(z) is invertible, then the series in F( t) have been shown to be K-summable, with a suitable multisummability type Kj see, e.g., Braaksma and Faber [14]. Analogous results hold even for nonlinear systems as well, so in a sense, in this case the theory is completely analogous to the one for differential equations. However, a new phenomenon arises when the leading term A0 fails to be invertible. This case, usually referred to as one with level 1 +, is still not fully understood; for a very readable presentation of this phenomenon and existing results, see the dissertation of B. Faber [19].

5. Singular Perturbation Problems. Let us consider a nonlinear system of ODE. of the form

ca x' = g(z, x, c),

20 W. BALSER

where g(z, x, c) additionally depends holomorphically on a parameter c, for small complex values of c, and u is a natural number. Analysis of the dependence of solutions on c is usually referred to as a singular perturbation problem. Under suitable assumptions on the right-hand side, such a sys-

oo

tern will have a formal solution x(z, c) = I: Xn(z) en, with coefficients Xn(z) n=O

given by differential recursion relations. In general, the series is divergent, and classically one has tried to show existence of solutions of the above system that are asymptotic to x(z, c) when c -+ 0 in some sectorial region. Very recently, one has begun to investigate Gevrey properties of x, or discuss its (multi-)summability. Of the recent articles containing results in this direction, we mention G. Wallet [35, 36], M. Canalis-Durant [15], and Canalis-Durant, Ramis, Schiifke, and Sibuya [16].

Since here we meet power series whose coefficients are functions of another variable, the situation calls for a generalization of the theory of multisummability to power series with coefficients in a Banach space, resp. even a Banach algebra. This point of view has been taken in [10], and the theory has been shown to carry over without any difficulties. So in this setting, a

,,. power series in several variables will be treated by viewing all but one of the variables as parameters, which most of the time may even be replaced by constant values. However, as becomes more obvious for the case of PDE, an even more general approach shall be necessary in general, applying summability methods which affect all variables more or less equally.

6. Partial Differential Equations. Besides the classical theory discussing convergence of power series solutions for Cauchy problems for certain classes of PDE, there are recent results concerning the Gevrey order of such power series; e.g., see [20, 24, 25, 26, 27, 28, 30] and the literature cited there. There are also articles, e.g., by Ouchy [29, 31], showing that such formal solutions are asymptotic representations of proper solutions of the underlying equation. Only few papers exist so far about multisummability of formal solutions of such problems: Lutz, Miyake, and Schiifke [23] obtained a first result on the Cauchy problem for the complex heat equation as in Example 4. Their theorem has been generalized in [9] and will be presented below. Very recent work by Balser and Miyake [11] treats a more general case, showing that the results obtained are not so much dependent on having a formal solution of a partial differential equation, but carry over to series whose coefficients are given by certain differential recursions.

Here we briefly describe a typical result on k-summability (in the sense of

SUMMABILITY OF FORMAL POWER SERJES SOLUTIONS 21

00

Ramis) for the formal solution u( t, z) = I: tn 'P(zn) (z) jn! in Example 4. In an n;:;:::Q

arbitrary compact subset K c D, one can show, using Cauchy's formula for derivatives, that II'P(Zn) \IK = supzEK I'P(Zn) (z) I ::; en f(1 + 2n ), for sufficiently large c. Hence u can be regarded as a formal power series in t with coefficients in the Banach space JEK of functions that are bounded on K and analytic in its interior. The above estimate implies u E JEK [[t]h, meaning that we have a power series in t with coefficients Un in JEK satisfying Jlun\1::; cKnr(1+n). Thus, it is natural to ask for 1-summability of u. If 'P happens to be entire, however, the above estimate upon its derivatives may be improved, and then u may be summable of another type. In detail, the following has been shown in [9]:

00

THEOREM 1. For an arbitrary ~.p(z) = l::'PnZn, z E D, define u as 0

above, and set joT 0 ::; j ::; 2:

• ~ (2n+ j)! n '1./;j(t) = ~ 'P2n+j I t ,

o n.

00

{;;±(t) = L 'Pn f(l + n/2) (±t)n. 0

Given any k :2': 1, and any direction d E IR, the following statements aTe equivalent:

(a) For every K c D(O, p) the series u, regarded as a power series in t with coefficients in lEoc, is k-summable in direction d.

(b) For 0::; j ::; 1, {pi is k-summable in direction d. (c) {;;± are both 2k-summable in direction d/2.

Using a decomposition result from [6], the above theorem can easily be generalized to multisummability; for details, see [9] or [10]. In general it is difficult to check (b) or (c) directly through investigation of ~.p(z), except for the special case of k = 1. In this situation, the above theorem essentially coincides with the result obtained by Lutz, Miyake, and Schiifke [23]: conditions (b) or (c) then are equivalent to 'P(z) admitting holomorphic continuation into small sectors bisected by rays argz = d/2 and argz = d/2+w, and satisfying an estimate of form (2.2) with k = 2 there.

In [1] the author has presented a new method of summability for power series in several variables, affecting all variables at the same time. This new method also applies to the above situation of the Cauchy problem for the heat equation, but with an initial condition 'P which is holornorphic in a sector instead of a full neighbourhood of the origin. Roughly speaking, this

22 W. BALSER

method is as follows. Let a formal power series

](zl, ... 'Zm) = L fnl,···,nm zfl ..... z~m n1, ... ,nm

in several variables be given, and take a vector s = ( 81, . .. , sm) of nonnegative real entries, not all of them equal to 0. Define

assuming that this series converges in some polydisc about the origin. Then it is natural to define the corresponding "sum" of / by the integral

00

(6.1) f(zi, ... ,zm) = J e-x g(x81 ZJ, ... ,x8 mzm) dx.

0

In order to give sense to this integral, one has to assume that the function g can be holomorphically continued into an appropriate region and satisfies a corresponding growth estimate; for details, see [1]. However, observe that for s2 = ... = sm = 0 and k = 1/s1 a change of variable in (6.1) shows that the above definition coincides with Ramis' notion of k-summability of /, when regarded as a power series in z1 with coefficients depending upon the remaining variables. In [1], this new method has been successfully applied in some situations of Examples 3 and 4, where the previously described results failed.

REFERENCES

[1] W. Balser, Summability of power series in several variables, with applications to singular perturbation problems and partial differential equations, Manuscript.

[2] W. Balser, Einige Beitriige zur Invariantentheorie meromorpher Differentialgleichungen, Habilitationsschrift, Universitii.t Ulm, 1978.

[3] W. Balser, Growth estimates for the coefficients of generalized formal solutions, and representation of solutions using Laplace integrals and factorial series, Hiroshima Math. J., 12 (1982), 11-42.

[4] W. Balser, Solutions of first level of meromorphic differential equations, Proc. Edinburgh Math. Soc., 25 (1982), 183-207.

[5] W. Balser, A constructive existence proof for first level formal solutions of meromorphic differential equations, Hiroshima Math. J., 15 (1985), 411-427.

[6] W. Balser, A different characterization of multisummable power series, Analysis, 12 (1992), 57-65.

[7] W. Balser, Summation of formal power series through iterated Laplace integrals, Math. Scandinavica, 70 (1992), 161-171.

SUMMABILITY OF FORMAL POWER SERlES SOLUTIONS 23

[8] W. Balser, Prom Divergent Power Series to Analytic Functions, Lecture Notes in Math., 1582, Springer Verlag, New York, 1994.

[9] W. Balser, Divergent solutions of the heat equation: on an article of Lutz, Miyake and Schiifke, Pacific J. of Math., 188 (1999), 53-63.

[10] W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations, - book to appear in the UNI TEXTS series of Springer Verlag, 1999.

[11 J W. Balser and M. Miyake, Summability of formal solutions of certain partial differential equations, Ulmer Seminare - Funktionalanalysis und Differentialgleichungen, University of Ulm, 1999, (to appear in Acta. Sc. Math. Szeged).

[12) B. L. J. Braaksma, Multisummability and Stokes multipliers of linear meromorphic differential equations, J. of Differential Equ., 92 (1991), 45-75.

[13] B. L. J. Braaksma, Multisummability of formal power series solutions of nonlinear meromorphic differential equations, Ann. Inst. Fourier Grenoble, 42 (1992), 517-540.

[14] B. L. J. Braaksma and B. F. Faber, Multisummability for some classes of difference equations, Ann. Inst. Fourier Grenoble, 46 (1996), 183-217.

[15) M. Canalis-Durand, Solution formelle Gevrey d'une equation differentielle singulierement perturbee, Asymptotic Analysis, 8 (1994), 185-216.

[16) M. Canalis-Durand, J.-P. Ramis, R. Schafke, andY. Sibuya, Gevrey solutions of singularly perturbed differential and difference equations, tech. rep., IRMA Strasbourg, 1999,- accepted by J. reine und angew. Math.

[17) J. Ecal!e, Les fonctions resurgentes I-II, Pub!. Math. d'Orsay, Universite Paris Sud, 1981.

[18) J. Ecalle, Les fonctions resurgentes III, Pub!. Math. d'Orsay, Universite Paris Sud, 1985.

[19) B. F. Faber, Summability theory for analytic difference and differential-difference equations, PhD thesis, Rijksuniversiteit Groningen, 1998.

[20) R. Gerard and H. Tahara, Formal power series solutions of nonlinear first order partial differential equations, Funkcialaj Ekvacioj, 41 (1998), 133-166.

[21) I. J. Good, Note on the summation of a classical divergent series, J. London Math. Soc., 16 (1941), 180-182.

[22) G. H. Hardy, On the summability of series by Borel's and Mittag-Leffier's methods, J. London Math. Soc., 9 (1934), 153-157.

[23) D. A. Lutz, M. Miyake, and R. Schafke, On the Borel summability of divergent solutions of the heat equation, Nagoya Math. J., 154 (1999), 1-29.

[24) M. Miyake, Relations of equations of Euler, Hermite and Weber via the heat equation, Funkcialaj Ekvacioj, 36 (1993), 251-273.

[25) M. Miyake andY. Hashimoto, Newton polygons and Gevrey indices for linear partial differential operators, Nagoya Math. J., 128 (1992), 15-47.

[26) M. Miyake and M. Yoshino, Fredholm property for differential operators on formal Gevrey space and Toeplitz operator method, C. R. Acad. Bulgare de Sciences, 47 (1994), 21-26.

[27) M. Miyake and M. Yoshino, Wiener-Hopf equation and Fredholm property of the Goursat problem in Gevrey space, Nagoya Math. J., 135 (1994), 165-196.

[28) M. Miyake and M. Yoshino, Toeplitz operators and an index theorem for differential operators on Gevrey spaces, Funkcialaj Ekvacioj, 38 (1995), 329--342.

[29) S. Ouchi, Asymptotic expansion of singular solutions and the characteristic polygon of linear partial differential equations in the complex domain, Manuscript.

24 W. BALSER

[30] S. Ouchi, Formal solutions with Gevrey type estimates of nonlinear partial differential equations, J. Math. Sc. Univ. Tokyo, 1 (1994), 205-237.

[31] S. Ouchi, Singular solutions with asymptotic expansion of linear partial differential equations in the complex domain, Publ. R1MS Kyoto University, 34 (1998), 291-311.

[32] J.-P. Ramis, Les seriesk-sommables et leurs applications, in Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Lecture Notes in Physics, 126, Springer Verlag, New York, 1980, 178-199. .

[33] J.-P. Ramis, Series Divergentes et Theories Asymptotiques, Panoramas et syntheses, 121, Soc. Math. France, Paris, 1993.

[34] J.-P. Ramis andY. Sibuya, A new proof of multisummability of formal solutions of non linear meromorphic differential equations, Annal. Inst. Fourier Grenoble, 44 (1994), 811-848.

[35] G. Wallet, Surstabilite pour une equation differentielle analytique en dimension 1, Annales Institute Fourier, 40 (1990), 557-595.

[36] G. Wallet, Singularite analytique et perturbation singuliere en dimension 2, Bullet. Soc. Math. France, 122 (1994), 185-208.


VOLUME 8 2001, NO 1-2 PP. 25-33

ABOUT CANONICAL TYPES OF THE DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT*

L.A. BEKLARYAN t

Introduction. Differential equations with deviating argument are considered

(1) i::(t) = f(t, x(q1(t)), ... , x(qs(t))), t E lR,

where f : lR x JE.n·s ---+ JE.n is a continuous map, q; : lR ---+ JR, i = 1, s are orientation-preserving homeomorphisms. A classification of such differential equations is related to selection of the equations with a canonical type of deviations of argument, to which equation (1) with the help of replacements of time is reduced [1]. This problem is equivalent to the classification of finite generated groups Q < q~o ... , q8 > of homeomorphisms of JR, which were generated by the functions of deviating of argument.

Problem of classification of groups (not only finite generated) of orientation-preserving homeomorphisms of lR is closely connected with the problem of existence of metric invariants (the invariant measure, thew- projectivelyinvariant measure). In turn, the properties of metric invariants are closely connected with the various topological (the minimum sets, the orbits of points, the set of fixed points of elements of group), algebraic (the group properties of chosen subsets), and combinatorial (the amenablence of a quotient group of initial groups on the canonically chosen subgroups) performances of the group [2]-[4].

* The work is maintained by the grant N 96-15-96135 of the state program for Support of scientific schools and Russian Foundation of Basic Researches (project N 97-01-00625)

t Central Economics and Mathematics Institute, 117418 Russia, Moscow, ul. Krasikova, 32

25

26 L.A. BEKLARYAN

In a considered classification problem equations are divided into the following canonical types of deviations of argument:

• ordinary differential equations with constant deviations of argument (there is an invariant measure for the group Q);

• ordinary differential equations with affine deviations of argument (there is a projectively-invariant measure for the group Q);

• differential equation with the partial derivatives of the first order in an infinite number of variables with affine deviations of arguments (there is an w-projectively-invariant measure for the group Q with w =/= 1).

Within the framework of the represented work the conditions of reduction of initial differential equation (1) to the first two canonical types will be described.

For the description of an offered classification, it is necessary to enter the concept of semiconjugacy, which is a canonical object of the theory of the metric invariants for groups of homeomorphisms of JR. [2]-[3].

DEFINITION 1. Suppose that G, .G are groups of orientation-preserving homeomorphisms of JR. The group G is named semiconjugate to the group .G if there is a monotonically growing map 71 : JR. -----.> JR. with an image consisting of more than one point, and an epimorphism 71~ : G -----.> * G such that, for any q E G, the diagram

is commutative, i.e. 71~(9)71 = 719·

If in the definition of the semiconjugacy the map 71 is continuous, then we speak about topological semiconjugacy. If the map 71 is a homeomorphism, and as a corollary the epimorphism 71~ is an isomorphism, then the group G is named topologically conjugate to the group .G or simply conjugate.

In the present work we shall use notations: " H omeo+ (JR.) is the group of all orientation-preserving homeomor

phisms of JR.; • G(t) = {g(t) : g E G} -is the orbit of a point t E JR.

Other canonical objects of the theory of metric in~ariants for groups of homeomorphisms of JR. are minimum sets.

DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENT 27

DEFINITION 2. Let G ~ Homeo+(!R.). A nonempty subset E(G) ~JR. is named minimum if it is closed, G invariant, and does not contain proper closed G invariant subsets. If a minimum set does not exist, then we shall assume in the definition that E(G) = f/J.

It appears that the minimum sets have a completely defined canonical structure, for which the description is given by the following theorem.

THEOREM 2 ([2]). Let for G ~ Homeo+(JR.) there exists the minimum set E( G). Then one of the following assertions is valid:

AO) E(G) is a discrete set {possibly not unique); Al) E( G) is a perfect subset not dense anywhere, it is a unique minimum

set contained in closure of orbit G(t) of any point t E JR.; A2) E( G) JR. {that is equivalent to a minimality condition of group G).

For finite generated groups the theorem of existence of the minimum set is valid.

THEOREM 3 ([3]). For a finite generated group G C Homeo+(JR.) a minimum set E( G) is not empty.

The main "obstacles" in the problem of reduction of a differential equation with deviating argument to an equation with canonical type of deviations of argument will be described further. The description of main "obstacles" in the investigated problem will be given in the terms of semiconjugacy.

1. About Reducibility of Differential Equations with Deviating Argument to the First Canonical Type. Here we shall describe main "obstacles" in the problem of reduction of initial differential equation with deviating argument to an equation with the constant deviations of argument.

DEFINITION 3. A a--final Borel measure 11 is named invariant with respect to a group G ~ Homeo+(JR.) if for any g E G, B E I: (2:: is the algebra of Borel subsets of JR.) the condition 11(B) = J1(q- 1 (B)) is satisfied.

THEOREM 4 ([3]). Let G ~ Homeo+(JR.). Then the following assertions are equivalent:

(1) there is a a--final Borel measure J1, which is invariant w. r. t. the group G;

(2) group G is semiconjugate to some group .G ~ H omeo+(JR.) of shifts on JR..

28 L.A. BEKLARYAN

NOTE 1. In theorem 4, the measure fl. and the appropriate map 'rJ from the definition of semiconjugacy are connected by a relation

(t) = { +J.L([t, t)) 'rJ -J.L((t,~)

if t 2 t, if t < t,

where t E lE. is some fixed point. Moreover, if a point t E lE. is fixed, then for a certain group a<;;; Homeo+(JE.) the correspondence between w-projectivelyinvariant measures fl. and maps rJ will be one-to-one. From continuity of map rJ follows the continuity of measure fl. and vice versa.

As we are interested in replacements of time, we should clarify when the condition of semiconjugacy becomes a condition of conjugacy.

THEOREM 5 ([2]). Let a <;;; Homeo+(JE.). For conjugacy of group a to the group of shifts .a it is necessary and sufficient that the following conditions are fulfilled:

1) there is a continuous u-final Bor·el measure J.1. 1, which is invariant w.r.t. to the group a;

2) the support supp fl. of the measure fl. coincides with alllE..

Proof. Necessity. From the condition of conjugancy of the group a to the group of shifts .a and theorem 4 follows the existence of the invariant measure fl.· As the map rJ is a homeomorphism, by virtue of note 1, the measure fl. is continuous, and its support coincides with alllE..

Sufficiency. If the group a is trivial, i.e. a =< e >, then the proof is obvious.

Let a #< e >. From the existence of an invariant measure fl. and theorem 4 it follows that the group a is serniconjugate to some group of shifts .a. Continuity of the measure fl. and note 1 imply that the map rJ is also continuous. Since the support of the measure fl. coincides with all JE., we deduce that rJ sets a one-to-one map lE. onto the image ry(JE.), and the epimorphism rJ~ (from the definition of semiconjugacy) is an isomorphism. It is necessary to show that TJ(lE.) coincides with alllE..

From invariancy of the measure fl. and conditions supp fl. = JE., a #< e > it follows that any element g E a\e acts on lE. freely, i.e. for any t E lE. the condition g( t) # t is valid.

From the existence of a freely acting element g E a\e and the invariancy of the measure fl. it follows that J.L(] - oo, OJ) = +oo, J.L([O, +oo[) = +oo. Therefore TJ(lE.) = lE.. D

By virtue of condition 2) of theorem 5, it is important to know how to describe the supports of the invariant measures. In fact, the supports of the


invariant measures are described in topological performances of groups of homeomorphisms lit Such description is given by the following theorem.

THEOREM 6 ([1]). Suppose that G ~ Homeo+(JR) and there exists a a-final Borel measure J.l, which is invariant w.r.t. the group G. Then one of the following conditions is valid:

1) the support of the measure J.l consists of an association of discrete minimum sets {discrete orbits). Moreover, there is a a-final measure f.l1, which is invariant w.r.t. the group G, for which the support supp J.ll coincides with association of all discrete minimum sets;

2) the support supp J.l of the measure J.l coincides with the unique minimum set E(G), and the measure J.l is continuous.

Now we can formulate the conditions of a topological character, which will guarantee fulfilment of conditions 1) and 2) of theorem 5.

PROPOSITION 1. Let G ~ H omeo+ (JR), and let there exist a a - final Borel measure J.l, which is invariant w.r.t. the group G. If the association of minimum sets coincides with JR, then there is a a-final Borel continuous measure f./,1, which is invariant w.r.t. the group G and for which the support supp p, of the measure J.l also coincides with all JR.

Proof. Let a minimum set for the group G be discrete. Then, by theorem 2, all minimum sets are also discrete. Since the association of minimum sets coincides with alllR, the elements g E G\e act on lR freely, i.e. for any t E lR condition g(t) #tis satisfied. From a discretization the minimum sets (the discretization of orbits of any point t E JR) it follows that the group G is cyclic. For the cyclic group G with freely acting elements, there is a a-final Borel measure f.l 1 , which is invariant w.r.t. the group G (see [1]).

Let a minimum set for the group G be not discrete and, consequently, unique. By assumption of proposition 1, the minimum set E(G) coincides with JR. Then, by theorem 6, the condition supp J.l = lR is valid. It is necessary to select a measure f.l 1 equal to the measure f.l· D

Thus, in the problem of reduction of the differential equation with deviating argument to an equation with constant deviations of argument, the main "obstacles" have the metric and topological character:

"' there should exist a a-final Borel measure J.l, which is invariant w.r.t. the group G;

"' the association of minimum sets should coincide with R The works of Bogoliubov and Krilov concern early outcomes on study

of an invariant measure, and the works of Denjoy concern early outcomes on

30 L.A. BEKLARYAN

study of a structure of a minimum set [5]. In these works the homeomorphism of a circle was investigated. The research of a homeomorphism of a circle can be replaced by research of an object which is equivalent to it. It is a commutative group of homeomorphisms of lR with two generators, in which one element is a covering of the initial homeomorphism of a circle, and the second element is a unit shift of R

Thus, the considered problem is connected with the establishment of analogues of the theorems of Bogoliubov-Krilov about existence of an invariant measure and the theorems of Denjoy about the structure of a minimum set. In papers [2], [4] one can find various criteria of existence of an invariant measure.

Actually, for complete replacement of time in initial differential equation (1), a map rJ (from the definition of semiconjugacy) should be not only continuous, but also smooth. Problem of smoothness of the map rJ even for a homeomorphism of a circle is complicated [5] and is an independent nontrivial problem.

2. About Reducibility of Differential Equations with Deviating Arguments to the Second Canonical Type. Here we shall describe main "obstacles" in a problem of reduction of the initial differential equation with deviating argument to an equation with affine deviations of argument.

DEFINITION 4. The u-final Borel measure J.t is named projectivelyinvariant w.r.t. a group G s;;· Homeo+(JR) if for any g E G there exists a number c(g) > 0 such that for every BE L: (L: is an algebra of the Borel subsets of JR) the condition J.t(B) = c(g)J.t(g-1(B)) is satisfied.

THEOREM 7 ([3]). Let G s;; Homeo+(JR). Then the following assertions are equivalent:

(1) there is a u-final Borel measure J.t, which is projectively-invariant w. r. t. the group G;

(2) the group G is semiconjugate to some group .G s;; H omeo+(JR) of affine transformations of JR.

NOTE 2. In theorem 7 the measure J.t and the appropriate map rJ from definition of semiconjugacy are connected by the formula indicated in note 1.

As we are interested in replacements of time, we should clarify when the condition of semiconjugacy becomes a condition of conjugacy.

THEOREM 8. Let G s;; Homeo+(JR). For conjugacy of the group G to the group of affine transformations .G it is necessary and sufficient that the following conditions hold:


(1) there is a a-final Borel measure Jt which is projectively-invariant w.r. t. the group G;

(2) the support supp Jt of the measure Jt coincides with all lit

Proof. Necessity. The condition of conjugacy of a group G to the group of affine transformations .G and theorem 7 imply the existence of a projectivelyinvariant measure Jt. As the map 7J is a homeomorphism, by virtue of notes 1 and 2, the measure Jt is continuous, and its support coincides with all lit

Sufficiency. If the group G is trivial, i.e. G =< e >, then the proof is obvious.

Let G #< e >. From the existence of a projectively-invariant measure Jt and theorem 7 it follows that the group G is semiconjugate to some group of affine transformations .G. From the continuity of the measure Jt and notes 1 and 2 it follows that the map 7J is also continuous. Since the support of the measure Jt coincides with all lR, we deduce that 7J sets a one-to-one map JR onto an image TJ(lR), and the epimorphism 7Ju (from the definition of semiconjugacy) is an isomorphism. It is necessary to show that 7J(lR) coincides with all lit Since supp Jt = lR and Jt is a projectively-invariant measure, we have that for any g E G\e the set of fixed points {t: g(t) = t} of an element g consists of no more than one point. Consider two cases.

The first case. There is an element g E G\e such that g(t) = t for some point t E lit We remark that for such element g the coefficient of expansion c(g) from the definition of a projectively-invariant measure is not equal to unit. From a property of the projectively-invariant measure Jt it follows that J,L([t, +oo[) = +oo, J,L(j - oo, t[) = +oo. In that case, 7J(lR) coincides with all JR.

The second case. All elements g E G\e act on lR freely, i.e. for any g E G\e the set of fixed points {t: g(t) = t} of an element g is not empty. We remark that for such groups the coefficients of expansion c(g), g E G\e are either all equal to unit and the measure Jt is invariant, or all not equal to unit.

If for any g E G\e the coefficient of expansion c(g) is equal to unit and, consequently, the measure Jt is invariant, it is obvious that 7J(lR) coincides with alllR, and the affine group .G is the group of shifts of JR.

If for any g E G\e coefficients of expansion c(g) are not equal to unit, then one of the following conditions holds: either J,L(j - oo, 0[) < +oo, Jt([O, + oo[) = + oo; or J,L(j - oo, 0[) = + oo, J,L([O, + oo, 0[) < +oo. For determinacy, we shall suppose that the first group of conditions is fulfilled. Then the image 7J(lR) coincides with the half-line JR+, and the group .G (from the definition of semiconjugacy) is the group of linear transformations, i.e.

32 L.A. BEKLARYAN

.G =< at, 0! E A >, where A is a subgroup of the multiplicative group of JR+. Consider a map ry1 : lR -+ JR, determined as follows: ry1(t) = ln t. Define the group of shifts .G1 = { t + ln a, a E A} and the isomorphism f/1 : .G -+ .G1 , for which the linear map at passes into a shift t + ln 0!.

Then a homeomorphism 'fj = ry1 fJ (rj : lR -+ JR) and isomorphism ryU = ryl'rJU (ryU : G -+ .G1) determine the conjugancy of the group G to the group of shifts .G1. 0

By virtue of conditions 1) and 2) of theorem 8, it is important to know how to describe the supports of projectively-invariant measures and the property of continuity of such measures.

THEOREM 9 ([3]). Let G ~ Homeo+(JR), and let it have no a-final Borel invariant measures. If there is a a-final Borel measure fJ., which is projectively-invariant w. r. t. the group G, then for the group G there exists a not discrete unique minimum set E(G), suppfJ. = E(G), and the measure fJ. is continuous.

Thus, in the problem of reduction of the differential equation with deviating argument to an equation with affine deviations of argument, main "obstacles" have metric and topological character:

" there should be a a-final Borel measure fJ., which is projectivelyinvariant w.r.t. the group G;

" in case of absence of an invariant measure, the unique minimum set should coincide with JR;

" in case of existence of an invariant measure, the support of the projectively-invariant measure should coincide with JR.

Thus, the considered problem is connected with the establishment of existence of more general metric invariants such as projectively-invariant measures and Denjoy's theorems about the structure of minimum sets. Papers [2]-[4] contain various criteria of existence of a projectively-invariant measure. Problems of smoothness of map fJ were already discussed in the end of the previous section.

REFERENCES

[1] L.A. Beklaryan, About reducibility of differential equation with deviating argument to the equation with the constant commensurable deviations, Mathematical Notes, 44, 5 (1988), 561-566.

[2] L.A. Beklaryan, On the classification of groups of orientation-preserving homeomorphisms of r. i. invariant measures, Sbornik: Mathematics, 187, 3 (1996), 335-364.


[3] L.A. Beklaryan, On the classification of groups of orientation-preserving homeomorphisms of llt II. Projectively-invariant measures, Sbornik: Mathematics. 187, 4 (1996), 469-494.

[4] L.A. Beklaryan, On the classification of groups of orientation-preserving homeomorphisms of llt III. w-projectively-invariant measures, Sbornik: Mathematics, 190, 4 (1999), 521-538.


VOLUME 8 2001, NO 1-2 PP. 35-48

LONG TIME VANISHING PROPERTY OF SOLUTIONS OF SUBLINEAR PARABOLIC EQUATIONS AND

SEMI-CLASSICAL LIMIT OF SCHRODINGER OPERATORS

Y. BELAUD, L. VERON • AND B. HELFFER t

Introduction. Let 12 be a bounded domain in JRN, b(x) a nonnegative, non-identically zero function in n and 0 ::; q < 1. Consider the following equation:

{

OtU- .6.u + b(x)[u[q-lu = 0 inn X (O,oo), (1) Bvu =Oon8Stx(O,oo),

u(x, 0) = uo(x) inn.

If b(x) 2: (3 > 0, the comparison principle with the solutions of

(2) { Bt9>+ (J[cf>lq-lcf> = 0 in (O,oo),

¢(0) = lluollL=•

implies that u vanishes for t 2: T = T ((J[[u0[[L= ). We denote this property by (TCS-property) Time-Compact-Support Property. If b(x) = 0 on some subdomain w C n, and if

u0 2:0 with ess infxEwuo(x) =a> 0,

we can bound from below the function u on w x (0, oo) by the solution v of

{

Btv- .6.v = 0 in w·x (0, oo), v =0on8wx(O,oo),

v(x, 0) =a in w;

• Laboratoire de Mathematiques et Physique. Theorique, CNRS ESA 6083, Faculte des Sciences et Techniques, Universite Fran,ois Rabelais, F 37200 Tours

t Departemen£ de Mathematiques, CNRS UMR 8628, Bat. 425, Universite Paris Sud, F 91405 Orsay-Cedex

35

36 Y. BELAUD, B. HELFFER, AND L. VERON

then v has exponential decay as t -+ oo, but it never vanishes. In fact, the exponential decay always holds, but the TCS-property is much more delicate and linked to well-known questions in the semi-classical limit of Schrodinger operators of the following type:

(3) -h2 .0.. + b(.),

that is, the analysis of the behaviour or the eigenvalues of (5) as h -+ 0.

2. The Exponential Decay. In the sequel we denote by 1-l(a) the following expression:

the lowest eigenvalue of the Neumann realization of the Schrodinger operator -.0.. + aq- 1b(.) in L2 (Q). The next easy-to-establish result is the key-stone of all the theory which will be developed below.

LEMMA 1. Suppose that v is a continuous function defined in 0 X (0, oo), which satisfies weakly

{

OtV- .0..v + b(x) + aHb(x)v ~ 0 in Q x (O,oo), (5) OvV = 0 on f)Q X (0, oo),

v(x, 0) = v0 (x) in Q.

Then the positive part v+ of v satisfies

(6)

and there exists C = C (N, I Ill) such that

(7)

for any t 2: 0.

Proof. With no loss of generality, let us assume that v is nonnegative. The proof of (8) is obvious by the spectral theorem and the definition of 1-l(a). For (9), we have

llv+(., t) ll£oo ~ C1 (1 + T-l )N/4IIv+( ., t- T) 11£2,

with C1 = C1 (N) > 0, by using the regularizing effect of the heat semigroup, and

SEMI-CLASSICAL LIMIT OF SCHRODINGER OPERATORS 37

for 0 :::; T < t. Then

Optimizing over T, using the fact that, for any p E [1, oo], s >--+ llv+(., t)IILP is decreasing (this is due to the nonnegativity of b), and Holder's inequality yields (9).

As a consequence, we have the exponential decay in t for any solution of (1).

PROPOSITION 1. Let u be a solution of (1) such that lluoiiLoo =a> 0. Then

(8)

for any t 2: 0.

Proof. Assume that u is nonnegative. Since 0 < u(x, t) :::; a, we have uq-l(x, t) :::; aq-l and

(9)

We get (13) from Lemma 1.

3. The Time-Compact-Support Property. The first result in this direction is due to Kondratiev and Veron. Up to a small correction, their result reads as follows.

THEOREM 1. Let q be such that 0 :::; q < 1, and denote

ftn = J.t(2nf(q-l))

= inf {£ (IIV'1/JII 2 + J.t(2n)b(x)1/12) dx: 1j; E W 1

•2(Q), [1/J2dx = 1}.

00

If I: J.t;; 1 lnJ.tn < oo, then any solution of (1) satisfies the TCS-property. n=O

REMARK 1. If b(x) 2: j3 > 0, then ftn 2: j32n and the series is convergent. If b(x) = 0 on some subdomain w c n, then ftn :::; .A1,w, the first eigenvalue of -D. in W~'2(w), and the series is divergent.

The first extension is to replace the sequence {2nf(q-l)} by any sequence going to zero.


THEOREM 2. Let {an} be a decreasing sequence of positive numbers tending to zero and such that

Then any solution of (1) satisfies the TCS-property.

Proof. Without any loss of generality, we can assume that 0::; u(x, 0) ::; JJu(., O)JILoo(n) by changing b into >.b, and set a0 = 1. Then

JJu(., t) JILoo(n) :S min (1, fl(l )N/4) e-'"(1).

If t:::: to= In (C(~(WJNI4

J), then JJu(.,t)IILoo(nJ ::; Cf1(1)NI4e-'"(1l. We

define t1 > to by

a1 = Cf1(1)N/4e-tJ"(1) ~ t1 = f1(1)-1Jn (C((fl~~))N/4).

(Notice that we can always assume that a 1 < 1.) Since JJu(., t)JILoo(n) :S 0<1, fort;::: t1 there holds o,u- Ll.u + ag-1b(x)u ::; 0, and again

JJu( ., t) JILoo(n) :S min ( 1, 11( al)N14) e-(t-tl)!'(<>t) a 1

inn X (t1,oo). We now define t2 by

a2 =min (1, fl(a1)N/4) e-(t2-t1l~<(<>1la1 ~

t2- t1 = fl(al)-1ln ( C((f1(a1))N/4:J. Iterating this process, we construct an increasing sequence tn such that llu(.,t)JJLoo(O) :S O<n-1 fort :S tn-1 and

O<n =min (1, fl(O!n-1)Nf4) e-(tn-tn-t)i'(<>n-t)an-1 ~

tn tn-1 = f1(0<n_l)-1ln ( C((fl(O<n-1))N/4 a~: 1 ).

Hence JJu(., t)IILoo(n) :S O!n fort;::: tn, and therefore

JJu(., t)JJLoo(nJ :S min (1,f1(an)NI4) e-(t-tnll'(<>nlJJu(., tn)JILoo(nJ·


It follows from the above inequalities that

By assumption, the right-hand side is bounded. Therefore lim = T and tn......)-00

the following inequality holds:

for t :::0: T and any integer n. Letting n go to infinity implies that u vanishes identically on n X [T, oo).

REMARK 2. In equation (1), we can replace the Neumann boundary condition on an by a homogeneous Dirichlet condition. In that case p( a) has to be replaced by p0(a), which is defined by the same expression as p(a), except that the infimum is taken over the set of normalized test functions belonging to the space W5•2 (n). Proposition 1 still holds without any regularity requirement assumed on an. Consequently, if there exists a decreasing positive sequence {!3n} such that

then any solution of (1) with homogeneous Dirichlet boundary conditions is identically zero in finite time.

The assumptions of Theorem 2 admit a simpler form.

THEOREM 3. The existence of a decreasing sequence {an} satisfying the assumptions of Theorem 2 is implied by

(12)

Moreover, it implies

(13)

1

I ln(p(t)) dt

( ) < oo.

J1 t t 0

1

I dt tp(t) < oo.

0


4. The Semi-Classical Analysis. The semi-classical· analysis deals with the description of the behavior of the spectrum of the operator -h2t:. + b(.) ash goes to zero. If we write-!:.. +h-2b(.) = -t:. +aq-1b(.), then the first eigenvalue A1 (h) of this operator can be written as

A1(h) = A1(a(1-q)/2l = J.t(a).

We denote by a ( -!:. + h-2b(.)) the spectrum of the Neumann realization of the Schriidinger operator -!:.. + h-2b( .) in L 2 (f.2). One way for giving lower estimates on A1 (h) is to use the counting number defined for 0 > 0 by

Nn(O) =card {A E a ( -t:. + h-2b(.)) : A::; 0}.

Notice that this has a meaning only if a ( -!:. + h-2b(.)) n ( -oo, OJ is finite. When b is nonnegative in JRN (N 2: 3) and satisfies liminfb(x) 2: co> 0, the

llxll->oo following estimate due to Cwikel, Lieb and Rozenblyum holds.

THEOREM 4. Suppose, N 2: 3. Then there exists a positive constant LN depending only on N such that

(14) NKI.N(O)::;LN ( dxd( J {11<11 2+h-2 b(x)s;o}

In particular,

(15) 1 ::; LN r dxdf; } { 11<112 +h- 2b(x)s;5:, (h)}

if we call ~ 1 (h) the first eigenvalue of-t:. + h-2b(.).

If b(x) is nonnegative and continuous inn, positive on an, we can extend it in JRN in some function b(x) satisfying liminfb(x) 2: co > 0; this will

llxll-+oo affect the behavior of A1 (h) by an exponentially small term in the sense that A1(h) = ~ 1 (h)+O (exp( -cjh)) for some c > 0. In our case, the last inequality yields

1 ::; LN ( dxdf; } { ll<ll2+a'-1b(x):SI'(a)+O( exp( -ca(q-1)/2))}

since n is bounded. From this estimate it is possible to derive a lower estimate on J.t(a).

REMARK 3. In dimensions 1 and 2, the Cwickel-Lieb-Rozenblyum formula does not hold. However, the use of another "moment-type" formula due


to Lieb and Thirring allows us to derive a sharp enough estimate on p(a). This formula (valid in any space dimension) asserts that for any 'Y such that 'Y + N /2 > 1, there exists a positive constant L'Y,N such that

L (p- Aj)'Y :<::: L'Y,N

{ ~;SP} I (

_ )'Y+N/2 p- h-2 b(x) dx

{ x:h- 2b(x)SP}

for any p such that p < inf a (-D.+ h-2b(.)).

Thanks to those previous estimates, we prove the following result.

THEOREM 5. Suppose that N :;:: 1 and that b is a nonnegative function, continuous in fl and positive on ofl. If In (1/b) E V(fl) for some p > N/2, then the TCS-property holds.

Proof. We restrict ourselves to the case N :;:: 3. From the above inequality we get

1- o(1) :<::: LN f dxdt; j {11<11 2 +a•- 1b(x)9,(<>)}

Since {x E fl: b(x) :<::: a 1-qp(a)} = {x E fl: In 1/b(x)):;:: In (aq-1 fp(a))}, then for a > 0 small enough,

Using the fact that In (1/b) E V(fl) (actually the Marcinkiewicz space MP(fl) would have been sufficient) yields

1/2 :<:::eN (!t(a))N12 (in (aq-l/p(a))rp I lin(1/b(x)JPdx.

n

We solve this inequality with respect to J-L( a) and get

p(a):::: CN (Jn(1/a))2p/N

1 lnp(a)da for any a E (0, a 1], for some a 1 > 0. If 2p/ N > 1, J ( ) < oo, and

o pa a the TCS-property follows from Theorem 3.


REMARK 4. The essential positivity assumed on b near an can be weakened if we assume that the function b is continuous in n and has only isolated zeroes on an. In such a case, the integrability assumption of Theorem 5 is sufficient to insure the TCS-property.

COROLLARY 1. Suppose that N ;::: 1, that b is continuous in n and - J

positive in n\C where C = u Cj and the Cj are Cl, d-dimensional compact J~l

submanifolds ofn with (0:::; d:::; N 1). Suppose that there exist C > 0 and 0 < O" < 2(N- d)/N such that

(16) b(x) :2: Cexp(-1/J(;(x))

for any x En, where b"c(x) = dist(x, C). Then the TCS-property holds.

Proof. By estimating the volume of small enough tubular neighbourhood of the Ci and using (15), it is easily seen that the integrability property of Theorem 5 holds.

As an interesting particular case, we have:

COROLLARY 2. Suppose that N ;::: 1, b is nonnegative and analytic in · n, continuous in n and positive on an. Then the TCS-property holds.

Proof. The set F = {x En: b(x) = 0} is a semi-analytic set in the sense given by Lojaciewicz. Moreover, it is a compact subset of n. Therefore, by a result of De Rham, it is a finite union of analytic submanifolds Cj with dimension dj, each of them being the graph of a function \l">j which satisfies IID\I">j II :::; M for some M ;::: 0. By Lojaciewicz's inequalities and the compactness ofF, there exist positive constants C and K such that

where b"p(x) = dist(x, F). The remaining part of the proof is as in Corollary 1. D

In the case of Dirichlet homogeneous boundary conditions, we have a stronger result since it is no longer necessary to assume that b is positive on the boundary.

THEOREM 6. Suppose that N :2: 1, and that b is a nonnegative and continuous function inn. /fin (1/b) E V'(n) for some p > N/2, then any solution of

{

atu- Llu + b(x)lulq-lu = 0 inn x (0, oo), (17) u =Oonanx(O,oo),

u(x, 0) = uo(x) inn,


satisfies the TCS-property. Clearly, Corollaries 1 and 2 still hold for the solutions of (17).

5. The Non-Compact Case. If p(x) is a bounded, nonnegative and measurable function defined in some smooth subdomain w C n, we denote

1/Jp,w the corresponding eigenfunction, and

(19) ( )

1-q V p,w J 1f; p,wdx

1+ w

( £ b1/(1-q)1f;p,wdx) 1 q

The following result is the key to the study of the non-vanishing property.

THEOREM 7. Suppose that SUPp,w Tp,w = oo. Then any solution of (1) such that ess inf uo(x) > 0 does not satisfy the TCS-property.

xErl

Proof. With no loss of generality, we can assume that u0 = 1. Multiplying (1) by 1f;p,w and integrating over w yields

! J u1f;p,wdx + Vp,w J u1f;p,wdx + J b(x)uq1f;p,wdx w w w

= J p(x)u1f;p,wdx- J UOv1f;p,wdS, w ow

and the right-hand side is nonnegative. By Holder's inequality,

(remember that 0::; q < 1). Setting y(t) = J u1f;p,wdx and w


we derive the following Bernoulli-type differential inequality:

(20) y' + vp,wY + Kyq :2': 0.

If z = yl-q, then

:! + (1- q)vp,wZ + K(1- q) :2': 0,

or

Consequently,

K z(t)e(l-q)vp,wt :2': z(O)-- (e(l-q)vp,wt- 1). ZJp,w

As long as

(21) t < 1 ln (1 + z(O)vp,w) = T*, (1- q)vp,w K

the right-hand side of (21) remains positive. Because z(O) = J 7/Jp,wdx and w

u0 = 1, we have T* = (1-q)- 1Tp,w· By assumption supp,w Tp,w = oo therefore for any t > 0, there exist some p and some w (depending on t) such that J u(x, t)!f;p,w(x)dx > 0. w

It is possible to replace the local Dirichlet eigenvalue problem associated to a weight function p by a Neumann eigenvalue problem in n (where the nonnegative function p is now defined). Setting

and 7/;p the corresponding eigenfunction, we have that the following result holds.

THEOREM 8. Suppose that

1 sup= -ln p?_O Vp

= oo,


then the conclusion of Theorem 7 is still valid.

The proof follows the same Jines as the one of Theorem 7. This criterion may be useful when b is degenerate near the boundary.

REMARK 5. Theorem 7 shows essentially the local non-vanishing property over a domain w c fl. In the particular case where p = 0, one has Vp,w = A!,w and '1/Jp,w =<!>wE W~'2 (w). It is clear that the relation

(23) 1

-In A1,w

= 00

implies that supp,w Tp,w = oo.

When b is degenerate only at isolated points, it may be useful to localize them by using balls instead of general subdomains w C fl. If y E fl, we denote

Ry = sup{r > 0: Br(Y) C fl}. Then .A1,B,(y) = cr-2 , k ::0: </>B,I(y)(x) I ::; k-1

r- y-x with k fixed provided the normalization </>B,(y)(Y) = r is imposed. Finally

and

j <!>B,(y)b1f(!-q)dx::; k- 1r j b1f(!-q)dx.

&M &M

The following result is a straightforward consequence of (23).

THEOREM 9. Assume that

sup sup r2

ln ( J bl~(I- )d ) = oo. yE(J O<r<Ry q X

B,(y)

(24)

Then the TCS-property does not hold.

We can also introduce other criteria, namely

(25) ln(l/a)

sup = oo, 0<<>9 ft(a)


(26)

THEOREM 10. The following implications hold:

(26) =} (25) =}sup Tp,w = oo. p,w

Proof. Step 1. (26) =;. (25). We recall that ¢w is a positive first eigenfunction of -ll in W~·2 (w). Then

!f.( a) I ¢~dx::; I (IIY'¢wll2 + aq- 1b(x)¢~) dx

w w

::; ( >'l,w + o;q-lllbiiL=(w)) I cp~dx, w

for any a E (0, 1]. Therefore !f.( a) ::; )'l,w + aq-lllbiiL=(w)' and

The particular choice of aq-l = .>'l,w/llbiiL=(w) (which is valid since Al,w/llbiiL=(w) is not bounded from above) yields

Therefore (26) =;. (25).

Step 2. (25) =} supp,w Tp,w = oo. For 'Y :::0: 1, we denote

Then


and (25) implies

ln 1/a lll'y oo = sup -- =sup--.

aE(O,I] p,( a) 7 ?1 v('Y)

Now

1 ( ( q J-q) ( ) lwy ln v('y) -In 1 + v 'Y) 'Y ::?: 1- q -( ) + q ( ) . ')' V')' V')'

Therefore

(27) In 'Y 1 ) sup-(-) = oo =?sup -In (1 + v('Y)q'YJ-q = oo. 1'?1 v 'Y 1'?1 'Y

Set ,(!;7 = 7/J7b'/Cl-ql, the eigenfunction corresponding to v('Y). Since

(28)

and

'Y J b1/(H),(j;pdx = v('Y) J ,(f;pdx,

f! f!

the right-hand side of (28) is just sup -(1

) In (1 + v('Y)q'YI-q), which is infinite ')'>! v 'Y

by (27). Therefore the condition supTp,w = oo follows. p,w

We end this section by quoting a result, which emphasizes the point-wise character of the non-vanishing property.

THEOREM 11. Suppose that b is a continuous and nonnegative function defined in 0, which satisfies for some Xo E [l

~~ r2ln (1/llbiiLoo(B,(xo)) = 00.

Then any nonnegative weak solution u of (1) whose initial value uo is such that u0 (x) ::?: c: > 0 a. e. in some neighbourhood of xo satisfies u(xo, t) > 0 for any t > 0.


6. The Exponential Border-Line Case. As we hitve seen, nonnegative analytic potentials fall into the scope of the TCS-property, and therefore the limit case lies with potentials with non-analytic type fiat minima. We give below a short review of such type of flatness.

CoROLLARY 3. Suppose that b(x) = exp( -1/llxllfl), for some (3 > 0 (assuming that 0 E fl and N 2 1}, then (i} if 0 < (3 :'0 2, the TCS-property does not hold, (ii} if (3 2 2, the TCS-property holds.

Proof. If 0 < (3 :s; 2, the claim is just Corollary 1. If (3 2 2, we have for some K > 0

2 ( 1 sup r In J bl/(l D<r<R

B,(O)

and we conclude by Theorem 9.

2 K sup r2-fl = oo, O<r<R

It must be noticed that the case (3 = 2 is not reached by our techniques.

REFERENCES

[1] Y. Belaud, B. Helffer, and L. Veron, Long-time vanishing properties of the solutions of some semilinear parabolic equations, preprint, 1999.

[2] V. Arnold, A. Varchenko, and S. Goussein-Zade, Singularites des applications differentiables, 2. Monodromie et comportement asymptotique des integrales, ed. Mir, Moscou (1986).

[3] M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrodinger operators, Ann. Math. 106 (1977), 93-100.

[4] B. Helffer, Semi-classical analysis for the Schrodinger operator and applications, Lecture Notes in Math. 1336, Springer-Verlag, (1989).

[5] B. Helffer, and J. Sjostrand, Multiple wells in the semi-classical limit I, Comm. Part. Diff. Equ., 9 (1984), 337-408.

[6] V. Kondratiev, and L. Veron, Asymptotic behaviour of the ·solutions of some parabolic or elliptic equations, Asymptotic Analysis, 14 (1997), 117-156.

[7] E. Lieb, and W. Thirring, Inequalities for the moments of the eigenvalues of the Schrodinger hamiltonian and their relations to Sobolev inequalities, in Studies in Math. Phys. essays in Honnor of V. Bargmann, Princeton Univ. Press, 1976.

[8] S. Lojaciewicz, Ensembles semi-analytiques, Institut des Hautes Etudes Scientifiques, Bures sur Yvette, France 1964.

[9] V. Mazya, Sobolev Spaces, Springer-Verlag, 1985. [10] G. Rozenblyum, Distribution of the dicrete spectrum of singular differenial opera

tors, Doklady Akad. Nauk. USSR, 202 (1972), 1012-1015 (in Russian).


VOLUME 8 2001, NO 1-2 PP. 49-56

SOME SEMILINEAR ELLIPTIC EQUATIONS WITH SINGULARITIES AT THE BOUNDARY

M.F. BIDAUT-VERON " AND L. VIVIER t

Abstract. We study the behaviour near the boundary of the nonnegative solutions of the semilinear elliptic problem in a bounded regular domain n of rrr'

{ -Au=u• inO, u = p, on 80,

where p, is a Radon measure on 80. We give a priori estimates and existence results in the subcritical case 1 < q < ~!\. They lie on regularity properties of superharmonic functions in weighted spaces.

1. Introduction. Let !1 be a bounded domain in JRN with a C2 boundary 8!1. Here we consider the behaviour near the boundary of nonnegative solutions of the semilinear equation

(1.1) inn,

when 1 < q < (N + 1)/(N- 1). By solution we mean any function u such that uq E L£oc(!l) and satisfying the equation in D'(!l).

We extend the results of [11 J and [10] concerning an interior isolated singularity x 0 E !l :

(1.2) in !l\ {xo},

when 1 < q < Nf(N- 2). Recall that in that case, uq E Lioc(!l), and there exists some 1 2 0 such that

(1.3) in D' (!l).

' Laboratoire de MatMmatiques et Physique TMorique, CNRS UPRES-A 6083, Faculte des Sciences, Pare de Grandmont, 37200 Tours, France

t Universite de Toulon-Var, UFR Sciences et Techniques, Laboratoire ANLA, BP 132, Avenue de l'Universite, 83957 La Garde Cedex, France

49

50 M.F. BIDAUT-VERON AND L. VIVIER

Concerning the existence of solutions of (1.3) for a given 1, there exists 1' > 0 such that equation (1.3) admits a solution ·u 2: 0, with u = 0 on an, if and only if 1 E (0, 1']. Moreover, any solution of equation (1.2) satisfies the a priori estimates

(1.4) 1 E(xo,x):::; u(x):::; 1 E(xo,x) (1 + o(l))

near x0 when 1 > 0, where E is the fundamental solution of the Laplace equation. For proving (1.4), the method of [lO]lies on the fact that u is in the Marcinkiewicz space M1~f(N-2) (D.), hence u E Lj0~'(D.) for some c > 0. Then one can estimate the difference u- 1 E from Vllf~~+'(D.) regularity results and Sobolev injection theorem, by using a bootstrap technique.

Let us come back to the boundary problem. The first question is to define a trace f1 of u on an in a suitable sense, which will play the role of 1 50 for the interior problem. The second one is to construct solutions for a given trace fl· The third one, which is the most important, is to find a priori estimates of u with respect to f1 analogous to (1.4).

For simplicity, we suppose N 2: 3, but our main results are true when N = 2. We denote by p(x) the distance from any point XED. to an. Let g be the Green function of the Laplacian in D. defined on the set { (x, y) E D. x D. I X # y} . Let p be the Poisson kernel defined on D. X an by p (X' z) = -aQ(x, z)jan. We call M+(Q) and M+(aD.) the sets of nonnegative Radon measures on D. and an, respectively.

Recall that any superharmonic function U 2: 0 in D. satisfies U E Lfoc(D.). From the Herglotz theorem, there exist some unique <p E M+(Q) and v E M+(aD.) such that U admits the integral representation

(1.5) U = G(<p) + P(v),

where, for almost any x E D.,

(1.6) G(<p)(x) = j Q(x,y)d<p(y), P(v)(x) = J P(x, z)dv(z); 80 n

moreover, I pd<p < +oo. Thus U admits a trace v in the sense of measures. n

Conversely, for any <p E M+(Q) such that I pd<p < +oo and v E M+(aD.), n

the function U defined by (1.5) lies in Lfoc(D.), and satisfies the equation

SOME SEMILINEAR ELLIPTIC EQUATIONS 51

-t::.U = cp in 1J'(fl), see for example [6]. We shall say that U is the integral solution of the problem

(1.7) { -t::.U = cp in fl, U = v on {)fl.

This answers to the first question: any solution u of (1.1) in fl satisfies J pu9dx < +oo, and there exists a measure f.l E M+(afl) such that n

(1.8) { -t::.u = uq in fl, u = ,u on {)fl.

In order to solve the two other questions, we need some regularity results for the function u. In Section 2, we give precise regularity properties for the solutions of linear problem (1.7) in Marcinkiewicz spaces Mk(fl, rydx) (k 2: 1) with a weight 77 of the form rf3 ((3 E JR) (see [3] for the definition of those spaces). Most of them are new and can present an interest in themselves. As a consequence, we get corresponding estimates for u. In Section 3, we give existence results of solutions of (1.8) for a given measure f.l on {)fl.

In Section 4, we solve the question of a priori estimates near the boundary. We refer to [3] for complete proofs.

2. Regularity results. First we give estimates of the functions P(v) and G(cp). They are based on precise estimates of the Green function and the Poisson kernel. First we can show that the definition of solutions U of (1.7) in an integral sense coincides with the notion of weak solution introduced in [9]: it means that U E L1 (fl) and

(2.1) J U( -t::.~)dx = J ~dcp + J ~~ dv n n an

for any~ E C 1 (fl) vanishing on {)fl with \7~ is Lipschitz continuous. Then we give estimates for P(v).

THEOREM 2.1. For any v E M+(afl), let 1J! = P(v) be the solution of the problem

(2.2)

Then

(2.3)

{ -t::.\l! = 0 1J!=v

in fl, on {)fl.

for any (3 > -1,


(2.4) for any 'Y > 0.

This completes the estimates of [9] given for the weight p. Then we give estimates for G ( 'P), which appear to be new, more especially as the measure = G(<p) be the solution of problem

(2.5) { -~= 'P inn, = 0 on on.

Then

(2.6) E M(N+fJl/(N-z+c.l(st, /dx)

forany(J E (-N/(N+a-1),aNj(N-2)) ifa # 0, andforany(J E

( -N/(N- 1), OJ if a= 0. And

(2.7) I'VI E M(N+"!)/(N-l+a)(O,p"~dx),

for any 'Y E [O,aN/(N -1)) if a E (0,1), any')' E (O,Nj(N -1)) if a= 1, and 'Y = 0 if a = 0.

As a consequence, for any 'P E M+(st) such that J pd<p < +oo and any n

v E M+(ost), the solution U of (1.7) satisfies in particular

(2.8) U E M(N+fJ)/(N-I)(O, I dx) for any (3 E ( -1, Nj(N- 2)),

(2.9) IDUI E M(N+"!)/N (11, p"~ dx) for any 'Y E (0, Nj(N- 1)),

and this applies to u. By using an interior bootstrap, it implies also that u E C""(ll). But we cannot hope to use W 2·P regularity results up to the boundary as before.

3. Existence Results. Our first main theorem gives existence results. It lies on an a priori estimate for the function G(Pq(v) ), for any v E M+(oll), which is proved by using again upper estimates of the Poisson kernel.

THEOREM 3.1. Assume that 1 < q < z=i· Then Pq(v) E L 1(Sl,pdx) for any v E M+(oll), and there exists a constant K = K(N, n, q) such that

(3.1) in st.


If, moreover, v = Oa for some point a of 80., then there exists a constant C = C(N,0.,q) such that

(3.2) Pq(v)(x) ::::; C P(x, a) lx- aiN+!-(N-l)q in 0..

As a consequence, we get existence results which extend those of [10] to the boundary value problem.

THEOREM 3.2. Assume 1 < q < ~~i. Let p, E M+(80.) with p,(80.) = 1, and a ::0: 0. Then there exists some finite positive a* such that the problem

(3.3) { -f:lu = uq in 0., u = o-p, on 80.

admits a solution if and only if a E [0, a'].

Proof. The existence of solutions for small a > 0 is a direct consequence of Theorem 3.1. The function aP(p,) is a subsolution of (3.3). We can construct a supersolution of (3.3) of the form y =a P(p,) +a G [Pq(a p,)] for some a > 0, and get the existence of a solution. The existence of an interval [0, a*] is an adaptation of some results of ([4]). D

REMARK 3.3. Observe that the condition q < ~~i is sharp: if q ::0: ~~i, then problem (3.3) has no solution for any a > 0 when p, is a Dirac mass Oa at some point a of 80.. Indeed the existence would imply P(oa) E Lq(0., pdx). Now using lower estimates for the Poisson kernel, we can show that for any (3 > -1, P(oa) E £k(fJ, pi3dx) if and only if k < ~~q.

REMARK 3.4. Recently and independently of our work, Amann and Quittner [1] have proved existence results in the case q < NN_ 1. They do not use spaces with weights, but duality techniques. They show the existence of at least two solutions when a< a*. They also prove that u E W 1-'·1(0.) for any c E (0,1).

4. A Priori Estimates. Our second and main result is an a priori estimate of any solution of (1.8) in terms of the solution P(p,) of the associated linear problem, up to the value ~~i. It lies essentially on the regularity theorems 2.1 and 2.2. It also uses estimate (3.1), which in fact can be shown almost as a necessary condition of existence of solutions, by using recent techniques of [4].


THEOREM 4.1. Assume that 1 < q < ~~i. Let fl. E M+(iJ12), and let u be any nonnegative solution of (1.8). Then there exists C -C(N,q,n,f.J.(an)) > 0 such that

( 4.1) P(f.l.) ::; u::; C (P(f.l.) + p) in n.

More precisely, if fl. = u ba for some a E an and rY > 0, then

(4.2) rY P(ba)(x)::; u(x)::; rY P(ba)(x) [1 +C(!x- a!N+l-(N-1)q)].

Proof. First we show as in [7] that for any q > 1, for any solution u of (1.8), l!uq!!u(n,pdx) is majorized independently of u. Now we distinguish two cases.

i) The case q < ;!..1

. Here we do not need to use regularity in spaces with weights. Let us set

where u 1 = G(uq). Now u E MN/(N-1)(n) from (2.8). Since q < NN_ 1, we have uq E Lko(n) for some k0 > 1. Now uq ::; 2q-1(Pq(f.l.) + u'f), hence from the maximum principle and from estimate (3.1) we obtain

where u2 = G(u'i.), hence u::; C;( P(f.l.) + u2). By induction for any n;:::: 2, we can define Vn = G(v~_ 1 ), and we get

where Cn, C~ depend only on n, N, q, nand fl.( an). And Un E Lk•(n) with

kn = N kn-1/q (N- 2 kn-1) > kn-1>

till kn < N/2. But if kn < N/2 for any n EN, then kn -'t R. = N(q-1)/2q < 1, which is impossible. Then changing slightly k0 if necessary, we find some no= no(N, q) such that kn0 > N/2, hence Vn0 E C0 (n). Then

(4.3) inn.

with Vno+l = 0 on an, hence there exists a constant Co > 0 such that

(4.4) Vno+I(x)::; Co p(x) inn,


and C0 depends on N, q, rJ, J-t(afl) and lluqiiP(n,pdx)· Then (4.1) follows from (4.3) and (4.4). In the case J-! = a8a , we deduce (4.2) from (3.2).

ii) The case /..1 :::; q < ~~i. Here only we use spaces with weights. Let r 2:: 2 be some fixed integer such that

1/r < N + 1- (N- 1)q,

and for any n E [0, r], let

f3n = 1 - njr E [0, 1 J. Now we start from the fact that u E M(N+1)/(N-1l(fl, pdx) from (2.8). Let uo = u, hence

ug E L''(rl,/'dx), with 1 < r0 < (N + (30)/(N -1)q.

Here again we define u1 = G(uq). From Theorem 2.2, for any E > 0 small enough we have

for any (3 E (-1,N/(N- 2)). Taking (3 = (31 = 1-1/p E (0, 1), we get

(4.5) uj E L''(rl,/'dx), with 1 < r 1 < (N +(31)/(N- 2+f3o)q,

since N + (31 - (N- 2 + (30 )q = N + 1- (N- 1)q- 1/p > 0 . By induction, for any n:::; p, we can define Un = G(u~_ 1 ) in V(fl), and we get

Un E L(N+f3)/(N-2+f3n-l)-<(rJ, /dx)

for any (3 E (-1,(3n_ 1N/(N- 2)). Taking {3 = f3n 2::0, we have

(N + f3n)- (N- 2 + f3n-1)q > (n- 1)(q- 1)/p > 0,

hence

(4.6) U~ E £Cn(fl,/ndx), with 1 < rn < (N + f3n)/(N- 2 + f3n-!)q.

Now in case n = p, we have (Jp = 0. This proves that u~ E £"P(rl), with rp > 1 and we are reduced to the first case: there exists an integer n0 = n0 ( N, q) such that ·uno+p E C0 (rl). We deduce (4.1) and (4.2) as above.D

REMARK 4.2. In the case q 2:: ~~i, let u be a solution of (1.2) with a possible singularity at the point a E an. Its trace I' is zero by Remark 3.3.


Following the results of [8] for the interior problem, we conjecture an estimate of the type

u(x)::; Cp(x) lx- al-(q+l)/(q-l)

near a, at least when q < ~~~-The problem is open.

REFERENCES

[1] H. Amann and P. Quittner, Elliptic boundary value problems involving measures: existence, regularity and multiplicity, Advances in Diff. Equ., 3, 6 (1998), 753-813.

[2] M.F. Bidaut-Veron, Local behaviour of solutions of a class of nonlinear elliptic systems, Advances in Diff. Equ.,5, 1-3 (2000), 147-192.

[3] M.F. Bidaut-Veron and L. Vivier, An elliptic semilinear equation with source term involving boundary measures: the subcritical case, Revista Matematica Iberoamericana (to appear).

[4] H. Brezis and X. Cabre, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Italiana, 8 (1998), 223-262.

[5] H. Brezis and P.L. Lions, A note on isolated singularities for linear elliptic equations, Math. Anal. Appl. Adv. Math., Suppl. Stud., 7 A (1981), 263-266.

[6] R. Dautray and J.L. Lions, Analyse mathematique et calcul numerique, Masson, 1987.

[7] D.G. De Figuereido, P.L. Lions and R.D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures et Appl., 61 (1982), 41-63.

[8] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598.

[9] A. Gmira and L. Veron, Boundary singularities of solutions of some nonlinear elliptic equations, Duke Math. Journal, 64 (1991), 271-324.

[10] P.L. Lions, Isolated singularities in semilinear problems, J. Diff. Equ., 38 (1980), 441-450.

[11] J. Serrin, Isolated singularities of solutions of quasilinear equations, Acta Math. 1134 (1964), 247-302.


VOLUME 8 2001, NO 1-2 PP. 57-68

ON THE EXISTENCE OF THE SOLUTION OF MAXWELL EQUATIONS IN A WAVEGUIDE

A.N. BOGOLYUBOV, A.N. DELITSYN, AND A.G. SVESCHNIKOV'

Abstract. The problem of uniqueness and existence for Maxwell equations in cylinder is considered. The coefficients of equations don't change along axis variable and depend only on the transverse coordinates. The problem of radiation conditions is considered. The existence in some functional space is found.

Introduction . The consideration of the problem of existence and uniqueness of Maxwell equations in the case of harmonic dependence on time e-iwt or the problem of waveguide excitation is considered. The coefficients of equations permittivity and permeability depend only on transverse coordinates and have no dependence on axis variable z. This problem was considered by different authors [1-3]. The problem is considered in a cylinder and formulation of radiation conditions is an important question. For such formulation the homogeneous equations are considered. The consideration of mode type solutions reduce the problem to the two-dimensional spectral problem. The homogeneous Maxwell equations make a system of eight equations of first order for six unknown functions. This system is not studied directly and is reduced to the equations of second order. There are different approaches for formulation of the spectral problem. It is necessary to emphasize that a mathematically correct solution of the problem of waveguide excitation have been found by A.A. Samarskiy and A.N. Tikhonov only for the case of constant coefficients. By using Hertz potential they reduce the problem to the Helmholtz equation. In the case of variable coefficients such method can not be applied. In the study of P.E. Krasnushkin and E.I. Moiseev [2] the problem is reduced to a J - symmetric operator of second order. This approach permits to prove the completeness of eigenvectors

• Moscow State University, Faculty of Physics, 119899 Moscow, Russia

57

58 A.N. BOGOLYUBOV, A.N. DELITSYN, AND A.G. SVESCHNIKOV

and the existence of the approximation of the solution, but n.ot to solve the problem of the existence of the solution. Strong requirements for regularity of coefficients have been formulated in their method. Another approach has been applied by Yu.G. Smirnov [3]. In this study the problem is reduced to the system of equations for longitudinal components of fields. The spectral problem is nonlinear now and there are no general results for .such problem. In the case of piecewise constant coefficients the problem is reduced to the Helmholtz equation but spectral parameter presents in equation as well as in conjugation conditions. For a particular case the completeness of eigenvectors has been proved. Many other approaches have been considered by physicists, but the mathematical consideration has not been given. All these approaches were based on the consideration of curl equations of Maxwell as the main one. The equations for div operator were considered as supplementary ones. We are considering the equations with the derivative along the waveguide axis as the main ones and two other equations as supplementary ones. This permits us to reduce the problem to a problem of Keldysh type operator pencil [4]. The completeness of the generalized eigenvectors is a consequence of the well-known Keldysh theorem [5] now. This allows us

~ to formulate the radiation conditions. The existence and uniqueness of the solution of Maxwell equations in some functional space is proved.

We consider the system of Maxwell equations in the case of variable coefficients:

(1)

(2)

't=

'i7 X E =ikE, . 41T .

'i7 x H = -zkD + -J, c

'i7 D = 47rp, 'ii'B = 0,

D = e(x, y)E, B = Ji(x, y)H,

( c!J CJ2

c33 ) '

( ~11 ~12

~3J CJ2 C22 Ji = ~i2 ~22

't=E*, Ji=Ji*, al<c:<(Jl, al<~<(Jl.

We will use the designations :

c = Eij, ~ = ~ij, i,j = 1,2.

SOLUTION OF MAXWELL EQUATIONS IN A WAVEGUIDE 59

The standard approach for consideration of Maxwell equations consists in investigation of equations (1) of curl type. Equations (2) are considered as differential consequenses of equations ( 1).

We suppose that current j and charge p have a finite support and are connected by the equation of continuity

(3) divj - iwp = 0.

The problem is considered in cylinder Q = ((x,y) E O,z E (-oo,oo)). The functions of dielectric permitivity c and magnetic permitivity p, are

piecewise continuous. On the boundary of domain 8Q and on the surfaces of discontinuity we must use boundary and matching conditions

(4) Ex nlaQ = 0, BnlaQ = 0

Problem (1)--(5) must be completed by radiation conditions. This is the main problem. The formulation of radiation conditions permits us to consider a boundary problem and prove the existence and uniquness of solution.

Our approach consists in the consideration of another formulation of the problem, which differs from the ordinary one. We use the equations for div operator (2) and equations for rot operator with %z derivative and represent equations (1)-(2) in the form

(=~: ik

') (Bx) _ c=:):1 (c-1 hz ) 0 ( Dx) (6) 0 &y By - (c: )12 (c:-1)zz az Dy '

_.§.. 0 Ez P33 H. 8x 8y

(~~8 -ik ~) (Dx) _ 0 ay Dy -_.§.. 0 H.

(7) ox IJy

(,.-1)11 (p,-1)12 ) [) (Bx) 4n (-jy) (P-1)iz (p,-1 b 7) By +- ix , 1':33 z Ez c -cp

(8) 8 ( -1 ) 8 ( -1 ) .k 4n . ""i'l p B y - -;:) fJ, B x + l c33Ez = -J., uX uy C


( ) a ( -!D) a ( -!D) . 9 ax E y - ay E x + ~k/L33Hz = 0.

We will use designations

&x & )

~ '

-ik .!}_ ) &x 0 .!}_

&y ' _.!}_ 0

&y

J 47l' ( . . )T =- -]y,]x, -cp ' c

rot A.l = aAy _ a Ax. ax ay

We will consider equations (6)-(7) as the main and equations (8)-(9) as supplementing differential conditions. In this notation, equations ( 6)-(7) take the form

(10)

(11)

The peculiarity of operators a1 and a2 is the presence of infinite dimensional kernel of the kind

N1 =(rot¢, ik<f;) for a1,

N2 = 3D(rot ¢, -ik¢) for a2 •

with any substitute ¢. This leads to the following result.

LEMMA 1. Let vector J have the form

J = (rot¢, ik¢),


where the vector z2 .

Then there exists a solution of (1) -( 5), which has the form

Proof. For z > z1 we have

z z

A1 = J d21Jdz = j(rot<jJ,ik<Pfdz = (rot'ljJ,ik'I/J)T, -00 Zl

and by direct eomputation it is easy to show that A1 = 0, z > z2 .

We consider the problem further in the case when the parameter k is not equal to the eigenvalue of the problems

(12)

and

(13)

rot C 1 rot X = k2 J.lssX, (x, y) E !:1,

f-1 rot x x ni = 0

an

xi =0. an

THEOREM 1. Let k be not equal to the eigenvalues of problems (12) -(13). The vector J E L~(n), !:1 is a domain with a Lipschitz continuous


boundary J.tlnfan = 0, lzfan = 0. Then there exist</; E ifl, FE H(div, !1) Ell H 1(!1) such that

(14)

Proof. Let </; E H 1 be the solution of the problem

Under our assumptions this problem has a unique solution. Let us take this equation in the form

We use designations A= (A1., Azf, where

Vector A is the solution of the equation

with boundary condition

because (J.!Al.n) fan = 0, <Plan = 0 and that is why rot </;nfan = ~~ = 0. Let us decompose J.!A1. as

Using the definition of vector A1., we obtain

(grad '1/;, rot"¢)!_, + (J.t-1 rot x, rot"¢)£, + ik(A, ;f)L, = 0, \/</; E H 1

or


From this result we deduce Jt-1 rot x E H(rot) and the following equality takes place

- i -1 Az = -;;;rotJt rotx.

Let us take Az in the form

Because A_L =grad 't/J + Jt-1 :1 grad x, we have

We use designations

(F_L, Fz) E H(div) EB H 1 . As a result, we obtain

) ( rot</> ) ( -ik:T

egg ik</> + - div

The proof of Theorem 1 is finished. D

In this case the vector J = J1 + Jz.

Function </> is a solution of the problem

(15) (x, y) E !1,


<Plan= 0.

We consider in Lemma 1 a solution of problem (1)-(2) with vector J = J2. Now we will consider the case J = J1. The consideration of equations (10)-(11) may be reduced to one vector equation for vector potential A. We consider the problem

(16)

or

(- grad c 3} di v + k2 .J f.JJ

ik div f.JJ

( ~1 ~).

ik.Jf.J ) (Aj_) = d2.!!_ (Aj_) + (Ju) - div f.J grad Az 8z2 Az J1z '

div Aj_ = 8A, + aAy grad A = ( 8A, 8A,) .J = ax 8y' z 8x'8y'

We must add to equations (16) the equation

(17)

and conditions

(18)

(19) [Aj_nJj

8 = 0, [AzJis = 0,

[f.J331 div Aj_Jj8

= 0, [c(gradAz + ik(A X ez))nJj8

= 0.

Vectors A1 , A2 are determined as

(20)

(21)

It is easy to see that if A is a solution of problem (16)-(19), vectors A1 , A2 are solutions of problem (1)-(5). We will consider now the problem of radiation conditions. We consider homogeneous problem (16)-(19) when J1 = 0 and we deal with solutions of the form A= A(x,y)ehz. After cancelling the exponential multiplier ehz, we obtain the problem


with conditions (17)-(19), which is a two-dimensional problem. Problem (22), (17)-(19) may be reduced to a form satisfying the assumptions of the Keldysh theorem. We define Hilbert space V = (A E H0 (div, n) Ell iii (n), (rot J.L-I A)z + ikc33Az = 0)).

We recall our results from [4]. The imbedding of V in to L~(n) is compact. The system of generalized eigenvectors of problem (22), (17) - (19) is complete in the functional space V.

All eigenvalues 'Y~, except maybe a finite number, are in the sector larg('Y;;.ax- 'Y~)I < /3, where f3 is arbitrary small, 'Ymax is a constant. The eigenvalues 'Y~ have asymptotic behavior 'Y~ = O(n).

There exists only a finite number of 'Y~ for which 'Y~ > 0. Let AI, ... AN be a system of eigenvectors, for which 'Y;; > 0. Then the system of vectors

(23)

is biorthogonal to all generalized eigenvectors of problem ( 22), ( 17)-( 19). Now we will formulate radiation conditions. We will say that a vector A is under radiation conditions if it has the form

(24) A= L CnAn(X, y)e±hnz +A, n=l,N

(25) A= L dnAn(X, y)e±i'Ynz +A, n=l,N

where

00

(26) J IIAll~dz < oo, (A, An) = 0, n= l, ... ,N. -00

First of all, we consider the calculation of coefficients c,., dn. Let the vector Jf be equal to

(27) J{" = L (h An)An. n=l,N

For the vector J[', we consider the problem with radiation conditions

(28) A= L CnAn(X, y)e±hnZ, Z > Z2,

n=l,N


(29) A= L dnAn(X, y)e±hnZ, Z < Zt.

n=l,N

We will consider the solution of this problem in the form

N

(30) A= L Zn(z)An(X, y). n;;;;;;l

As a result, for coefficients Zn we obtain the equations

(31)

(32)

(33)

where Jn(z) = (J1, An). From these equations coefficients Zn(z) are determined uniquely.

Now we consider the problem of determination of the vector A, when the vector J 1 = J1 - Jf. We must give a definition of a weak solution. Let us consider semilinear forms

We will consider the subspace V1 of the space V, which is determined by conditions (A, An) = 0, n = 1, ... , N. We must find a vector A E V1 which is the solution of the equation

(34) - - a2 - -~- - -

a(A,A) +b(A,A) = az2 c(A,A) +c(d2 J 1,A),\IA E V1

It is simple to show that the solution of problem (34) is also the solution of the same problem for any vector A E H0(div, fl) EB H1. Problem (34) is equivalent to the problem

a2 (I +C) = az2 H A + 7J,


where operators C, H and vector 7J are determined from the equations

a(CA,A) = b(A,A),

a(H A, A) = c(A, A),

Because of the compactness of the imbedding of the space V1 into £~(0) one can prove the compactness of the operators C, H. The operator H is selfadjoint. Let us consider the problem

where 00

1JF = 2~ J e-hz1J(X, y, z)dz. -oo

THEOREM 2. The solution of problem (34) has the form

-oo J ehz (I+ C + ,.,/ Htl1JF d"f.

00

Proof. By formal computations we get

00

-oo -oo J r(ehzd!- J (I+ C + 1 2H)-17JFehzd'Y = 1J --(I+ C)A.

00 00

We must show that vectors A and :;, H A are from the space:

-oo

-j IIAII~dz < oo, 00 00


Indeed,

-oo -oo

j IIAII~dz = 21r j [[AF[[~dr = 00 00

-oo -oo

21r j [[(I+C+r2Ht1r/"[[~dr<M j [[rt[[~dr<oo, 00 00

and r2 [[HAF[[v E Lz(-oo,oo). Furthermore,

[[~HAF[[v = [[r2H(I +C +r2HtV"liv =

We use the well-known fact [5] that the norm of an operator is bounded on V1 for r E ( -oo, oo).

So we get the solution of problem (34). The sum of solutions of the problems with conditions (28)-(29) and problem (34) will be a weak solution of problem (16)-(19) with radiation conditions (24)-(25).

REFERENCES

[1] A.A. Samarskiy and A.N. Tikhonov, About excitement of radiowaveguide. Journ. of Techn. Phys., 11 (1947), 1283-1296 (in Russian).

[2] P.E. Krasnushkin, and E.I. Moiseev, Soviet Acad. Sci. Dpkl. Math., 264, 5 (1982), 1123-1127 (in Russian).

[3] Yu.G. Smirnov, Soviet Acad. Sci. Dpkl. Math., 271 (1987), 511-514 (in Russian). [4] A.N. Bogolyubov, A.L. Delitsyn, and A. G. Sveshnikov. Journ. of Comp. Math., 39,

11 (1999), 1791-1813. [5] I.Tc. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint

operators, M. Nauka, 1965 (In Russian).


VOLUME 8 2001, NO 1-2 PP. 69-88

AN APPLICATION OF POWER GEOMETRY TO FINDING SELF-SIMILAR SOLUTIONS*

A. BRUNO t

Abstract. Here ideas and algorithms of Power Geometry are applied to a study of a partial differential equation. We put in correspondence to each differential monomial a point in JR:n, that is, its vector power exponent. To the differential equation there corresponds its support, that is, a set of the vector power exponents of its monomials. The affine hull of the support is called a super-support, and its dimension is called the dimension of the equation. If the dimension is smaller than n, then the equation is quasihomogeneous, and it has quasihomogeneous (self-similar) solutions. Such a solution is defined by a function of a smaller number of independent variables. Here it is shown how to calculate all kinds of self-similar solutions of the equation by means of the methods of linear algebra using the support of the equation. Equations of the combustion process without a source and with a source are considered as examples.

Introduction. The Power Geometry is a new calculus developing the differential calculus and aimed at nonlinear problems. Its main concept consists in the study of nonlinear problems not in the original coordinates, but in the logarithms of these coordinates. Then many properties and relations, which are nonlinear in the original coordinates, become linear. The algorithms of the Power Geometry are based on these linear relations; they allow to simplify equations, to resolve their singularities, to isolate their first approximations, and to find either their solutions or the asymptotics of the solutions. These algorithms are applicable to equations of various types: algebraic, ordinary differential and partial differential and also to systems of such equations. An elementary introduction to the Power Geometry for the algebraic and ordinary differential equations is expounded in Chapters I and

' Supported by RFBR, Grant 99-01-01063 t Department of Mathematics, Keldysh Institute of Applied Mathematics, Miusskaja

Sq. 4, Moscow 125047, Russia

69

70 A. BRUNO

II of the book [1]. The book [2] contains a more advanced presentation for all types of equations.

Here some ideas and algorithms of the Power Geometry are applied to a single partial differential equation without parameters. Namely, we put in correspondence to each differential monomial a point in !Rn, which is its vector power exponent. To a differential polynomial, which is a finite sum of differential monomials, we put in correspondence its support, i.e. the set of vector power exponents of its monomials. The affine hull of the support is called the super-support, and its dimension is called the dimension of the differential polynomial. To each differential polynomial there corresponds a · differential equation obtained by equating of the polynomial to zero. Such an equation has a support, a super-support, and a dimension defined by the polynomial. If the dimension is smaller than n, then the differential polynomial is a quasi-homogeneous one, and the corresponding equation has quasi-homogeneous (self-similar) solutions. Such a solution is determined by a function of the smaller number of independent variables. Here it is shown how to compute all types of self-similar solutions with the help of linear algebra using the support of the equation. As examples, the equations of the combustion process without a source and with a source are considered.

1. Supports of a Function. Let X = (xb ... , Xn) E ocn, where lK is c or lR, and Q = ( q1 , •.• , qn) E !Rn; we denote by XQ the monomial xi' ... x~n. To the sum

(1.1) f(X) = 'f:.JqXQ over Q E S,

in which the coefficients f Q E lK and all the similar terms are collected, we put in correspondence the set

(1.2) S(f) = { Q: fq # 0}

in the space !Rn. That set S (f) is called the support of sum ( 1.1). Let SS (f) be the smallest linear manifold in !Rn containing the set S (!); we call

it the super-support of (1.1). The number d(f) '1gf dimSS(f) is called the dimension of sum (1.1).

If the function f(X) is not a sum of form (1.1), then it may not have a pointwise support (1.2), but it may have a super-support SS(f). Let d < n, vectors B~o ... , Bd E !Rn be linearly independent, and the vector R E !Rn. Let g(y~o ... , Yd) be an arbitrary smooth function in its arguments,

AN APPLICATION OF POWER GEOMETRY 71

and agj[}yj i= 0, j = 1, ... , d. Then the linear manifold in ~n

d

(1.3) {Q: Q = R+ LPjBj, {Lj E ~n} j=l

is called the super-support of the function

(1.4)

and we denote it by SS(f). For sum (1.1), both definitions of the super

support SS(f) coincide. The number d(f) ~ dim SS(f) is called the dimension of function (1.4).

EXAMPLE 1. Let n = 2 and f = x1 +x2. Then the support S(f) consists of two points E1 = (1, 0) and E2 = (0, 1), and the super-support according to the first definition is the straight line

(1.5) Ql + q2 = 1

passing through points E1 and E 2 , hence d(f) = 1. Its normal vector is N = (1,1). If we write fin the form (1.4) as f = x1(1+x}1x2), then d= 1, B1 = N, and according to the second definition (1.3), the super-support SS(f) is the straight line

(1.6) SS(f) = {Q: Q = (1, 0) + p( -1, 1), p E ~}

with the directing vector B1 = (-1, 1). When p = 0 and p = 1, it passes through the points E 1 and E2 respectively, i.e. it coincides with the straight line (1.5). Notations (1.5) and (1.6) are dual descriptions of the same straight line.

We consider now the space JR;: that is dual to the space ~n. Thus for P = (p1, ... , Pn) E JR:: and Q = (q1, ... , qn) E ~n the scalar product (P, Q) =

P1q1 + ... + Pnqn is defined. Let K be a linear manifold in ~n, and N = N(K) be the maximal

linear subspace in JR::, which is normal to the manifold K For a function f of form (1.4), together with the super-support SS(J), we will consider its

normal subspace N(J) ~ N(SS(f)) C JR::. Let vectors N1 , ... , Nn-d E JR:: be linearly independent and normal to vectors B1 , ... , Bd. Then they form the basis of the normal subspace N(f) of function (1.4).

72 A. BRUNO

2. Supports of a Differential Polynomial. The priine here marks vectors of length n-1. Thus, X'= (x1 , ... ,xn_1). The differential monomial a( X) is the product of powers of coordinates X and the derivatives of the form

(2.1) <>IIL'IIx /"'xl' "'xln-l v n u 1 ... u n-1'

where L' = (h, ... , ln-1) E z~-\ i.e. all li 2: 0 and integer, and IlL' II = 11 + .. . +ln- 1 • To each differential monomial a( X), we put in correspondence the point Q(a) E JRn (its vector exponent) by the following rule: the point Q corresponds to the monomial constXQ; the point .

(2.2) Q = (-£',1), i.e. Q' = -L', qn = 1,

corresponds to derivative (2.1); and the point Q(ab) = Q(a) + Q(b) is put in correspondence to the product of two differential monomials a( X) and b(X). The finite sum of differential monomials

s

(2.3) f(X) = I>i(X) i=l

is called a differential polynomial. The set of points S(f) ~ {Q(ai), i -1, ... , s} is called the support of differential polynomial (2.3). The minimal linear manifold SS(f) containing the support S(f) is called the super-support

of polynomial (2.3). The dimension d(f) ~ dimSS(f) and normal space

N(f) ~ N(SS(f)) C JE;.'" are defined in a similar way.

ExAMPLE 2. We consider the one-dimensional equation descri.bing the combustion process without sources

(2.4)

for the temperature u, the timet, and the unique space coordinate x [6,7]. Here n = 3,

(2.5) X1 = t, X2 = X, X3 = U.

The supportS consists of two points Q1 = ( -1, 0, 1) and Q2 = (0, -2, 1 +O"). The super-support SS here is the straight line passing through the point Q1

with the directing vector B = Q2 - Q1 = (1, -2, O"), i.e. the straight line {Q : Q1 + p.B, p. E JR}. Consequently, d = 1. The normal subspace N is two-dimensional with the basis N1 = (2, 1, 0) and N2 = (0, O", 2).


3. The Lie Operators. The differential equation f(X) = 0 corresponding to polynomial (2.3) admits the Lie operator

(3.1) n [)

"\:""' AiXi-L.. 8x· i=l t

if the equation is invariant under the change of coordinates

(3.2) Xi =Xi!/'', {t E JR, {t > 0, i = 1, ... , n,

i.e. it is a quasi-homogeneous equation (see [5]).

THEOREM 1. The differential equation f(X) = 0 corresponding to polynomial (2.3) admits the Lie operator (3.1) if the vector A = (>.1, .•. , >.n) E N(f).

Proof. Under substitution (3.2) we have that the monomial XQ -ft-(A,Q) i(Q, derivative (2.1)

[JIIL'IIxnf8xi' ... ox~~{ = tt-(A,Q)&IIL'IIxnfoxi' ... ox~~i

with the vector Q cited in (2.2), and a product of monomials and derivatives acquires as a factor such a power of tt which is equal to the sum of powers of ft acquired by each of the factors. Hence after substitution (3.2) the differential monomial a(X) turns into the monomial a(X)tt-(A,Q), where Q = Q(a), and the differential polynomial (2.3) turns into

s

(3.3) L ai(X)tt-(A,Q,), i=l

where Qi = Q(ai)· If the vector A E N(f), then all scalar products -(A, Q;) in (3.3) are equal to each other fori= 1, ... , s. Hence f(X) = f(X)tt-(A,Q,), and the equation f(X) = 0 turns into itself under substitution (3.2), i.e. it admits operator (3.1). The proof is finished. D

If d ~ d(f) < n, then dim N = n - d > 0, and the equation f (X) = 0 admits n- d linearly independent operators (3.1), which are easily computed from S(f).

The expression Xn = <p(X') is called the solution to the equation f(X) = 0 if j(X',<p(X')) = 0. The solution is called self-similar if it is invariant under the changes of coordinates forming the Lie group [5]. In particular, the solution is self-similar if it admits an operator of form (3.1),

74 A. BRUNO

i.e. it is invariant under changes (3.2). One can seek such self-similar solutions Xn = <p(X') to the equation j(X) = 0, which admit some of operators (3.1) admissible by the equation. If the solution Xn = <p(X') admits operator (3.1), then the corresponding super-support SS(xn- <p(X')) is normal to the vector A. For the remainder Xn- <p(X'), the super-support necessarily passes through the point Q =En= (0, ... , 0, 1) corresponding to the term Xn, and the super-support SS(<p(X')) is placed in the hyperplane qn = 0. Hence the super-support SS(xn - <p(X')) is placed in the hyperplane

(3.4) {Q: (A, Q) =An}·

It intersects the hyperplane qn = 0 by the linear manifold

(3.5) {Q': (A', Q') =An},

in which the super-support SS(<p(X')) is placed. If solution Xn = <p(X') admits several operators of form (3.1) with vectors Ai = (A1i, ... , Ani), i = 1, ... , m, then the super-support SS( <p(X')) lies in the linear manifold

(3.6) {Q': (A;, Q') =Ani, i = 1, ... , m}.

EXAMPLE 3 (a continuation of Example 2). Equation (2.4) admits operators (3.1) with A= Nt and A= Nz.

4. Self-Similar Solutions. For sum (2.3), the sum of some number of its addends ai(X), where 0 < s1 ::; s, is called its proper subsum j.

THEOREM 2. Let Xn = <p(X') be such a solution to the equation j(X) = 0 corresponding to polynomial (2.3) that no proper subsum j of sum (2.3) with dim(}) < dim(!) vanishes identically on the solution. Then

(4.1) N(xn- <p(X')) c N(f).

Proof. Let the vector A E N(xn- <p(X')), then, according to (3.5), An= (A', Q'), where Q' runs the super-support of the function <p(X'). Under substitution (3.2) differential polynomial (2.3) assumes form (3.3). Since Xn = <p(X') is a solution to the equation f(X) = 0 that is invariant under the substitution (3.2), i.e. Xn = <p(X') is the same solution, then on it we have

' (4.2) L ai(X)tL-(A,Q,) = 0.

i::::l


Let the different values of scalar products (A, Q;) be c1 < C:l < ... < ct· We denote as lr the collection of all indices i such that (A, Q;) = Cr. r = 1, ... , t. We define the differential polynomials

fr(X) ~ L a;(X) over i E lr. r = 1, ... , t.

Then equality (4.2) takes the form

t

L p-c, fr(.X) = 0. r=l

Since the equality is valid for all p > 0, then on the solution the equalities

f,.(X) = 0, r = 1, ... , t

are satisfied, and it is the solution to this system of equations. According to the construction of polynomials fr(X), the vector A E N(fr) for all r = 1, ... , t. Consequently,

(4.3) A E N(ft) n ... n N(ft)·

If d(jr) = d(f) for some r, then N(fr) = N(f), and property (4.1) follows from ( 4.3). By the hypothesis of Theorem, d(fr) = d(j) for all r = 1, ... , t. Hence property (4.1) is satisfied. But if d(fr) < d(j) for all rand intersection (4.3) does not lie in N(f), then inclusion (4.1) may be violated (see Example 6 below). The proof is finished. D

EXAMPLE 4 (a continuation of Example 2). We consider equation (2.4) in coordinates (2.5). If we draw through the point Es = (0, 0, 1) a straight line parallel to SS(f), i.e. with the directing vector B = (1, -2, 1Y), then this straight line will cross the plane q3 = 0 in the point Q4 = ( -1/0", 2/0", 0). Property (4.1) means that for the solution x3 = cp(x1 , x2) to equation (2.4), the super-support of the function <p in the plane q3 = 0 either consists of the point Q4 , or is the straight line passing through the point, or coincides with the whole plane. Therefore solutions related to the first two cases may be found in the form

( 4.4) -1/u 2/u x3 = cw1 w2 ,

where w; = e; + li;x;, c; = const, li; = ±1, i = 1, 2. In the first case c = const, and we obtain for can "algebraic" equation -151 = 2(2+1Y)C0 jO". In the second case c is a function of an arbitrary monomial w~w2 • In the plane q3 = 0, the

76 A. BRUNO

straight line passing through the point Q4 with the directing vector (b, 1, 0) has the form (q1 , qz) = ( -1/IY, 2/0') + t-t(b, 1), J.t E JR.. It intersects the axis q1 when J.t = -2/0' in the point with q1 = -(1 + 2b)j0'. Hence that straight line can be also written in the form

(qi,qz) = (-(1 + 2b)/0',0) + t-t(b, 1),

and the self-similar solution (4.4), which has the super-support of the righthand side placed in that straight line, can be written in the form

-(1+2b)/a (C) c b X a = W 1 V s , s = w1 Wz.

Substituting that expression in equation (2.4) and cancelling w;-J-(!+Zb)/a,

we obtain for the function v(~) the ordinary differential equation

6I[bv'~- (1 + 2b)vj0'] = v"v" + O'V"-1v'2 ,

where the prime denotes the derivative with respect to ~. In both cases, solution (4.4) is a self-similar one; but it has different dimension d(x3 - <p): the unit in the first case and two in the second one. When 61 = -1 and c1 > 0, solution ( 4.4) in the first case is a blow-up solution, i.e. it goes to infinity in a finite time [7]. These solutions were studied in [7, Chap. III] in § 1 and § 2 for the first and the second cases, respectively.

EXAMPLE 5. We consider the equation of the combustion process with a source [6,7]

(4.5) Ut = (u"ux)x + u13 , 0' = const > 0, fJ = const > 0.

Here n = 3. In notation (2.5), the supportS for equation (4.5) consists of three points Q 1 = ( -1, 0, 1), Q2 = (0, -2, 0'+ 1), Q3 = (0, 0, fJ). We compose remainders

Qa- Ql = (1,0,{} -1) ~ Bz

and compute their vector product N ~ [B1 , B2] = (2-2{}, 1+0'- {J, 2). Since N # 0 with any values of exponents 0' and {J, it follows that the vectors B1

and B 2 are linearly independent, and the dimension of the equation d = 2. Its super-support in JF.3 is the plane with the normal vector N. Property ( 4.1) means that the super-support of the solution i-3 = 'P( WI> w2) either


coincides with JR3 or lies in the plane with the normal vector N. Since the plane SS(x3 - 'P) passes through the point E3 = (0, 0, 1), then its equation (3.4) is 2(1- ,8)q1 + (1 +a- ,8)q2 + 2q3 = 2, and it intersects the plane q3 = 0 by the straight line (3.5), i.e.

(4.6) 2(1 - ,8)q1 + (1 +a- ,8)q2 = 2.

We consider at first the case

(4.7) ,8/=1.

In this case straight line (4.6) intersects the axis q1 at the point q1 = 1/(1-,8). Consequently, equation ( 4.5) has a self-similar solution of the form

(4.8) _ 1/(HJ) x _ 1 + a - ,8

u- w1 7j;(w1 w2), u-- 2(1

_ ,8) ,

where the vector (u, 1,0) is the directing one for straight line (4.6), w1 = c1 + <ht, w2 = c2 + 82x, c1, c2 = const, 8i = ±1. The function 7/J(~) satisfies the ordinary differential equation

where~= wj'w2 , and the prime denotes differentiation with respect to~. In [6,7] the asymptotics and local properties of bounded solutions to equation ( 4.9) are either long studied or given without explanation for various values of exponents a > 0 and ,8 > 1. But these properties may be simply studied by methods of Chapter 6 of the book [2] (see the end of this Example). If the function 7/J(O does not tend to zero when w1 --> 0, then solution ( 4.8) tends to infinity when (3 > 1, i.e. when c1 > 0 and 81 = -1, it is a blow-up solution [6,7]. When (3 < 1 and the function 7/J(O is bounded, solution (4.8) does not tend to infinity for finite t.

Now we consider the case

(4.10) (3=1.

In this case the equation of straight line (4.6) is q2 = 2/a. Hence equation (4.5) in case (4.10) has a self-similar solution

(4.11)

Equation ( 4.5) for the function v gives the ordinary differential equation

(4.12) v = wva+! + v, with w = 2(2 + a)/a2,

78 A. BRUNO

where the dot denotes differentiation with respect to t. Its solutions are v

log ( )II = t + c, wv" + 1 "

where c is an arbitrary constant. When v -+ oo, the left-hand side of the last equality has the asymptotics

_I_ (logw + - 1-). a wvu

When v-+ oo, we have v-"""' -awt+c1, c1 = const, i.e. v""' (c1-11wt)-11". That means that in case (4.10) solution (4.11) to equation (4.5) is a blow-up solution.

That result can be obtained by other means: by methods of Chapter 6 of [2]. In this case n = 2. In the plane qr, q2 the support of equation ( 4.12) consists of three points ( -1, 1), (0, a+ 1), (0, 1). Their convex hull is the triangle with vertices in the points. The boundary of the triangle consists of three vertices and three edges. The edge of the triangle that connects the first two vertices has the ray as the normal cone, which is spanned on the vector (-a, 1). To the edge there corresponds the truncated equation iJ = wv"+1 . It has a power solution v = aw~Ifu, where -or/ a = wa, i.e. a= -or/(aw). This solution to the truncated equation is the asymptotics of the solution to the complete equation when w1 -+ 0 and v -+ oo, since in the basis vector (-a, 1) of the normal cone the first coordinate is negative and the second one is positive.

The third way is to apply the methods of Chapter 3 [2]. In this case n = 1. We write equation (4.12) in the form (logv) = wv" + 1. Then the support of the system consists of two points q1 = a and q1 = 0. Their convex hull is the segment [0, a], since a > 0. To solutions with v -+ oo there corresponds the first vertex of the segment, i.e. the point q1 = a. To this vertex, in turn, there corresponds the truncated system (log v) = wv". Further, it is as it was in the second method.

The fourth method to obtain blow-up solutions was suggested by S.A. Pososhkov. When f3 = 1, the change u = uct, i = (e"t- 1)/a reduces equation (4.5) into equation (2.4) (see [7, Chap. II,§ 7, Subsection 1]). Here to blow-up solutions to equation (2.4) there correspond blow-up solutions to equation (4.5). In particular, solution (4.4) related to the first (algebraic) case may be taken.

Theorems 1 and 2 give such an algorithm for finding all self-similar solutions, which is substantially simpler than the traditional method of computation of the Lie operators admitted by the equation [5]. The test of inclusion


(4.1) for the results cited in [5] showed there a number of misprints and inaccuracies. One of them is discussed below in Example 6. According to [5], among all symmetries of differential equations approximately 70% are the symmetries corresponding to operators of form (3.1), i.e. power and logarithmic ones.

EXAMPLE 6. The test of inclusion ( 4.1) for the results cited in [5] showed the following case of its failure. Page 120, line 12. The equation Ut = (k( u)ux)x with k = u-413 , the solution ·u = x-3 f(t). Here n = 3, x1 = t, X2 = x, x3 = u. The support of the equation consists of two points Q1 = ( -1, 0, 1) and Q2 = (0, -2, -1/3), its dimension d = 1. The directing vector

of the super-support of the equation is B ~ Q2 -Q1 = (1, -2, -4/3). If f(t) is an arbitrary function, then the super-support of the solution is parallel to the vector E1 = (1, 0, 0); besides, it passes through points E 3 = (0, 0, 1) and

Q3 = (0, -3, 0), i.e. it is parallel to the vector E3 - Q3 = (0, 3, 1) ~ B2 . The

vector product [E1 , B2] = (0, -1, 3) ~ N is the normal vector to the supersupport of the solution. The scalar product (N, B) = -2 # 0, i.e. property (4.1) is not satisfied. Find the equation for the function f(t). Substituting the mentioned solution into the original equation, we obtain the equation f' = 0 for f(t). Consequently, f(t) = const. In this case the support of the solution consists of two points E3 and Q3 , and the directing vector of the super-support of the solution is B2 . It is not collinear to the vector B, and property ( 4.1) of Theorem 4.1 is not satisfied. On this solution, property (4.1) of Theorem 2 is not satisfied either, since Ut = 0 and (u- 413ux)x = 0, i.e. both part of the original equation vanish. Each of them has the zero dimension, and for each of them property (4.1) is satisfied. That shows that in Theorem 2 the condition on the solution is essential.

5. Power Transformation. Form the vector log X = (Jog x1 , ... ,

Jogxn)', where the asterisk means transposition. Let A= (a;J) be a square non-singular n-matrix. The change of coordinates

(5.1) logY= AlogX

is called power transformation [1]. The inverse change log X= A-·1 logY is also a power transformation.

THEOREM 3. Under the power transformation (5.1) : 1) sum (1.1) tums into the sum ](Y) = 'f:,fqYQ, wher-e Q = A'-1Q; 2) the function f(X) of form (1.4) tums into the function ](Y) = f(X)

of form (1.4), besides, SS(j) = A*- 1SS(f);

80 A. BRUNO

3) the dijJerential polynomial (2.3) turns into the ratio ojtwo differential polynomials g(Y)/h(Y) = f(X); besides, the support S(h) consists of one point Q = 0, and the support S(g) = A*-1S(f);

4) the normals to supports are transformed linearly by the rule N = AN.

For the proof of the Theorem, auxiliary statements are needed. Let p( X) be the product of logarithmic derivatives of the form

aiiJ/IIlogxn def iJIIL'IIlog Xn -

(8logxi)l! ... (ologxn_Jln-1 (8logX')L'' (5.2)

where L' = (l 1 , ••. ,ln-1), li;:: 0. Such a derivative may be written in the ordinary form as a differential polynomial. Consequently, the product p(X) of such derivatives is also a differential polynomial. It is obvious that S(p) = 0. We call p(X) a differential logarithmic monomial. The sum

s

(5.3) LXQ'Pi(X), i=l

where Pi( X) are differential logarithmic monomials, is called the logarithmic form. It is obvious that every logarithmic form (5.3) may be written in the form of the differential polynomial (2.3). The reverse is also true.

LEMMA 1. Every differential polynomial (2.3) may be written as the logarithmic form (5.3).

Proof. It would suffice to make it for one differential monomial a(X). As it was shown in the proof of Theorem 2.2 of Chapter 6 [2], the equality

(5.4) 81xn = X'L' Xn Pt (a log Xn, ... '81log Xn) 8x11J ••• OXn_1

1n- 1 (8logxi)l! ... (8logxn-1)1n- 1

is valid, where l = h + .. . +ln-l and .Pt(6, ... ,.;t) are some polynomials with constant coefficients containing only the terms of the form

const ,;~ 1 .•• .;1k' with k1 + 2k2 + ... + l kt = l, ki ::: 0.

Since the differential monomial a( X) is the product of an ordinary monomial and some derivatives of form (5.4), then, according to (5.4), it is written in the form

t

a(X) = XQLPi(X), i=l


where Q = Q(a) and Pi are differential logarithmic monomials. The proof is finished. D

LEMMA 2. Let

(5.5)

be a local expansion with arbitrary coefficients 'PQ'. Let, as a result of the affine transformation

(5.6) X= WY, det W =J 0, W = (w;j),

the expansion

(5.7) Yn = 7J>(Y') ~ I: 7f'R! Y'R' over R' E Z~1

be obtained from expansion (5.5). Then its coefficients have the form

'lj;R! = /3n•({'PQ•})/a211R'II-1,

where ,Bn' ( { 'PQ'}) are polynomials in coefficients 'PQ' with !IQ'll ::; IIR'II, and

(5.8) n-1

a = Wnn - L 'PEi Win,

i=l

where E; denotes the i-th unit vector.

Proof. We write expansion (5.5) in the form f(X) ~ Xn - rp(X') = 0 and make change (5.6) in the equation. Then it takes the form

n

(5.9) def "\:""' ') f(X) = g(Y) = L. WniYi- rp((AY) = 0. i=l

We consider it as an equation for implicit function (5.7). It is clear from

(5.9) that a~ ag I OYn has form (5.8) in the origin. Applying Theorem 1.1 of Chapter 2 [2] to (5.9), we obtain the existence and uniqueness of expansion (5.7) and the mentioned form of its coefficients. The proof is finished. D

COROLLARY 1. Under' change (5.6), paTtial derivatives f)!!R!IIyn/(fJY')R! are rational functions of partial derivatives fJIIQ'IIxn/(fJX')Q' with IIQ'II ::; IIR'II, besides, denominators of these rational functions are powers of the sum

n-1

a~ Wnn- L Wn;OXnfox;. i=l

82 A. BRUNO

Indeed, according to the Taylor formula

1 &IIQ'IIxn 1 &IIR'IIyn 'PQ' = Q'! (&X')Q'' '1/JR' = R'! (&Y')R''

where Q'! = q1! ... qn_1 !, i.e. the derivatives differ from the coeffi.cients of expansions (5.5) and (5.7) by constant factors.

COROLLARY 2. Let A= W-1 = (aij)· Then in the situation of Lemma 1, partial derivatives &IIQ'IIxn/(&X')Q' are rational functions of derivatives &IIR'IIynf(&Y')R' with IIR'II::; IIQ'II, besides, their denominators are powers of the sum

(5.10) n-1

ann- L ainOYn/OYi· i=l

REMARK 1. If the matrix W has the block-triangular form cited in Section 2 of Chapter 6 [2], i.e. w1n = ... = Wn- 1n = 0, then, according to (5.8), a = Wnn = const, and derivatives &IIR'IIynf(8Y')R' are polynomials in derivatives &IIQ'IIxn/(&X')Q'. The reverse is also true, i.e. the second derivatives are polynomials in the first ones, since in that case a1n = ... = an-1n = 0, and sum (5.10) is equal to the constant ann·

Proof of Theorem 3. Statement 1 is Property 1 of the power transformation from Section 3 of Chapter 2.

Statement 2 of the Theorem follows from the equality X 8 • = Y ii,, where Bi = A ·-1 Bi·

Let us prove Statement 3 of the Theorem. Firstly, according to Lemma 1, we write the differential polynomial in logarithmic form (5.3), where Pi(X) are products of derivatives of form (5.2). After transformation (5.1), XQ• = yQ•, where Qi = A*-1Qi. We denote

.;i =log xi, 7li =logy;, i = 1, ... , n.

In that notation, derivative (5.2) is

(5.11)

and transformation (5.1) is the affine transformation H = A2. According to Corollary 2 from Lemma 2, derivatives (5.11) are rational functions of


derivatives i)IIM'II1Jn/(DH')M' with \\M'II ::; IIL'\1, besides, the denominators are powers of the sum

n-1 a = ann - L. a;n01Jnl 01];.

i=1

Consequently, the differential logarithmic monomial Pi(X) is the ratio

- (Y)/-m h - . h I . I . d . . f h c i)IIM'II]og Yn Pi a , w ere Pi 1st e po ynom1a m envat!Ves o t e 1orm (DlogY')M'

n-1

and ii =ann- :E a;n81ogyn/8logyi, and the whole sum (5.3) is g(Y)/iim, i=l

where g(Y) is the logarithmic form of some differential polynomial, and ih :;:: 0 is integer. Since the supports of differential polynomials ,ii;(Y) and a(Y) consist of one point (the origin), then the support S(i'im) = 0 and the support S(g) = A*-1S(f). That completes the proof of the third statement of Theorem 3.

The fourth statement follows from the first three, since in order to preserve the scalar product, transformation (5.1) induces the transformation N = AN in the dual space ~. The proof is finished. 0

To the multiplication of the differential polynomial f(X) by the monomial XT there corresponds in JRn the parallel translation of supports S(J) and SS(J) by the vector T. Hence if d(J) = d < n, then by a linear transformation of the form Q = A*-1(Q + T) the set S(J) can be placed in the coordinate subspace

(5.12) iiJ = · · · = iJn-d = 0.

THEOREM 4. For a differential polynomial f(X) with d(J) ~ d < n, there exist a vector T E JRn and a matrix A such that power transformation (4.1) reduces the differ-ential equation XT f(X) = 0 to the for-m g(Y) = 0, where pr·operty ( 5.12) holds for all points Q of the support S (g).

Proof follows from Statement 3 of Theorem 3 and the solution of Problem 3 in Section 11 of Chapter 1 [2]. As a vector - T, any vector from the super-support SS(J) can be taken.

REMARK 2. If the coordinate x; is present in polynomial (2.3) only under the differentiation sign, then the power changes of the form dxi = Y B; dy; can be made as well, where B; = (b;1 , ... bin) and b;i = 0.

84 A. BRUNO

EXAMPLE 7 (a continuation of Examples 2-4). Fo~ equation (2.4), n = 3 and d = 1. In notation (2.5), we make the powe~ t~ansfo~mation (5.1)

with the matrix

(5.13)

Y1 = X1,

Y2 = X2,

Y - X 8 1o- - x1/o-x- 21o-x a- - 1 2 3

( 1 0 0)

A= 0 1 0 . 1/CY -2/CY 1

In the original coordinates t, x, u, this transformation is u = vr 11o-x21o-. After cancelling r 1- 1/a- x21o-, equation (2.4) turns into the equation

(vtt)- v/CY = (Vxxx2 )vo- + CY(vxx) 2vo--l (5.14)

+ 4(CY + 1)CY-1(vxx)v,. + 2(2 + CY)C5- 2vo-+l.

The support of the equation consists of two points

(5.15) (0, 0, 1) and (0, 0,1 + CY).

Here the vector T = (1 + 1/CY, -2/CY, 0), and the matrix A is given by formula (5.13). Instead oft and x, w1 and w2 can be taken as x1 , x 2 .

6. The Logarithmic Transformation. Let all points Q of the support of a differential polynomial g(Y) have the coordinate iii = 0. Then the coordinate Yi belongs to the g(Y) only as a power of the differential a log Yi· Hence if we make the logarithmic transformation Zi = log Yi, then g(Y) becomes the differential polynomial in YI> ... , Yi-1, zi, Yi+b ... , Yn·

THEOREM 5. If for all points Q of the support S(g) of the differential polynomial g(Y) property (5.12) is satisfied, then after the logarithmic transformation

(6.1) zi=logyi, i=1, ... ,n-d, Zj=yj, j=n-d+1, ... ,n

the differential polynomial g(Z) = g(Y) is obtained.

Proof. According to Lemma 1, we write the differential polynomial g(Y) in the logarithmic form

t

(6.2) g(Y) = LY0'pk(Y), k=l

,---~·· --··


where Pk(Y) are differential logarithmic monomials. By the hypothesis of the Theorem, for all vector exponents Q; property (5.12) is satisfied, i.e. Y1> ... , Yn-d are absent in the monomials yQ•, which are present in (6.2). These coordinates are present only in Pk(Y) in the form logy;. Hence after the logarithmic change (6.1), ez' do not appear in logarithmic form (6.2), i.e. form (6.2) remains a differential polynomial. The proof is finished. D

EXAMPLE 8 (a continuation of Example 7). We put z1 = logt, z2 = logx. Since Vtt = av;az1, VxX = av;azz, VxxX2 = a2vjaz~- av;azz, we deduce that equation ( 5.14) takes the form

(6 3) av - ':!_- a2v u ( av ) 2 u-1 3a + 4 av u 2(2 +a) u+l . a -a 2v +a a v + a v + 2 v . z1 a z2 z2 a zz a

The support of the equation consists of points

(6.4) ( -1, 0, 1), (0, 0, 1), (0, -2, 1 +a), (0, -1, 1 +a), (0, 0,1 +a).

Consequently, two points (5.15) of the support of equation (5.14) are blown up into five points (6.4) of the support of equation (6.3).

EXAMPLE 9 (a continuation of Example 5). We consider equation (4.5) first in case (4.7). We make the power transformation

t = t, (6.5) y = t"x,

v =X B,/(13-1) = t1/(13-1)u

where the number x is defined in (4.8). Here vectors (x, 1, 0) and (1/((3 - 1), 0, 1) form the basis of a linear subspace in IR3 , which is parallel to the super-support of equation (4.5). The inverse transformation to (6.5) is

t = t, (6.6) X= t-"y,

u = t 1f(l-(3)v.

We compute the derivatives using (6.5) aud (6.6)

1 Ut =

1_ (3tlf(1-(3)-1V + t 1/(1-(3)(v1 + Vyxt-1y), Ux = t 1f(I-(3)Vyt",

86 A. BRUNO

Substituting these values into equation ( 4.5) and cancelling tM1-f3), we obtain the equation

(6.7) 1 + t + _ u-1 2 + " + f3 1

_ f3 V Vt UYVy - CJV Vy V Vyy V ..

Now we make the logarithmic transformation T = logt (i.e. t = e7). Equa

tion ( 6. 7) transforms into

(6.8) _1_ + + _ u-1 2 + " + {3

1 _ f3 V V7 uyvy - CJV Vy V Vyy V .

If v does not depend on T, i.e. V7 = 0, then the equation becomes equation (4.9). However, now we can seek a solution to complete equation (4.5) or (6.8) in the form of a series in negative powers ofT:

or in the form of a polynomial in T:

where¢(~) is a solution to equation (4.9). This leads to such expansions of solutions u(t,x) to equation (4.5) in powers of (logt)- 1, which begin with a self-similar solution.

We consider now equation (4.5) in the case (4.10). We make the power transformation

(6.9) t = t, x=x, v = x-2/"u,

where the vectors (1, 0, 0) and (0, -2/CJ, 1) form a basis of the two-dimensional linear subspace parallel to the super-support of equation (4.5). Here u = x 21"v. We compute derivatives:


Substituting this expression in equation ( 4.5) and cancelling x21", we obtain the equation

After the logarithmic transformation log x = 71 (i.e. x = e~) the equation takes the form

(6.10)

since Vx = v~fx, Vxx = (v~ry - v~)fx2 . If v does not depend on 7J, then the last equation coincides with equation (4.12). We can seek a solution to equation (6.10) in the form of a polynomial in 7J or as a series in 71-1

; here the coefficient at the leading term is the solution to equation (4.12). In the power transformations (6.5), (6.9) and so on, one can use w1 and w2 instead oft and x.

7. The Asymptotics of Solutions. On the basis of the Power Geometry the algorithms for the computation of asymptotics of solutions to the equation f(X) = 0 corresponding to the differential polynomial (2.3) are developed. For this purpose, the convex hull M(f) of the support S(f) that is a polyhedron is considered. The boundary aM of the polyhedron M(f) consists of faces of various dimensions. We put in correspondence to each its face r the truncated polynomial

(7.1) /(X) = 'E ai(X) over i: Q(ai) E r,

including those and only those differential monomials ai(X) from (2.3), the vector power exponents Q(ai) of which are in the facer. Truncated polynomial (7.1) is the first approximation to polynomial (2.3) in some domain of the X-space. Let Xn = <p(X') be a solution to the equation f(X) = 0. It turns out that in this domain the first approximation to the solution Xn = <j>(X') is the solution to the truncated equation /(X) = 0, i.e. /(X', <j>(X')) = 0. Since the support of truncated polynomial (7.1) lies in some hyperplane, we have d(/) = dim r < n. Hence Theorem 3 is applicable to the truncated equation .f = 0, which allows to find all its self-similar solutions. In order to find its other solutions, one can apply Theorems 4 and 5; and to find the logarithmic asymptotics of its solutions, we must build a polyhedron for the polynomial g(Z) of Theorem 5 and isolate its truncations. This procedure is described in detail in [2]. The method is extended there also on a system of equations and on problems with parameters.

88 A. BRUNO

With the help of the Power Geometry in § 6 of Chapter VI in [2], a rigorous mathematical justification is given for the theory of the boundary layer in the simplest streamline problem. More complicated problems with the boundary layer are considered with methods of the Power Geometry in [3,4]. Until now, most applications of the Power Geometry relate to problems described by ordinary differential equations (see [2]).

REFERENCES

[1] A.D. Bruno, LOCAL METHODS IN NONLINEAR DIFFERENTIAL EQUATIONS, Springer-Verlag, Berlin, 1989.

[2] A.D. Bruno, POWER GEOMETRY IN ALGEBRAIC AND DIFFERENTIAL EQUATIONS, Elsevier, Amsterdam, 2000.

[3] M.M. Vasiliev, About the asymptotic analysis of the viscous heat conductive gas flow equations, Preprint No. 65, Keldysh Institute of Applied Mathematics of RAS, Moscow, 1998.

[4] M.M. Vasiliev, About the computation of the boundary layer on the semi-infinite fiat plate in the viscous heat conductive gas flow, Preprint No. 43, Keldysh Institute of Applied Mathematics of RAS, Moscow, 1999.

[5] N.H. Ibragimov, Lie Group Analysis of Differential Equations, CRC Press, Boca Ration, 1994, Vol. 1.

[6] Yu.A. Klokov, A.P. Mikhailov, and M.M. Adjutov, Nonlinear mathematical models and nonclassical boundary-value problems for the ordinary differential equations, Fundamental Basis of the Mathematical Modelling, Nauka, Moscow, (1997), 98-197 (in Russian).

[7] A.A. Samarskii, V.A. Galaktionov, S.P. Kurdyumov, and A.P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, 1995.


VOLUME 8 2001, NO 1-2 PP. 89- 100

ON WELL-POSEDNESS OF BOUNDARY VALUE PROBLEMS FOR SOME CLASS OF GENERAL PDES IN A

GENERALIZED SETTING

V.P. BURSKH'

Abstract. The Dirichlet problem, the Neumann problem, other boundary value problems in a general setting for the equation £+ £u = f with some linear general differential operation £ with smooth matrix coefficients in a general bounded domain n and its formally adjoint operation £+ are introduced and studied.

Introduction. The basis of the theory of general boundary value problems for the equation Cu = f with some linear general differential operation £ with smooth coefficients in a general bounded domain n was laid down in the known work of M.l.Vishik [16]. The point of view [16] is that a boundary value problem reveals itself in the domain of the operator generated by the primary differential operation was elaborated further by L. Hiirmander in [9], where one can find an explicit definition of a boundary value problem in L 2 (n) as a subspace of the Cauchy space of the primary operation. The general boundary value problems generated by smooth functional subspaces are considered in the works of Yu.M.Berezanskii [1] and A.A.Dezin [4]. Results which we have now for some classes of differential operators are stated below in s.l.2.

There are also other generalized settings of boundary value problems arisen in works of O.A. Ladyzhenskaja and N.N.Uraltseva, where the space £ 2 (n) plays the role of a central space and the operator generated by the operation with domain X generated by a boundary value problem is acting from some space X into its dual space X' (see for example [12, 13]). These last notions for the equation £+ Lu = f with some linear general differential operation £ with smooth matrix coefficients in a general bounded domain n

'Institute Appl. Math. Mech. NASU, Luxemburg str.,74, 340114, Donetsk, Ukraine

89

90 V.P. BURSKII

and its formally adjoint operation ;:+ are examined in the present paper. The Dirichlet problem, the Neumann problem, other boundary value problems for such equations are studied on the basis of the theory of expansion definitions. Note that some papers, in which such equations are considered, were already published, see for instance [11], but the connections with the theory of general boundary value problems were not explicitly observed. A part of results of this paper was announced in the work [3].

Note that, considering direct or generally posed problems, we shall often use the term correct for the term well-posed.

1. Expansions of PDO and Boundary Value Problems.

1.1. Definitions. We call to mind general facts about extensions of a differential operator and boundary value problems in a domain (see [16, 9, 4]). Let 0 be an arbitrary bounded domain in the space Rn with the boundary 80 = 0\0,

£ = L aa(x)D'", D"' = ( -iiJ)Ial jax~1 ••• 8x~", a E Z~, Ia I = L ak

jaj:<;m k

be some differential operation with smooth complex j x k-matrix coefficients aa(x), i.e. its elements belong to C 00 (0), ;:+. = Ljaj:SmD"(a~(x)·), a~ = a,1 be the formally adjoint differential operation. The closure of the operator, which is defined on the space (C0 (0))k by means of the operation £, in the norm of the graph II ulli = II ull~~(fl) + ll£ull~{(n), is called the min

imal expansion of the operator£ in the space L~(O) or simply a minimal operator L 0 . Below we shall often miss out vector indices for ease of writing, but one can easily restore them.

A restriction of an operator, which is generated by the operation £ in the space 7J'(O), onto a domain D(L) = {u E Lz(O)I £u E Lz(O)}, L = £1v(LJ is said to be the maximal expansion of the operator £1c8"(fl) or simply a maximal operator L. Note that the space D(L) is some Hilbert space with a scalar product of the norm II · IlL as well as its closed subspace D(Lo), which is the domain of the operator L 0 . The kernel ker L is closed in the spaces D(L) and £ 2(0), the kernel ker L0 is closed in the spaces D(L) and ker L. Consider another expansion of the operator £1c=(IT)• which we denote L. This is an operator with a domain D(L), which is the closure of the space C 00 (0) = {u E C 00 (0)13U E C00(Rn), Uln = u}, in the norm of the graph

II·IIL · We shall consider the following conditions:

(1.1) the operator L0 : D(Lo) -r Lz(O) has a continuous left inverse;

BOUNDARY VALUE PROBLEMS FOR SOME CLASS OF GENERAL PDEs 91

(1.2) the operator Lt : D(Lt) -+ L2 (0) has a continuous left inverse;

(1.3) L = (Lt)*;

(1.4) L+ = (Lo)*.

Note that the first condition evidently means that the following estimate holds: 3C > 0, 't/<p E C0 (0), II'PI!L,(n) ::; CJIC'PIIr,2 (n) . It is well known that L = (Lt)* and£+ = (£0)*, so that conditions (1.3),(1.4) mean the equalities D(L) = D(L), D(£+) = D(L+), i.e. the possibility to approximate each function from D(L) or D(£+) by functions from C00 (0). Conditions (1.1), (1.2) imply respectively the conditions: ker L0 = 0, ker Lt = 0. Conditions (1.1), (1.2), (1.3), (1.4) were introduced in connection with the study of the concept of a well-posed boundary value problem, which we also recall here (see [16,9]). We define a Cauchy space C(L) as D(L) I D(L0 ). In paper [9] the Cauchy space was introduced as a factor G(L) I G(L0 ), where G(L ), G(£0 ) are the graphs of the operators L and £ 0 respectively. It is not difficult to see that this definition is equivalent to the one introduced before. A homogeneous linear boundary value problem is by definition ([9]) the problem of finding a solution u E D(L) of the relations

(1.5) Lu= j, fu E B,

where r : D(L) -+ C(L) is the factorization mapping, B is a linear set in C(L). The boundary condition ru E B generates a subspace D(LB) = r- 1 (B) of the space D(L) and an operator L8 , which is a restriction of the operator L on the space D(L8 ) and which is an expansion of the operator L 0 . This operator L 8 is closed if and only if the linear space B is closed in C(L) or the space D(L8 ) is closed in D(L) [9]. A boundary value problem is called well-posed or correctly posed or simply correct and the operator L 8 is called a solvable expansion of the operator L0 if the operator L 8 :

D(L8 ) -+ £ 2 (0) has a continuous two-sided inverse. The operator L1 :

D(£1) -> L2(0), which is a restriction of the operator L (i.e. D(L1) <:;: D(L)), is called a solvable restriction if it has a two-sided inverse.

THEOREM 1.1. There exists a solvable expansion of the operator L 0

and there exists a corr·ectly posed boundary value problem for the equation Lu = f if and only if conditions ( 1.1) and ( 1. 2) are fulfilled.

See a proof of this theorem in the works of M.Vishik [16] and L.Hormander [9]. Note that the same conditions (1.1) and (1.2) are equivalent to the

92 V.P. BURSKII

existence of some correctly posed boundary value problem for the equation £+u =f.

We shall also consider the following conditions:

(1.6) the operator L: D(L) --+ L2 (11) is surjective;

(1.7) the operator L+: D(L+)--+ L2 (11) is surjective;

(1.8) the operator Lo : D(Lo) --+ Lz(\1) is normally solvable,

i.e. Im Lo = Im Lo.

THEOREM 1.2 [9]. Condition (1.1) is equivalent to condition (1.7), and condition (1.2) is equivalent to condition (1.6).

1.2. Examples of PDE with well-posed boundary value problems. Consider known examples of operators, for which a correct boundary value problem exists. It is known that the Dirichlet problem for any scalar equation, which is properly elliptic in n (in another terminilogy, right elliptic), is correct in a bounded domain with a smooth boundary [ 13]. The fulfilment of conditions (1.1), (1.2) in a bounded domain for any scalar operator with constant coefficients is proved by L.Hiirmander in [9]. See another proof of this fact in [4]. We draw two more examples of scalar operators, for which correct boundary value problems exist. In both cases the correctness follows from results of work [8], see also [10]. We recall that a differential operator L = L(x, D) has a constant strength in the domain n [10] if Vx E n, Vy E n, 3C > 0, v~ E Rn, l (x, ~)I l (y, ~) :::; c, where l (x, ~) 2 = :L IDf L(x, ~W, and that an operator with constant coefficients

l"l:o;m H(D) is weaker than the same Pz(D) (P1 -< Pz) if F1 (~) I Fz(~) :::; C. The operator of constant strength can be represented as

N

(1.9) L(x, D)= P0(D) + L, Cj(x)Pi(D), j=l

where the coefficients Cj E C 00 (\1); pi compose a basis of a finite-dimensional space of operators with constant coefficients, which are weaker than the operator P0 (D), pJ-< P0 , where P-< Q means D(Po) :::J D(Q0 ) ([9]). Formula


(1.9), in particular, means that D(L0 ) = D(Pg). The results of work [8] (see also [10], theorem 13.5.2) imply that for operators of constant strength conditions (1.6), (1.7) are fulfilled in some neighbourhood of each point, and for the case of analytic coefficients in some 0' :J n they are fulfilled in the whole domain n. The same is correct in the whole domain for operators of real principal type of form (1.9) (i.e. P 0 E R[~], '\7 P0 (~) # 0) (locally [9], in domain [8,10]). The following theorems follow from Theorems 1.1 and 1.2.

THEOREM 1.3. Any operator of constant strength admits some correct boundary value problem in some neighbourhood of each point. If its coefficients are analytic functions in the domain i1 , then such operator admits a correct boundary value problem in any domain n such that n C fl.

THEOREM 1.4. Any operator of real principal type of form (1.9), where Pi are arbitrary operators of orders less than m = deg Po = deg L, admits a correct boundary value problem.

For the case of matrix differential operators, we have, firstly, properly elliptic (correctly elliptic) inn systems by Douglis-Nirenberg [5] in the bounded domain with a smooth boundary, for which there exist correctly posed boundary value problems, and secondly, square systems of differential equations with constant coefficients, satisfying the property ofB.P.Panejach-B.Fuglede: each minor q of order N - 1 of the characteristic N x N matrix p( ~) satisfies the inequality !q(~) !2 ::; C I: !D" det p(~W, which is nesessary and sufficient

" for the validity of properties (1.1), (1.2) [15,7]. Concerning conditions (1.3), (1.4) one could see works [14], [9], [4]. We give here an example (see [2]) of a correctly posed boundary value

problem (1.5) with a maximal operator, which is generated by the wave

operator L = D = 0

°~ , n = K = {x E R2!!x! < 1} in the disk K and X1 X2

which will used later.

THEOREM 1.5 [2]. The following boundary value pr·oblem is correctly posed in the space L2 ( K):

(1.10)

(1.11) ulr1 = 0, u~lr2 = 0,

where r1 = H:::: 7:::: 21r}, r2 = {1r:::: 7:::: 3;}, 7 is the angular variable.

94 V.P. BURSKII

2. Generalized Solutions of Linear Equations. in this section we will observe connections between conditions (1.1),(1.2) and properties of generalized solutions of boundary value problems for the equations

(2.1} c+c u = J,

(2.2) c c+u= f.

Recall that, altilough we most commonly omit vector indices and consider the scalar case, alleonstructions can be carried with obvious modifications in the case of j x k-matrix coefficients. In addition, one should have in mind that there are different spaces L2 (0). If, for example, u is a column k x 1, then Lu E L?x 1(0) is a column j x 1 but D(L) C L~x 1 (0), L~x k(O) C D'(L).

I . However, L' act§Jrom the space L2 x 3 (0).

2.1. Dirichlet problem. The function u E D(L0 ) satisfying the integral identity

(2.3) <Lou, Cv >=< j,v >

for each function v E CQ"(n) will be called a generalized solution of the Dirichlet problem in the domain 0 for equation (2.1) with f E D'(L0 ).

Note that integral identity (2.3) is equivalent to the identity

. "'·<Lou, Lov >= < j, v >, Vv E D(Lo),

which may be written as the equality

(2.4) L~ ·Lou= j,

where L~ : L2 (0) -> D'(L0 ) is the dual operator to the operator L0 :

D(Lo) --+ L2 (0). This equality is, as a matter of fact, a realisation of equation (2.1). Note also that, by virtue of density of the embedding D(Lo) c L2 (0), the space L2 (0) is densely embedded into the space D'(L0 ). Therefore the solvability of problem (2.3) with each function f E D'(L) implies its solvability with f E L2(0). The generalized setting of the Dirichlet problem in the domain 0 for equation (2.2) is analogous. The function u E C2m(O) n cm-1 (0) satisfying equation (2.1) with a function f E C(O) and the boundary value conditions ui&n = u~i&n = ... = usm-l)l&n = 0 is called a classical solution of the Dirichlet problem in the bounded domain 0 with a smooth boundary for equation (2.1). It is obvious that the following theorem is correct.


THEOREM 2.1. Each classical solution u E D(Lo) of the Dirichlet problem for equation (2.1) in a bounded domain with a smooth boundary and f E D' (L0 ) is a generalized solution of this problem.

A generalized Dirichlet problem (2.3) will be called correctly posed or simply correct if the operator L0 · L0 : D(L0) -0 D'(Lo) has a continuous two-sided inverse operator M: D'(Lo) -0 D(L0).

THEOREM 2.2. A generalized Dirichlet problem (2.3) is correct if and only if the condition ( 1.1) is fulfilled.

Proof. Condition (1.1) is equivalent to the triviality of the kernel ker L0

and the closeness of the subspace Im L0 in the space L2 (0). Therefore condition (1.1) implies the existence of a continuous inverse operator M1 : Im L 0 -0

D(L0 ) to the operator L0 : D(L0 ) -0 Im L0 . Hence it follows that the operator L~ as acting in the spaces (Im L0 )' -0 D' ( L0 ) has a continuous inverse M;. By virtue of the closeness of the subspace Im L0 , its dual space in the scheme of equipped spaces [1 J can be identified with itself (or there is an isomorphism Im'Lo ~ Im£0 in the case of matrices j x k,j =I k). Hence condition (1.1) implies the existence of a continuous inverse operator M = M1 • M;. Conversely, the existence of a continuous inverse operator M implies that the operator M Lb is a left inverse to L0 . D

In the same way one can prove the following theorem.

THEOREM 2.3. The generalized Dirichlet problem for equation (2.2) is correct if and only if condition (1.2) is fulfilled.

From the theorems of subsection 1.2 the correctness of the following facts follows.

THEOREM 2.4. The gener-alized Dirichlet problem for equation (2.1) is cor-rect in the bounded domain 0 if the operator C is one of those indicated below:

1) C is a scalar operator with constant coefficients;

2) C is an operator of real principal type of form (1.9), where Pj are operators of orders less than m = deg P0 ;

3) L is an operator of constant strength of form (1.9) with analytical coefficients in a domain Sl' ~ 0 , where Pj are operators with constant coefficients of strength less than that of the operator P0 ;

4) C is a matrix operator with constant coefficients satisfying the condition of Panejach-Fuglede;

96 V.P. BURSKII

5) £ is a matrix operator, uniformly elliptic by Douglis-Nirenberg in a domain with smooth boundary.

EXAMPLE 2 .1. Consider the generalized Dirichlet problem for the Poisson equation ~u = f: £=grad,£+= div, D(L) = H 1(0), D(Lo) = ifl(O), f E [ifl(n)]', iJl(n) is a subspace of the the Sobolev space H 1(0) consisting of such functions that vanish on an. Theorem 2.2 establishes, in particular, that such problem is correctly posed if and only if in the domain n there holds the Friedrichs inequality: Vi.p E cgo(n), IIV''PI!L,(n) ~ C\I'PIIL,(n), which is the Vishik condition (1.1) for the operator \7. It is known that the Friedrichs inequality is fulfilled in each bounded domain (see [6]). Therefore this problem is correct in such domain.

EXAMPLE 2.2. A more interesting example can be obtained if one replaces the derivatives \7 with some set of differential operators £J, j = 1, 2, ... , N, one of which satisfies condition (1.1). In this case the operator £ = (£1, ... , £N) also satisfies condition (1.1). Indeed, II u IIi = lluiJi,(n) + I:: IJ£Ju1Ji,(n) and the inequality II'PIIi,(n) :S Cdl£1 'P II'L(n) im-

J

plies II'PIIi,(n) :S CJI£'PIIL(n) , i.e. condition (1.1) for£. Now we have, by theorem 2.2, correctness of the generalized Dirichlet problem for the equation L;£j £Ju =f.

j

2.2. Neumann problem. A function u E D(L) satisfying the integral identity

(2.5) < Lu,Lv >=< j,v >

for any function v E D(L) will be called a generalized solution of the Neumann problem in the domain n for equation (2.1) with an arbitrary function f E D'(L).

If condition (1.3) is fulfilled, then it is sufficient to require the fulfilment of integral identity (2.5) for each function v E C 00 (0). Note that integral identity (2.5) which may be written as the equality

(2.6) L' L U= /,

where L' : £ 2 (0) -+ D'(L) is the dual operator to the maximal operator L: D(L) -+ £ 2(0). This equality is, as a matter of fact, a consequence of equation (2.1).

REMARK 2.1. Note that the classical Neumann problem in the domain with smooth boundary for equation (2.1) has the form: r+ Lu = 0. One may


prove it using the Green formula:

m-1

(Lu, v)n- (u, L+v)n = L < Lm-j-1u, of,v >11n. i~O

In addition, the conjugate problem r+u E B+ for the equation L+u = g to the problem ru E B for the equation Lu = f is defined by the space B+ c C(L+) consisting of all such r+v that for all u E D(L8 ) the righthand side of the Green formula vanishes. Let us for simplicity assume that all functions are smooth. For the operator L + L the Green formula can be

m-1

written in the form (L+Lu,v)n- (u,L+Lv)n = L: (< Lm-j-1Lu,atv >an i~O

+ < Lm-j-1 u, QtL +v >an), whence we obtain that if one understands the Dirichlet problem ru = 0 for L+ Lu = f as Lm-j-lu = 0, Vj = 0, ... , m- 1, then the Neumann problem can be written as Lm-j-1Lu = 0, and it is the conjugate problem to the Dirichlet problem.

A generalized Neumann problem (2.5) will be named normally correct if for each function f E D'(L), which is orthogonal to the space ker L, there exists a unique (up to an additive component hE ker L) function u E D(L), which is a solution of equation (2.6) and which continuously depends on f.

THEOREM 2.5. A generalized Neumann problem (2.5) is normally correct if and only if the operator L is normally solvable. In particular, it is so if condition (1.2) is fulfilled.

The proof is analogous to the proof of theorem 2.2 with the only difference that the space D(L) requires a factorization by the space ker L. The operator Lis normally solvable, in particular, if condition (1.6) holds, which is equivalent to (1.2) by theorem 1.2.

Theorems 2.5 and 2.2 imply the following theorem.

THEOREM 2.6./fthe Dirichlet problem for equation (2.2) is correct, then the Ne·umann problem for equation (2.1) is normally correct.

The theorems of subsection 1.2 imply the validity of the following theorem.

THEOREM 2.7. The generalized Neumann problem for equation (2.1)

in the bounded domain n is normally correct if the operator£. is one of those listed in theorem 2.4.

EXAMPLE. 2.3. Consider the generalized Neumann problem for the Poisson equation flu= f (see example 2.1). Theorem 2.5 permits to state, in

98 V.P. BURSKII

particular, that such problem is normally correct in the connected domain 0 with a finite-dimensional space of first cohomologies H 1(0, R), for example, in a bounded domain with a smooth boundary. (Here we have the same operators and spaces as example 2.1, and the kernel and the cokernel of the operator of this problem are one-dimensional if the domain is connected.) Indeed, by the de Rham theorem [17], the closed in L~(Q) kernel of the operator rot ( = the exterior differential d2) includes the image of the operator grad ( = the exterior differential d1), and their difference is a finitedimensional space. Therefore the space of potential vector fields \1 H 1(0) is closed. Note that various. generalizations of this example were studied beginning with D.C.Spencer, see for instance [11].

On the other hand, as one could show, the normal solvability of the operator L is equivalent to the fulfilment of the inequality :JC > 0, Vu E

D(L), II uiiL(rl)- II PkeruiiL(n) .:::; CIILuiiL(n)' where Pker: L2(0) -+ ker M is the orthogonal projector. For L = \1, we have kerL = {const},Pker : u -+ meisn f u(x) dx, and the last inequality in this case has the form

n of the well known Poincare inequality: :JC > 0, Vu E coo(o), II uiiL(n) :::; mek n (f u dx) 2 + C ll\7ui1L(n) . Thus, theorem 2.5 asserts that the general

n ized Neumann problem for Poisson equation is normally correct in a bounded domain 0 if and only if in this domain the Poincare inequality is fulfilled.

2.3. Other linear boundary value problems. Now let us consider the generalized setting of other general boundary value problems. Assume that the space B C C(L) gives boundary value problem (1.5) and hence it gives an expansion Ls of the minimal operator £ 0 • By B+ E C(£+), denote a subspace giving the conjugate problem L+v = g, r+v E B+ to problem (1.5), i.e. D(Ls+) = (r+)-1B+ = {v E D(£+)1Vu E D(Ls), (Lu,v) = ('u,L+v)} is the domain of the conjugate operator to Ls (see remark 2.1).

The function u E D(L) satisfying the integral identity

(2.7) < Lsu, Lsv >=< j,v >

for each function v E D(Ls) will be called a generalized solution of the problem fu E B, r+ Lu E B+ , generated by problem (1.5), in the domain 0 for equation (2.1) with any function f E D'(Ls).

Note besides that integral identity (2. 7) means the validity of the equality

(2.8)

where L~ : £ 2 (0) -+ D'(Ls) is the dual operator to the operator Ls D(Ls)-+ £ 2 (0), which defines equation (2.1) more exactly. Note also that,


by virtue of density of the embedding D(L8 ) c L2(n), the space L2(n) is densely embedded into the space D'(L8 ). Therefore solvability of problem (2.7) with each function f E D'(Ls) implies its solvability with f E L2(n).

Problem (2.7) will be named normally correct if for each function f E D'(L8 ), which is orthogonal to the space ker L8 , there exists a function u E D(Ls) (unique up to an additive component h E ker L8 ), which is a solution of equation (2.8) and continuously depends on f. Generalized boundary value problem (2.7) for equation (2.1) will be called correct if the operator L~ · L8 : D(L8 ) -7 D'(L8 ) has a continuous two-sided inverse M: D'(Ls) -7 D(Ls). These definitions imply the following theorem.

THEOREM 2.8. Generalized boundary value problem (2.7) for equation (2.1) in a domain n is normally correct if and only if the operator Ls is normally solvable, and correct if and only if the operator L 8 is normally solvable and has a trivial kernel. If the kernel of the normally solvable operator Ls is finite-dimensional, then problem (2. 7) is a Fredholm problem, and its index is equal to zero.

EXAMPLE 2.4. Consider generalized problem (2.7) with the wave operator L = 0 of theorem 1.5 and the space B generated by boundary conditions (1.11). Namely, we denote by D(D8 ) the closure of the set of smooth functions satisfying boundary conditions ( 1.11) in the graph norm \1 • llo· For a smooth function u, we have the following boundary value problem: D2 u = f, ulr, = 0, u~lr, = 0, Dulcr1 = 0, (Du)~lcr2 = 0, where crj = BK\fj are the complements, and the second pair of conditions are the conjugate boundary conditions to (1.11) for the function Du. From theorem 1.5 we have that Im Ds = L2 (K), and theorem 2.8 guarantees well-posedness of this generalized boundary value problem.

REFERENCES

[1] Yu. M. Berezanskii, Expansion by eigenfunctions of selfadjoint operators, Naukova Dumka, Kiev, 1965 (in Russian).

[2] V. P. Burskii, About fundamental solutions and correct boundary value problems for general differential equations, Nonlinear boundary value problems 7 (1996), 67-73.

[3] V. P. Burskii, Boundary properties of L2-solutions of linear differential equations and equation-domain duality, Soviet Math. Dokl. 40, 3 (1990), 592-595.

[4] A. A. Dezin, General questions of bo1mdary value problems theory, Nauka, Moscow, 1980 (in Russian).

[5] A. DouglL,, and L. Nirenberg, Interior estimates for elliptic systems of partial differentia,! equations, Comm. Pure and Appl. Math., 8 , 4 (1955), 503--538.

[6] Yu. V. Egorov, and M. A. Shubin, Linear partial differential equations. Funda-

100 V.P. BURSKII

mentals of classical theory. Differential equations-1, Results of science and technology. Modern problems of mathematics. Fundamental directions, VINITI, Moscow, 30 1988 (in Russian).

[7] B. Fuglede, A priori inequalities connected with systems of partial differential equations, Acta Math., 1961, 105.

[8] G. Gudmundsdottir, Global properties of differential operators of constant strength, Ark. Mat., 15 (1977), 169-198.

[9] L. Hormander, On the theory of general partial differential operators, Acta Math., 94 (1955), 161-248.

[10] L. Hormander, The analysis of linear partial differential operators II. Differential operators with constant coefficients, Springer-Verlag, 1983.

[11] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure and Appl. Math, 18 (1965), 443-492.

[12] 0. A. Ladyzhenskaja and N. N. Ural'tseva, Linear and quasilinear eqMtions of elliptic type, Nauka, Moscow, 1973 (in Russian).

[13] J. L. Lions and E. Magenes, Problemes aux limites non homogenes et applications, I-III, Dunod, Paris, 1968-1970.

[14] V.Maz'ya, Sobolev spaces, St. Petersburg University, 1994. [15] B. P. Panejach, On general systems of differential equations with constant coeffi

cients, Doklady Ak. Nauk USSR, 138, 2 (1961), 297-300 (in Russian). [16] M. I. Vishik, On general boundary value problems for elliptic differential equations,

Trudy Mosk. Math. Obschestva, 1 (1952), 187-246 (in Russian). [17] H. Whitney, Geometric integration theory, Princeton, 1957.


VOLUME 8 2001, NO 1-2 PP. 101-109

SOME GEOMETRIC ASPECTS OF DRILLING MODELliNG *

G. CHAKVETADZE t

Abstract. A one-dimensional dynamical drilling model is studied. The parameter set of this model is characterized. This description can be used in the solution of optimization problem posed for a functional expressing the efficiency of drilling.

Introduction. In the present paper some geometric questions related to the mathematical model of drilling are discussed. This model was suggested by A. Lasota and P. Rusek in [10] (see also the discussion in [3]). It is presented by a family of piecewise monotonic mappings (denoted below by v) depending on a functional and scalar parameters. A transformation w of the closed interval I = [c, d] is called C1-piecewise monotonic if there are points So, S), ... , Sn-), Sn, C = So < S) < ... < Sn-! < Sn = d, SUCh that for any k, k = 1, n, the restriction of w to the open interval h = (sk_1, sk) is a C1-diffeomorphism on w(h). Sometimes, the statistical properties of dynamics be understood via the spectral properties of the Perron-Frobenius operator, acting on a suitable space of observables (see [8], [2], [9]). In case of a C1-piecewise monotonic transformation w this operator is defined by the formula

n

Pwf(r) = Lf(wi[k1(r))i(wi[k1)'(r)IXw(Ik)(r), f E BV(I), rEI k~!

(here BV(I) denotes the space of equivalence classes of summable complex-valued functions on I with bounded essential variation, XA is the characteristic function of a set A).

• The researsh was partially supported by the RFBR grants No. 99-01-01104 and No. 00-15-96107

I Moscow State Aviation Institute, Dept. of Differential Equations, Russia, 125871, Moscow, Volokolamskoe shosse, 4.

101

102 G. CHAKVETADZE

The spectral analysis of the operator Pv shows that the model suggested possesses strong stochastic properties (see sect. 1). The question arises: to what extent this information is useful for the investigation of real .drilling process? One aspect of the problem, in fact, probabilistic, was considered in [6]. On the other hand, the features of the model put certain restrictions on the form of the drilling bit which can be explicitely formulated. More precisely, the drilling bit may be viewed as a toothed gear. The natural definition looks as follows. An £-toothed gear is a closed bounded subset \!: of the plane such that

(1) \!: has the "center" 0 and a symmetry axis passing through 0; (2) \!:is invariant with respect to the rotation by the angle f around 0; (3) the boundary 8\!: of the set \!:is a starlike contour (this means that

every ray beginning at 0 meets {)\£ at unique point different from 0);

(4) the set 8\£ is given by the graphs of convex functions (see sect. 1).

The restrictions mentioned above can be expressed as the conditions on the set 8\£. This paper is devoted, in particular, to the formulation of these conditions in the analytic form. Furthermore, in the paper the range of the functional parameter of the model is characterized. The description obtained can be useful for solving a related optimization problem. VVe mean the extremal problem for a functional expressing the efficiency of drilling stated in [7].

The paper is organized as follows. The first section contains the necessary definitions, basic facts about the dynamical properties of the model, and formulation of the main result. Proofs are presented in the second section.

The author expresses his deep gratitude to A. Stepin for stating the problem and helpful discussions.

1. Definitions, Background, and the Result. Let J be the set of graphs r(g) of functions g : lR >-+ lR having the following properties:

(a) each g is 1-periodic and twice continuously differentiable on [0, 1]; (b) g(l- r) = g(r), r E lR, g m = 1.

Denote by 'Jconv C J the subset consisting of the graphs of the functions which are convex on [0, 1] (throughout the paper the convexity of a function g means that g" < 0). Given a function p with the graph from 'Jconv, a number b E JR, 0 < b < 1, and a number N E N, define the transformation v of the interval [O,N) as follows. Set m = inf lv"(r)i, M = sup lv"(r)l,

rE[O,l) rE[O,l)

SOME GEOMETR1C ASPECTS OF DRILLING MODELING 103

a = ;:! . For s E lR, the graph of the function

bm h8 (r) = p(s) + p'(s + O)(r- s)- 2 (r- s)2

, r > s,

intersects f(p) in a sequence of points with abscissas rl> r2 , ••• , rq, s < r 1 < r 2 < ... < rq. Put r(s) = r 1. The transformation v = vp,b,N is defined by the formulae

Vp,b,l(s) = r(s) mod 1, s E [0,1),

vp,b,N(s) = vp,b,l(s mod 1) + I[r(s)] mod N, s E [0, N), N ;::: 2

(I[s] is the integer part of s). The transformation v was studied in [10], [1], [4], [5], [7]. It is piecewise monotonic and in a sense expanding (see Prop. 4 in [7]). Recall what is known concerning the iterations of v.

LEMMA 1. Assume that b > ~· Then for any natural number N the transformation v has the unique periodic orbit with period N, attracting all the phase space [0, N).

THEOREM 1. Assume that p" is of bounded variation on [0, 1]. Then for any positive integer N the transformation v has an absolutely continuous invariant probability p, provided that the inequality

(1

- W > (a- b) 2 +~(a- 1)(1- b) b 8

holds.

THEOREM 2. Assume that p" is of bounded variation on [0, 1] and the parameters a and b satisfy the condition

(1 ~ b)3 > 2(a- W.

Then for any natural number N the absolutely continuous invariant probability p, of v is unique and weak mixing.

From the last theorem one can derive that the dynamical system (v, p,) has strong stochastic properties (see [7]). Note that the theorems stated above are in fact assertions about the spectral properties of the operator Pv·

It was suggested in [10] to consider the transformation v as the mathematical model of drilling. Let L E N, L > 2, 4> = f. Note that only the convex hull of an £-toothed gear (see Introduction) is essential for the model under consideration. This set is completely determined by the subset of the plane defined ·as follows. A contact curve of type L is a graph f(f:l) of a function TJ = f:l(O : [-~0 , ~0 ] r-t JR, ~0 > 0, having the following properties:

104 G.CHAKVETADZE

(c) S) is an even twice continuously differentiable convex function on the interval [-.;o, .;o];

(d) S) satisfies the conditions

(1)

(2)

The object of the study is a map TI from the set of all contact curves into J, defined as follows. Consider a contact curve It of type L, It = f(SJ). Given .; E [-.;0 ,.;0], denote by f3 = (3(.;) the inclination of the line 1r = 1r(.;) tangent to f(SJ) at the point (.;,SJ(.;)). Let(!= Q(.;) be the distance between 1r and the origin. Denote by.;= .;((3) : [-¢, ¢] 1-t [-.;0 , .;0] the inverse function for (3(.;). The graph f(q:!) of the function q:l = q:l(fJ) = (!(.;((3)) : [-¢, ¢] 1-t lR is said to be the profile curve corresponding to the contact curve <!:. The set TI(<t) is the graph f(p) E J of the function p defined by the equality

1 p(r) = q:l(O)q:!(-¢+2¢{r}), r E lR

( {r} is the fractional part of r). Assume that f(p) E Jconv, and assume also that the drilling bit has the form determined by the contact curve <!:. Then (see [7]) the study of the iterations of the transformation v = Vp,b,L

(with properly chosen b, 0 < b < 1) gives the possibility to make certain conclusions concerning the process of drilling under consideration.

Let !Jt(L) be the set of contact curves of type L, Jmod(L) = U(!Jt(L)), J(L) = Jconv n Jmod(L). The elements of the set J(L) are said to be basic profiles. Denote by !Jlconv(L) the set of contact curves f(SJ) of type L satisfying the condition

(3) (1 + (fl'(.;)) 2)

2 + fl"(.;)(SJ(.;)- .;SJ'(.;)) < 0, .; E [-.;o,.;o].

Let 1!:1 ~ 1!:2, lt1o 1!:2 E !Jlconv(L), denote the relation of being hornothetic. Write !Jlconv ( L) / ~ for the corresponding quotient. The following theorem establishes some properties of the map TI.

THEOREM A. For any L the following assertions hold:

(i) the map TI is a bijection between the sets J(L) and <Jlconv(L)/ ~;

(ii) the functions g with the graphs in the set Jmod(L) are characterized by the condition

(4) . ( g"(r)) rnm g(r), g(r) + 4¢2 > 0, r E [0, 1], g'(O + 0) > 0.

SOME GEOMETRIC ASPECTS OF DRILLING MODELING 105

REMARK 1. The transformation v is defined correctly only in case of piecewise convex parameter p. Thus the first part of theorem A may be referred to as a result on the applicability limits of the model.

REMARK 2. The second part of the theorem relates to the inverse problem: given basic profile, to restore the drilling bit form. In the paper [7] a functional rl, defined on the parameter set of the model (i. e. depending on the basic profile and the scalar parameter b), was introduced. This functional can be interpreted as the velocity of drilling. A. Stepin posed a variational problem for rl considered as a function of basic profile (with b fixed). The characterization of the set :Jmod(L) is important in context of this nonclassical extremal problem.

2. Proof of Theorem A. In this section the notation introduced in section 1 is used. Fixing L, denote by II1 the map transforming contact curves of type L into the corresponding profile curves (see sect. 1). The properties of the map II are essentially determined by the properties of the map II1 • Thus theorem A will be derived from the following results (lemmas 2 and 3) concerning II 1 •

LEMMA 2. The graph of an even and twice continuously differentiable function f : [-¢, .P] >-+ lR is a profile curve if and only if

(5) min(f(j3), f(j3) + f"(j3)) > 0, j3 E [-¢, ¢], f'( -¢) > 0.

Proof. ( .;=) Since the inequality ip > 0 is obvious, it suffices to show that ip"(j3) + ip(j3) > 0, j3 E [-¢, ¢], ip'(-.P) > 0. For every j3 E [-¢,¢],the tangent 1r(~(j3)) touches the contact curve f(SJ) at the point (~(j3), SJ(~(j3))). The coordinates of points of 1r(~(j3)) in the system (~, 17) satisfy the equation

(6) ip(j3) = -~sin j3 + rJ cos j3.

Hence

(7) ip(j3) = -~(j3)sinj3+SJ(~(j3))cosj3, j3 E [-¢,¢].

Note that

(8) SJ'(~(j3)) = tgj3, j3 E [-¢, <{1].

106 G.CHAKVETADZE

Differentiating the last two equalities with respect to fJ, we get

(9) ( f, cos fJ + Sj ( f,) sin fJ) I<=WJ) ,

(10) S)"(f,(fJ)) df, = _1_. dfJ cos2fJ

Using expressions (7), (8), (9), (10) and standard trigonometry, we write

(11) d\IJ(fJ) . dfJ = - (f, cos fJ + SJ(f,) sm fJ) I<=Wl),

SJ(f,)cosfJ)I<=W3J =-Sj}(") (1 + (SJ'(£,))2)~~ - \IJ(fJ). ~ <=e<f!)

(12)

Note that formula (11) implies the validity of the smoothness condition in the statement of the lemma. Taking into account equalities (11), (1) and inequalities (2), we derive

~~~fl=-4> = -(f,ocos(-4>) +SJ(f,o)sin(-4>)) =

-Sj(f,o)cos</> (S)f~o)- tg¢) > 0.

From the convexity of the function Sj and also from equality (12) we obtain that '+!" (fJ) + \IJ(fJ) > 0. ( =*) Let 2l be the envelope of the family of lines

(13) f(O) = -f,sin0+7]cos0, (} E [-</>,¢].

To prove the second part of the lemma, we establish the following two facts:


(e) the set 2l is a contact curve of type L; (f) the graph of the function f coincides with the profile curve corre

sponding to Ql. To identify 2l, we append to (13) the equations

(14) j'(fJ) = -~cosfJ-1)sinfJ, fJ E [-¢,¢].

Solving system (13)-(14) with respect to ~ and 1), we obtain

(15) ~(B)=- f(B) sinO- f'(B) cosfJ,

(16) 7J( B) = j (B) cos e - f' (B) sin fJ.

Formulae (15) and (16) provide a parametric representation of the envelope Ql. Since ~ = -(f(fJ) + f"(B)) cosO< 0 on[-¢,¢] (L ;::>: 3), this representation defines a function f) : [~(¢), ~(-¢)] H R depending on~- Let us prove assertion (e). We have 2l = f(f:J). To verify the property (c) in the definition of contact curve, we write

(17) df) = d1)! dB= -(f(B) + 1::(8)) sinO I = tgB(O, d~ dfJ M(eJ d~ -(f(B) + f (B)) cosO o=o(e)

~f) d 1 d~2 = d~(tgB(~)) = -cos3 1J(~)(f(B(~)) + f"(B(~))) < 0

for ~ E W-¢, ¢]). Hence the function f:J(-) is convex. Since f is even, we conclude that~(·) is odd and 1J(·) is even. Therefore, f) is even and is defined on a symmetric interval. Write ~0 = ~( -¢) for its right endpoint. Let us check property (d) of contact curve. Expression (17) implies that

f:l'(~o) = tg( -¢).

Taking into account representation (15)-(16), we write

0 <__§____=~(-¢)=sin¢ f(-¢) + f'(-¢)ctg(-¢) < tg¢ SJ(~o) 1)(-¢) cos¢ f( -¢)- !'(-¢) tg( -¢) ·

Here we have used the following inequalities:

f( -¢) > 0, !'( -¢) > 0,

!(-¢)- f'(-¢)tg(-¢) > 0, ~(-¢) > 0.

108 G. CHAKVETADZE

This completes the proof of (e). To prove assertion (f), it suffices to verify that f = \13, where r(qJ) =

I11(Qt). Applying equalities (7), (17) and (13), we get

\13(;3) = -~( 8) le=;3 sin ;3 + S:i( ~( 0)) le=;3 cos ;3

=-~(;3)sin;3+ry(;3)cos;3=f(;3), ;3E [-¢,¢]. 0

LEMMA 3. Let r(S:J) be a contact curve of type L. A function \13 such that r(qJ) = I11(r(S:J)) is convex on the interval[-¢,¢] if and only if

(1 + (S:i'(~))2 ) 2 +S:i"(O(S:i(~)- ~S:i'(~)) < 0, ~ E [-~o,M

Proof. Using equalities (12), (7) and (8), we have

~\13(;3) = (--1-. (1 + (S:i'(~))2) ~ + ~S:i'(~) 1

d;32 S:i"(~) . (1 + (S:i'(~))2)2

S:i(~) ) (1 + (S:i'(OJ2)! {=((;3}.

Solving the inequality d'd~\fl < 0 and taking into account convexity of the function S:J( ·), we get the desired result. 0

Let .C(L) = I11('.n(L)). To finish the proof of theorem A, let us consider the map !12 : .C( L) >-+ 'J given by the formula

IT2(r(J)) = r(9),

where 1

g(r) = f(0/(-¢+2¢{r}), r E JR.

Obviously, !12 o I11 = IT. The graphs of the functions h and f2 defined on the interval [ -¢, ¢] are said to be similar if fi differs from h by a constant nonzero factor. It is easy to see that !12 moves two similar profile curves into the same element of the set 'J. Also, the convexity property is preserved, and condition (5) transforms into the condition

. ( g"(r)) , mm g(r), g(r) + 4¢2 > 0, r E [0, 1], g (0 + 0) > 0.

Together with lemmas 2 and 3, this implies theorem A.


REFERENCES

[1 J Sh. Akhalaya, Invariant measures of noninvertible mappings, PhD Thesis, Moscow State University, 1984.

[2] V. Bakhtin, A direct method for the constructing of invariant measures on the hyperbolic attractor, Izvestiya RAN, ser. matem., 56, 5 (1992), 934-957.

[3] A. Boyarsky and P. G6ra, Laws of Chaos. Invariant measures and Dynamical Systems in One Dimension, Birkhiiuser, Boston-Basel, 1997.

[4] G. Chakvetadze, On the ergodicity of a model of drilling, J. Dyn. and Cont. Sys., 2, 4 (1996), 485-502.

[5] G. Chakvetadze, One-dimensional dynamics and a model of drilling, PhD Thesis, Moscow State lnst. of Electronics and Math., 1998.

[6] G. Chakvetadze, Stochastic stability in a model of drilling, J. Dyn. and Cont. Sys., 6, 1 (2000), 75-95.

[7] G. Chakvetadze and A. Stepin, On the dynamical model of drilling, Int. J. of Bijurc. and Chaos, 9, 9 (1999), 1705-1718.

[8] F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180 (1982), 119-140.

[9] G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Preprint, 1998.

[10] A. Lasota and P. Rusek, Applications of ergodic theory to the determining of cogged bits efficiency, Archiwum Gornictwa, 19, 3 (1974), 281-295.


VOLUME 8 2001, NO 1-2 PP. 111-122

ON THE ASYMPTOTIC§ OF THE SPECTRUM OF A BOUNDARY VALUE PROBLEM WITH NONPERIODIC RAPIDLY ALTERNATING BOUNDARY CONDITIONS

G.A. CHECHKIN AND E.L DORONINA •

Introduction. In this paper we study boundary value problems with non periodic rapidly alternating boundary conditions for second order elliptic equation. Namely, letting the boundary of the domain to be divided in two fine-grained parts, we pose Dirichlet boundary condition on the first part and Fourier (Neumann) condition on the other one and then, under the assumption that the part that presents Dirichlet boundary condition has a low concentration, we investigate the asymptotic behavior of the spectrum of this boundary value problem.

Previously boundary value problems with rapidly changing type of boundary conditions were considered in many papers. In case of periodic alternation of the boundary conditions the asymptotic behavior of solutions was investigated in [1], [2], [3]), while nonperiodic case was treated in [4]. The work [5] deals with boundary homogenization for random boundary structure.

In the present paper we estimate the deviation of solution to the original problem from the solution to the limit problem in H 1(0)-norm and then construct the asymptotics of the eigenelements of the original problem, in the framework of the theorem on the limit behavior of the spectrum of operators defined in different Hilbert spaces (see [8]).

Statement of the Problem. Let 0 c JR.n be a smooth bounded domain, n ~ 2. We suppose that its boundary CIO consists of two parts r. and

• Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia

111

112 G.A. CHECHKIN AND E.!. DORONINA

N, .

"fe and, moreover, that fe = U f~, where diam (f:) ::0 E and the distance k;J

between any two sets r~ is greater or equal than 2c (c is a small positive parameter). From now on we denote by Ne the number of connected components of re.

Denote by t: the conormal derivative aii (x) ~~:vi> where .v = (1/1> v2 , ... ,

vn) is the outward normal to the boundary o!l. Consider the boundary value problem

(1)

o ( ·· OUe) L[ue] = OXj a'3(x) ox; = f(x)

Ue = 0 On f.,

in n,

OUe ( ) Of..L +ax Ue = 0 on

Here and throughout the paper we use the usual convention of repeated indices.

Regarding the coefficients and the data of this problem we assume: f(x) E £ 2 (0); aii(x) and a(x) are bounded measurable functions; a(x) ~ 0, aii(x) = aii(x) and x;J!~I 2 ::; aii(x)~i~i ::; Kzl~l 2 for some Kr and Kz, 0 < Kj < Kz.

The existence and the uniqueness of a weak solution Ue to problem (1) can be obtained from the Lax-Milgram lemma in a standard way (see, for instance, [9]).

We use the notation H 1(0, fe) for the space of functions from H 1(fJ) with zero trace on r e•

The Asymptotics of Solutions to Problem (1).

The case a(x) '¥= 0. Let u(x) be a weak solution of the problem

(2) {

o ( .. ou(x)) L[u]- OXj a'3 (x) ox; = f(x)

ou(x) &;;:" + a(x)u(x) = 0 on o!l.

in n,

In the further analysis we will use the Sobolev imbedding theorem in the following form (see, for instance, [7]):

THEOREM 1. For a domain fJ which can be represented as a finite union of star-shaped sets, the space H 1(!l) is continuously imbedded into the space Lq(!l) for all q ::0 n

2":2 •

THE SPECTRUM OF A BOUNDARY VALUE PROBLEM 113

The first result of the paper is given by the following statement:

THEOREM 2. Suppose N, = 0(\lnc:\n-l-'lf) as c: -t 0, where 0 < 8 < 2 - ~, then the inequality

(3)

holds true, where the constant cl does not depend on c.

Proof. We begin by constructing the cut-off function vanishing in the ~::

neighborhood ofr,. Let 'lj;(s) E C 00 (lR1) be such a function that 0::; 'lj; :S 1,

'lj;(s) = 0 for \s\ ::; 1 and 'lj;(s) = 1 for \s\ > 1 + u, 0 < u < 1/2. For each r:, we fix an arbitrary point p; E r:, k = 1, ... , N:. Denote by 'lj;, the product

It! 'lj;: with 'lj;:(x) := 'lj; u::JI) , where rk = \x- P:\.

FIG. 1. The functions ,P, are equal to zero in the "black" neighbourhood off~, k = 1, 2 ... N,, and equal to one in the "white" part of fl C JR2 .

Taking the difference of the integral identities of problems (1) and (2) and then substituting cp = (u,- u)'lj;; E H 1 (Q, r,) as a test function into the resulting formula, we obtain

J ij( )a(u,- u) a[(u,- u)\b;J d + J ( ) ( _ )2,1,2d = 0 a X a a X a X Ue U 'l"e S . Xi Xj

n an

Using the equivalence of the norm \lv\1 2 = J aiJ(x) ff:, t; dx + J a(x)v2 ds n ' &n

and the standard norm in H 1 (Q) (see, for instance, [6]), after proper rearrangements we obtain

( 4) ·jj(u,- u)'l/J,\I~fl(n) ::; Cz J (u,- u)2 \V''l/J,\2 dx. n

114 G.A. CHECHKIN AND E.I. DORONINA

Now we estimate the right-hand side of (4):

1

where Q~ is the intersection oU1 and the ball of the radius c:~+o, centered at the point p: E r:, k = 1, ... , N,.

From the definition of '1/J: we derive IY''I/J:I :::; C3 llnc:l (1 1 )'. By the

Tk firk

Holder inequality for p1 and p2 such that l + l = 1, we have P1 P2

j ( u, - u)2 IY''I/J:I 2 dx :::; Ca j ( u, - u) 2 (lin c:llln rk l-2 rJ; 1)

2 dx :::;

Q~ Q~

:::; Callnc:l 2 ( j (u, - u) 2

P1 dx) Pi' ( j (lin rkl-4 rJ;2)P' dx) P~ dx.

Q; Q~

If we set p1 = § = n~2 , p2 = ~, then the last term on the right-hand side admits the estimate

J (lin rkl-4 rJ;2) ll dx = J (lin rkl-2n r;;n) dx :::; C4IIn c:l(1

-2n)

Q~ Q~

and thus,

By Theorem 1 and the Holder inequality with p1 = ~ and P2 = n~2 , we get

Finaiy, combining the previous inequalities, we have


The next step is to prove the uniform upper bound in H 1(Q)-norm for solutions u, to problem (1).

LEMMA 1. If esssup a(x) > 0, then llu,- uiiH'(n) ::; Cs, where u, and u are weak solutions to problem (1) and (2), respectively and the constant C8

does not depend on c.

Proof. Substituting cp = u, E H 1(Q, r,) into the integral identity of (1), using the equivalence of norms in H 1(n) and the Cauchy-Schwarz inequality, we obtain

llu,ll~'(n)::; C9 \ J aij(x) ~~: ~:; dx + j a(x) u,(x)2

ds\ ::; Cg\ J f u, dx\::; n ~ n

::;c9(j f 2 (x) dx)~(j u~(x) dx)~ =C!OIIu,IIH'(!l)· n n

Thus, llu,IIH'(n) ::; C10• Taking into account the regularity of the solution u to problem (2) and using the latter estimate, we obtain the desired bound. Lemma 1 is proved. D

From (6) and Lemma 1 we obtain (3). This completes the proof of Theorem 2.

The case a(x) = 0 . In this subsection we study problem (1) with alternating Dirichlet and Neumann boundary conditions. We are going to show that the limit problem takes the form

(7) {

8 ( .. ou(x)) L[u] = OXj a'3 (x) oxi = f(x)

ou(x) = 0 on on. Ofl

in n,

In this case uniform bounds for solutions to problem (1) and the existence of the solutions to the limit problem are ensured by the following compatibility condition J f(x) dx = 0. From now on we assume this condition to be

n satisfied for problem (1).

Solutions to this problem are defined up to an additive constant that can be fixed by the following normalization condition: J u(x) dx = 0.

n


Let us define<u,>= l~l j u,(x) dx. n

LEMMA 2. If a(x) = 0, then for a solution u, to problem (1) the estimate llu,- < u, > IIH'(n) ::; Cn holds, where Cn does not depend on c:.

Proof. Substituting 'P = UE E H 1(0, r,) into the integral identity of (1), using the ellipticity of the equation, by means of the Cauchy-Schwarz and Poincare inequalities, we obtain

J 2 rJ·· 8u,8u, Ill. I KJ IV'u,l dx :S a'1 (x) 8

xi 8

xi dx = f(x)u,(x) dx = n n n

= [ j f(x)u,(x) dx- j f(x) < u, > dx[ = [ j f(x)(u,(x)- < u, >) dx[ ::;

n n n

::; (j f2 (x)dx)<,(j(u,(x)-<u,>)2 dx)~ :S:Cdfi!L,cn) (j1V'uel2 dx)~. n n n

Thus,

j IV'uEI 2dx ::; C13 n

and by the Poincare inequality llu<- < ·u, > 111-'cn) ::; C14. Lemma 2 is proved. D

Due to the Rellich Theorem (see, for instance, [8]), there exists a subsequence such {c:k} that u<• -<u<• > -" u weakly in H 1(0) as c:k -t 0.

THEOREM 3. If a(x) = 0 and N< = O(llnc:ln-J-~') as c: -t 0 with 0 < 8 < 2- ~, then for solutions u< and u to problem (1) and (7) respectively, the estimate

(8)

is valid, where c15 does not depend on c.

Proof. Subtracting the integral identity of (1) from the integral identity of (7), substituting 'P = (u£- < UE > -u)'l/!; E H 1(0, r£) as a test function, by means of the inequality 2ab ::; e a2 + ~ b2

, where

·- ii( )8(u,-<u,>-u) ·- ( )81/J< a.- a x 8

1/!<, b.- u<-<u£>-u ,--, Xi UXj


we obtain

+~ j(u,-<u,>-u)2 \\J'!f;,\ 2 dx = ~ jcu,-<u,>-u)2 \\J'!f;,\ 2 dx+ 0 0

+(h22 j \\J(u,- <u, > -uW '!(;; dx.

0

Taking sufficiently small 0, we deduce

(9) j \\J(u,- <u, > -uW '!(;; dx::; 0 16 j (u,- <u, > -u)2 \\J'!f;,J2 dx. 0 0

By the same arguments as in the previous subsection, we estimate the integral

j (u,- <u, > -u) 2 \\J'!f;,\ 2 dx::; 011(N,)~ \ln£\*-2 \Ju,- < u, > -u\\~'(o)· 0

From the smoothness of u and Lemma 2 it follows that the norm JJu,- < 1 n8

u, > -u\\H'(O) is bounded uniformly in E. Taking N, = 0(\ln£\n- -.,) as E --+ 0 with 0 < i5 < 2 - ~, we obtain the inequality

(10) j(u,-<u,>-u)2 \\J'!f;,\ 2 dx::; 017\lnc\-8

0

Now estimate the norm

= J \\J(u,- < u, > -u)'!f;,) + (u,- <u, > -u) \J'!f;,\ 2 dx+ 0

+ j(u,"-<u,>-u)2 '!f;;dx::; 2 j \\J(u,-<u,>-uW'!fJ,2) dx+

0 0


(11) +2 j(u,-<u,>-u)2 l\l'-/J,I 2 dx+ j(u,-<u,>-u)2 '-f;;dx. n n

The Poincare inequality for the last integral in (11) gives

[(u,-<u,>-u)2 '-fJ;dx-::;.cls ([(u,-<u,>-u)'-fJ,dxr +

+C19 j 1\l((u,-<u,>-u)'-/J,Wdx. n

Keeping in mind the definition of <u,> and the.condition J u(x) da; = 0, we n

obtain

and then by the Cauchy-Schwarz inequality we have:

( j(u,- <u,> -u)(l- 1./J,) dx f -5. j (u,- <u, > -u)2 da; j (1- '-f;,) 2 dx = n n n

= llu,-<ue> -uiiL(n) I:; J(l-1./J,?dx -5. N,llu,- <u,> -ull~'(f!) J 1 dx. k=l Q~ Q~

Note that

as c -+ 0 with 0 < 8 < 2- ~· By (9), we deduce

ll(u,-<u,>-u)'-/J,II~'(n)-::;. c22 j(u,-<u,>-u)2 IV'-/J,I 2 dx-::;. c23llncl-", n

where N, = O(llncln-l- n;) as c -+ 0 and 0 < 8 < 2 - ~· Theorem 3 is proved. D


The Limit Behavior of the Spectrum. In this section we compare the spectra of original problem (1) and limit problem (2) or (7).

Consider the spectral problems

L[u,] +A, u, = 0 m 0,

(12) { u," 0 on r" ou, Of.L + a(x)u, = 0 on f<

and

{ L[u] + Ao u = 0 m 0, (13) ou

80. Of.L + a(x)u = 0 on

Let { u~} and { uk}, k = 1, 2, ... be orthonormal eigenbases in £ 2 (0) of problem (12) and (13), respectively, and denote by {A~}, {A~}, k = 1, 2, ... the corresponding eigenvalues such that A~ :<:; A~ :<::: ... :<::: A~ :<::: ... , A6 :<::: A5 :<:::

•.• :<:; A~ :<:; .•• and they are repeated with respect to their multiplicities. Denote by A, : £ 2 (0) -+ H 1 (0, r,) the resolvent operator of (1):

Ad = -u8 • This definition should be modified in the case a(x) = 0. Namely, in this case A,f = -u,+<u,>.

The operator A0 : £ 2 (0) -+ H 1(0) is defined by the formula Aof = -u, where u is a solution to problem (2) in the case a(x) 'I= 0 and u is a solution to problem (7) in the case a(x) = 0.

Suppose H 6 = H0 = L2(0), V = H 1(0) and R,: £ 2 (0) -+ £ 2 (0) is the identical operator.

Below we use the theorem from [8] on the limit behavior of the spectrum of operators defined in different Hilbert spaces. Let us verify conditions C1-C4 of this theorem.

Condition C1 is obviously satisfied. It is easy to check the positiveness and self-adjointness of the operators A, and A0 . The compactness of A61 Ao follows from the compactness of the imbedding of H 1(0) into £ 2 (0). The uniform boundedness of the norms JJA,JI follows from Lemma 1 if a(x) 'I= 0 and Lemma 2 if a(x) = 0. Thus condition C2 holds. Condition C3 can be verified by virtue of Theorem 2, Theorem 3 and the following assertion:

LEMMA 3. If a(x) 'I= 0, then for the solutions u, to problem (1) and the solution u to problem (2) the estimate

(14)


is valid, where the constant C24 does not depend on c. If a(x) = 0, then for the solutions u, to problem (1) and the solution u

to problem (7) the estimate

(15)

holds true, where the constant C25 does not depend on c.

Proof. First we prove (14). Using the definition of 7/Je by the Holder inequality with p1 = ~ and P2 = n:2 , we deduce

:S: C26€ 1~" !l(u,- u)lli;/'.",(!1)·

Finally, by Theorem 1 we get

To complete the proof, it remains to apply Lemma 1. The estimate (15) can be obtained in the same way. Lemma 3 is proved. D

Now we check the condition C3. • a(x) =J- 0. Using the definitions of A., A0 and the imbedding theorem, we obtain

for all f E L2(0)

!lA. Ref- R.Aofll£2(n) = llu,- uli£2(!1) :S: ll(u,- u)7/J,IIL2(n)+

+ll(u,- u)(l-7/J,)II£2(n)::; ll(u,- u)'I/J,IIH'(n) + ll(u,- u)(1 -7/J,)IIL2(n)·

• a(x) _ 0. Similarly we have for any f E L2(0)


:S IJ(ue-<Ue> -u)7/Jei!H'(Il) + IJ(ue-<Ue> -u)(1-7/Je)!!L2(n)·

Keeping in mind estimates (3), (8), (14), and (15), we obtain

asc:-+0

if N6 = O(Jlnc:Jn-l-no/2 ) and 0 < 8 < 2- ~-In order to verify condition C4, consider an arbitrary sequence {16 }

bounded in L 2 (S1) and denote v6 = Ade· Since { V6 } is uniformly bounded in H 1(S1) and therefore is compact in L 2 (S1), the sequence {Ade} is compact in L2 (S1), and condition C4 holds.

Denote by f.t~ and f.t~ the eigenvalues of problems A6u~ = J.t~u: and A0u~ = ft~U~, respectively. The said theorem from [8] gives the following estimate:

!J.t!- !L~I :S C3o sup IIAeRd- RoAofiiH,, k = 1, 2, ... , uEN(Jt~,Ao),

lluiiH0 ~1

where N(J.t~, Ao) = { u E I:lo, A0u = ft~U} is the corresponding eigensubspace of Ao.

Finally, >.: = .j;, >.~ = -';;, and we arrive at the following theorem Me J.to

THEOREM 4. If 0 < 8 < 2 - ~' Ne = O(llnc:Jn-l- '1'), then for sufficiently small c: the estimate

I~~ -~~I ::; C3J!lnc:J-of2

holds true with c31 independent of c and 8.

REFERENCES

[1 J M. Lobo and E. Perez, Asymptotic behavior of an elastic body with a surface having small stuck regions, MMAN, 22, 4 (1988), 609-624.

[2] A. Damlamian and Li Ta-Tsien (Li Daqian), Boundary homogenization for elliptic problems, J. Math. Pure et Appl., 66 (1987), 351·-361.

[3] G .A. Chechkin, Averaging of boundary value problems with a singular perturbation of the boundary conditions, Russian Academy of Sciences. Sbornik. Mathematics, 79, 1 (1994), 191-222 (in Russian).

[4] O.A. Oleinik and G.A. Chechkin, Solutions and eigenvalues of the boundary value problems with rapidly alternating boundary conditions for the system of elasticity, Rendiconti Lincei: Mathematica e Applicazioni. Serie IX, 7, 1 (1996), 5-15.


[5] A.Yu. Beliaev and G.A. Chechkin, Homogenization of operators with finescale structure of the boundary conditions, Math. Zametki, 65, 4 (1999), 496--510 (in Russian).

[6] V.P. Mikhailov, Lectures on partial differential equations, Nauka, Moskow, 1984 (in Russian).

[7] S.L. Sobolev, Some applications of functional analysis in mathematical physics, AMS Press, Providence, 1991.

[8] O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization, North-Holland, Amsterdam, 1992.

[9] K. Iosida, Functional analysis, Springer-Verlag, New York, 1965.


VOLUME 8 2001, NO 1-2 PP. 123-·140

AVERAGING OF TRAJECTORY ATTRACTORS OF EVOLUTION EQUATIONS WITH RAPIDLY OSCILLATING

COEFFICIENTS *

V.V. CHEPYZHOV AND M.I. VISHIK t

Abstract. Evolution equations with rapidly oscillating terms are considered. It is assumed that the rapidly oscillating coeficients have averages in the corresponding function spaces. The following theorem is proved. The trajectory attractors of these equations tend to the trajectory attractors of the corresponding averaged equations. This result is applied to the 3D Navier-Stokes system, the reaction-diffusion system, the GinzburgLandau equation, and the damped wave equation.

1. Averaging of Rapidly Oscillating Functions. In this section we briefly consider some questions concerning rapidly oscillating functions. We study the averaging of functions of the form b ( ~) and b ( x, ~) as c: -+ 0 + . The variable z = ~ is called rapid and the variable x is called slow. Here z and x are spatial variables, that is, z E JRn and x E !1 cs JRn.

Let we be given a family of real functions {f,(x), c: > 0} in a domain !1 cs JRn depending on a small positive parameter c:. Let also j,(x) E Lp(rl), where p > 1. We say that the functions j,(x) have an average ](x) E Lp(!l) as c: -+ 0+ in the space Lp,w(!l) if

(1) f.,(x)., J(x) as c:-+ 0 + weakly in Lp(!l),

* The research described in this publication was made possible in part by Grant No. 99-01-00304 of Russian Foundation of Fundamental Researches and by Award of the U.S. Civilian Research & Development Foundation for the Independent States of the Former Soviet Union (CRDF). The talk was given by M.I. Vishik.

t Institute for Information Transmission Problems, Russian Academy of Sciences, B.Karetniy 19, Moscow 101447,GSP-4, Russia.

123

124 V.V. CHEPYZHOV AND M.I. VISHIK

that is, for every function <p(x) E Lq(Q), 1/p + 1/q = 1, we have

(2) I f,(x)<p(x)dx -t I J(x)<p(x)dx as c: -t 0 +. () ()

Similarly we define the averaging of functions f,(x) E L00(Q) as c: -t 0+ in Loo,•w(Q). In this case average /(x) E £ 00 (!2) and (2) should hold for every function <p(x) E L1(Q) (L00 (Q) = L1(Q)*), that is,

f,(x).., J(x) as 6 -t 0 + *-weakly in L00 (Q).

If the functions f. ( x) have an average J ( x) in the space Loo,•w ( Q), then they clearly have the same average in the space Lp,w(Q) for each p > 1.

EXAMPLE 1.1. Let f,(x) =sin(~), x E [0, 27r]. It is easy to see that sin(~) .., 0 as c: -t 0+ weakly in £ 2 (0, 27r) and * -weakly in £ 00 (0, 27r). Thus, sin ( ~) has the average 0 in these spaces.

Let we be given a real function b(x,z), x En <S JRn, z E JRn. In many cases it is possible to define the function f,(x) = b (x, il1). For example, if b(x, z) E C(QxJRn), then the function b (x, ~)is well defin'ed for X En. Below we shall consider other examples of such functions. We study the averaging of functions b (x, ~) as c: -t 0+ in the spaces Lp,w(Q) and Loo,•w(Q). For simplicity, we assume that 0 E n.

To begin with, we consider functions of the form b (~), where b(z) E L~c(JRn) or b(z) E L~0c(JRn) for some p > 1. It is not hard to prove the following statement.

PROPOSITION 1.1. Let the function b ( i!1) have the average b(x) in - . Lp,w(Q). Then the function b(x) is homogeneous of degree 0, i.e., for every .\>0

(3) b(.\x) = b(x) \:fx E ]Rn.

Therefore if the averaged function b(x) is not a constant, then the origin is a discontinuity point for b(x). We consider in more detail the case when the function b(x) is continuous at the origin, that is, b(x) - const.

We denote P!;:::~~ = [a1 , bl] x ... x [an, bn] = P a parallelepiped in JRn n

with volume IPI = n (bi- ai)· We assume that the function b (il1) has the bl •

average bE lR in Lp,w(n). We claim that for any parallelepiped P!;:::~~ = P

(4) 1 I -I.\PI b(z)dz -t bas.\ -t +oo. ).p

TRAJECTORY ATTRACTORS OF EVOLUTION EQUATIONS 125

n Here I API = An !PI = AnTI (b;- a;). Indeed, we set in (2) cp(x) = '1/Jp(x), is

i=l the characteristic function of P, that is, '1/Jp(x) = 1 if x E P and 'lj;p(x) = 0 otherwise. Then, by (2), for A= ~ we have

(5) J b(Ax)dx -t j bdx = IPiii as A-t +oo. p p

Changing the variable AX = z in the left-hand side of (5), we obtain

1 J -A" b(z)dz -t IPib as A-t +oo, AP

and ( 4) is proved. The converse statement is also true. We have the following criterion.

THEOREM 1.1. Let b(z) E L~c(JRn). The function b (~) has the average bE lR in Lp,w(D) if and only if:

(i) the family {b (~), 1 > c: > 0} is bounded in Lp(D); (ii) limit (4) holds for every parallelepiped P%{:::~~ such that a1b1 =

... = anbn = 0.

REMARK 1.1. Property (i) is valid, for example, if the function b( z) is uniformly bounded in L~c(JRn) in the following sense:

sup j lf(z + zo)!Pdz < +oo, zoEJR.n

?,1 ... 1 o ... o

where p,L .. l is the unite cube [0, 1]n in lRn. 0 ... 0

Similarly, we have

THEOREM 1.2. Let b(z) E L~c(JRn). The function b (~) has the average b E lR in Lp,w (D) if and only if:

(i) b(z) E L 00 (lRn); (ii) limit (4) holds for every parallelepiped P%;:::~~ such that a1b1

... = anbn = 0.

REMARK 1.2. In the case n = 1 the condition (ii) reads:

A 0

(6) 1 J 1 J -lim \ b(z)dz = lim \ b(z)dz =b. A-too A >.-too A

0 -A


The following sufficient condition is also useful. We denote K;.(x0 ) a cube in JR.n with centre x0 and edge ..\ :

(7)

THEOREM 1.3. Let b(z) E Loo(fl). We assume that there exists a real number b such that for all ..\ > 0 and x0 E JRn we have

(8)

where f3(u)--+ 0 as u--+ +oo. Then for every fl IS JRn the function b (~) has the average b as c --+ 0+ in Loo,•w(fl).

The proof of Theorems 1, 2, and 3 will be given in [2].

EXAMPLE 1.2. Let b(z) be a continuous periodic function with respect to each z3, i.e., b(z) E C(JRn) and

b(z1o ... , Zj + 1, ... , Zn) = b(z!o ... , Zj, ... , Zn), j = 1, ... , n

for all z E JRn. It is easy to verify condition (8) with

b = j b(z)dz.

11'"

(Here 'f" = (JRmod 1 t is the n-dimensional torus.) Hence the function b( z) has the average b as c--+ 0+ in Loo,•w(fl).

We now consider the averaging of functions depending on rapid and slow variables of the form b (x, ~) , X E fl @ JRn.

EXAMPLE 1.3. Let b (x, ") = b1 (x) b2 ("), where b1(x) E Lp(fl) and e e _ b2 (z) E L00 (1Rn). We assume that the function b2 (~) has the average b2 E .JR. in Loo,•w(fl). Then, evidently, the function b (x, ~) has the average b1 (x) b2 in Lp,w ( fl). Similarly we construct functions of the form

(9)


where b1i (x) E Lp(S1), b2i (z) E L00 (1Rn), and the functions b2i (~) have the averages ~i E JR. in Loo,*w(S1). It follows easily that bN (~, x) has the average

N

/)N (x) = 2:': bli (x) b2i i=l

in Lp,w(S1). We also note that analogous results are valid for the limit of functions bN ( x, ~) of form (9) as N -+ oo in a suitable function space.

ExAMPLE 1.4. Let the function b(x, z) E C(IJ x JR.") be quasiperiodic with respect to each Zj, j = 1, ... , n (z = (z1, ... , zn)). More precisely, there exists a continuous function B(x, Wn, ... , wlk, ... , Wnl> ... , WnkJ E C (n x 1fk1 x ... x 'Jfkn) which is 1-periodic with respect to each Wij such that

(10) VzElR.n.

Here { D<ij} ;~};.'.'.',%, are rationally independent real numbers. Using the Kronecker-Weyl theorem, we can prove that the function b (x, ~) has the average

(11) b(x)= J .. ·J B(x,wb ... ,wn)J.L(dwt ... dwn), 'I['klx ... x'!fkn

where wi = (wil, ... , Wik,) E 'Jfk,, i = 1, ... , n, J.L(dw 1 ... dwn) is the Lebesgue measure on 1fK = 1fk1 x ... x 'Jfkn, K = k1 + ... + kn, and J.L(1fK) = 1.

2. Trajectory Attractors of Evolution Equations. In this section we give an abstract scheme for the construction of a trajectory attractor of an evolution equation. In the next section we shall apply this scheme to the study of trajectory attractors of concrete evolution equations with rapidly oscillating coefficients and their limits.

We consider an abstract evolution equation

(12) o1u = A(u), t 2: 0.

We are given a (nonlinear) operator A(·) : E-+ E0 , where E, Eo are Banach spaces and E r,:; E0 . For instance, A(u) = ab.u- f(u) + g (see Section 3).


We are going to study solutions u(s) of equation (12) being functions of s E JR.;. as a whole. Here s = t denotes the time variable. A set of solutions of (12) is said to be a trajectory space x;+ of equation (12). Describe the trajectory space x;+ in more detail.

First we consider solutions u(s) of (12) defined on a fixed segment [tll t2] from JR. We are looking for solutions of (12) in a Banach space :F1,,1,. The space :F1,,12 consists of functions f(s), s E [tll t2], such that f(s) E E for almost all s E [t1, t2]. For example, :Ft,,t, can be the space C([tll t2]; E), or Lp(t1, t2; E), p E [1, oo], or an intersection of such spaces (see Section 3). We assume that llt,t,:Ft3 ,t4 = :Ft,,t2 if [t1, t2] C [ta, t4], where llt,,t2 denotes the restriction operator onto the segment [t1, t2].

We make the following assumption. If f(s) E :F1,,1, then A(f(s)) E V1,,1, where V1,,1, is a larger Banach space, :F1,,1, <;; V1,,1,. The derivative otf(t) is a distribution with values in Eo, otf(s) E D'(]tll t2[; Eo): V11 ,t2 <;; D'(]t1, t2[; Eo). A function u(s) E :F11 ,t2 is said to be a solution of (12) from :F11 ,t2 (on the interval ]t1 , t2 [) if 81u(s) = A(u(s)) in the distribution sense.

We also define the space

(13) :F!:c = {f(s), s E JR.;. I llt,,tzf(s) E :Ft,t2 'if [tllt2] <;;JR.;.}.

For example, if :F11 ,t2 = C([tll t2]; E), then :F!:c = C(JR;.; E), and if :Ft,,t, = Lp(t1, t2; E), then :F!:c = L~oc(JR;.; E).

A function u(s) E :Ffc is said to be asolutionof(12) in :Ffc ifll11 ,12u(s) is a solution of (12) in :F1,,1, for every [t1 , t2] C JR.;..

We denote by x;+ a set of solutions of (12) from :Ffc. (Notice that x;+ is not necessarily the set of all solutions from :Ffc.) The elements of x;+ are called trajectories, and the set x;+ is called the trajectory space of the equation (12).

We assume that the trajectory space x;+ is translation invariant in the following sense: if u(s) Ex;+, then u(h + s) Ex;+ for every h :2: 0. This is a very natural assumption for solutions of an autonomous equation.

We now consider the translation operators T(h) in :Ffc:

T(h)f(s) = f(s +h) for h :2: 0.

It is clear that the mappings {T(h), h :2: 0} form a semigroup in :Ffc T(hl)T(h2) = T(h1 + h2) for hll h2 :2: 0 and T(O) is the identity operator. The semigroup {T(t), t :2: 0} is called the translation semigroup. By our assumptions, the translation semigroup maps the trajectory space x;+ to itself:

(14) T(t)K+ <;; x;+ for all t :2: 0.


We shall study attracting properties of the translation semigroup {T(t)} acting on the trajectory space JC+ c Ffc. We define a topology in Ffc.

Let we be given a convergence topology in Ft,,t,· Denote by Gt,,t, the topological space Ft,,t, endowed with this topology. We suppose that Gt,,t, is a Hausdorff topological space. For example, Gt, ,t, can be Ft,,t, itself with the strong or weak (or even *-weak) convergence topology of a Banach space. We denote by e~c the space F~oc with local convergence topology on Gt,,t, for every [t1, t 2] c;;; ll4 i.e., by definition, a sequence Un(s)} C Ffc converges to f(s) E Ffc as n-+ oo in e~c ifi1t,t,fn(s)-+ I1t,,t,f(s) (n -> oo) in Gt,,t, for any [ t 1 , t 2] c;;; JJ4 . It is not hard to prove that e~c is a Hausdorff topological space. We note that the translation semigroup {T( t)} is continuous in e~c. This assertion follows directly from the definition of the topological space eloc

+' We also consider the following space

F! : F! = {f(s) E Ffc IIIJIIF't < +oo },

where the norm is defined as

(15) IIJIIF' =sup IITio,d(t + s)IIFo,,· + t?O

Evidently, Ft with norm (15) is a Banach space. Notice that Ft t;;; e~ac. We need the space Ft to define bounded sets in JC+.

We suppose that JC+ c;;; F!, i.e., any trajectory u(s) E JC+ of equation (12) has finite norm (15). We define a trajectory attractor of the translation semigroup {T(t)} acting on JC+.

DEFINITION 2.1. A set P c;;; e~c is said to be an attracting set for the trajectory space JC+ in the topology e~c if for any set B c;;; JC+ bounded in B c;;; JC+ the set P attracts T(t)B as t -+ +oo in the topology e~c, i.e., for any neighbourhood O(P) in e~c there exists t1 2 0 such that T(t)B c;;; O(P) for any t 2 t1 .

DEFINITION 2.2. A set A c;;; JC+ is said to be the trajectory attract.or of the translation semigroup {T(t)} on Jc+ in the topology e~c if A is compact in e~c, A is strictly invariant with respect to the translation semigroup: T(t)A =A for all t 2 0, and A is an attracting set for JC+

REMARK 2.1. Using the terminology from [1], we can say that the trajectory attractor A is a global (F!, e~c) -attractor of the translation semigroup {T(t)} acting in JC+, that is, A attracts T(t)B as t-+ +oo in the topology e~c for any bounded (in F! ) set B from JC+.


Now we have the following result on the trajectory attractor of equation (12).

THEOREM 2.1. Let the trajectory space JC+, JC+ ~ Fi corresponding to equation (12) be closed in e~c. We assume that there is an attracting set P of {T(t)} for JC+ in e~c such that P is compact in e~c and bounded in F.t. Then the translation semigroup {T(t), t ;::: 0} acting on JC+ has a trajectory attractor A~ JC+ n P. The set A is compact in e~c and bounded in F.t.

Proof. Indeed, the semigroup {T(t)} is continuous on JC+ with respect to e~c. The set Pis (F!, e~c)-attracting, compact in e~c, and bounded in F.t. Then the semigroup {T(t)} has the global (F!, e~c)-attractor A, which is evidently a trajectory attractor (see [3] for the complete proof). 0

Now describe the structure of the trajectory attractor A of the equation (12) in terms of complete trajectories of this equation.

Consider equation (12) on the whole time axis

(16) 81u = A(u), t E JR.

We have defined the trajectory space JC+ of equation (16) on~- We now extend this definition on the whole JR. If a function f(8), 8 E JR is defined on the whole time axis, then translations T(h)f(8) = f(8 +h) are defined for negative h as well.

A function u(8), 8 E JR is said to be a complete trajectory of equation (16) if II+ u( 8+ h) E JC+ for all h E JR. Here II+ = IIo,oo denotes the restriction operator on the semiaxis ~.

We have introduced the spaces Ffc, Fi, and e~c. In the same way we define spaces Floc, Fb, and e 1oc :

where

(17)

Fb = {!(8) E Floc 111/ll.r• < +oo},

11/ll.r• =sup IIIIo,d(h + 8)ll.ro,,· hEIR

Topological space eloc coincides (as a set) with Floc and, by definition, fn(8) -+ /(8) (n -+ oo) in etoc if IIt,,tJn(8) -+ IIt,,t,/(8) (n -+ oo) in et,,t, for any [til t2] ~ JR.


DEFINITION 2.3. The kernel K in the space Fb of equation (16) is the union of all complete trajectories u(B), s E JR. of equation (16) bounded in Fb with respect to norm (17):

(18) IITio,Iu(h + s)ll.ro.1 :S:: Cu Vh E R

THEOREM 2.2. Assume that the conditions of Theorem 2.1 hold. Then

(19)

the set K is compact in etoc and bounded in Fb.

The proof can be found in [3]. Let us briefly clarify the nature of attraction of bounded set B from K+

to the trajectory attractor A.

CoROLLARY 2.1. Under the assumptions of Theorem 2.1, let B be a set from K+ bounded in Ft. Then, for any M > 0, the set ITo,MT(t)B tends to I1o,MK in the topological space Go,M as t -+ oo. For example, if Go,M is a metrizable space, then

diste0,M(I1o,MT(t)B, I1o,MA) = diste0,M(I1o,MT(t)B, I1o,MK)-+ 0 (t-+ oo).

Here, as usual, the distance from a set X to a set Y in a metric space M is defined as follows:

distM(X, Y) =sup distM(x, Y) =sup inf PM(x, y), XEX xEX yEY

where PM(x, y) denotes the metric in M. Theorems 2.1 and 2.2 show that to construct the trajectory at tractor we

need an attracting set P compact in e~c and bounded in F!. Usually in applications a large ball BR = {llfll.rt :S:: R} in F! (R » 1) serves as such an attracting set. The attraction to B R follows from the inequality

(20)

for any u E K+ and any t ~ 0, where C(R) depends on R and R 0 does not depend on u. Usually inequality (20) follows from a priori estimates for solutions of equation (12). If, in addition, a ball B R in F! is compact in e~c, then B2Ro is the required compact uniformly attracting set.

In the next section we shall consider evolution equations and their trajectory attractors depending on a small parameter c: > 0.


DEFINITION 2.4. We say that trajectory attractors A, converge to a trajectory attractor A as e -t 0+ in the topological space e~c if for any neighbourhood O(A) in e~c there exists e1 2:: 0 such that A.~ O(A) for any e < e1• In particular, if 6 1,,12 are metric spaces, then

diste,,,,2 (II,,,,

2A., II,,,,

2A) -t 0 (t -t oo ).

3. Trajectory Attractors of Equations with Rapidly Oscillating Terms and their Limits .

3.1. Averaging of the trajectory attractor of 3D Navier-Stokes system with rapidly oscillating external force. Consider 3D NavierStokes system in the domain n <S JR3 :

(21) OtU + vLu + B(u) = 9 ( x, ~), (V', u) = 0, ulan= 0,

where X= (x1,x2,X3) E Q, U = u(x,t) = (ul,u2,u3),and 9 = (9!,92,93).

Here Lis the 3D Stokes operator: Lu = -Pl:,.u; B(u) = B(u,u), B(u,v) = , 3

P(u, V')v =PI: u;&x,v. By Hand V we denote the closure in (L2 (Q))3 and i=l

(H1(Q))3 of the set V0 = {v I v E (CQ'(Q))S, (V',v) = 0}. The P denotes the orthogonal projector in (L2(Q))3 onto the Hilbert space H. The scalar products in Hand in V are (u, v) = J(u(x), v(x))dx and ((u, v)) = (Lu, v) =

n f(V'u(x), V'v(x))dx, and the norms are lui= (u,u) 112 and !lull= (Lu,u) 112 , n respectively.

We suppose that 9 (x, ;) E H for every e > 0 and the function 9 (x, ;) has the average g(x) as e -t 0+ in the space Hw, that is,

(22) (9 ( x, ~), <p(x)) -t (g (x), <p(x)) as e -t 0 + V<p E H.

REMARK 3.1. Using Section 1, we can construct many examples of function 9 (x, ;) that satisfy (22). For instance, 9 (x, ;) = 91 (x) 92(;), where 91 E H and 92 (z) is a periodic or quasiperiodic function w.r.t. z, or just a function having the average 92 E lR in Loo,•w(n) n H.

To describe the trajectory space JC; of equation (21), we consider weak solutions of this equation in the space L~0<(][4; V) n £~<(][4; H). If u(s) E

L~<(Jl4; V) n L~c(Rt; H), then the equation {21) makes sense in the distribution space D'(Rt ;V'), where V' is the dual space of V (see [6]).


DEFINITION 3.1. The trajectory space Kt is the union of all solutions u(s) E L~oc(J14; V)nL~c(J14; H) of equation (21) with external force g (x, ~) that satisfy the following inequality:

(23) 1 d 2dtlu(tW + vllu(t)W::; (g,u(t)), t E J14.

Inequality (23) should be read as follows: for any function 1/;(s) E

C8"(]0, +oo[), 1/J 2: 0,

+oo +oo +oo

(24) -~ j lu(sW'l/J'(s)ds + v j llu(s)ll 21/!(s)ds::; j (g, u(s)) 1/!(s)ds. 0 0 0

If u0 E H, then there exists a weak solution u(s) of equation (21) belonging to the space L~0c(ll4; V) n L~c(ll4; H) such that u(O) = u0 and u(s) satisfies inequality (24). For the proof, see [6], [3].

REMARK 3.2. For the 3D Navier-Stokes system, the uniqueness problem is still open. It is not known either whether any weak solution of (21) satisfies inequality (23). Nevertheless, the solutions u(t), t 2: 0 resulting from the Galerkin approximation method satisfy (23).

It is known that for any weak solution u(s) E L~oc(ll4; V) n L;:(J14; H) of equation (21) the derivative OtU E L;(~(J14; V').

Consider the space

Ft;'c = L~0c(ll4; V) n £~C(ll4; H) n { v I OtV E L~~~~ (lRo-; V')}

endowed with the following convergence topology. A sequence { vn} C Ft;'c converges to v E Ffc as t --+ oo if Vn(s) ---, v(s) (n --+ oo) weakly in L2(t1, t2;V), *-weakly in L00 (t1, t2; H), and OtVn(s) ---, Otv(s) (n --+ oo) weakly in L4; 3 (t1 , t2; V') for every [t1 , t2] C ll4. The space Ffc with the above weak topology is denoted by e~c. We shall use also the space

F! = L~(J14; V) n L~(J14; H) n { v I 8tv E L~;3 (J14 ;V')}

that is a subspace of Ffc. Recall that

PROPOSITION 3.1. For any u(s) E Kt,

(25) IIT(t)u(- )li.F';.::; Cllu(. )IIL(o,l;HJ exp(->.t) + Ro Vt 2: o,


where >. is the first eigenvalue of the operator v L; C depends on >. and Ro depends on>. and !lull~ (see [3]).

The translation semigroup {T(t) I t ?.: 0} acts on ICi :

T(t)u(s) = u(t + s), s?.: 0.

Evidently,

T(t)u(s) E JCi VuE JCi, t 2: 0,

Therefore

T(t)ICi ~ JCi 'It 2: 0.

It follows from (25) that the ball B0 = llvll.:r~ ::; 2Ro is an absorbing set of the translation semigroup {T(t)} acting on ICi. The set B0 is bounded in :F! and compact in e~c.

· PROPOSITION 3.2. The trajectory space ICi is closed in e~c (see [3]).

The kerneliC, of equation (21) consists of all weak solutions u(s), s E JR. that satisfy inequality (24) for '1/J(s) E C0 (JR.), 'lj! ?.: 0 and that are bounded in the space

:Fb = L~(JR; V) n Loo(lR.; H) n {v l8tv E L~;3 (JR.; V')}.

By Propositions 3.1 and 3.2, Theorems 2.1 and 2.2 are applicable. Equation (21) has a trajectory attractor A,. The set A, is bounded in :F! and compact in e~c. Moreover,

(26)

The kerneliC, is non-empty, bounded in :Fb, and compact in 6 10<.

We note that the following embedding is continuous: e~c c L~oc(Rr; H1- 5), 0 < 8 ::; 1. Therefore for any set B C ICi bounded in :F!

where M > 0. Along with equation (21) we consider the averaged equation

(27) Otft + vLu + B(u) = g (x), (\7, u) = 0, ulan= 0.


Clearly, equation (27) has the trajectory attractor A in the trajectory space JC+ corresponding to equation (27) (see Definition 3.1) and

(28)

Now we can formulate the main theorem.

THEOREM 3.1. The following limit holds in the topological space 8~oc :

(29)

Moreover,

(30)

It is clear that (30) implies (29). The proof will be given in [2].

COROLLARY 3.1. For every 0 < !i :S 1 and for any M > 0

(31) distL,(O,M;H'-'l (ITo,MA, ITo,MA) -+ 0 (c:-+ 0+ ).

3.2. Averaging of trajectory attractors of reaction-diffusion systems with rapidly oscillating terms. We consider the reaction-diffusion system .with rapidly oscillating terms of the form

(32) OtU = ailu- b (x, ~) f(u) + g ( x, ~), ulan= 0,

where X En~ Rn, u = (u\ ... ,uN), a is anN X N with positive symmetric part. For simplicity, we consider the case N = 1. We note that all the results can be applied to the systems with nonlinear terms of the form m :>-; bj (x, :Z) fj(u), where bj are matrices and f'J(u) are polynomial vectors. j=l

We assume that the function f(v) E C(R; JR.) satisfies the following inequalities:

(33) f(v)v 2': /'lviP- C, lf'(v)l :S C1 (lviP + 1), p 2': 2.

We also assume that b(x, z) E Cb(Q x JR.), (J1 2': b(x, z) 2': (30 > 0, and the function b (x, :Z) has the average b(x) as c:-+ 0+ in Loo,*w(12), i.e.,

(34) J b ( x, ~) <p(x)dx-+ J b (x) <p(x)dx (c:-+ 0+)

n n


for any function cp E L1(n). For the function g (x, ~),we assume that it has the average g(x) in the space V' = H-1(n) :

g ( x, ~) -, g(x) (c: -+ 0+) weakly in V',

that is,

(35)

for any cp E V = HJ(n). In particular, the following functions are available:

n

g (x, ~)=Yo (x, ~) + L:ox,Yi (x, ~), c i;l c

where the functions g; (x, ~) have the averages Di(x) E L2(n) in H = L2(n) :

(g; ( x, ~), cp(x)) -+ (.9 (x), cp(x)} (c:-+ 0+ ), i = 1, ... , n.

We note that the H-norms of the functions 8x,9i (x, ~) = Yix; (x, ~) + }Yiz; (x, ~) can tend to infinity as c:-+ 0+ and are bounded in V' only.

Similarly to [4], we study weak solutions of equation (32), that is, the functions u(x, s) E L!,"c(JI4; Lp(Sl)) n L~c(JI4; H) n L~0c(JI4; V) which satisfy equation (32) in the distribution sense of the space D'JI4; H-r(st)), where r = rnax {1, n(l/2- 1/p)}. For every u0 E H, there exists at least one weak solution u(x, s) of equation (32) such that u(O) = u0 . This solution is not necessarily unique because we do not assume the Lipschitz condition for f(v) with respect to v. We denote K"t the set of all weak solutions of (32).

We study the trajectory attractor Ac of equation (32), that is, the global (:rt, e~c)-attractor of the translation semigroup {T(t)} acting in Kd- Here we denote

etoc = Ltoc (1t> . L (n)) n Ltoc (1t> . H) n Ltoc (1t> . V)n + p,w ll.~' P oo,*W .u:~.+' 2,w .u.~ 1

Along with equation (32) we consider the averaged equation

(36) O(il = aAu- b (x) f(u) + g (x), ulan= 0,


for which the trajectory attractor A can also be constructed in the corresponding trajectory space K+.

PROPOSITION 3.3. Under assumptions (33), (34), and (35), equations (32) and (36) have trajectory attractors Ac and A respectively. Moreover,

Ae = IT+Ke and A= IT+K,

where Ke and K are kernels of these equations in the space Fb

The proof can be found in [4]. We note that e~c c L~oc(Jt4; H 1- 8(0)), 0 < o ::: 1, and therefore the

trajectory attractors Ae and A attract bounded sets of trajectories of the equations in the strong topology of the spaces L&0c(Rt-; H1- 8).

THEOREM 3.2. Under the above assumptions,

(37) Ac -+ A as c; -+ o + in e~c

and

(38) Ke-+ K as c;-+ 0 + in etoc.

The proof will be given in [2].

COROLLARY 3.2. For every 0 < o ::= 1 and for any M > 0,

distr,,(o,M;H'-') (ITo,MAe, ITo,MA) -+ 0 (c;-+ 0+ ).

These results are applicable to the averaging of the complex Ginzburg-Landau equation

with periodic boundary conditions. Here x E [0, 27r]n, u = u1 + iu2 E IC, g(z) E L&oc(JE.; IC), and R(z), (J(z) E C&(lE.). We assume that functions g (~), R (~),and (3 (~) have the constant averages g, R, and {J. We claim that the trajectory attractors Ae of equation (39) converge as c; -+ 0+ to the trajectory attractor A of the averaged equation


3.3. Averaging of the trajectory attractor of the dissipative wave equation with rapidly oscillating coefficients. We consider the hyperbolic equation with damping

(40) ~u + ')' ( x, ~) 8tu = ~u- b ( x, ~) f(u) + g ( x, ~), ulan= 0.

Here X En@ JR_n, Let 'Y(x,z),b(x,z) E Cb(n X JR.), and let the following inequalities hold:

(41)

(42)

0 < 1'1 ::; ')'(x, z) ::; ')'2,

0 < fh ::; b(x, z) ::; /32, Vx E fl x Rn.

The function f(v) satisfies the inequalities

(43) lf(v)l < 'Yo(lvlp-l + 1), p > 2, (44) F(v) > 'Y1IviP- C1, f(v)v 2': 'Y2F(v)- C2 Vv E JR.,

v

where F(v) = f f(w)dw. 0

We assume that the functions 'Y (x, ~) and b (x, ~) have the averages 'Y (x) and b (x) as£ -t 0+ in the space Loo,•w(fl), that is,

( 'Y ( x, ~) , <p(x)) -t ('Y (x), <p(x)),

(45) (b(x,~),<p(x)) -t (b(x),<p(x)) (e-tO+)

for any function (g(x),<p(x)) (e-tO+)V<pEL2(fl).

We also consider the averaged equation

(47) 8fu + 'i' (x) 8tu = ~u- b (x) f(u) + g (x), ulan= 0.

It is clear that the functions 'Y (x) and b (x) satisfy inequalities (41) and (42). It follows from assumptions (41)-(46) that equations (40) for £ > 0

and equation ( 47) have trajectory attractors A, and A, respectively. The construction and the properties of these attractors were described in [3]. Recall that a function u(x, s) is the weak solution of (40) (or (47)) if u(x, s) E L:c(~; Lp(fl) n HJ(fl)), 8tu E L~(~; L2(fl)), and u satisfies the equation


in the distribution sense of the space D'(Rr; H-r(f:l)), where r = max{1, n(l/2 1/p)}. Then, clearly, a[u E L~c(JI4; H-r(f:l)) (see [6]).

We set

Jc,a(v, v1) = J (IY'v(xW + lv1 (x) + av(xW + 2b ( x, ~) F(v(x))) dx. n

Let u(x, s) be a weak solution of (40). We denote

z(s) = Jc,a(u(s),a,u(s)).

The trajectory space K;(N), N 2: 0 of equation (40) consists of the weak solutions of ( 40) that satisfy the inequality

(48) z(t) ::; R, + N exp ( -li,t) l;f t 2: 0

for every a such that

Here the positive constants Ra and li" are defined similarly to those given in [3] and do not depend on E. Analogously we define the trajectory space K+ (N) of equation ( 47). The translation semigroup {T(t)} acting on K; (N)

-+ -and K ( N) has the trajectory attractors Ac and A, respectively. The spaces e~c and :Fi are defined as follows:

e~c = L~~*w(l!4; Lp(f:l) n HJ(f:l)) n {vI OtV E L~~*w(l!4;1:J(O))} n {vI a[v E L~~*W(JI4;Wr(O))},

:Fi = L::0(JI4;Lp(f:l)nHJ(O))n

{vI a,v E L::0(JI4;Lz(O))} n {vI azv E L::0(JI4;Wr(O))}.

The sets A, and A are bounded in :Fi. Finally, we obtain that

(49)

where K, and K are kernels of equations (40) and (47), respectively. It is easy to see from (48) that Kc consists of all weak solutions u(x, s), s E lR of ( 40) such that

(50) z(t) = J,,,(u(t),a,u(t))::; R" l;ft E JR.


Similarly, the kernel K of equation ( 4 7) consists of all its weak solutions u(x, s) that satisfy

(51) z(t) = J,.(u(t), 8tu(t)) s R,. Vt E lR,

where to define Ja(v, vi) we have to replace b (x, ~) in (48) by b (x). Notice that the trajectory attractors A, and A are independent of the constant N from the definition of the trajectory spaces JC:(N) and K! (N).

THEOREM 3.3. The trajectory attractors A, = IT+ICs of equation (40) tend to the trajectory attract or A = IT+IC of ( 47) as c: ---+ 0+ in the space eloc

+'

REMARK 3.3. In this paper we studied averaging of trajectory attractors of autonomous evolution equations with rapidly oscillating terms of the form b ( x, ~) or b ( ~) having the averages with respect to the spatial variable X E n. We proved that the trajectory attractors A, of these equations converge to the trajectory attractor A of the corresponding averaged equation. We prove an analogous result for non-autonomous evolution equations with terms that depend on the rapid time variable of the form b (t, D or even b (x, ~' t, D . For instance, the trajectory attractor of non-autonomous 3D Navier-Stokes system with time dependent external force g (x, ~) tends to the trajectory attractor of autonomous 3D Navier-Stokes system with the external force g(x) if it is known that the function g(x) is the average of the function g (x, D as c:---+ 0+ in the space L2,w(~; H) (see [2]).

REFERENCES

[1] A.V. Babin and M.I. Vishik, Attractors of evolution equations, North Holland, 1992. [2] V.V. Chepyzhov and M.I. Vishik, Trajectory attractors of evolution equations with

rapidly oscillating terms, to appear. [3] V.V. Chepyzhov and M.I. Vishik, Evolution equations and their trajectory attrac

tors, J.Math.Pures Appl., 76, 10 (1997), 913-964. [4] V.V. Chepyzhov and M.I. Vishik, Trajectory attractors for reaction-diffusion sys

tems, Topol.Methods Nonlinear Anal., 7, 1 (1996), 49-76. [5] V.V. Chepyzhov and M.I. Vishik, Trajectory attractors for evolution equations,

C.R.Acad.Sci.Paris, 321, Series I (1995), 1309-1314. [6] J.-L. Lions, Quelques methodes de resolution des problemes aux limites non

lineaires. Dunod, Gauthier-Villars, Paris, 1969.


VOLUME 8

2001, NO 1-2 PP. 141-146

THE BEST PARAMETRIZATION AND NUMERICAL SOLUTION OF THE CAUCHY PROBLEM FOR A SYSTEM

OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER '

A.N. DANILIN, E.B. KUZNETSOV AND V.I. SHALASHILIN t

Abstract. This work shows the results of using the best continuation method (the best parametrization) to build numerical solution algorithms for the problems of applied dynamics. As a result of discretization by spatial coordinates, the dynamic problem is reduced to a coupled system of nonlinear ODEs with predetermined initial conditions. Time is the natural parameter in dynamic problems. The best parametrization can be used for effective solution of such problems. This paper describes the parametrization procedure of Cauchy problem for ODEs of the second order. We prove a statement about the convergence of simple iterations of numeric solutions to the equations after their transformation to the best argument.

1. We consider the Cauchy problem for a system of ordinary differential equations of the second order

(1) ii = f(t, u, u), u(O) = Uo, u(O) = Uo,

where u lR -t Rn, f : JRn+n+l -t Rn, t E JR. The dots denote derivatives with respect to t.

Many problems of science and engineering can be reduced to problem (1). For example, such problems arise in modeling of dynamic processes.

Applying the continuation method with respect to the best parameter [1], we study an integral curve of (1) with the help of the equation d)-, 2 = dt2 + dudu + dvdv. Here we introduce a new argument A as the length of the integral curve in the Euclidean space JRn+n+l : { t, u, v = u}.

' This work was supported by the RFBR Grant 00-01-00072. t Moscow State Aviation Institute (Technical University)

141

142 A.N. DANILIN, E.B. KUZNETSOV AND V.I. SHALASHILIN

The transition to the new argument ..\ allows to transform problem (1) to the form

(2) v' = t'f(t,u,v),

(3) u' = t'v,

(4) t' = 1/V1 +vv+ If,

(5) u(O) = uo, v(O) = uo = Vo, t(O) =to

and to consider it as a differential problem of continuation along the solution curve with respect to the new argument ..\. Primes denote derivatives with respect to ..\ (see [2]).

For obtaining integral curves of problem (2)-(5), consider implicit numerical methods involving iterative procedures. One can apply to our problem different methods for numerical solution of the first order ODE's, such as Runge-Kutta, Adams-Molton, Milne, etc. (e.g., see [3,4]). Numerical solution of (2)-(5) can be also obtained by methods for integration of the second order. Among these there are methods of central differences, Habolt, Newmark, Wilson and the method of linear acceleration, which are widely used in computational mechanics [5,6].

Consider, for example, the method of linear acceleration. It is a one-step method of the third order that runs as follows. It is agreed that the vector v'(>.) is a linear function of..\ within the integration step ,6.,\ = ..\i+1 - Ai· Therefore for the interval 0 ::; T ::; ,6.,\ we have

(6)

where i is the number of integration step. From this and other algorithms it follows, that we must solve a system of nonlinear equations in implicit form for each integration step. Here we can use different iterative procedures. But experience of calculations showed that simple iterations are well acceptable for solving the problem in transformed form (2)-(5) and make it possible to economize the computing processes.

Studies of this fact demonstrated that, when applying implicit schemata to solve initial value problems as stated in (2)-(5), one may always formulate the conditions, which ensure the convergence of simple iterations.

THE BEST PARAMETRIZATION OF THE CAUCHY PROBLEM 143

2. Introduce vectors x = (v,t)Y,g = (J,1)Y. Then equations (2), (4) can be joined together and written as

(7) x' = t'g.

Formula ( 4) for t' may be written in another form:

(8) t' = J(l- u'u')/(1 + JJ),

whence it follows that llu'll = ..JV!V! :S: 1. With regard to (8), rewrite equation (7) in the following manner:

(9) x' = y1- u'·u'e,

whe-re-e = g{ifij9 is a unit vector. For a one-point iterative vector function cp, which transforms the m()tric

space of approximations to itself, we study the iterative convergence given by Xk+l = 'Pk = cp(xk), where k is the iteration number [7,8].

We will impose the following restriction on the iteration function cp. Suppose that iterative processes under consideration satisfy the equality

(10) x;,. - X~ = p( 'Pm - 'Pn) / ~.\,

where m and n are arbitrary iteration numbers, ~.\ is a discrete step along a continuation curve, and p > 1 is a quantity. Many well-known algorithms meet condition (10), such as implicit Euler method, Adams-Molton, RungeKutta, Wilson, Newmark, and others. For example, for linear acceleration method (6) p = 2 [5], and for the Adams-Mol ton method of the fourth order p = 24/9 [4].

In fact, for such methods the procedure of integration of equation x' = f(t, x) can be described by the formula x(i+l) = x<il + ~.\(j(i+l) + I:; a1j<Hl)fp (j = 0, 1, ... , s). Here i is a number of the integration step; a 0 , 0!1, ... , Dis are real values. Writing out this formula for iterations numbers m and nand then subtracting one from the other, we obtain x~~~) -x~:Jl = cp~+l)- cp~+l) = ~.\(J:/;+Il- /n_i+

1))/p. The upper index (i + 1) will be suppressed in the sequel. Since fm = x;,., we get 'Pm- 'Pn = ~.\(x;,.- .T~)fp, which implies (10).

Now we will prove the following statement.

THEOREM. Let cp be a one-point iterative function satisfying restriction ( 10). Then for choosing integration step there exists a condition such that cp


is a contracting mapping of the set of approximations of x into itself, which satisfies the Lipschitz condition

(11)

with a constant C < 1.

Proof. Using the notation

(12)

where k is the number of iteration, consider the difference x:,. - x~ -amem - anen. Taking into account (10), the last equality may be written as 'Pm- 'Pn = .6...\(amem- anen)fp, whence it follows the equality of norms

(13)

(14)

We can rewrite the norm on right of (13) in an expanded form:

llamem- anenll = v'a;,. +a~- 2aman(emen) =

Consider separately the quantity lam - ani and find its value. Since u!,u!, :::; 1, the power series for ak (12) converges and the following equality is valid:

where 8k is a value depending on the number of iteration and meeting the condition 0.5 :::; 8k :::; 1. We can check this by analyzing the values of the continuous function 8 = (1 -~)/~given~= u'u' varying from 0 to 1. Therefore we get

where 8mn is a certain mean value 8 between Om and On. Due to the Cauchy-Schwarz inequality, the last notation can be continued

as

THE BEST PARAMETRIZATION OF THE CAUCHY PROBLEM 145

Since uk and Vk are related to each other by (3), where the coefficient t' :::; 1, we have

(15)

where t:r,n is an average value oft' between t~, and t~. Hence from (14) together with (15) follows the inequality

llamern- anenll < J4(omnt~,,)211xm- Xnll 2 + 2aman(1- emen) =

Let D<mn be the angle between unit vectors em and en. Then we have

and, following the cosine theorem, we obtain 1 - cos D<mn = ( a;,.n - (am -an) 2 )/(2aman), where amn is the length of the vector connecting the end points of the vectors x:r, and x~. Therefore, for the second summand in (16), we have

Now consider the norm llxm- Xnll· Since

where x(.\;) is a known value of x from the previous i-th integration step, introducing a notation x~ = 6.xk/ 6..\, we obtain

Recalling that ak = llxkll and amn = llx:n- x~ll, from (17) we get

(18)


Hence, using (16), it follows that

llamem- anenll < llxm- Xnllv4(8mnt:, .. )2 + 1/(~A)2

and, according to (13), we obtain the following inequality: II'Pm- 'Pnll < Amnllxm- Xnll, where

(19)

Require that Amn ::; C < 1, where C is a constant to be independent of the number of iteration. From this restriction there follows the condition for choosing an integration step ~A. In fact, the substitution of (19) into the condition Amn::; C gives yl4(8mnt:, .. llA) 2 + 1::; Cp. Hence it follows that

(20)

Thus, if the integration step satisfies inequality (20), then the iterative function <p is a contracting mapping of the set of approximations of x into itself, satisfying the Lipschitz condition (11) with a constant C < 1. The theorem is proved. D

In the course of algorithmization, Omn and t;, .. may be substituted, e.g., by their values from the previous integration step.

From (20), we obtain

COROLLARY. Since (Cp)2- 1 must be a positive value and C < 1, we have 1/p ::; C < 1.

REFERENCES

[1] V.I. Shalashilin and E.B. Kuznetsov, The continuation method with respect to a parameter and the best parametrization in applied mathematics and mechanics, Editorial URSS, Moscow, 1999 (in Russian).

[2] V.I. Shalashilin and E.B. Kuznetsov, Cauchy problem as a continuation problem with respect of the best argument, Diff. Eqs., 30, 6 (1994), 964-971 (in Russian).

[3] N.S. Bakhvalov, Numerical methods, Nauka, Moscow, 1973 (in Russian). [4] J.M. Ortega and W.G. Poole, An introduction to numerical methods for differential

equations, Pitman Publishing Inc., Marshfield, 1981. [5] K.J. Bathe and E.L. Wilson, Numerical methods in the finite element analysis,

Prentice-Hall Inc., 1976. [6] K.J. Bathe, Finite element procedures in engineering analysis, Prentice-Hall Inc.,

Englewood Cliffs, New Jersey, 1982. [7] J .F. Traub, Iterative methods for the solution of equations, Chelsea Publishing

Company, New York, 1982. [8] J.M. Ortega and W.O. Rheinboldt, Iterative solution of nonlinear equations in

several variables, Academic Press, New York-London, 1970.


VOLUME 8

2001, NO 1-2 PP. 147-161

NEWTON'S POLYGON IN THE THEORY OF SINGULAR PERTURBATIONS OF BOUNDARY VALUE PROBLEMS

R. DENK ' AND L. VOLEVICH 1

Abstract. In this paper we discuss ellipticity conditions for some parameter-dependent boundary value problems, which do not satisfy the Agmon-Agranovich-Vishik condition of ellipticity with parameter. The appropriate definition of ellipticity uses the concept of the Newton polygon. For the corresponding boundary value problems with small parameter we construct a formal asymptotic solution, thus explaining the nature of the Shapiro-Lopatinskii condition for these problems.

1. Introduction. Starting with the papers of Agmon [1 J and Agranovich-Vishik [2], the theory of ellipticity with parameter was developed, which treats boundary value problems depending on a complex parameter.:\. Under some ellipticity conditions, uniform (with respect to the parameter) a priori estimates and solvability results were obtained. In the proofs of these results, a key point was the fact that the corresponding .:\-dependent principal symbols were (quasi-)homogeneous with respect to the covariables and the parameter .:\.

However, there are some situations where the Agmon-Agranovich-Vishik theory cannot be applied. For instance, let us consider a matrix differential operator

(1.1) A(x, D)= (Aij(x, D))i,j=l, ,N

having Douglis-Nirenberg structure; i.e. suppose that there exist nonnegative integers sh ... , SN and t 1 , ... , tN with ord A;j ::; si+tj such that the principal symbol (in the sense of Douglis-Nirenberg) A0(x,l;) = (A?j(x,l;))i,j=l, ... ,N is

' Departement of Mathematics, University of Regensburg, D 93053 Regensburg, Germany

t Keldysh Institute Applied Math. RAS, 125047 Russia, Moscow, Miusskaya pl., 4

147

148 R. DENK AND L. VOLEVICH

invertible for ~ # 0. If we consider A(x, D) - >.IN where IN stands for the unit matrix, the symbol A0(x, ~)->.IN is, in general, not quasi-homog(:)neous with respect to ~ and >.. This is due to the fact that the operators on the diagonal may have different orders.

In the case of "constant order", i.e. if s1 = ... = s N = 0 and t 1 = ... = tN = 2m, the Agmon-Agranovich-Vishik (AAV) condition says that there exists a ray C in the complex plane (starting at the origin) and a constant C, independent of x,~ and>., such that

holds for all x, ~ and >. E C. The AAV theory also includes a parameterdependent version of the Shapiro-Lopatinskii condition which ensures solvability for boundary value problems connected with the operator A(x, D) ->.IN. What is the analogue of the AAV condition and of the ShapiroLopatinskii condition for general operators of the form (1.1)?

The same question can be posed for scalar operator pencils given by

(1.2) P(x, D, >.) = l::>ak(x)>.k D" a,k

or for general non-stationary operators

(1.3) P(x, Dx, Dt) . L aak(x, t)D~ D~. a,k

In the present paper we concentrate ourselves on parameter-dependent operator matrices of the form

(1.4) A( D >.) = (An(x,D) A12(x,D) ) x, ' A21 (x, D) A22(x, D) - >. .

Such matrices may be considered as a mixture of a parameter-independent system (possibly of Douglis-Nirenberg structure) and of a parameter-dependent system as discussed above. Operators of the form (1.4) may serve as model problems in the theory of general ellipticity with parameter as it will be clear below (see Remark 2.5). We also want to mention that there are direct applications of operators of the form (1.4) to transmission problems, which we want to discuss in a forthcoming paper.

Replacing in (1.4) the parameter >. by c 1 , we obtain a problem with small parameter e; this leads to the theory of singular perturbations of boundary value problems as started by Vishik-Lyusternik [13]. The aim

NEWTON'S POLYGON AND SINGULAR PERTURBATIONS 149

of the present paper is to introduce and study a general notion of ellipticity with parameter for (1.4) and the related problem with small parameter. We will construct (in Section 4) the so-called formal asymptotic solution for this problem; in particular, this construction allows us to understand the appearance of non-standard ellipticity conditions.

2. Newton's Polygon and Ellipticity with Parameter. In this section we will sketch the fundamental concepts and results in the theory of N-ellipticity with parameter which is based on the so-called Newton polygon (see, e.g., [9] and [3]-[5]). We start with operator pencils of the form (1.2) and consider for fixed x the polynomial

(2.1) P(~, A) = P(x, ~'A) := L a,k(x)Ak~a. a,k

Define Newton's polygon N(P) of the symbol (2.1) as the convex hull inlR2

of all points (lo:l, k) with aak(x) f 0, their projections (Ia:!, 0) and (0, k) and the origin (0, 0). As a simple example, consider

P(~, A) - (A+ Pz"(O)(A + Pzm-21'(~))

- A2 + (Pz"(O + Pzm-z1,(~))A + Pz"(0P2m-2!'(~).

where ? 21' and Pzm-z" are polynomials of degree 2f.J. and 2m-2f.J., respectively, with f.1. > m - JL The Newton polygon of this example can be found in Figure 1.

DEFINITION 2.1. The operator (1.2) is called N-elliptic with parameter in a ray £ in the complex plane if there exist positive constants C and Ao such that

(2.2) IP(x,~, A)l ::0: C (i,k)EN(P)nZ 2

holds for all x and all ~ E lRn, A E £ with IAI ::0: A0•

This definition is closely related to the notion of N-parabolicity for operators of the form (1.3).

DEFINITION 2.2 ([8]). The operator (1.3) is called N-parabolic if the following conditions hold: (i) There exists a real constant Ao such that estimate

(2.2) holds for all x, ~and all A with Im A:<::: Ao. (ii) The polygon N(P) does

not contain edges parallel to the coordinate axes and not belonging to them.


k

2

1

2m- 2j.t 2j.t 2m

FIG. 1. The Newton polygon for the example above.

If condition (i) holds, then (ii) means hypoellipticity of the corresponding differential operator (see also Figure 2).

The definition of N-ellipticity for Douglis-Nirenberg systems is reduced to that of scalar pencils:

DEFINITION 2.3 ([3]). The matrix differential operator (1.1) is called N-elliptic in C if the determinant of its symbol

satisfies the condition of Definition 2.1.

If (1.1) acts on a closed manifold (i.e. on a compact manifold without boundary), the property of N-ellipticity leads to (and is even equivalent to) unique solvability of the system

(A(x, D) >.IN )u = f

for large 1-'1 and uniform estimates in terms of parameter-dependent norms (see [3]).

The class of N-elliptic Douglis-Nirenberg systems coincides with the class of systems previously introduced by Kozhevnikov [11]. Set s; + t; = 2r; and assume without loss of generality that r 1 ;:::: .•. ;:::: rN ;:::: 0. For simplicity, let us consider the case where these inequalities are strict:

(2.3) r1 > ... > rN > 0.


k k

Condition (ii) satisfied Condition (ii) not satisfied ~

FIG. 2. Cond·ition (ii) in Definition 2.2.

J:<'or ""= 1, ... , N denote by A('")(x, ~) the"" x ""matrix

and by E ( "") the matrix, which differs from the "" x "" zero matrix only by the element at position ( ""' K), which equals 1. We will write A 0 ( K) for the principal symbol (in the sense of Douglis-Nirenberg) of A(K).

THEOREM 2.4 ([3]). Assume that (2.3) is satisfied. Operator (1.1) is N-elliptic in .C, if and only if the K ozhevnikov conditions are satisfied: (i)

For"'= 1, ... , N, the symbols A(K)(x, 0 are elliptic in the sense of DouglisNir-enberg. (ii) For""= 1, ... , N, we have

REMARK 2.5. Let A(.1:, D) be a matrix of form (1.1) and consider the operator A(x, D)- )JN. Using the theory of Newton polygons, it is possible to see that there exists a covering of JE.n x .C by neighbourhoods Uj and Vj (j = 1, ... , J) and a subordinated partition of unity

.J J

I: (M~, .\)+I: 7/Jj(~, .\) = 1 j=l


such that the following statements hold.

(i) The operator (A(x, D) ->..IN )l/>j(D, >..) is a small regular perturbation of

(A(,.)(x, D) o ) ..~,·(D >..)

0 >..IN-" 'i'J ' •

Here and in the following, we use the pseudodifferential operator notation tPi(D, >..) := F-1 </Jj(~, >..)F where F stands for the Fourier transform in JRn.

(ii) The operator (A(x, D)- >..IN )'lj;j(D, >..)is a small regular perturbation of

(See [3] for details.) This shows that the study of operators of form (1.4) is of particular interest in the theory of N-ellipticity.

REMARK 2.6. In the same way the study of scalar pencils (1.2) can be reduced to the study of small regular perturbations of homogeneous pencils of the form

P(x, D, >..) = P2m(x, D)+ A.P2m-1 (x, D)+ ... + >..2m-2'" P2'"(x, D),

where Pj(s, D) are differential operators of order j. Such pencils are discussed in [4], [5] on manifolds with boundary.

Setting >.. = €-1 and multiplying the operator by €2m-2'", we obtain a traditional operator with small parameter:

P,(x, D)= €2m-2'" P2m(x, D)+ ... + P2p,(x, D)

as studied, e.g., in [7], [12]. The condition of N-ellipticity means that

3. N-ellipticity for Boundary Value Problems. Now let us come back to operator matrices of form (1.4) acting on a smooth compact manifold M with boundary &M. For simplicity, we assume that Aij(x, D) is a scalar differential operator of order 2m; the case of general Douglis-Nirenberg systems is treated in [6]. We set A(x, D):= (Aij(X, D))i,j~ 1 ,2 , thus A(x, D, >..) = A(x, D).:... >..E(2). The operator matrix A(x, D, >..)will be supplemented with general boundary conditions


for j = 1, ... , 2m. Here Bj(x, D) is a differential operator of order mj, where we assume that

holds. Let A(x, D, .\) be N-elliptic with parameter in the ray £ = [0, oo). What is the proper formulation of the Shapiro-Lopatinskii condition for the boundary value problem (A(x, D)- .\E(2), B1(x, D), ... , B2m(x, D))?

As usual, the boundary value problem on a manifold with boundary is reduced, using local coordinates, to a boundary value problem in the half space lR'l- := {x = (x', Xn) E JR:n : Xn > 0} with boundary &lR'l- = JR:n-·l For simplicity of notation, we shall consider only the corresponding model problem, i.e. we assume that the operators A and Bj have constant coefficients and no lower-order terms. So we consider the 2 x 2 system

(3.1) An(D) u 1 + A1z(D) Uz it in lR'l-,

Az1 (D) u1 + (A22 (D)- .\) Uz = .fz in lR'l-

with boundary conditions

(3.2) Bj(D)u(x', 0) = g(x') (j = 1, ... , 2m) on JR:n-l.

The Newton polygon of det (A(O- .\E(2)) has the form indicated in Figure 1, and theN-ellipticity condition means that

(3.3) I(AnAzz- A12A21)(~)- .\An(OI 2: C !~1 2m(.\+ !~1 2m)

((~, .\) E JR:n X £).

Inequality (3.3) implies (cf. [4]) that A11 (D) and A(D) are elliptic. To formulate the Shapiro-Lopatinskii condition for (3.1)-(3.2), we consider, as usual, a problem on the half-line t 2: 0:

(3.4)

(A((, Dt)- .\E(2))v(t)

Bj(~', Dt)v(O)

v(t)

0 (t>O),

9} (j=1, ... ,2m),

0 (t-+oo).

DEFINITION 3.1. The boundary value problem

(A(x, D)- .\E(2), B1 (x, D), ... , Bzm(x, D))

in lR'l- is called N-elliptic with parameter in £ if A( D) - .\E(2) is N-elliptic (i.e. if (3.3) holds) and if the following conditions are satisfied:


k

1

FIG. 3. The Newton polygon for system (3.1).

(i) For every E,' E R''-1 \{0}, every A E £and every (gt, ... ,g2m) E ((5lm, problem (3.4) is uniquely solvable.

(ii) For every E,' E JR.n-1 \{0} and every (g1 , ... ,gm) E em, the problem

(3.5)

An(E,', Dt)v1(t)

Bjl (E,', Dt)Vt (0)

V1 (t)

0 (t > 0),

9i (j=l, ... ,m),

0 (t-+oo)

is uniquely solvable (i.e. (A11 (D), B 11 (D), ... , Bm1 (D)) satisfies the standard Shapiro-Lopatinskii condition).

(iii) For every (9m+b ... , 92m) E em, the problem

(3.6)

(A(O, D1)- E(2))w(t)

Bi(O, D1)w(O)

w(t)

has a unique solution.

0 (t > 0),

9i (j=m+l, ... ,2m),

0 (t---+ oo)

In [6] it is shown that under the condition of N-ellipticity uniform a priori estimates in parameter-dependent norms hold. The definition of the norms and the proof of the a priori estimate again use the concept of the Newton polygon.

To explain the appearance of the non-standard conditions (ii) and (iii) in the above definition, we will discuss in the next section the construction of the formal asymptotic solution for the problem with small parameter connected with (3.1)-(3.2).


4. Construction of the Formal Asymptotic Solution. Let us consider boundary value problem (3.1)-(3.2) with h = f2 = 0. Setting,\= c 2m,

we can rewrite (3.1) as

0 ( 4.1)

· mm lll ~+,

· mm Ill ""+.

Let us assume that (4.1), (3.2) is N-elliptic with parameter in the ray [0, oo). Then, due to conditions 3.1 (ii) and (iii), respectively, boundary value problem (An, En, ... , Em1) is elliptic, and system (3.6) is uniquely solvable. To avoid technical difficulties, let us assume that (An, Eu, ... , Em1) is uniquely solvable, too. If this is not the case, one has to deal with kernels and cokernels of the operator related to this boundary value problem.

Similar to the notation in [12], let us call (An, En, ... , Em1) the first limit problem and (3.6) the second limit problem. The aim of this section is to construct a formal asymptotic solution (FAS) of boundary value problem (4.1), (3.2), i.e. a formal series

00

w(x, E)= :L>kw(kl(x, c), k=O

(

(k) )) (k)( ) _ W 1 (x,E W X, E - (k)( ) ,

w2 X,E

N for which the partial sums :Z:::: Ekw(k) satisfy (4.1), (3.2) up to order O(EN).

k=D Following Vishik-Lyusternik [13] (see also [10]), we seek the solution in the form

w(x, E) = u(x, E)+ v(x, E),

where

00

(4.2) u(x, E)= I: Eku(kl(x), k=O

is a so-called exterior expansion and

(4.3) 00

v(x, E) =I: Em+kv(k) ( x', XEn), k=O

is a so-called interior expansion or boundary layer. Our aim is to find partial differential equations and boundary conditions determining the functions u(k), v(k). We start with the equations in the interior of our domain JR';..

\


(i) Differential equations for u(k). Substituting (4.2) into (4.1), we obtain

= f: Ek [(An(D) k=O 0

0 ) ( (k-2m) )] A22(D) ~~k-2m) '

where we have set

(4.4) u(k) := 0 fork= -2m, -2m+ 1, ... , -1.

Thus we obtain the recurrence relations

(4.5) An (D)u\k) ~A12(D) ( A21 (D) A22(D)) u<k-2m) (k = 0, 1, 2, ... ),

In order to determine u(k) (with starting values (4.4)), we have to impose m boundary conditions on u\k)' see below.

(ii) Differential equations for v(k). To find the corresponding equations for v(k), we note that

A( D) [v(k) ( x', XEn)] = E-2m [A( ED', Dn)v(k)] ( x', ~n),

due to homogeneity. Here D' = (D1 , ... , Dn-d· Substituting (4.3) into (4.1), we get

Now we expand A( ED', Dn) in a Taylor series with respect to £,

2m

A(ED1,Dn) = I:>lA(l)(D',Dn) l=O


with A(0l(D', Dn) = A(O, Dn) and A(Zm)(D1, Dn) being a constant complex

2 x 2 matrix. Substituting this expansion into the last sum in (4.7), we see that this

sum equals

where we have set

(4.8) vUl := 0 (.j =-2m, -2m+ 1, ... , -1).

Therefore we obtain the recurrence relations

(iii) Boundary conditions. Now we want to find boundary conditions for the functions u\k) and v(k) with k = 0, 1, .... Setting

(

Bt(D) ) B(D) := : ,

Bzm(D)

we trivially have

00 00

B(D) :~::.>ku(kl(x1 , 0) = 2::.>k B(D)u(kl(x1, 0)

k=O k=O

and, by homogeneity,

B (D) (k) ( 1 Xn) I _ -m; B ( Dl D ) (k) ( I ) I j V X 1 --;;.- Xn=D - E j E 1 n V X 1 Xn Xn=O ·

Therefore,


Again we use the Taylor expansion with respect to t,

m;

Bi(<D',Dn) = :~:::>1 BJ1)(D',Dn), l;Q

where Bj0)(D', Dn) = Bj(O, Dn)· We obtain

Bj(D)(u + v) =fa [<k Bj(D)u(k) + E-m;+m+k ~ t 1 Bj')(D)v(k)]

(4.10) m;

L €k[Bj(D)u(k) + l:BJ')(D)v(k+m;-m-1)]. k=min{m-mj ,0} l=O

Note that here negative powers of < may appear; for negative values of k in u(k) and v<k) we use (4.4) and (4.8).

From the conditions Bi(u + v) = gi for j = 1, ... , 2m we obtain the boundary conditions ·

m; I; BY) (D)v(k+m;-m-l) = Oko gj l;Q

fork= min{m- mi, O},min{m- mj,O} + 1, ... (k,j)

andj = 1, ... ,2m.

This set of boundary conditions is numbered using the indices k and j. At the first moment it seems to be unclear how conditions (k, j) determine the functions u(k) and v(k). This is the essence of the following theorem.

THEOREM 4.1. Assume that boundary value problem (1.1)- (1.2) is Nelliptic and that the first and the second limit problem are uniquely solvable. Then recursion formulas (4.4)- (4.6), (4.8) (4.9) with boundary conditions (k, j) uniquely determine the functions u(k) and v<k) for k = 0, 1, 2.... The boundary conditions for v.\k) and v(k) have the form

(4.11) BdD)u\k\x',O)=gik(x') (j=1, ... ,m)

and

(4.12) Bi(O, Dn)v(kl(x', 0) = 9ik(x') (j = m + 1, ... , 2m),

respectively, where the right-hand sides of ( 4.11) and ( 4.12) can be determined recursively and contain only functions u (l) and v<lJ with l < k.

NEWTON'S POLYGON AND SINGULAR PEBTURBATTONS 159

Proof. Due to recursion formulas ( 4.5)-( 4.6) and ( 4.9) with starting values (4.4) and (4.8), we only have to show that the boundary condition (k,j) determine ulk) and v(k). For simplicity, we restrict ourselves to the particular case that

(4.13) mJ := ord BJ = j -1 (j = 1, ... , 2m).

The general case can be treated with the same idea, but is somewhat more delicate. Figure 4 shows for each index pair (k,j), which appears in the above boundary condition, the function for which this condition is used. (The index

pairs marked with []] do not appear). The main question is in what order the formula (k,j) has to be applied.

1 2

'I

-m+1

-1

FIG. 4. Usage of formula (k,j) in the case (4.13}.

We will use the formulas (k,j) with j :S mas boundary conditions for u(k) and the formulas (k- j + 1- m,j) with j = m + 1, ... , 2m as boundary conditions for v(k). We still have to show that this can be done in such a way that all functions appearing in formnla (k,j) (except u(k) and v(k),

respectively) are already known. Let us assume that in step k we already know the functions u(l) and v(l)

with l < k. We want to find u(k) and v(k). First let j :S m. Then condition


(k, j) contains the functions

u(k), v<k-m+j-1), v<k-m+j-2), . .. , v<k-m) .

As k + j- m -1 < k and u~k) is defined by ( 4.6) which only contains u<k-2m),

the only unknown function in condition (k,j) is ulk). We get (4.11) with

j-1

9ik := 8ko Yi- Bi2(D)u~k)(x', 0)- L Bjl)(D)v<k-m+i-1-ll(x', 0). 1=0

Due to the condition of unique solvability of the first limit problem, the function ulk) is uniquely determined by (4.5) and (4.11).

Now let j > m. The boundary condition (k- j + 1- m,j) contains the functions

v(k)' v<k-1)' ... 'v<k-j+l) and u<k-i+m+1).

As k - j + m + 1 :::; k and as we already know u<l) for l :::; k, this gives m boundary conditions for v<k). Due to unique solvability of the second limit problem, the function v<k) is determined by these boundary conditions, and we can continue with step k + 1. The boundary conditions for v<k) have the form ( 4.12) with

j-1

9jk = 8k-j+m+l,O 9j - Bj(D)u<k-j+m+!) - L BJ!) (D)v(k-l)

1=1

for j = m + 1, ... , 2m. Summarizing, we see that we use formula (k, j) as boundary conditions

for u(k) and v(k) in the way indicated in Figure 4. Here we compute u(k) and vCk) in the order

u(o), vCO), u(l), v<1J, u<2), . . . . D

REMARK 4.2. One can see that the recursion formula for u~) (see equations ( 4.5) and ( 4.11)) is given by the first limit problem with appropriate right-hand side. Similarly, the recursion formula for v(k) (cf. equations (4.9) and (4.12)) is exactly the second limit problem with appropriate right-hand side. So we can see that non-standard ellipticity conditions (ii) and (iii) in Definition 3.1lead (under the additional assumption that the boundary value problem in 3.1 (ii) is not only elliptic but uniquely solvable) to the existence of a formal asymptotic solution. Here the first limit problem corresponds to the exterior expansion and the second limit problem to the boundary layer. Therefore from the point of view of singular perturbation theory the non-standard ellipticity conditions are very natural.


REFERENCES

[1] S. Agmon, On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15 (1962), 119-147.

[2] M. S. Agranovich and M. I. Vishik, Elliptic problems with parameter and parabolic problems of general form, Russian Math. Surv., 19, 3 (1964), 53-157.

[3] R. Denk, R. Mennicken, and L. Volevich, The Newton polygon and elliptic problems with parameter, Math. Nachr., 192 (1998), 125-157.

[4] R. Denk, R. Mennicken, and L. Volevich, Boundary value problems for a class of elliptic operator pencils, Preprint 58, Keldysh Inst. Appl. Math., 1998.

[5] R. Denk, R. Mennicken, and L. Volevich, On elliptic operator pencils with general boundary conditions, Preprint 37, Keldysh Inst. Appl. Math., 1999.

[6] R. Denk, and L. Volevich, Some mixed order boundary value problem. I. A priori estimates, Preprint 58, Keldysh Inst. Appl. Math., 1999.

[7] L. Frank, Coercive singular perturbations. I. A priori estimates, Ann. Mat. Pura Appl., 119, 4 (1979), 41-113.

[8] S.G. Gindikin, and L.R. Volrvich, Pseudodifferential operators and the Cauchy problem for differential equations with variable coefficients, Functional Anal. Appl., 1, 4 (1967), 262-277.

[9] S.G. Gindikin, and L.R. Volevich, The Method of Newton's Polyhedron in the Theory of Partial Differential Equations, Math. Appl. (Soviet Ser.), 86, Kluwer Academic, Dordrecht, 1992.

[10] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, 102, American Mathematical Society, Providence, RI, 1992.

[11] A. Kozhevnikov, Asymptotics of the spectrum of Douglis-Nirenberg elliptic operators on a closed manifold, Math. Nachr., 182 (1996), 261-293.

[12] S.A. Nazarov, The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. I. The problem in a cone, Siberian Math. J., 22 (1982), 594-611.

[13] M. I. Vishik, and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Amer. Math. Soc. Trans!., 20, 2 (1962), 239-364.


VOLUME 8 2001, NO 1-2 PP. 163-181

ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A SEMILINEAR ELLIPTIC BOUNDARY PROBLEM

YU.V. EGOROV • AND V.A. KONDRATIEV t

Abstract. We consider solutions to elliptic linear equation (1) of second order in an unbounded domain Q in IRn supposing that Q is contained in the cone

K = {x = (x',xn) : lx'l < Axn + B, 0 < Xn < oo},

and contains the cylinder

C = {x = (x',xn) : lx'l < 1, 0 < Xn < oo}.

We study the asymptotic behavior of the solutions of (1) satisfying nonlinear boundary condition (2) as Xn -+ oo. We obtain more precise results, which depend on the structure of Q. In general we assume that Q is contained in the domain

{x = (x',xn) : lx'l < '1'(Xn), 0 < Xn < oo},

where 1 ::; '1'(t) ::; At+ B. We show that any solution of the problem growing moderately as Xn -+ oo is bounded and tends to 0 as Xn -+ oo. In our notes [2] ,[3] we proved such a theorem for the case '1'(Xn) = B, i.e. for a cylindrical domain Q = !1 X (0, 00) ' n c JRn-l.

1. Introduction. We study the solutions to the elliptic second order linear equation

n 8 au Lu : = "' -;::;--- ( aij ( x)-;::;---) = 0 L... uX· ux·

i,j=l t J

(1)

in an unbounded domain Q in IR" supposing that Q is contained in the cone

K={x=(x',xn) :ix'I<Axn+B, O<xn<oo},

' Univ. Paul Sabatier, 118, route de Narbonne, bat. IR2, 310 Toulouse Cedex 4, France t Moscow State University, Faculty of Mechanics and Mathematics, 199899 Russia,

Moscow, Leninskie gori, MGU, GZ 163

164 YU.V. EGOROV AND V.A. KONDRATIEV

and contains the cylinder

C = {x = (x',xn) : jx'l < 1, 0 < Xn < oo}.

Under additional assumptions on the structure of Q, we obtain more precise results. In general we assume that Q is contained in the domain

{x = (x',xn) : jx'l < ')'(Xn), 0 < Xn < oo},

where 1 ::; ')'(t) ::; At+ B, and that u satisfies the boundary condition

(2) 8u oN+ b(x)ju(x)IP-lu(x) = 0

on the lateral surface

S = {x E 8Q, 0 < Xn < oo},

where p > 0, b(x) 2': bo > 0, ')' E 0 1 (0, oo ), and

8u ~ 8u aN= ~a;J(x) -

0 . cosO;,

i=l XJ

0; is the angle between the axis x; and the outer normal vector. Suppose that

n

L a;J(x)~i~J 2': coi~I 2 ,Co > 0, x E Q, ilj=l

and that laiJ(x)l::; C for i,j = 1, ... ,nand for all x E Q. We don't assume that aij are continuous.

Denote by !:lr and I:r the sections of the domain Q and the boundary S by the plane Xn = T, and by Qr and Sr, respectively, the parts of Q and S between the planes Xn = 1 and Xn = T.

We consider weak solutions u satisfying (1) and (2). It means that u E H1~c(Q) n Lp+l,loc(S) and

! ~ 8u8<p J ~ aiJ(x)-0

. -8

dx + b(x)lu(x)IP-1u(x)<p(x)dS = 0 . . XJ X,

Q z,J=l 8

(3)

for all functions <p(x) E H 1(Q) equal to 0 for Xn = 0 and in a neighbourhood of Xn = 00.

We will show that any solution of our problem growing moderately at infinity is bounded and tends to 0 as Xn --+ oo. In our notes [2], [3] we proved such a theorem for the case 'Y(xn) = B, i.e. for a cylindrical domain Q = !:1 X (O,oo), !:1 C !Rn-l.

SEMILINEAR ELLIPTIC BOUNDARY PROBLEM 165

2. Auxiliary Results. LEMMA 1 (MAXIMUM PRINCIPLE). Let D be a bounded subdomain in Q, u be a function from H 1(Q), satisfying the inequality Lu ;;:: 0 in D weakly, i.e.

J ~ au a<p L.. aij(x)--dx::; 0 . . OXj OXi

Q ,,J=l

for all positive functions <p from HJ(D). Ifu ::C: 0 on the boundary of D, then u;;:: 0 in D.

Proof. Let <p = min(u(x), 0). Then <p E HJ(D) and

J n fJ<p fJ<p L aij(x)-;:;-:~dx::; 0. . . ux1 ux~ Q t,]=l

Therefore grad <p(x) = 0 in D and thus <p(x) = 0 in D, i .e. u;;:: 0. 0

LEMMA 2. Let K 8 be a ball of radius R and Lu = 0 in KR. Then

and the constant C does not depend on R.

Proof. See [1], Theorem 5.1, p. 217 for the case R = 1. In the general case we use the homothety x = Ry. 0

LEMMA 3. Let K 8 be a ball of radius R centered at the point 0 and K~ be its part situated in the domain x1 > 0 , let Lu = 0 in K~ and u fJuj fJN ::; 0 as x 1 = 0. Then

and the constant C does not depend on R.

Proof. The proof follows the same scheme as that of Theorem 5.1 in [1], pp. 217-223. This proof uses as its base the following inequality (5.10) from this work :

Jo?v2dx<Kj(o?+o?)v2dx aEC1(Q) X - X l 0 l

(] (]


which holds in n = Kfi because of the boundary conditions u 8uf8N ~ 0 and can be proved by substitution of the function <p = a 2u in the definition of weak solutions. 0

LEMMA 4. Let Lu = 0 in Q, 8u/8N ~ 0 on S. lfO < p < 1, we assume that lui ~ M. If

J !Vu! 2x~-ndx + J !u!P+1x~-ndS < oo,

Q s

then u(x) -t 0 as Xn -too uniformly in Q.

Proof. Let Xo E r, Xon = T, R = min('y(T), T/2), and let K be a ball centered at x 0 of radius R. Let K1 = K n Q, 8 1 = S n Q and Kz be a ball of radius R/2 concentric with K.

If p ~ 1, then iul 2 ~ M 1-PiuiP+1 and therefore,

J iui 2x~-ndS ~ C. s

If p > 1, then

x uz < x21uiP+l + xl-2/{p-1) n - n n

and therefore,

J iui 2x;.-ndS ~ J iuiP+lx~-ndS + J x;;n+l-Z/(p-l)dS ~C. s s s

By the Sobolev inequality,

f iui 2x;;ndx ~ C1 ( f !Vui 2x~-ndx + f iui 2x;.-nds) ~ Cz. Q Q s

It is clear that for any c > 0 there exists a N such that

J iui 2x;;ndx < c/2, K

if T > N, and therefore,

R-n J iui 2dx < c.

K

If 81 = 0, then our statement follows from Lemma 4, and if 81 =f 0, then it can be obtained from Lemma 5 using a partition of the unity. D


3. Conical Domains . Consider firstly conical domains corresponding to the function J'(Xn) = Axn +B.

THEOREM 1. Let J'(Xn) = Axn + B, p > 1. There exist constants a0 , Ao > 0 such that any function u, satisfying (1) and (2) and the inequality lu(x)l :S bx~ in the domain Q = {x E lRn, lx'l :S J'(Xn), 0 < Xn < oo}, with some constants b > 0, 0 < a < a0 , 0 < A< A0 tends to 0 as Xn -+ oo

' uniformly in Q.

Proof. Let h(xn) be a smooth function such that h(xn) = 1 as 1 < Xn < T, h(xn) = 0 for Xn > 3T /2 and for Xn < 1/2. We can assume that ih'(.xn)l :S CjT and lh"(xn)l :S C/T2 as Xn > T'. Set

J(T) = J t aii(x)8au aau h(xn)x;,-ndx + J h(xn)b(x)iu(x)iPnx;;-ndS.

. . XJ Xz Q z,;=l s

Substituting in (3) the function rp(x) = h(xn)x;;-nu(x), we obtain that

Therefore, if lu(x)l::; bx~, then J(T') s C3T'2". Set now

If lu(x)l ::; bx~, then I(T) ::; kJ(T) s C4T'2". Substituting in (2) the function rp(x) = u(x)x;;-nh,(xn), where h£(xn) is a smooth function, equal to 1 for Xn < T'- e and vanishing for Xn > T, and passing to the limit as e -+ 0, we obtain that


where

C1 =-J i::Uin(x) ~u udx' . . 1 ux,

fh $=

We have

J u(x)2x:;ndx::; c6 (A~ J IVu(xWx~-ndx + J u(x) 2x~-nds) Qr Qr Sr

::; c6[A~ J IVu(xWx;';-ndx+ JcA~Iu(x)IP+l Qr Sr

+A-4/(v-1)x-1-2/(v-1l)x2-nds] < C A2J(T) + C r-2/(p-1) 0 n n -: 6 0 7 '

since

as Xn > 0. Using the Sobolev inequality, we obtain

J u(x)2x;';-ndx' S C8 [A~T2 J IVu(xWx~-ndx' + T J u(x) 2x;';-nds]

~ ~ ~

::; Cg [A~T2 J IVu(xWx~-ndx' + T J (Tiu(x)IP+l + r-2/(p- 1))x'?,-nds] '

Or Er


since

as T > 0. If A0 is so small that C9A6 ::; 1/2, then

J u(x) 2 x~-ndx'::; C10[T2 I'(T) + y(p-3)/(p-l)J,

nr

and therefore

Thus,

I(T)::; CnTI'(T) + Cl3·

Integrating this inequality, we see that either I(T) :S C13 , or I(T) ?. C13 + C12T1/Cu. Since the latter is impossible when 2aC11 < 1, we obtain that I (T) :S C13, i.e.

I i'Vui2dx +I iuiP+ldx < oo. Q s

Now our statement follows from Lemma 4. D

COROLLARY 1. Two solutions of the equation Lu = 0, satisfying boundar-y condition (2) on the lateml surface of the domain Q and the estimate lu(x)l ::; bx~ with a constant a depending on Q, coincide in Q, ~f they coincide at Xn = 0.

EXAMPLE 1. If Q = { (x1,x2) : 0 < x1 < x2 < 00} and u (.T 1,x2) = r 4 cos 4cp + r 16 sin 16<p, where r, 1. Let now Q = fl x (O,oo), where fl is a domain in !Rn-l, i.e. Q is a cylinder in !Rn.

THEOREM 2. Let p ?. 1. Suppose that the coefficients ai1(x) fori, j = 1, ... , n-1 do not depend on Xn· Let >.2 be the first eigenvalue of the Dirichlet


problem inn, i.e.

n-!

J I; a,i(x)aw(x)faxiaw(x)faxidx \2 . ffl '-"-"i,Je...=..:;cl __ ---;>--;--,-;:-:-----A = m -

wEC8"(fl) J w(x)2dx fl

Let !'I > 0, ')'2 > 0 be such constants that

n 2 n n-1 n

'"Yf "l>nj(x)~j ::::; L aij(x)~i~i• ')'~ L aij(x)~i~i ::::; L aij(x)~i~i• j=l i,j=l i,j=l i,j=l

for all ~i E R Set p = ')'j')'2 • For any c: E]O, c:o(, there exists a constant a > 0 such that if iu(x)i ::::; aeP(>.-<)xn/2 , then u(x) -+ 0 as Xn -+ oo uniformly in Q.

Proof. Let QT = n X [0, T], ST = r X [0, T]. Set

I(T) = 1 t aii(x) a~;;) a~~~) dx +I b(x)iu(x)iP+IdS. QT ,,J=! ST

LethE C 00(R), h(xn) = 0 for Xn 2: T + 1, h(xn) = 1 for 0::::; Xn ::::; T. Putting in (2) the function cp(x) = h(xn)u(x), we obtain that

1 h(xn) t aij(x) a~(x) a~(x) dx + 1 h(xn)b(x)iu(x)IP+ldS . . ux1 ux,

Q hJ=l 8

1 1( ) ( )~ ( )au(x)d I ( 1 )~ ( )au(x1

,0) 1 = h Xn U X ~ anj X ~ X - U X , 0 L.....- anj X ax. dx · Q j=l J fl j=l J

We have 2

I h1(xn)u(x) tani(x)a~~x) dx Q j=! J

If iu(x)l::::; aebxn, then

2 1 h1(Xn)u(x) t anj(x) a~~x) dx ::::; C1I(T)e2bT Q j=! J


and therefore,

Putting in (3) the function rp(x) = u(x)h,(xn), where h,(xn) is a smooth function, equal to 1 for Xn < T- E and to 0 for Xn > T, and passing to the limit as E -+ 0, we obtain that

J 1 )~ ( )OU(X1,T) 1 J ( 1 )~ ( )OU(X

1,0) 1 + u(x , T L.. anj x ox dx - u x , 0 L.. anj x ox dx .

n .i=1 'J n .i=1 1

Let

n-1 J :L ai.i(x)ow(x)fox.iow(x)foxidx' + T J w2dS

, 2 . f n i,.i=l r "' = m 7

wEC=(n) J w(x)2dx (J

Since A7 -+ A as r -+ oo, we can choose r so large that A7 :": A - E /3. Let C3 be a constant depending on T such that for every real 11 and x E n

By the definition of the number A~o we have

(JL;n-l ( )ou(x)ou(x)d,~ j' 2ds) aijX----x+r u ox· ox . . 1 J ' n "'!;;:::;. r

1 ( j ~ ou(x) ou(x) :'S 2(A- c/3)2 L.. aij(x)-,-. ~dx

/2 .. _1

ux1 ux, n t,J-

+ j b(x)Ju(x)JP+ldS+C3r). r


On the other hand,

I ( 1 T) Ln ·( )8u(x1

, T)d 1 ux, an3 X" X vx· n J=I J

( )

1/2

::; (1/yi) [ u(x1, T) 2dx1 I'(T) 112

•

Using the obtained estimates, we see that

where

I ~ 8u(x1,0) c= u(x,O)L._..anJ(x) ox· dx1 +C4

n J=I J

and the constant C4 depends on T and c:, but does not depend on T. The function I(T) is increasing. If I(TI) 2 c for some T1 > 0, then for

T>T1

11(T) "!(>..- 2c:/3)::; I(T) _ c·

It follows that I(T) 2 c + C5eP(>.-2'13lT. But if ju(x)l ::; aebxn and b = p(>.. - c:/2), then I(T) ::; C2e2bT, as we have shown before. Since this is impossible for large T, we see that I(T) < c, i.e.

I i'Vuj 2dx +I iuiP+ldx < oo.

Q s

Now we may apply Lemma 4, and the proof is complete. D


REMARK 1. If the coefficients aij(.r) depend on Xn, then the Theorem is true in the following sense: "There exist constants a, b such that if a solution to problem (1), (2) satisfies the inequality

lu(x)l :S aebxn,

then u(x) -+ 0 as Xn-+ oo."

CoROLLARY 2. Two solutions to the equation Lu = 0, satisfying boundary condition (2) on the lateral surface of the cylinder and the estimate lu(x)l ::; aeP(>--<)xn/2 with a constant a, depending on Sl, coincide in Q, if they coincide at Xn = 0.

ExAMPLE 2. The following example shows that our hypothesis on the growth of solutions is essential. Let n = 2 and Q = { (x1 , x2 ) : -271' < x1 < 271',0 < x 2 < oo}, u(.r1 ,x2 ) = ex'sinx1 - ex2/

4 sin(.rJ/4). It is easy to see that the function u is harmonic in Q and 8u/8u + ulul 3 = 0 for x1 = ±271'.

5. Other Domains, p > 1. Let now l.r'l ::; 'J'(Xn), 0 < Xn < oo, x = (x',xn) E Q. Let

t

J ds F(t) = 'Y (s).

0

Suppose that 'Y ( s) = o( s) as 8 -+ oo, 'Y( s) -+ oo as s -+ oo and 'J'(T + 8) ::; C'Y(T) if 0 < s ::; Cn(T). Note that F(t) -+ oo as t-+ oo.

THEOREM 3. If 1t is a solution of equation (1) in Q, satisfying (2), and

lu(x)l::; beaF(xn)

in Q with a small enough constant a, then u(x) -+ 0 as Xn -+ oo uniformly in Q.

Proof. Let h(xn) be a smooth function such that h(xn) = 1 for 1 < Xn < T, h(xn) = 0 for Xn > 3T/2. We may assume that lh'(xn)l::; C/T and lh'' (xn) I ::; C /T2

, if Xn > T. Put

J(T) = J t ai1(x) ~u ~u h(xn)x;-ndx + ;· h(xn)b(x)lu(x)IP+lx;-ndS. . . ux1 uX,

Q Z,J=l S

Putting in (3) the function <p(x) = h(xn)x~-nu(x), we obtain that

J(T) = J t aij(x) ~u 8(x~-;h(xn)) u(x)dx . . ux1 x,

Q Z1J=l


~ cl + Cd(T)112( I x;nu(x)2dx) 112

•

QaT/2

Therefore, if lu(x)l ~ be•F(xn), then J(T) ~ C3e2•F(T).

Let now

If lu(x)l ~ be•F(xn), then I(T) ~ J(T) ~ C4e2•F(T). Putting in (2) the function <p(x) = u(x)x~-nh,(xn), where h,(xn) is a smooth function, equal to 1 for Xn < T - E and to 0 for Xn > T, and passing to the limit as s --+ 0, we obtain that

where

I~ ( ) &u 2-n , CJ = - L.J a;n x &x. uxn dx.

fh .i=l J


I u(x)2x~-ndx' ~ Cs [r(T)2 I IY"u(xWx~-ndx' + r(T) I u(x) 2x~-nds] ~ ~ ~

~ c6 [r(T)2 I IY"u(xWx~-ndx' nT

+r(T) I x~-n(r(T)iu(x)IP+l + 1(T)-;:, )ds] , BT

i.e.

I u(x)2x~-ndx' ~ C7[r(T) 2I'(T)+ r(T)~], nT


and therefore

On the other hand, using translation Xn -+ Xn + T, we can suppose that r(Xn) :'0 EXn for Xn ::: 0. Using the inequality

2 1-n < p+l 2-n + 1-n-2/(p-1) U Xn _ U Xn Xn ,

we see that

~! n ,;, <=> 2-n

uU uXn "C""' aij(x)u---dx L....- ax. ax . . 1 J z

T Z1.1=

:'0 c:C13(I(T) + 1).

If E is SO small that C13E :'0 1/2, we obtain the inequality

I(T) :'0 c + C141(T)I'(T).

Integrating this inequality, we see that either I (T) :S c, or I (T) ::0: c + cl.,eF(T)/Cl4. Since the latter is impossible when 2aC14 < 1, we obtain that I (T) :S c, i.e.

j !'Vu! 2dx + j !u!P+ldx < oo.

Q s

Applying Lemma 4, we can complete the proof. D


6. Case 0 < p < 1. Let us consider now the case 0 < p < 1.

THEOREM 4. LetO <p< 1 andO <a< 1/(1-p). Let[(xn):::; Axn+B and A < Ao with Ao small enough. If lu(x)l :::; bx~ with some b, then u(x) -7 0 as Xn -7 oo uniformly in Q.

Proof. 1. Let h(xn) be such a smooth function that h(xn) = 1 for 1 < Xn < 2T, h(xn) = 0 for Xn < 1/2 and for Xn > 5T/2. We can assume that lh'(xn)l:::; C/T and lh"(xn)l:::; C/T2 if Xn > 2T. Put

Putting the function <p(x) = h(xn)x;;-nu(x) in (3), we obtain that

Therefore, if lu(x)l:::; bx~, then J(T):::; C3T 2".

2. Put

I(T) = J t a;J(x) ~u ~u x~-ndx + J b(x)lu(x)IP+lx~-ndS . . .

1 ux3 ux,

QT z,J= ST

If lu(x)l :::; bx~, then J(T) :::; kJ(T) :::; C4T 2". Putting in (3) the function <p(x) = u(x)x;;-nh,(xn), where h,(xn) is a smooth function, equal to 1 for Xn < T- c and to 0 for Xn > T, and passing to the limit as c -+ 0, we obtain that

where n _ J "\:"""' ( ) OU 2-n I

C1-- L...Jain X ~uxn dx . . _

1 ux3 .

fh Z-



J u(x) 2x;;-ndx'::; c5 [T2 J IY'u(xWx;;-nd.T1 + T J u(.T)2x;-nds] nr nr Er

::; c6 [T2 J IY'u(xWx;-ndx' + (bT")l-p J lu(x)ll+Px;;-nds] , Dr ~T

1.€.

J u(x) 2x;;-ndx'::; C7[T2 + (bT") 1-P]J'(T),

nr

and therefore,

Thus

I(T) ::; c + C9T I'(T).

Integrating this inequality, we see that either I(T) ::; c, or I(T) ~ c+C2T 1/C9 •

Since the latter is impossible if 2aC9 < 1, we obtain that I(T) ::; c. 3. Now we show that the last inequality implies that u is bounded in Q.

Let M = max!u(x))!. Put Xn=l

w(x) = max(O, u(x)- M).

We put in (2) the function <p(x) = O(xn)x;;-nw(x), where e(xn) is a continuous function, linear for T < Xn < 2T and such that e(xn) = 1 for 0 < Xn < T, e(xn) = 0 for Xn > 2T. Then IB'(xn)l ::; r-l Since w(x', 0) = 0, we obtain that

+ j b(x)lu(x)IP-lu(x)w(x)e(xn)x;;-ndS s


--1 ( )~ ·( )8u8[x~-n8(xn)]d - W X L.,. an3 X

8 8 X.

·-1 Xj Xn Q J-

It is clear that

Thus,

< _ 1 ( ) ~ ·( ) 8w 8[x~-ne(xn)]d _ wxL...an3 x8 8

x ·-1 Xj Xn

QT J-

+ ; 1 [A~T2 1 IV'wl 28(xn)x~-ndx Q2T


:<; 012 [ J IY'wl 2x;-ndx + ~(bTa)l-p J x;-nlw(x)IP+ldS] + cl2Aol.

Q,·\Qr S,r\Sr

Since

J x;-ndS :<; C13T,

Szr\Sr

we see that

J IY'wl 2x;-ndx -t 0, J x;-nw(x)P+ 1dS -t 0

Q,r\Qr S2r\Sr

as T --+ oo, and if A0 is small enough, we have

if T -t oo. This means that w(x) = 0, i.e. u(x) :<; M. In the same way one can see that u(x) 2: -M. Since

j i'Vui2dx + j iulp+ldx < oo,

Q s

the proof is complete after application of Lemma 4. D

ExAMPLE 3. The function u(xJ, X2) = xr + 3xl(l- x§) is harmonic for x 1 > 0 and satisfies the condition au joN+ 6u113 = 0 as x2 = ±1. Moreover, u(O, x 2) = 0, u(x1, x 2) > 0 as x1 > 0. It is obvious that u(x1 , x2) :<; 2xy for XJ > 2.

1. Existence of Positive Solutions. THEOREM 5. Let Q be a general domain described in Introduction. There exists a function u(x), positive in Q, satisfying the equation Lu = 0 and boundary condition (2), and such that u(x', 0) = 1.


Proof. Let ur be a solution to the equation Lu = 0 in Qr, satisfying boundary condition (2) and such that

(4) ur(x', 0) = 1, ur(x', T) = 0.

Such a solution can be found by minimizing the functional

in the class of positive functions u from C 00 (Qr), satisfying condition (4). The minimizing function uris positive, 0 < ur(x) < 1 in Qr.

Moreover,

(5)

for all functions <p(x) E H1(Q), equal to 0 for Xn = 0 and for Xn = T. The function ur(x) is continuous in Qr, see [1].

Put ur(x) = 0 in Q for Xn > T. Let K be a compact subset in Q, and let To be such that K c QTo.

Let h(xn) be a piece-like function such that h(xn) = 1 for 1 < Xn < To -1, h(xn) = Xn for 0 < Xn < 1, h(xn) = 0 for Xn > To.

Put in (5) the function <p(x) = h(xn)ur(x), where T > T0 • We obtain

I .;;;.... our our I +1 J(T, To)= L... aij(x)~~h(xn)dx + h(xn)b(x)iur(x)IP dS . , vXJ UXt

Q t,J=l s

Therefore J(T, To) :'0 C1 (To).


Therefore the set of bounded functions ur on K is weakly compact, i.e. there exists a subsequence { urk}, weakly converging in H 1(K) nLr+I (Sn K) to a function u. Choosing a sequence of compact sets Km tending to Q and using diagonalization, one can find a subsequence which will be denoted as {uk}, converging everywhere in Q and in the space H1~JQ) n Lp+I,loc(S) to a function u from this space.

We have

for all functions <p(x) E H 1 (Q), equal to 0 for Xn = 0 and for Xn Tk.

Passing to the limit, we obtain that

for all functions <p(x) E H 1(Q), equal to 0 for Xn = 0 and in a neighbourhood of infinity, i.e. u is a weak solution to problem (1)-(2). Moreover, u ;::: 0 in Q and u(x', 0) = 1, so that u ¢ 0.

The proof is complete. 0

REFERENCES

[1] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus, Ann. Inst. Fourier Grenoble, 15 (1965), 189-258.

[2] Yu.V. Egorov and V.A. Kondratiev, On an Oleinik's problem, Uspehi Mat.Nauk, 57 (1997), 159-160.

[3] Yu.V. Egorov and V.A. Kondratiev, On asymptotic behavior in an infinite cylinder of solutions to an elliptic solution of second order, Applicable Analysis, 71 (1999), 25-41.

(1)


VOLUME 8

2001, NO 1-2 PP. 183-·194

ON AN INITIAL BOUNDARY VALUE PROBLEM IN A BOUNDED DOMAIN FOR THE GENERALIZED

KORTEWEG - DE VRIES EQUATION '

A.V. FAMINSKII t

The Kortewcg - de Vries (KdV) equation

Ut + Uxxx + aUx + UUx = 0

is a model equation for description of one-dimensional nonlinear waves in a dispersive media. Surface water waves, propagating in a narrow and shallow channel, give an example of such a process. So initial boundary value problems for the KdV equation naturally arise in the case of a bounded channel.

In this paper we consider nonlocal solvability and well-posedness of an initial boundary value problem in a rectangle Qr = (0, T) x (0, 1) (T is an arbitrary positive number) for the generalized KdV equation

(2) Ut + Uxxx + g(u)ux + 91 (t, x)ux + 92(t, .T)u = f(t, x)

with initial and boundary conditions

The function g satisfies the following growth restriction: for a certain constant c 2: 0 and all u E lR

(4) ig'(u)l :S c.

Equations of type (2) naturally arise in the case of inhomogeneous media.

' The work was supported by RFBR (grant 99-01-01139). t Russia Peoples' Friendship University, 117198 Moscow, Russia, Miklukho-Maklaya 6

183

184 A.V. FAMINSKII

First results on existence and uniqueness of nonlocal solutions of problem (1), (3) were established in [1] for zero boundary data u1 , u2, u3 • In [2] and [3], certain initial boundary value problems in QT for equation (1) with other than (3) types of boundary conditions were considered. An initial boundary value problem in QT for the linearized KdV equation with conditions (3) was studied in [4] and [5].

The types of boundary conditions for x = 0 and x = 1 are different in the considered problem. It is an implication of anisotropy of the KdV equation in different directions. Note that initial boundary value problems in right and left half-strips (0, T) x (0, +oo) and (0, T) x ( -oo, 0) have different settings and different properties (see, for instance, [6] and bibliography in it). The main results of the present paper have been previously published in [7].

The nonlocal theory of the KdV equation is based on certain conservation laws. For example, for a sufficiently smooth decreasing solution u of the initial value problem for equation (1)

I u2(t, x) dx =I u~(x) dx. IF1 II!.

In the case of initial boundary value problems one must take into account boundary conditions in order to establish similar estimates (see, for instance, [6]).

Note that the KdV equation possesses the property of smoothing of solutions in comparison with initial data. For the first time it was discovered just for problems in a bounded domain in [1] and [2].

In the present paper we consider solutions of problem (2), (3) in functional spaces

XT = { u(t, x) : u E C([O, T]; L2), Ux E L2(QT)}

and for natural k

XT,k = { u(t, x) : Df'u E C([O, T]; H3(k-ml), 0 :S m :S k- 1, Dfu E XT}

(here and further we use the notations Lp = Lp(O, 1), W~ = W~(O, 1), H 1 = WJ). Let also

XT = {u(t,x): u E L00(0,T;L2),ux E L2(QT)},

XT,! = {u(t,x): u E L00 (0,T;H3 ),Utx E L2(QT)}.

GENERALIZED KDV EQUATION 185

DEFINITION. A function u(t,x) E Xr is called a generalized solution of problem (2), (3) if for any function rp(t, x) E Xr,1 such that 'Pit~r= 0, 'Pix~o= 'Pxlx=o= 'Pix~1 = 0 there holds

J k}u('Pt + 'Pxxx)- (g(u)ux + .9!Ux + g2u- J)rp]dxdt+

(5) 1 T

+ J Uo<p(O, x) dx + J (ul'Pxx(t, 0)- U2'Pxx(t, 1) + U3'Px(t, 1))dt = 0. 0 0

For a description of properties of boundary data we use the Slobodetskii spaces

where k is a nonnegative integer, s E (0, 1), p E [1, +oo). ForsE (0, 1/6) we put

rt+,(o, T) = w,;'\,(0, T) n w;+I 16(o, T),

r- (o T) = wk+S/6+•(o T) n Wk+ 113 (o T) k+e ' 1 ' 2 ' ·

In [6] there were studied in detail certain properties of a boundary potential for the linearized KdV equation

t

J(t,x;p,)=:~(1+3signx) J t~T ((t-XT)!/3)f1(T)dT, 0

where A is the Airy function

A(O) =: 2~ J ei(('+B<) d~, IR

introduced for the first time in [4] for x > 0. Let

where ry(x) is a certain nondecreasing infinitely smooth function such that ry(x) = 0 for x < 1/4 and ry(x) = 1 for x > 3/4.

186 A.V. FAMINSKII

First of all, we establish two auxiliary lemmas concerning an initial boundary value problem for the linearized KdV equation

(6) Ut + Uxxx = j(t, X).

LEMMA 1. Let u1 E rt(O, T), u2 E f;(O, T). Then there exist junctions J E L1(0,T;L2) andua E L2(0,T) such that the function '1j;(t,x;u1 ,u2) is a generalized solution of linear problem (6), (3) for f = J, uo = 0, ua = ua and given u1, u2. Furthermore,

If, in addition, Uj E wf13 (0, T), then

Proof. This lemma is a simple corollary of Lemmas 2.1 and 2.4-2.8 from [6].

LEMMA 2. Let Uo E £2, U1 E ft(O,T), U2 E f;(O,T), ua E L2(0,T), f E £ 1 (0, T; L2). Then there exists a unique solution u(t, x) of problem (6), (3), u E Xr and for any to E (0, T]

(12)


Furthermore, for a function U(t,x) = u(t,x) -1j;(t,x;u1,uz),

1 t 1

.I U2 (t,x)pdx+3 .I .I u;p'dxdr 0 0 0

1 t 1

(13) :::; .I u6pdx + 2 .I .I JU pdxdr + c(T) (iluzii~Ji'(o,T) + llu:J!IL(o,T)) 0 0 0

(

1 ) 1/2

+c(T,e) (11u1iiL,(o,T) + iiuzilw'i'+'(OT)) sup ;· U2 (r,x)pdx , 1 ' rE(O,t)

0

where t E [0, T] and either p(x) = 1, or p(x) = 1 + x. Ij, in addition, Uo E H3, UtE rt+,(O, T), Uz E rl+£(0, T), U3 E H 1(0, T),

f E Wl(O, T; L2 ), u0 (0) = u1(0), uo(1) = Uz(O), u~(1) = u3(0), then u E XT,l·

Proof. We put F(t,x) = f(t,x)- J(t,x), v3(t) = u3(t)- u3(t), where f and u3 are defined in Lemma 1, and consider in QT the following problem for the function U:

(14) Ut + Uxxx = F,

Let { \Oj(x ), j = 1, 2, ... } be a system of linearly independent functions such that the system {(1 + x)'Pj(x)} is complete in the space offunctions {'P E H 3 : <p(O) = <p'(O) = <p(1) = 0}. We construct an approximate solution Un of problem (14) in the form

n

Un(t, x) = l:::Cnj(t)<pj(X) j=l

via the conditions 1

(151[(1 + x)<pmUnt- ((1 + x)<pm)"'Un- (1 + x)<pmF]dx- 2<p;,(1)v3 = 0

0

fortE [0, T], m = 1, 2, ... , n; Un(O, x) -+ uo in L2 .

Multiplying (15) by 2Cnm and summing with respect tom, we find that

1 1 1

:t.f(l+x)U~dx+3 .I U~xdx+2U~xlx=1=2 .f(l+x)FUndx+4v3Unxlx=l 0 0 0 .

188 A.V. FAMINSKII

and therefore for to E (0, T]

According to (7),(8) aEd (16), we select a subsequence nk --7 oo such that Un• --7 U *-weakly in Xr. Using Lemma 1, we find that a function u = U +"lj; is a generalized solution of problem (6), (3) and satisfies inequality (12) (where [[ullx,

0 must be substituted by !lull.x, ).

Now suppose that the additional conditions of the lemma are satisfied. Consider an initial boundary value problem of (6), (3) type, where /, u0 , u1 ,

u2 , u3 are substituted by ft, flt=o-u~', u~, u;, u~. ~ccording to the previous result, there exists a solution of this problem v E Xr. Let

t

u(t,x) = uo(x) + 1 v(r,x)dx. 0

Using compatibility conditions, one can deduce that the function u is a generalized solut~n of the original problem and, in addition, due to equality (6), we have u E XT,1·

Note that, using existence of a solution of problem (6), (3) in_ Xr,1 , one can prove a uniqueness result for a solution of this problem in Xr via the standard argument.

Next, multiplying equation (14) by 2U(t, x)p(x) and integrating, we find that

1 1 1

:t 1 U2pdx + 31 u;j dx ~ 21 FU pdx + (u;p)L=1

0 0 0

and hence, using (7) and (8), we derive inequality (13) (for "smooth" solutions from Xr,1)·

The rest part of the lemma can be proved via the closure of a corresponding set of "smooth" solutions.

Now we consider a result on nonlocal well-posedness of the nonlinear problem itself in Xr.

THEOREM 1. Let g E C1 (1R) and satisfy condition (4). Let g1 E

L2(0,T;Loo), Y1x E L1(0,T;Loo), Y2 E L1(0,T;L00 ), f E L1(0,T;L2). Assume also thatuo E L2, u1 E rt(O,T)nW1

113 (0,T), u2 E r;(O,T), ua E L2 (0, T). Then there exists a unique solution u(t, x) of problem (2), (3) such


that u EXT. Furthermore, the m.ap (uo, u1 , u2 , u3, f)>-+ u is continuous fr-om.

£2 X rt(O, T) n W1113 (0, T) X r;:-(0, T) X £2(0, T) X £1(0, T; L2) into X.r.

Proof. First we prove the local (with respect to t) well-posedness. Let t 0 E (0, T]. We define a map A on Xto in such a way: u = Av for v E X 10

if u E X to and solves in Q10 a linear initial boundary value problem for the equation

(17)

with initial and boundary conditions (3). Using the well-known inequality

(18)

we find that, according to (4),

(19) to

C J (ilviiLoo + 1)llvxi!L, dt :S: c(T)tt(1 + llvll3;-,0

).

0

Besides that,

(20)

(21)

Therefore, according to Lemma 2, the desired solution u of problem (17), (3) exists and

(22)

where w(·) is a certain nondecreasing function, continuous on [0, +oo) and such that w(O) = 0. Then, choosing A ::C 2c0 and t0 ::; w- 1 (1/(2A)), we find that A takes Xto,A = { u E X 10 : llullx,

0 :S: A} into itself.

Next we consider two functions v and v from Xto,A· Let u = Av, u = Av, w = u - u, w = v - v. Then w solves in Qto the problem

190 A.V. FAMINSKII

Similarly to (19)-(22), one can derive that

(23) l!wl!x,0 ::S c(A)w(to)llwl!x,0

•

Thus, for sufficiently small t0 depending on lluoiiL,, the map A is a contraction on Xto,A and, therefore, in Q10 there exists a unique solution u E Xt0 of problem (2), (3).

The result on continuous dependence can be proved by a similar argu-ment.

Next we establish a relevant a priori estimate. Let u E X10 be a solution of problem (2), (3) for some t0 E (0, T]. Then, choosing in (13) p = 1, we find that for t E [0, to)

(24) Note that

11

U2(t, x) dx:::; c [1 + sup (11

u2 dx)

112

] rE[O,tJ

0 0

t 1

-2 J J (g(u)ux + 91Ux + g2u)U dxdr. 0 0

(25) g(u)uxU = Dx (! g(O + '1/>)0d(;l) + '1/>x! g(O + '1/>) dO

and therefore

(26)

t 1

J J g(u)uxU dxdr 0 0

t 1 u - 11 '1/>x 1 g(O+'l/J)dOdxdr 0 0 0

::S eft 11'1/>xiiLoodT [1 + sup 11

(U2 + 'l/>2 )dx] . TE[O,tj

0 0

Besides, gluxU = 91(U2)x/2 + 91'1/>xU, and hence,

(27)

t 1 11 g!uxU dxdT 0 0

t 1

::S cj(llg1xi!Loo + ll9iiiLoo)dT SUp 1 U2 dx

rE[O,tJ 0 0

t 1

+c j j 'lj>; dxdr. 0 0


Finally, we obtain

(28)

Thus, using (9)-(11), we derive from (24) that, for some constant c independent of t 0 ,

(29) llullcc[o,to];Lz) :S c.

This completes the proof. D

Next we consider solutions of problem (2), (3) in the space XT,t·

THEOREM 2. Let 9 E C2 (JR) and satisfy condition (4). Let 91 E

Wl(O, T; Loo) n Lt (0, T; w;,), 92 E Wl(O, T; Lz) n Lt (0, T; Loo), f E W{(O,T;Lz). Assume also that uo E H3, 7Lt E f{+<(O,T), Uz E I'!+e(O, T), u3 E lJl(O, T) and uo(O) = Ut(O), uo(1) = u2(0), u~(1) = n3 (0). Then the solntion u(t, x) of problem (2), (3) lies in the space XT,l·

Proof. Just as for the previous theorem, we first establish a local result. For t 0 E ( 0, T], we define a set

a value f3(u; to)= llullx,0

+ llu1 llx,0

, and consider the map A on Yi.0 just as in Theorem 1. Using (18) and another simple inequality

(30)

one can find similarly to (19)-(21) that

11Dt(9(v)vx)IIL,(O,to;L2 ) :S ct~14 (1 + f32 (v; to)),

Therefore, according to Lemma 2, there exists a solution u of problem (17), (3) in the space XT,t and, by analogy with (22),

192 A.V. FAMINSKII

Choosing A and to just as in Theorem 1, we find that the map A takes Yio,A = { u E Yio : !3( u; to) S A} into itself.

Next, similarly to (23), one can deduce that for v, v E Yio,A

j3(A(v- v); to) s c(A)w(to)f3(v- v; to),

and finally prove existence of a solution u E X10,1 of problem (2), (3) for sufficiently small to depending on lluoiiH3·

To conclude the proof, we establish a relevant a priori estimate for the solution u E X10 ,1 . First, choosing in (13) p = 1 + x, and taking into account (29), we derive that

tol tol

(31) f f u; dxdt s c- 2 f f (1 + x)(g(u)ux + g1ux + g2u)U dxdt. 0 0 0 0

With the help of (18),(25) and again (29), by analogy with (26) we find that

to 1 J J (1 + x)g(u)uxU dxdt 0 0

S C l (IIUIILoo + ll¢xiiLoo + 1) [ 1 +I (U2 + 'if;2)dx] dt

, ~ u 1 u: dx~ r + ••

Estimating other terms just as in (27) and (28), we derive from (31) that for some constant c independent on t0

(32)

Next we consider a function v = u1, which solves in Q10 the following linear problem


Let 1/Jl = '1/J(t,x;u~,u;). We put u1 = Ut(t,x) -1/J1(t,x). Then, using for this problem inequality (13) with p(x) = 1 + x, we find that

(33)

1 t 1

.I (1 + x)Uf(t, x) dx + 3 J .I Ufx dxdT 0 0 0

::; c [1 + sup (.11

(1 + x)Uf dx)

112

] TE[O,tj 0

t 1

-2 .I /(1 + x)DT(g(u.)ux + 91Ux + gzu)U1 dxdT. 0 0

According to (9), (10), (18), (30) and (32), we obtain

t

t 1

.I .I (1 + x)DAg(u)ux)U1 dxdT 0 0

t

:S C .I (lluTx IlL, + lluxliL,IIuTIIL,)IIU1IILoo dT 0

t 1 1

::; o jj Ufx dxdr + c(o)t113 sup /(1 + x)Uf dx + c1, TE[O,tj

0 0 0

t 1

.I .I (1 + x)DT(91Ux + gzu)U1 dxdr 0 0

:S c .I (llg1TII£oolluxiiL, + ll9zTIIL,IIuiiLoo + lluTxiiL, + lluTIIDoo)IIU111L, dr 0

t 1

:S 0 .I .I Ufx dxdr 0 0

1

+c(o)(II91TIIL(o,t;Loo) + ll9zTIIL(o,t;L2 ) + t) sup /(1 + x)Uf dx + C1, TE[O,tj 0

194 A.V. FAMINSKII

where li > 0 can be chosen arbitrary small. Hence from (33) we find that

and then, using equality (2) itself, that

llullc([o,T];H•) :'0 c,

where c does not depend on t 0 . This completes the proof. D

Using Theorem 2, one can establish existence of nonlocal solutions of problem (2), (3) in the space Xr,k for any natural k:;:: 2, when u0 E H 3k, u1 E rt+<(O, T), U2 E r;;+.(O, T), Ua E Hk(O, T), g, gr, 92 and f are sufficiently smooth and certain compatibility conditions are satisfied. One can also prove corresponding results on continuous dependence.

REFERENCES

[1] V.V. Khablov, Some well-posed boundary value problems for the Korteweg -de Vries equation, Preprint, Inst. Mat. Sibirsk. Otdel. Acad. Nauk SSSR, Novosibirsk, 1979.

[2] B.A. Bubnov, Solvability in the large of nonlinear boundary problems for the Korteweg- de Vries equation in a bounded domain, Diff. Uravn., 16 {1980), 34-41.

[3] T. Colin, and J.-M. Ghidaglia, Un probleme mixte pour !'equation de Korteweg - de Vries sur un intervalle borne, C. R. Acad. Sci. Paris, Ser.l, 324 {1997), 599-603.

[4] L. Cattabriga, Un problema a! contorno per una equazione parabolica di ordine dispari, Ann. Scuola Norm. Sup. Pisa , 13, 3 {1959), 163-203.

[5] T.D. Dzhuraev, Boundary value problems for equations of mixed and mixedcomposite types, "Fan", Tashkent, 1979 (in Russian)

[6] A.V. Faminskii, Initial boundary value problems for the Korteweg- de Vries equation, Matern. Sbornik, 190 {1999), 127-160 (in Russian)

[7] A.V. Faminskii, On initial boundary value problems for the Korteweg de Vries equation with irregular boundary data, Soviet Akad. Sci. Dokl. Math., 366 {1999), 28-29 (in Russian)


VOLUME 8 2001, NO 1-2 PP. 195-224

ON ONE-DIMENSIONAL UNSATURATED FLOW IN A POROUS MEDIUM WITH HYDROPHILE GRAINS '

A. FASANO t AND V. SOLONNIKOV !

1. Introduction. According to the model presented in [1], the flow in a porous medium with hydrophile grains is characterized by the following functions:

p(x, t) : pressure, c:(x, t) : the medium porosity, V(x, t) : the volume fraction occupied by the grains, S(x, t) : saturation.

These functions depend on the spatial variable x E (0, s(t)) and on time t > 0. The interval (0, s(t)) is a wet region and the curve x = s(t) on the (x, t)-plane represents a wetting front. There hold the following relationships between the above functions. S(x, t) depends on the pressure p(x, t):

(1.1) 1- So

S(x, t) =So+ p(x, t) =So+ cop Ps

for 0 < p < Ps,

S(x,t) = 1 for p 2: Ps

' This work was partially supported by the Italian MURST project "Mathematical Analysis of Phase Transitions" and by the Portuguese grant PRAXIS XXI/2 /2.1 /MAT /125/94.

t Dipartimento di Matematica "U.Dini", Universita di Firenze, Viale Morgagni 67/ A, 50134 Firenze, Italia

l V.A.Steklov Institute of Mathematics (S.Petersburg Department), Fontanka 27, 191011, S.Petersburg, Russia

195

196 A. FASANO AND V. SOLONNIKOV

where c0 = (1 - S0 )p$1, Ps = Const is the saturation pressure and S0 =

Const > 0 is the saturation on the wetting front. The relationship between V(x, t) and S(x, t) is given by

(1.2) 8V at = f(Vmax - V)(S- So),

where f(~) is a smooth function such that f(O) = 0, f'(t;) ~ 0, f(~) > 0 for ~ > 0, Vmax is the maximal admissible volume fraction of the grains and V0

is the volume fraction of the grains in a dry medium. Let

v dy <J>(V) = I f(V. - )

Vo max Y

and let \[1 be the inverse function of <J>. Integrating (1.2), we obtain

t t

V(x,t)=w( I(S(x,r)-S0)dr)=w( I c0p(x,r)dr), t ~ O(x), 8(x) 8(x)

~ where O(x) is the inverse function of s(t) (i.e., the time of passing a wetting front through the point x). Further, c(x, t) and V(x, t) are related by

(1.3)

or

c(x, t) + V(x, t) =co+ Vo

where co is a porosity of a dry medium. Hence,

t

c(x, t) =co+ Vo-w( I Cop(x,r)dr ). O(x)

(1.4)

Finally, the mass balance and the Darcy law give

(1.5) ap

q(x, t) = -k(S, c:) ox (q is the volumetric velocity, k(S,c) is the hydraulic conductivity). Eliminating q and taking account of (1.1) and (1.3), we arrive at the main equation for the pressure .

(1.6) ap 0c: a [ ap] cCo 8t - (1- S) at - OX k(S, c) OX = 0, 0 < x < s(t),

ON ONE-DIMENSIONAL UNSATURATED FLOW 197

where Sand care given by (1.1) and (1.4), respectively. As for the boundary conditions, we assume that the function q is pre

scribed for x = 0 and that it assumes strictly positive values. This means that p( x, t) satisfies the condition

(1.7) apl -k(S, c) ax x=O = q(t).

On the free boundary r = {x = s(t)} we have

(1.8) p(s(t), t)) = 0, s'(t)-- ap I t - JL ax x=s(t)

where JL = k0 j S0 ro 0 , k0 = k(S0 , c0). We assume finally that the wetting front is separated from the point x = 0 at the initial moment of time t = 0, i.e., that

(1.9) s(O) = 0.

Hence, no initial conditions are necessary; at the origin we have

(1.10) p(O, 0) = 0, Px(O, 0) = -q(O)/ko = p~, further, differentiating the condition p(s(t), t) = 0, we obtain

(1.11) Pt(O, 0) = -s'(O)p~ = JLP~2 ,

and if the derivatives of p(x, t) are continuous up to the point x = 0, t = 0, we can find Pxx from (1.6):

(1.12) Pxx(O, 0) = ~: (coJL- k~(So, c) )p~2

In [1 J an existence result was obtained, under some restrictions on the choice of the functions f and k. Under more general assumptions concerning k(S,c) the existence of a unique solution of the problem was proved in [2], under the assumption that at the initial moment of time the wetting front is already inside the medium. Here we prove that problem (1.6)-(1.9) has a unique classical solution in a certain finite time interval during which saturation is not yet attained. Exact formulation of this result will be given in the next section. The problem of continuing the solution after saturation has been reached at the inlet surface is still open and it is characterized by the presence of another free boundary, i.e. the saturation front. One could also consider the limiting case in which the flow is saturated from the very beginning (like in the well known Green-Ampt model), which is of course much simpler. We refer to [3] for the description of the latter case and for a more general discussion of the model.


2. Transformation of the Problem and Formulation of the Main Result. Let so(t) = lot where lo = s'(O) = q(O)/Soeo, and let 17(~) be a smooth monotone function of a real argument such that

17(~) = 0 for ~<it, 17(~) = 1 for ~ > l2,

where l; satisfy the condition 0 < 11 < l2 < lo. Consider the mapping given by

(2.1) z

x = z + 1/>(z, t) = z + 17( t )a(t) = Y(z, t),

where

(2.2) z

1/>(z, t) = 17( t )a(t), a(t) = s(t)- so(t).

Since a(t) vanishes for t = 0 together with its derivative, the mapping Y is invertible for small t, and it establishes one-to-one correspondence between the intervals 0 :::; z :::; s0 (t) and 0 :::; x :::; s(t). In the coordinates (z, t) equations (1.6)-(1.9) take the form

(2.3) A 8fjl -k- = q(t), az z;Q fJ(so(t), t)) = 0,

(2.4) da (ap , ) I -- -J.t - -p dt - . az 0 z;so(t)

where the hat denotes transformed functions: fJ(z, t) = p(Y(z, t), t) = p o Y, S =SoY, € =soY, €t = s1 o Y, k = k(S,€).

Next, we introduce a new unknown function

r(z, t) = fJ(z, t) - p~lj>(z, t)

satisfying, by virtue of (2.2), the equation


(2.5) 1 ak aft - ~

(1 + 1/J'z)Z az az - (1- S)Et = 0.

Since f(t) = r(s0 (t), t) = -p~o-(t), the boundary conditions take the form

(2.6) -ar I -kc;- = q(t), uz z=-0

1 dr Dr I 1 --,- + J..l- =/~Po· Po dt az z=so(t)

Hence, our free boundary problem reduces to a nonlinear problem (2.5), (2.6) in a given domain {0 < z < s0 (t), 0 < t < T}. By j5 in (2.5) we mean the function j5 = r+p~'ljJ where 1/J(z, t) is defined in (2.1) and o-(t) = -+r(t).

Po Let r0 (z, t) be an auxiliary function satisfying conditions (1.10)-(1.12),

i.e.,

ro(O, 0) = 0, 7'oz(O, 0) = p~, '2 rot(O, 0) = J..!Po,

We take it in the form of a polynomial

(2.7)

with

Co I ( '2 bo = 2ko (coJ..l- k8 So, Eo))p0 .

and introduce a new unknown function u(z, t) = r(z, t) - r0 (z, t). In virtue of (2.5), (2.6), we have

(2.8) CoEoUt- koUzz = F[u] + fo(z, t),

(2.9) au I - Du -oro I -ko- = (k- ko)- + k- + q(t), az z=O az Dz z=O

(2.10) 1 du au I 1 dfo oro I I ---+J..l- =---j..l- +J..!P Po dt Dz z=so(t) Po dt Dz z=so(t) 0

where


' 5 • _ ( 1 ak ap I ( ) 2 ) _ "' [ 1 (2.11) +(1- S)ct + (1 + 'lj;~)2 az az - ks So, co CoToz = 6 Fi u,

fo(z, t) = - (Coco Tot - korozz - k$(So, co)cor~z),

p(z, t) = u(z, t) + r0(z, t) + p~'lj;(z, t),

(2.12)

It follows from the definition of r 0 that f 0 (0, 0) = 0. Moreover, assuming that q(t) is continuously differentiable and putting

we obtain

b1 = -k01 ( cok$(So, £o)pp0

3 + q1(0)),

b2 = p~ Gb1tt - 2p?p~bo),

[ • au , ar0 Jl aro I (k- ko) az + k az z=O + q(t) = ko az z=O + q(t)+

- (au aro)l +(k- ko) az + az z=O = 9o(t) + G[u],

( aro 1 aro) \ . Yo(t) = ko-a + k8 (So,£o)coro-a + q(t)

Z z z=O


and

1 dfo 8ro I , 0 -,-- Jt- + J!Po = · Po dt fJz z=.so(t}

Hence, conditions (2.9), (2.10) can be written in the form

fJu I 1 du fJu I (2.14) -k0 "z z--o = go(t) + G[u], - 1 - + Jt- = 0. u Po dt fJz z=so(t)

Let us introduce the necessary functional spaces. By C1(J,) and C1(0, T) (l > 0, non-integer) we mean standard Holder spaces of functions given in the interval J, = (0, s(t)) or (0, T), respectively, with the norms

[IJ I dju(.T) I (I} l!ullc'(Jt) = _t;~~Y, dxj + [u]J,'

[I] ldju(t)l (I} llullc1(o,r) = L sup -d-. + [uJcor)'

j=O tE(O,T) tJ '

where

[ ](I}_ I _ 1111 _ 11d[llu(x) _ di1lu(y) I

u J, - sup x y d [1] d [IJ x,yE.lt X y

and [uJ/2,r) is defined in a similar way. Further, by C1•112 (!10r), !lor =

{x E J, t E (0, T)}, we mean an anisotropic Holder space of functions u(x, t), (x, t) E !10r, with the norm

_ " I fJHku(x, t) I (21,1} llulb'l'(n01·) - L, sup f) jf)tk + [u]nor

j+2k<l noT X •

where (2.15)

[ ](21,1) _ . [ ](I} h-1; 2+[1]/2 1 fJ['/2lu(x, t +h) _ fJ[I/Zlu(x, t) I

u noT -- sup u 1, +sup sup "t[l/2] "t[l/2] r<t noT hE(O,T-t) U U

(we assume that s'(t) > 0, so this expression has a sense). But a central role in this paper is played by a weighted Holder space C1•112 (!1or) in which the norm is given by

(2.16) llullcui'(n ) = sup [u]g'1112) +sup t-1lu(x, t) I,

OT rE(O,T) T 2,r noT


where [u]~·112l is a principal part of the norm in C1•112 (IJ7 ; 2 r) (see (2.15)). Urf2,T l

Moreover, we introduce a family of spaces C~1/2 (1J0r), {3;:::: 0, with the norms

!lull0·•.•t'<n ) = sup T11 [u]g'1112) + suprl+/Jiu(x, t)i,

11 uor TE(O,T) T 2

'T nor

so that 6~·1 12 (1JOT) = C1•112(1J0r). We need also similar weighted spaces of functions u(t), t E (0, T), Cb(O, T) and 01(0, T) = 66(0, T), where the norm is defined as follows:

(2.17) !lullc•(oT) = sup T11 [u]~~;2 r) + sup t-2l+i1iu(t)i. p ' TE(O,T) ' tE(O,T)

It is easily seen that the expressions sup [u]g•jl and sup [u]~2 r) in rE(O,T) ~·T rE(O,T) 2 '

(2.16) and (2.17) (with {3 = 0) may be replaced with [uJg;~~ and [u]~~.T)• respectively, which leads to equivalent norms.

Main results of the paper are contained in the following two theorems.

THEOREM 2.1. Assume that k(S,t:) is three times continuously differentiable function of positive arguments, that q E 0 312+<>1/2 (0, T), a 1 < 1 and that q(t) > q0 > 0. Then problem (2.8), (2.14) has a unique solution u E (j2+a,l+af2(1Jo,T) with Ut E (ja,af2(1Jor), Uz E (jl+a/2,(l+a)/2(1Jor),

Uzz E (jaf2,af2(1Jor), UtE (j(l+a)f2(0,T), a E (O,al), in a certain finite time interval (0, T), and this solution satisfies the inequality

(2.18)

+lliitllc<>+olf'(O,T) :::; c(llfollco.of'(Oor) + ll9ollc<>+o)f'(O,T)) ·

It is easily seen that the functions

(2.19) p(z, t) = r0(z, t) + p~7J( f)a(t) + u(z, t),

Pz(z, t) = roz(z, t) + P~7J'(f) a~t) + Uz(z, t),

ON ONE-DIMENSIONAL UNSATURATED FLOW

belong to C"·"l2(floT) and the functions p(::c, t), Px(x, t)

A ( t) I ( t) - A ( t) + A ( t) ,v;(z,t) I Xpz z, , z=Y-l(x,t)' Pt X, - Pt z, , Pz z, l+,V,(z,t) z=Y-l(x,t)

C""'I2 (11T ). The second derivative

Pzz(z, t) = rozz(z, t) + p~'l/lzz(z, t) + Uzz(z, t)

is bounded but it has no definite limit as t -+ 0, however,

Pxx(x, t) = (1 +1'1/J~) 2 (Pzz(z, t)- pz(z, t) 1 ~z~J lz=Y-l(x,t)

203

1 X l+,V,(z,t)

belong to

tends to cok01 (E:o{l- k'(So,E:o))p~2 as t-+ 0 and Pxx E C"·"I2 (11T)· Hence, it follows that problem (L6)-(L8) has a solution p E C2+a,l+a/2(~), s E

C312+<>12 (0, T). We show that this solution is unique:

THEOREM 2.2. Problem (L6)- (L9) can have only one solution s(t), p(x, t) possessing the pmperties s E C312+"/2 (0, T), p E C2+<>,H<>/2 (0T ). Every such solution can be represented in form (2.19) with u E C2+<>,l+a/2 (110T ).

3. The Scheme of the Proof of Theorem 2.1. The proof of Theorems 2.1 and 2.2 is based on the analysis of a linear problem

(3.1) ow 82

1JJ Coco iJt - ko OX2 = f(z, t),

(3.2) owl -ko- = g(t), OZ z=O

1 dw awl ---+JL- =ht. Po dt oz z=so(t) ( )

THEOREM 3.1. Let T be an arbitrary positive number. For arbitrat·y Aaa/2 '(1+<>)/2 f E Cp' (~loT), g,h E Cp (O,T) with o: E (0.1), (3 E [0, 1 +o:] problem

(3.1), (3.2) has a unique solution with the following properties:

E CA 2+a,l+a/2 (" ) E CA a,a/2 (" ) W {3 "O,T , Wt {3 "O,T ,

E CA l+a,(l+a)/2 (" ) E c' a,aj2(n ·) A E c' (l+a)/2 (0 'T') Wx {3 "OT 1 Wxx {3 "OT , Wt {3 , " •

This solution satisfies the inequality


(3.3)

llwtllc~l+Q)'2(o,r) ::; c(11fllc;·Q'2 <nor) + llullc~l+Q)'2 <o,r) + llhllc~>+Q)'2(o,r)) · where c(T) is an increasing function ofT.

Inequality (3.3) was proved in [4] in the case f3 = 0, and the same arguments apply in a general case. First of all, using estimates of solutions of model problems on the half-line z > 0 with boundary conditions (3.2) at the point z = 0 and some local estimates and interpolation inequalities, it is possible to show ( cf. (3.3) in [4]) that

sup [wt(·, t)J}~l + sup [w(·, t)]~+a) + 7-2-" sup lw(z, t)l

t€(7(2,7) tE(7(2,7) flr/2,r

+c([f](a) + 7-a sup lf(z t)l + [g]((l+a)/2) + 7-l-a sup lg(t)l

n.,.-/4,'1' n ' rf4,r /4 HTj4,T T ,T

(3.4)

with arbitrarily small o > 0, and from (3.6) in [4] it follows that the last term is less than

C7_1_"( sup lg(t)l +sup lh(t)l + suptlf(z, t)l)

t<r t<r !10 .,.

::; c7-t3 (sup cl-a+f'llg(t)l+ sup cl-a+f'llh(t)l +sup c<>+!'llf(z, t)l). t<r t<-r Oo.,.

Hence, multiplying (3.4) by 7!'1, taking supremum with respect to 7 < T and choosing o small enough, we arrive at

sup tf'l[w1(·, t)J}~l + sup tf'l[w(·, t)]~+a) + supc2-"+f'llw(z, t)l tE(O,T) tE(O,T) flo,T


Furthermore, interpolation inequalities (2.1) in [4] and the estimate

Coco sup sup It- t'l-a/2 lut(z, t)- ut'(z, t')l T j2<t<t1 <-r Jt

::; ko sup sup It- t'l-a/2 luzz(z, t)- Uzz(z, t')l+ T j2<t<t1 <r Jt

+sup It- t'l-a/2 lf(z, t)- f(z, t')l J,

multiplied by T-13 yield

sup T 13 [ut]ha,a/2) + sup T 13 [·uz]g+a,(l+a)/2 + sup rl-a+f31uz (z, t;) I T -r-(2,-r- <T r/2,-r- r.

T< T J~O,T

which completes the proof of (3.3).

205

The existence of the solution can be established by approximation of the functions j, g, h by functions vanishing for small t for which the solution of problem (3.1), (3.2) exists, and by subsequent passage to the limit (see [5]).

The idea of the proof of Theorem 2.1 is as follows. Let .C[f, g, h] be a linear operator which makes correspond the solution of problem (3.1),(3.2) to the data(!, g, h). According to Theorem 3.1, this is a continuous operator with the domain of definition (;a,<>/2 (!.10r) x C(l+a)/2 (0, T) x (;(l+a)/2 (0, T) and with the range C2+<>,1+<>/2 (!.10r ). The norm in the latter space which is denoted by N 2+a[w] is the sum of norms in the left hand side of inequality (3.3). It is clear that problem (2.8), (2.14) is equivalent to the equation

u = .c[F[u] + fo, G[u] +go, o] = M[u]

in the space cz+a,I+a/2 (S10r ). We show that in the case of small T M is a contraction operator in a certain ball

(3.5)

of this space, which is a consequence of the estimates

(3.6) It> 0,

•


(3.7) 1'2 > 0,

for arbitrary u, u1 , u2 from the ball (3.4). In their turn, these estimates follow from

(3.8) IJF[u]IJ0o,of2(0or) ::> caT73,

with positive 'f'i, because fo and g0 evidently satisfy the inequalities

with C7, cs depending on \\qiJc•i'+a,/'(o,T)·

4. Proof of Inequalities (3.8)-(3.11). We start with the following auxiliary proposition.

LEMMA 4.1. If !J E C"'·"'l2 (0or), !2 E C"'•"'/2 (0or), then fd2 E

C"'·"'/2 ( Oor), and

(4.1)

where

{J}a,Oor =sup !f(z, t)i + supT"'[f]/;'·;12) DoT -r<T r

2'-r

Proof. Indeed,

+sup I!J (z, t)iih(z, t)l ::> cii!JIIca,o/2(0or) {h}a,Oor• noT .

q.e.d.


We also put

11/lla,ilo'r = ll.fllc;o,af2(ilor)

and pass to the estimates of the expressions Fi[u] in (2,11), We choose o so small that 1 + 1/J~(z, t) E (1/2, 3/2), make use of the preceding lemma and of the identities

Op 1 1 (OU 07'0 1) 1 oz 1 + 1/J'z -Po = oz + oz -Po 1 + 'lj!'z'

and evaluate F;[u], i = 1, 2, 3, as follows:

(4.2) IIF![u]lla,ilor ::: cllt- colla,ilo1" ( {ut}a,ilor + {rot}",ilor),

( 4.4)

We estimate the norm lluzlla,ilor with the help of the inequalities

[uz]~) :C: tl-a sup luzz(z, t)l :C: tlluzzlla,ilcOt)> J,

sup sup h-a/2 lv-z(z, t +h)- Uz(z, t)l nr/2,r h<r-t

" 1

:C: 2 1.;:" (SUp h-(l+<>)/2 luz(z, t +h)- Uz(z, t)l) l+a (SUp luz(z, t)l) 1+o

nrj2,T i)Tj2,r


which yield

lluzlla,n0r ::; cT(IIuzzlla,n0r + lluziiHa,n0r ).

Other norms in (4.2)-(4.4) will be evaluated by the inequality

(4.5) II.PIIa,nor::; cT1-"( sup I.Pz(z, t)l +sup I.Pt(z, t)l)

!loT !loT

where <,b(z, t) is an arbitrary function with bounded first derivatives such that <,b(O, 0) = 0. This inequality can be applied to the functions

(4.6) l/;~, roz- p~, k(S, c)- ko , t(z, t) -co.

The derivatives of the function t =co+ V0 - \II(1(z, t)) where

t

1(z,t) = I eofi(X(z,t,r),r)dr, O(Y(z,t))

X(z,t,r) = y-1 (Y(z,t),r)

are given by

t. = -\II'(1(z,t))~!, €t = -\II'(I(z,t))~~,

81 t /Jz =Co I Pz(X(z, t, r), r)X.(z, t, r)dr,

O(Y(z,t))

81 t at =co I p.(X(z,t,r),r)Xt(z,t,r)dr+cofi(z,t),

O(Y(z,t))

( ) 1 + l/;~(z, t) ( ) l/;;(z, t) _ ( )

X. z,t,r = 1 +l/;'z(X,r)' Xt z,t,r = 1 +l/;'z(X,r)' X=X z,t,r

(two last formulas follow from (Y-1 (x, t))x = (1 + l/; 1 (~, t))-11 1 ). It e=Y- (x,t)

is clear that under the hypothesis N2+a[u] ::; 8 the first derivatives of c(z, t) and of other functions in ( 4.6) are bounded. All the norms { ·} in ( 4.2)-( 4.4) are also bounded, hence,

3

L IIF;[u]lla,noT ::; crl-a. i=l


Further, we write F4[u] and F5 [u] in the form

F4[u] = (1- So)(l- P!Ps)ft,

where

ft = -c0W'(I(z, t))p(z, t).

Since the derivatives of the functions k's(S, €) and k~(S, €) are bounded, we have

The norms in the right-hand side can be estimated by the same procedure as above, i.e., with the help of (4.5). The boundedness of the derivatives of t z follows from the equations

t = -W"(I) 0I al- i!i'(I) 02

! zt az at . 8z8t,

~:; = -coz3z(X, li(Y))Xz(z, t, li(Y))Ii'(Y)(l + 1/J:(z, t))

t

+co j (Pzz(X, r)X; + Pz(X, r)Xzz(z, t, r) )dr, O(Y(z,t))

::~t = CoPz(z, t)- CoPz(X, li(Y))Xz(z, t, li(Y))Ii'(Y)1/;;(z, t)


t

+co I (fizz( X, r)XzXt + Pz(X, r)Xzt)dr, O(Y(z,t))

X ( ) 1/J~.(z, t) zz z, t, r = 1 + 1/J~(X, r)

1/J~z(X, r)(1 + 1/J~(z, t))2

(1 +1/J~(X,r))3

1/Jzz(X, r)1/J"f(z, t) (1 +1/J~(X,r))3 ·

where 1/J~z(z, t) = r/' (zjt)a(t)jt2, 1/J~t = r!'(zjt)a'(t)jt- r/' (zjt)za'(t)jt3 are

bounded functions (see (2.12) which shows that a behaves like t2 ).

Hence, (3.8) holds with 'Ya = 1 -a. Estimate (3.10) is proved by similar arguments. Since

1

k- ko- k~(So,co)(S- So)= I (k~(So + .X(S- So),co + .X(€- co)) 0

1

-k~(So, co)) (S- So)d.X +I k~(So + .X(S- So), co+ .X(€- co))(€- co)d.X, 0

we have

sup r 1-"IG[u]J :::; c sup r 1-"(Ju(O, t)l + Jp(O, tW + J€(0, t)- col (O,T) (O,T)

and

[G[u]rH<>l/2

:::; cT(l-<>l/2 sup (lut(O, t)J + Jp(O, t)J + lit(O, t)J +sup lkt(O, t)J) (O,T) (O,T) (O,T)

+ sup Jk(O t)- k l[u ](H<>l/2 < cT(l+<>l/2 ' . 0 z (O,T) - '

~~ .

which shows that (3.10) holds with 'Ys = min(1- a, (1 + a)/2).


To prove (3.9), we use the following representation formulas for Fi[udFi[u2]:

( 8f;2 1 /) ( (A A )"''' A ("'' ,/, ) + oz 1 + 1f2z -Po X Co £1 - £2 'l'lt + co£2 'l'lt - '1'2t

+(1- So)cofi2(1- eofJ2) ( w'(h) - w'(h)),


Here p; = ro+Ifo'ljl;+u;, i = 1, 2, 'ljl(z, t) = TJ(zjt)a;(t), a;= -(u;+i'o)fp~, k; = k(S;, €;), S; = So+ CoP;, €; =co+ Vo- \]"!(!;),

l;(z, t) =co j p(X;(z, t, r)r)dr, O;(Y;(z,t))

Y;(z, t) = z + 'ljl;(z, t) = z + TJ(zjt)a;(t), X;(z, t, r) = y-1 (Y;(z, t)r),

and, as a consequence,

1

(4.7) €1(z, t)- €2(z, t) = (h(z, t) -I1(z, t)) I \]i'(lr + >.(!2 -lr))d>., 0

&I2 ) 11

, , (&h &I1) (4.8) = &z (!2 -/1 \]i (lr + >.(!2 -II))d>. + \]i (II) &z - &z . 0

Further, we introduce the function

Bmax(z, t) = max(B1(Y1(z, t)), B2(Y2(z, t))),

and write the differences of l; and of their derivatives in the form

Oma.a::(z,t)

l1(z,t)-I2(z,t)=co[ I p1(X1(z,t,r),r)dr 01(Y1(z,t))

Oma.a::(z,t) t

I P2(X2(z,t,r),r)dr+ I (P1(X1,r) -p2(X2,r))dr] O,(Y,(z,t)) Omax(z,t)

1

= co(Bmax- Bl(YI)) I P1(X1(z,t,r),r)lo1(Y1))+A(Omax-01(Yl))d>. 0

1

-co(Bmax - B2(Y2)) I P2(X2(z, t, r), r)lal(Y,))+A(Omax-02(¥,)) d>.+ 0

(4.9)


t

+co J (P1(X1,T) -pz(Xz,T))dT, Brna:e(z,t)

oil(z, t) _ oiz(z, t) _ (e _ e (Y, )) oz oz - Co max 1 1 X

1 .

X J P1z (X 1 (z, t, T ), T )X lz ( z, t, T) IT=B!(Yl)+A(Bmax-h(Yl)) d.\ 0

1

213

-co(Bmax- Bz(Yz)) J Pzz(Xz(z, t, T), T)Xzz(z, t, T)[T=Bz(Yz)+A(Bmax-Bz(Yz))d.\ 0

t

(4.10) +co j (:P1z(X1(z, t, T), T)X1z- Pzz(Xz(z, t, T), T)Xz, )dT. Om ax

We have

Let us get some estimates of B1(z,t)- Bz(z,t) and B1(Y1)- Bz(Yz). We start by observing that

Y1(z, t)- Yz(z, t) = 1/J1(z, t) -1/Jz(z, t) = 7)(z/t)(o1(t)- oz(t)), z E lt.


\Ylz(z, t)- Yzz(z, t)l + \Yit(z, t) Yzt(Z, t)\ :S csup \u~(T)- o~(T)\, T$.t

( 4.11)


Now, we prove the following lemma.

LEMMA 4.2 There hold the inequalities

(4.12)

(4.13)

where 1: = (O,min(s1(t),s2(t))), and

(4.14) 1101 (YI)- 02(Y2)1ia,n0T :::; cT1-" sup Ia; (t) - a~(t)l.

t<T

Proof. Evident equations si(Oi(x)) = x, i = 1, 2, imply

or

where

1

I(x) = j s; (02(x) + .A(01(x)- 02(x)) )d.A 0

is the function possessing the properties:

min s'(r):::; I(x):::; max s'(r), rE[O,tJ rE[O,tJ

II(x)- I(y)l:::; cix- vl"[s;]~~!t)'

Hence,

and (4.12), (4.13) are proved.


The difference 111 (Y1) -112 (Y2) can be represented in the form

= [iil(Ymin(z, t)) -112(Ymin(z, t))] + rJ(zjt)(o-min(t)- o-z(t))x

1

X J e;(Yz + >-(Ymin- Yz))d.X- n(z/t) X

0

1

(4.15) x(o-min(t)- o-1(t)) j e;(Yl + .X(Ymin- YI))d.X, 0

where

Ymin(Z, t) = z + n(z/t)o-min(t).

Clearly, these functions have bounded first derivatives with respect to t, and

Yz- Ymin = max(Yz- Yi., 0).

It follows from (4.15), (4.11)-(4.13) that

III1(Yl(z,t)) -llz(Yz(z,t))i :S ctsuplo-;(r) -o-~(r)l, T'S:t

hence, (4.14) also holds. The lemma is proved. 0

COROLLARY.

(4.16) :S cT1-"sup lo-;(r)- o-~(r)l. T5:T


We also need estimates of the differences X 1(z,t,r)- X 2 (z,t,r) and of their derivatives.

LEMMA 4.3. For arbitrary z, t, T satisfying the conditions Omax(z, t) ::; T ::; t ::; T there hold the inequalities

( 4.17) IX1z(z, t, r) - X2z(z, t, r) I+ IXJt(z, t, r)- X2t(z, t, r)l ::; csup lu; (0- u~(~)l.

{9

Proof Since Y;(X;(z, t, r)r) = Y;(z, t), we have

To be definite, assume that s2 (r) ::; s1(r). Then Y1(X2(z,t,r)r) is well defined, and

The left-hand side can be written in the form

I d (X1(z,t,r) -X2(z,t,r)) j d~Y~(~,t)i{=X2+A(X1 -x,)d)..

0

I

(4.19) = (X 1(z,t,r) -X2(z,t,T)) j(l+'I/J~t;(~,r))lt;=X2+>.(X1 -X2 )d).. 0

where the integral is bounded from above and from below by some positive constants. Hence,

For the difference X 1z- X 2z we have the representation formula


1 + 1/J~z (z, t) + ~( 1_+_u_J\~z(c--X-c-l-, ,· ) ) ( 1 + 1/J~z( X 2, r)) X

( 4.20)

and, as a concequence, the estimate

+lu1(r)- O'z(r)l + IX1- ~zl kz(r)l). T T T

The same kind of estimate holds for Xlt- X 2, so we arrive at (4.17). The lemma is proved. D

LEMMA 4.4. If r E (Omax(z, t), t) n (Omax(z', t), t), then

IX1z(z, t, r)- X1z(z', t, r)l

where X 1z = X1- Xz. If T E (Omax(z, t), t) n (Omax(z, t +h), t +h), hE (0, t), then

IX!2(z, t + h, r)- Xl.2(z, t, r) I ::= chif (c\l' lu1 (t)- O'z(t) I+ ,-\l'-lu1 ( r)- O'z( r) I)

( 4.23) +clu1(t +h)- O'z(t +h)- O'J.(t) + uz(t)l,

(4.24) IXlzz(z,t+h,r) -Xlzz(z,t,r)l ::= ch";z,-cx;z sup lu;(~) -0'~(01, T<c<t+h

Proof. Estimates (4.21), (4.23) follow from (4.18), (4.19), because the integral factor in (4.19) is not only bounded but also satisfies the Holder condition with respect to z and t (with the exponent a). To prove (4.22), we


estimate the difference of the function (4.20) at the points z and z' and note that f;(z, t) = [1 + 'lj!'(Xi(z, t, 7), 7)]-1, j = 1, 2, satisfy the inequality

!fi(z, t)- fj(z', t)!::; cr-"lz- z'l"·

We also use the formula

from which we obtain

( IXI2(z, t, 7- x!2(z', t, 7)1 IXI2(z, t, 7)1)

::;c + I+ . 7 7 a

As a result, we arrive at

which implies ( 4.22). Inequality ( 4.24) is obtained in the same way. The lemma is proved. D

The next step is the estimate of differences (4.9), (4.10).

LEMMA 4.5. The differences / 1 - h and hz- hz satisfy the inequalities

(4.25) ll/1- I2!la,n0T::; cT1-"'(Tsup jo-;{t)- O'~(t)! sup !lui- u21!co(J,)), t<T t<T

Proof. Let us prove (4.26). In virtue of (4.16),

II liz- 12zlla,!1oT ::; crl-a sup,(}'; (t)- O'~(t)i + IIKIIa,noT' t<T


t where K(z, t) = J P(z, t, r)dr is the last term in (4.10). The function

Oma:r;

1

+Xzz(P!z(Xl, r)- Pzz(Xl, r)) + Xzz(Xl- Xz) J P2zz(Xz + .A(X1- Xz), r)dr 0

satisfies the inequalities

IP(z, t, r)l :<:; c(IXl(z, f;, r)- Xz(z, t, r)l + IX1z(z, t, r)- Xzz(z, t, r)l+

+[ulz(·, r)- UzzC r)]~l),

and, if T E (Omax(z, t +h), t +h) n (Omax(z, t), t), hE (0, t), then

IP(z,t+h,r)- P(z,t,r)l :<:; cha/Z(r-a/Zsupicr;(~) -cr;(OI+ <<t

Hence,

t+h

h-a/2IK(z,t +h)- K(z,t)i :<:; h-a/Z j IP(z,t,r)drl + h-a/Zx t


Bma:~:(z,t+h) t

x/ I IP(z, t+ h, r)idr/ + I IP(z, t+ h, r)- P(z, t, r)idr 8maa:(z,t) e(z,t,h)

where 6(z, t, h) = max(Omax(z, t), Bmax(z, t + h). These inequalities yield (4.26), and (4.25) is obtained in the same way. The lemma is proved. D

As a corollary of lemma 4.5 and lemma 4.1, it is now easy to obtain the following estimates for functions (4.7), (4.8):

ll€1- €2lla,nor s cT1-"( sup llu1(·, t)- u2(·, t)ilca(Jr) + Tsup lo-;{i)- o~(t)l), t<T t<T

In their turn, these inequalities imply

Now we can finally pass to the estimates of the differences Fi[u1]-Fi[u2].

In virtue of lemma 4.1, we have


Since

and

+llulzz- U2zzllco,of2(f!or)) :::: cT112 N2+a[1t],

we conclude that the norms [[Fk[ut] - Fk[uz][[a,nor can be estimated by cT N2+a[ul - uz], if k = 1, 2, 4, and by cT1i 2 Nz+a[u1 - u2], if k = 3, 5. Hence, (3.9) holds with 14 = 1/2.


Let us finally estimate

(we took account of the fact that fh- fj2 = u1 - u2 for z = 0). We have

[ ] (Ha)/2 (l )/2 ' '

G[uJ] - G[u2] ) :S cT -a (sup !kJt(O, t) - k21(0, t) I (O,T t<T

+cT[UJz(O, ·)- U2z(O, ·)]~~;!r))/2 :S cT(l+a)/2N2;-a[UJ - u2],

which proves (3.11) with '}'6 = (1 + a)/2. Thus, (3.8)-(3.11) are established and Theorem 2.1 is proved. 0

5. Proof of Theorem 2.2. Let p 1(x, t) and p2 (x, t) be two solutions of problem (1.6)-(1.8) belonging to the Spaces C 2+<>,1+'} (!11T) and ~+a,1+'}(!12r), respectively. Then the function u1 - U2 where ui =Pip~'I/Ji- r0 E 6~+a,J+al2 (!10r) satisfies the equations

(z, t) E !tor,

k 8( UJ - u2) I - G[ ] G[ ] - 0 a - U1- U2, z z=O

We show that the norms IIF[ui]- F[u2]lla,noT and IIG[uJ]- G[u2JII. (11•l C

012 (O,T)

are bounded when Pi E C2+a,l+a/2(!1ir), U1- u2 E C~+a,l+a/2 (!1or). Then, by Theorem 3.1, u1 - u2 E 6!j2"'1+a/2(!10r ). This information gives us the possibility to show that the norm IIG[u1]- G[u2JIIc<>+•l/2(o,T) is also bounded. Then, by Theorem 3.1, u 1 - u2 E C2+<>,1+<>/2 (!10r), and if one of these functions, say, u1 , is constructed by the above procedure and belongs to C2+<>,1+<>/2 (!10r), then another one also should belong to the same space. By Theorem 2.1, they coincide, hence, Theorem 2.2 is proved. 0


We observe that the norm in C~·"I2 (Slor) coincides with {·}a,nor and we estimate li) [u1] - Fdu2] and Fz[ui] - f2[uz] as follows:

I1Fdu1]- F1[u2JIIa,n0r ::= c(llc'!- t2lla,00r{71!t + rot}a,nor

+c{1Lizz -712zz}a,nor(llk2- kolla,n0r + II7/J2zlla,n0r)·

Since 111/ijzlla,nor ::= cT-<> supt<T ICJj (t) I ::= r!r(l-a)/2, these norms are bounded. The boundedness of IIF4 [u!] - F4[uz]lla,no1' and of IIF5[u!]- F5[uz]lla,nor is clear from the above estimates. The difference F3[u1] - F3[u2] contains the second derivatives 7/Jizz which are singular in the case CJj E C 112+<>12(0, T): 17/Jjzz(z, t)l ::= ct-(1+<>)/2 , but this singularity is compensated by the factor q(z, t) = Pzz/(1 + 7/Jzz)- p~, so the norm of the product q7/Jjzz is bounded:

+sup sup 17/Jjzz(z, t) I [qJh"'·;12) < oo. r<T n.,;2,r r z,r

Other terms in F3[u!]- F 3[u2] are estimated in a similar way. Clearly, not only IIF[u1] - F2[u2]lla,nor but also the norms IIF[u1] -

Fz[u2JIIc;·"/'(nor) with arbitrary non-negative (3 are bounded.

Let us turn to the estimate of C[u!]- C1u2]. We have

t

(5.1) lu1(0, t)- 112(0, t)l ::= j lu!t(O, r)- uz(O, r)ldrl ::= ctl+afz, 0

so, repeating the above simple calculation we obtain:

IIC[ul]- C[uz]llc-(l+o!!'(oT) ::= suprl-a/2 sup IC[1t!](O, t)- C[u2](0, t)l+ -' aJ2 1 t<T t<T


At the second step, we have, instead of (5.1),

t

lu1(0,t)- u2(0,t)l S j iuJt(O,T)- u2(0,T)idT S ctl+"llult- U2tli6",oi'(OT)' a/2 1

0


IIG[u!]- G[u2]llc<t+aJ/'(O,T) S cllu1- u211c~;;·Haf'(noT) < oo,

which completes the proof. D

REFERENCES

[1] A. Fasano, A one-dimensional flow problem in a porous medium with hydrophile grains, Math. Meth. App. Sci., 22 (1999), 605-617.

[2] R.Gianni and R.Mannucci, A free boundary problem in absorbing porous material with saturation dependent permeability, Quad. Dip. Mat. Univ. Padova (1999).

[3] A.Fasano, Porous media with hydrophile granules, in "Complex Flows in Industrial Processes", MSSET, Birkhauser, Ch. 10 (1999), 308-332.

[4] A.Fasano and V .A.Solonnikov, On one-dimensional parabolic problem arising in the study of some free boundary problems, to appear in Zap. Nauchn. Semin. P.O.M.I.

[5] A.Fasano and M.Primicerio, Su un problema unidimensionale di diffusione in un mezzo a contorno mobile con condizioni ai limiti non lineari, Ann. Mat. Pura Appl., 93, 4 (1972), 333-357.


VOLUME 8

2001, NO 1-2 PP. 225--238

MULTIDIMENSIONAL DIFFUSION PROCESSES WITH NONLOCAL CONDITIONS

E.I. GALAKHOV •

Abstract. We consider elliptic integra-differential operators of the second order with nonlocal perturbations of mixed boundary conditions and prove existence of Feller semigroups generated by such operators.

Introduction. The analytic structure of one- and multidimensional diffusion processes in a bounded domain was studied in a number of papers (see [1-6] and references therein). It was shown that these processes can be described by means of Feller (i.e. positive and contractive) semigroups generated by elliptic operators of the second order with nonlocal conditions that contain, in general, not only the boundary values of a function and its derivatives, but also their integrals over the domain with respect to Borel measures.

In this paper, an elliptic integra-differential operator with nonlocal perturbations of mixed boundary conditions is considered. We establish sufficient conditions, under which the closure of such operator is the infinitesimal generator of a Feller semigroup. For the proof, we use a modification of the method developed for other types of conditions in [7 --- 12]. In particular, we consider the geometrical structure of the nonlocal terms in order to reduce nonlocal problems to "local" ones and to prove their unique solvability. From this result we derive density of domain for the corresponding operator in an appropriate functional space and apply the Hille-Yosida theorem.

Acknowledgement. This work was completed during a stay of the author at the Trieste University with a financial support through a grant of Consorzio per lo Sviluppo Internazionale dell'Universita di Trieste. The

• Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe shosse 4, 125781 Moscow Russia

225

226 E.l. GALAKHOV

author also thanks N.N. Uraltseva for the formulation of the problem and V.A. Kondratiev and A.L. Skubachevskii for useful discussions.

1. Feller Semigroups and their Generators. I. Let X be a closed linear subspace in C(Q) containing at least one nontrivial nonnegative function. Here n = Q or 8Q, and Q c JR" is a bounded domain with boundary 8Q E C 00

, n 2: 2.

DEFINITION 1.1. A linear bounded operator T : X --+ X is said to be nonnegative if T f 2: 0 for any f E X such that f 2: 0.

DEFINITION 1.2. A strongly continuous semigroup of operators Tt : X --+ X ( t 2: 0) is called a Feller semigroup (or contractive nonnegative semigroup) on X if it satisfies the following conditions:

(a) I!Ttll ::; 1 (t 2: 0), (b) 1t: X--+ X are nonnegative (t 2: 0).

DEFINITION 1.3. A linear operator A : V(A) C X --+ X is called the infinitesimal generator of a strongly continuous semigroup {Tt} if

(1) A I. TtU- u

u= 1m t->0+0 t

(u E V(A)),

V(A) = {u EX: the limit (1.1) exists in X}.

We now formulate a version of the Hille-Yosida theorem, which is the most convenient for us (see Theorem 9.3.1, [6, Chapter 9, Section 9.3]).

THEOREM 1.1. 1. Let {1lh::o be a Feller semigroup on X, and let A : V(A) c X --+ X be its infinitesimal generator. Then we have:

(a) The domain V(A) is everywhere dense in X. {b) For each)..> 0, the operator AI -A has a bounded inverse (AI -A)-1 :

X --+X with norm II (AI- AJ-1 11 ::; 1/ A. {c) For each)..> 0, the operator (AI- AJ-1 :X--+ X is nonnegative. 2. Conversely, if A is a linear operator from X into itself satisfying con

dition (a) and there is a constant Ao 2: 0 such that for all ).. > Ao conditions {b), (c) are satisfied, then A is the infinitesimal generator of some Feller semigroup {Tt}t~o on X, which is uniquely determined by A.

II. It is known [6, Chapter 9, Section 9.4, Theorem 9.4.1] that a generator A of any Feller semigroup has the form of elliptic integra-differential operator of the second order, possibly with degeneration. Therefore it is natural to consider an elliptic differential operator with perturbations having the form of integral operators.

DIFFUSION PROCESSES WITH NONLOCAL CONDITIONS 227

We define the unbounded linear operator A0 : D(Ao) C C(Q) -+ C(Q) by the formula

n n

(2) A0u(x) =I: aiJ(x)ux,x;(x) + I.:ai(x)ux,(x) +a(x)u(x) (x E Q) i=l

with domain D(Ao) = {u E C2 (Q) nC(Q): A0u E C(Q)}, where aij,ai,a are real-valued functions, aij = aji, aij, ai, a E C 0 (Q).

Let the following condition hold.

n -1.1. 2::: aij(x)~i~j > 0, a(x) ::; 0 for all x E Q and 0 # ~ E JRn.

id=l

We also consider an operator A1 satisfying the following condition:

1.2. A1 : C( Q) -+ C( Q) is a linear bounded operator such that if u E

C(Q) takes a positive maximum at a point x0 E Q, then A 1u(x0 )::; 0. One can find examples of such operators in [12]. In this paper we consider the linear unbounded operator A= Ao + A1 :

D(A) c C(Q)-+ C(Q) with domain D(A) = D(Ao) n D(A1).

III. Now we formulate an auxiliary result, which shall be used in the sequel.

LEMMA 1.1. Let conditions 1.1, 1.2 hold.

Then the operator A: D(A) c C(Q)-+ C(Q) has a closure A. Furthermore, if u E D(A) and a function u(x) takes a positive maximum at a point x 0 E Q, then Au(x0 ) ::; 0.

Proof. Clearly, C 2 (Q) c D(A). Therefore the domain D(A) is dense in C(Q). From condition 1.2 it follows that if u E D(A) has a positive maximum at the point x 0 E Q, then Au(x0 ) ::; 0. Thus Theorem 9.3.3, [6] implies that the operator A has a closure A in C(Q). Moreover, by virtue of inequality (8) [6, p. 345], if u E D(A) takes a positive maximum at a point x0 E Q, then Au(x0 ) ::; 0. D

2. Solvability of a Nonlocal Problem. In this section we consider a second order elliptic differential operator with bounded perturbation satisfying condition 1.2. A domain of operator is given by nonlocal perturbation of a mixed boundary condition.

I. We consider the operator A defined in the previous section with non-

228 E.I. GALAKHOV

local conditions

(3)

au(x) Bu(x) = 'Y(x)u(x)- rJ(x) Otn +

n-i 0 ( ) + L {3;(x) ~x + B1u(x) + B2u(x) = 0

i:::::::l ~

(x E aQ).

Here /', '1/, (3; E C2+u(aQ), 'Y(x) ;::: 0, rJ(x) ;::: 0, 'Y(x) + rJ(x) > 0 (x E aQ), and the linear bounded operators B1 : C2+u(Q) -+ Ci+u(aQ), B2 : c2+u(Q)-+ C2+u(aQ) are defined by the formulas

B;u(x) = j[u(x)- u(y)]Jt;(x,dy) (i = 1,2),

Q

where Jt;(x, ·) (i. = 1, 2) are nonnegative Borel measures on Q for each x E aQ. In order to specify assumptions concerning the operators B1 and B2, we need some additional notation.

Denote r 1 = {x E aQ: rJ(x) = 0}, r2 = {x E aQ: 'Y(x) = 0}. By virtue of our assumptions, these sets are closed and do not intersect. Therefore, they have neighbourhoods r; and r~, open in the topology of aQ and such that

Denote also Jt(X, G) = f.ti (x, G)+ Jt2(x, G) for any point X E aQ and for any Borel set G C Q.

Let the following conditions hold.

2.1. 'Y(x) + tt(x,Q) > 0 (x E rl)·

2.2. supp (3; C 8Q\I"1 (i = 1, ... ,n-1).

2.3. For any x E r; and u E C2+u(Q), B1u(x) = 0.

2.4. There exist 8 > 0 and r 1 > 0 such that for any 0 < r < r1 there are linear bounded operators Bir : C2+u(Q) -+ Cl+u(aQ) and B?r : C2+u(Q) -+ C2+u(8Q), B1 = Bir + Bf"' and

IIBirullcH•(&Q) S cl(r)llullcz+u("Q\fi)' !!B?rullc'+•(OQ) S c2(r)llullcz+u("Q\rj},

where r1 = {x E Q: dist(x,r1) < 8},c1(r), c2(r) > 0 and c1(r) -+ 0 as r-+ 0.


2.5. There exists a v > 0 such that

where c3 > 0 is a constant independent of u, and

Qv = {x E Q: dist(x, iJQ) 2: IJ }.

For examples of such nonlocal operators, see [8, 11, 12]. Similarly to [12], define Cs(Q) = {u E C(Q) : Bu = 0}. Consider the

unbounded operators A 0s, As : 'D(Aos) c Cs(Q) -t C(Q) given by the formulas

Aosu = Aou (u E 'D(.Aos) = {u E C2 (Q) n Cs(Q): Aou E C(Q)}),

As = Aos + A1, 'D(As) = 'D(Aos).

First, we formulate some auxiliary results.

LEMMA 2.1. Assume that conditions 1.1, 1.2, and 2.1 hold. Let u E

'D(.A8 ), and let u(x) have a positive maximum at a point x0 E Q. Then there exists such point x1 E Q that u(.T1) = u(x0 ) and Asn(x1) ::; 0.

Proof. If x0 E Q, then Lemma 2.1 follows from Lemma 1.1. We now suppose that x0 E iJQ \ r 1 and n(x0 ) > u(y) for y E Q. Then, by virtue

of Theorem 3.IV, [13], o~(xo) < 0. Hence Bu(x0 ) > 0. This contradiction tn

proves that ifx0 E 8Q\f1 , then there exists x1 E Q such thatn(x0 ) = n(x1).

Finally, consider the case that x0 E f 1 and u(x0 ) > u(y) for y E Q. If J-L(x 0

, Q) > 0, this assumption and the inclusion 'D(A8 ) c C8 (Q) imply that nonlocal condition (3) for the function u(x) is violated at a point .T0 E iJQ. Therefore we can assume that J-L(x 0

, Q) = 0. Then nonlocal condition can be rewritten in the form

1(x0 )u(x0) + J u(x0 )J-L(x0

, dy) =- J [u(x0)- u(y)]J-L(X0

, dy),

N1, BQ\N"

where N11 = {x E iJQ: J-L(x, Q) = 0}. On the other hand, this equality is impossible, since, by condition 2.1,

its left-hand side is positive and its right-hand side is not positive. This contradiction proves that if x0 E iJQ, then there exists x1 E Q such that

230 E.I. GALAKHOV

u(x0 ) = u(x1 ). To complete the proof, it remains again to apply Lemma 1.1. 0

From this statement immediately follows

LEMMA 2.2. Assume that conditions 1.1, 1.2, and 2.1 hold. Let u E V(AB) be a solution of equation (>.I- AB)u = f, where A> 0, f E C(Q).

(4)

Then

1 llullc(Q) :s; >;llfllc(Q)·

Proof. Assume that m~ju(x)l = u(x0 ) > 0. Then, by virtue of Lemma xEQ

2.1, there is a point x1 E Q such that u(x1) = u(x0 ) and A 8 u(x1) :s; 0. Therefore

II. Now we prove solvability of nonlocal problem for a second order elliptic differential operator with B2 = 0. For this purpose, we modify the proof of solvability of corresponding "local" problem from [14].

LEMMA 2.3. Let conditions 1.1, 2.1-2.5 hold, and let B 2 = 0. Then, for any A > 0, f E C"'(Q), and <p E C 2+"'(8Q), there exists a unique solution u E C 2 (Q) n C(Q) of the problem

(5) Au(x)- Aou(x) = f(x) (x E Q), } Bu(x) = cp(x) (x E aQ).

Moreover, for each compact set G C Q \ r 1 , we have

(6)

where the constant c > 0 does not depend on <p.

Proof. 1. First of all, introduce a function 7Jo E C2+"'(8Q) such that 7Jo(x) > 0 (x E 8Q), 7Jo(x) = 7J(x) (x E r~), and consider an auxiliary problem

(7)

Auo(x)- Aouo(x) = f(x) auo(x)

'Y(x)uo(x)- 7Jo(x) 0

+ B 1uo(x)+ tn +I: ,B;(x) auo(x) = cp(x)

i=l at;

(x E Q),

(x E 8Q).


By Lemma 5.2, [12], this problem has a unique solution uo E C2+"(Q). Denote v = u - u0 . Thus, from (5) and (7) it follows that the function v solves the problem

(8) .\v(x)- Aov(x) = 0 (x E Q), } Bv(x) = rp0 (x) (x E DQ),

where

rp0 (x) = rp(x)- 'Y(x)uo(x) + 7)o(x) 8~(x)

tn ~ 8uo(x)

- {;;t(3;(x) 8

t; - B 1u0 (x) (x E 8Q).

Moreover, by (7) we have

(9) rpo(x) = rp(x) ·- Buo(x) = 0 (x E r;).

2. We shall search the function v as a limit of solutions v, of auxiliary problems

(10) .\v,(x)- A0v,(x) = 0 (x E Q), } B,v,(x) = rp0 (x) (x E 8Q),

8v, -where 0 < c < 1, B,v,(x) = Bv,(x)- c-;:,-. The solutions v, E C2+"(Q) of

utn problems (10) also exist and are unique by Lemma 5.2, [12].

Now let us prove that the set {v,} is bounded in C(Q). Let x' be such point that Jv,(x')l = ma,_JC Jv,(x)J. From the maximum principle it follows

xEQ

that x' E 8Q. If v,(x') > 0, then we have

( ') 8v,(x') 8v,(x') (. ) B1v,x 2:0, 8

<0, 8

=0 2=1, ... ,n-1. tn t;

Therefore equality (5) implies that x' E 8Q \ r;. Using (10) and the definitions of nonlocal operators B and Be, we obtain

ve(xe) < I'Pot71 :S klii'PollccoQ) 'Y x'

where the constant k1 > 0 does not depend on rp0 and c, because the function 'Y has a positive infimum on 8Q \ r;. Similarly, in case if ve(xe) < 0, we get Ve(xe) 2: -klll'PollccoQ)> and conclude that

llvellccCJJ :S klll'Pollcc&Q)·

232 E.l. GALAKHOV

By Schauder estimates (see Theorems 35.II and 35.VI, (13]), for each compact set G c Q \ r 1 we have

(11)

and therefore the set {v,} is compact in C2(G). We choose a sequence of sets {Gm} such that Gm n r 1 = 0, lim Gm = Q \ r 1. By means of a

m->oo diagonalization process, we can find a subsequence of { v,} which converges in C2(Gm) for each m, and it is evident that its limit function v satisfies equation ,\v(x)- A0v(x) = 0 for x E Q, and the quantity

converges to B0v(x) for all x E 8Q \ r 1 . Finally, by condition 2.4 we have

IIB1v,- B1v1ic~+•(aQ) :<:; c311v,- vllc'+•(l"J\r1l -+ 0 as e-+ 0.

Therefore, nonlocal conditions (3) also hold for the function v on 8Q \ r 1 .

By assumptions 2.2 and 2.3, there exists a neighborhood r~ of r 1 such that in r~ these nonlocal conditions take the form

av(x) Bv(x) = f'(x)v(x)- 7J(x) Otn = 0 (x E r~).

Thus, proceeding as in (14], we can prove that v satisfies a homogeneous Dirichlet condition on r 1. Hence v (resp. u) is a classical solution of (8) (resp. of (5)).

3. It remains to prove estimate (6). For this purpose, let us pass to the limit as e -> 0 in (11). By definition of 'P1> we have

From this inequality and Theorems 35.Il, 35. VI (13] it follows that

llullc2+•(G) :<:; lluollc>+•(G) + llvllc'+•(G) :<:; (1 + k3)lluollc'+•(Q)::; 0 :<:; k4(II'PIIc~+•(Ql + 11/lic.(Q)).

III. Now let us prove solvability of problem (5) with an arbitrary B satisfying conditions 2.1-2.5.


LEMMA 2.4. Let conditions 1.1 and 2.1 - 2.5 be fulfilled. Then, for any.\ > 0, j E cu(Q), and rp E Cl+u(oQ), there exists a unique solution u E C2 (Q) n C(Q) of the problem

(12) .\u(x) - A0u(x) = f(x) (x E Q), } Bu(x) = rp(x) (x E 8Q).

Proof. We consider an auxiliary boundary value problem

(13) .\v(x)- A0v(x) = j(x) (x E Q), } v(x) = 0 (x E 8Q),

where .\ > 0. By virtue of Theorem 36.I, [13] for any f E cu ( Q) there is a unique

solution v E C2+u(Q) of this problem. Denote G>.f(x) = v(x) and w = u-v. Then from (12) and (13) we obtain

(14) .\w(x)- Aow(x) = 0 (x E Q), } Bw(x) = rp(x)- BG>.f(x) (x E 8Q),

where BG>.J E Cl+u(8Q) by our smoothness assumptions concerning /,'f/, and Bi (in particular, BiG>.] E Cl+u(8Q) thanks to Schauder estimates and conditions 2.4, 2.5).

Denote Bg(x) = Bg(x)- B2g(x). Consider a problem

A_g(x)- A0 g(x) = 0 (x E Q), } Bg(x) = 1/J(x) (x E 8Q).

From Lemma 2.3 it follows that for any 1/J E Cl+u(8Q) this problem has a unique solution vi E C2 (Q) n C(Q). Denote T>.'I/J(x) = g(x). Clearly, problem (14) is equivalent to the operator equation in Cl+u(8Q)

(15)

From condition 2.5 and estimate ( 6) we conclude that

IIB2T>.1/JIIc'+c(8Q) ::0: c211T>.1/JIIc'+"(Qvl ::0: 111/JIIc>+c(aQ)·

Therefore the operator B2T>.: Cl+u(8Q)-+ Cl+u(8Q) is compact. By Theorem 16.2, [15], the operator I+ B2T>. : Cl+u(8Q) -+ Cl+u(8Q) is Fredholm, and ind(J + B2T>.) = 0 for all .\ > 0. From Lemma 2.2 we conclude that problem (12) has a unique trivial solution if J = 0 and rp = 0. Hence operator

234 E.I. GALAKHOV

equation (15) has a unique trivial solution if cp-BG>.f = 0. Therefore the operator I +B2T>. has a bounded inverse (I +B2T>.)-t : C1+"(8Q) -t C1+"(8Q), and there exists a function

which satisfies (12). Its uniqueness follows from Lemma 2.2. 0

LEMMA 2.5. Let conditions 1.1, 1.2, and 2.1 - 2.5 hold. Then there exists >.o > 0 such that for every >. > >.o the operator >.I-An :

:D(An) C C(Q) -t C(Q) has a bounded inverse (>.I -An)-t : C(Q) -t C(Q) and

(16) - t 1 II(AI- An)- II ::0 -x·

Moreover, the operator (>.I- An)-1 : C(Q) -t C(Q) is nonnegative.

Proof. 1. By virtue of Lemma 2.1, there exists a closure Aon : :D(Aon) C

C(Q) -t C(Q). Lemmas 2.2 and 2.4 (with cp = 0) imply that for>.> 0 there is a bounded inverse operator (>.I- Aon)-t: C(Q) -t C(Q) with a norm

(17) - t 1 II(>.I- Aon)- II :::; -x·

Clearly,

- - --1 ->.I- An= AI- Aon- At = (I- At (AI- Aon) )(>.I- Aon).

Therefore inequality (17) implies that for >. > >.0 = 2IIA1 II the operator >.I- An has a bounded inverse (>.I- A8 )-t: C(Q) -t C(Q).

Estimate (16) follows from Lemma 2.2.

2. We now prove that for each >. > >.0 the operator (>.I- A 8 )-1 is nonnegative. Assume to the contrary that for some f ~ 0 a solution of the equation (>.I- AB)u = f takes negative values, i.e. mi!!u(x) = u(x0 ) < 0.

xEQ

Denote v(x) = -u(x). By virtue of Lemma 2.1, there exists a point xt E Q such that v(xt) = v(x0 ) and A 8 v(xt) :::; 0. Thus from the equality (>.I -AB)v = -f we obtain 0 < v(x0 ) = v(xt) = (A8 v(xt)- f(xt))j>.:::; 0. This contradiction proves that the operator (>.I- A 8 )-1 is nonnegative. 0


3. Existence of a Feller Semigroup. Lemmas 2.2 and 2.5 show that conditions b) and c) of the Hille-Yosida theorem hold under assumptions 1.1, 1.2, and 2.1- 2.5. It remains to verify condition a). For this purpose, denote by As the restriction of operator As to the set D(As) = { u E D(As) : Asu E Cs(Q)} C Cs(Q). Our next aim is to prove density of domain of the operator As.

LEMMA 3.1. Let conditions 1.1, 1.2, and 2.1 - 2.5 hold. Then the set D(As) is dense in Cs(Q).

Proof. 1. First of all, fix .\ > 0. For every 1/J E C(8Q), there exists a unique solution wE C2 (Q) n C(Q) of the problem

(18) .\w(x)- A0w(x) = 0 (x E Q), } w(x) = 1/J(x) (x E 8Q).

Denote P>.1/J(x) = w(x). From Theorem 5.1, [4], and from Lemma 2.5 of this paper it follows

that there exists an operator -BP>. : D(BP>.) c C(aQ) ~ C(aQ), which is the generator of a contractive nonnegative semigroup in C(8Q). Therefore Lemma 5.1, [4] implies that the operator BP>. has a bounded inverse (BP>.)- 1 : C(aQ) ~ C(aQ). Denote q>, = II(BP>.)-1 11·

Define a linear bounded operator L: C(Q) ~ C(aQ) by the formula

Lu(x) = !'(x)u(x) + B2u(x).

By continuity, for each u E Cs(Q), .\ > 0, and E: > 0 there exists a li > 0 such that rf := {x E aQ: dist(x, f1) :<:: li} C f~, and there holds the inequality

(19) ILu(x)l ::::c. min(1, -21

) (x E r~). q>,

Now introduce a linear operator B : C(Q) ~ C(aQ) with domain D(B) = C2+a(Q) by the formula

A au(x) n-1 au(x)

Bu(x) = 'Y(x)u(x)- r12(x)-,- + L.::i3i(x)-,-.- + B1u(x) + B2u(x) = utn i=1 ut,

ou(x) n-1 au(x)

= -ry1(x)-,- + L.::!3i(x)-,-. + B1u(x) + Lu(x) (x E 8Q), utn i=1 ut,

236 E.l. GALAKHOV

where 'f/1 E C2+"(8Q), 7J1{x) > 0 (x E 8Q), 7]1{x) = 7J(x) (x E 8Q \ r~/2). Since rf c r~, from conditions 2.2 and 2.3 it follows that

{20) , 8u(x) 6

Bu(x) = -'f}l(x) &tn +Lu(x) (x E r 1).

Lemma 5.9, [12] implies that for any u E C(Q), .A > 0, and e > 0 there exists a function u1 E C2+"(Q) such that Bu1(x) = 0 (x E 8Q) and

(21)

Now let us consider a function Bu1. By our smoothness assumptions on B, we have Bu1 E Cl+"(8Q). Furthermore, due to our choice of the function 'f/1> from {20) we get

(22)

{23)

Since 7]1(x) 2 7J(x) 2 0 and 7]1(x) > 0 (x E 8Q), making use of {19) and {21), we obtain

{24)

/Bu1(x)/::; /Lu1(x)/ :'0 /Lu(x)! + /Lu1(x)- Lu(x)! :'0 ::; !Lu(x)! + IlLII . l!u- uJ/ic(Q) ::; ~ (x E rf).

q>,

Taking into account the first equality in (22), we finally get

2. Define !J(x) = .Au1(x)- A0u1(x), <po(x) = Bu1(x). Condition 1.2 implies that f 1 E C"{Q). By the smoothness assumptions on B, 'Po E C1+"(8Q).

We can consider u1 as a solution of the problem

(25) .Au1(x)- Aou1(x) = !I(x) (x E Q), } Bu1(x) = 'PI(x) (x E 8Q).


Consider also a nonlocal problem

(26) ,\uz(x)- Aouz(x) = h(x) (x E Q), } Bu2(x) = 0 (x E 8Q).

By Lemma 2.4, this problem has a unique solution u2 E :D(A08 ). Denote w1 = u2 - u1 . From (25) and (26) it follows that w1 is a solution of the problem

(27) ,\w1(x)- A0w1(x) = 0 (x E Q), } Bw1(x) = 'Pl(x) (x E oQ).

Obviously, problem (27) can be reduced to an operator equation

('PIE C(8Q)),

where w1 = P>.1./J1• By Lemma 5.1 [4], there exists a unique solution of this equation

and by (24) we get

From the maximum principle it follows that

(28)

3. Now consider the operator equation

(29)

Lemma 2.5 implies that this equation has a unique solution us E :D(As). Denote wz =us -u2 . From (29) it follows that w2 is a solution of the equation

AWz - Aswz = Asuz.

By virtue of Lemma 2.2, for ,\ > II € II , we have Asuz C(Q)

(30) II II _ < IIAsuzllc(Q)

Wz C(Q) _ ,\ < €.

238 E.I. GALAKHOV

From inequalities (24), (28), and (30) it follows that for each u E CB(Q) there exists a function u3 E V(AB) such that

From Theorem 1.1 and Lemmas 2.5, 3.1, we obtain

THEOREM 3.1. Assume that conditions 1.1, 1.2, and 2.1 - 2.5 are fulfilled.

Then the operator AB : V(AB) C CB(Q) -t CB(Q) is the infinitesimal generator of a Feller semigroup, which is uniquely determined by AB.

REFERENCES

[1 J W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519.

[2] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-30.

[3] A.D. Ventsel, On boundary conditions for multidimensional diffusion processes, Teoriya veroyatnostei i ee primen., 4 (1959), 172-185 (in Russian).

[4] K. Sato, T. Ueno, Multi-dimensional diffusion and the Markov process on the boundary, J. Math. Kyoto Univ., 4 (1965), 529-605.

[5] J .M. Bony, P. Courrege, and P. Priouret, Semigroupes de Feller sur une variete a bord compacte et problemes aux limites integro-differentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble), 18 (1968), 369-521.

[6] K. Talra, Diffusion Processes and Partial Differential Equations, New-YorkLondon, Academic Press, 1988.

[7] A.L. Skubachevskii, On some problems for multidimensional diffusion processes, Soviet Acad. Sci. Dokl. Math., 307 (1989), 287-292 (in Russian).

[8] A.L. Skubachevskii, Nonlocal elliptic problems and multidimensional diffusion processes,Russian Journal of Mathematical Physics, 3 (1995), 327-360.

[9] A.L. Skubachevskii, On Feller semigroups for multidimensional diffusion processes, Soviet Acad. Sci. Dokl. Math., 341 (1995), 173-176 (in Russian).

[10] E.I. Galakhov, On sufficient conditions of existence of Feller semigroups, Mat. Zametki, 60 (1996), 442-444 (in Russian).

[11 J E.I. Galakhov and A.L. Skubachevskii, On non-negative contractive semigroups with nonlocal conditions, Mat. Sb., 189 (1998), 45-78 (in Russian).

[12] E. I. Galakhov and A.L. Skubachevskii, On Feller semigroups generated by operators with integra-differential boundary conditions, to appear in Journ. of Diff. Eq ..

[13] C. Miranda, Equazioni aile Derivate Parziali di Tipo Ellittico, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1955.

[14] T.M. Kerimov , On a mixed boundary value problem for a linear elliptic equation of the second order, Diff. uravn., 13 (1977), 477-480 (in Russian).

[15] S.G. Krein , Linear Equations in Banach Spaces, Birkhiiuser, Boston, 1982.

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