Girardi M., Mataloni S., Matzeu M. - Existence of classical solutions for fully non-linear elliptic equations via mountain-pass techniques(2005)(13).pdf

7/25/2019 Girardi M., Mataloni S., Matzeu M. - Existence of classical solutions for fully non-linear elliptic equations via mount

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Existence of classical solutions

for fully non-linear elliptic equations

via mountain-pass techniques

Mario Girardi

Dipartimento di Matematica

Universita degli Studi Roma Tre

Largo San L.Murialdo, 00146 Roma, Italia

Silvia MataloniDipartimento di Matematica

Universita degli Studi Roma Tre

Largo San L.Murialdo, 00146 Roma, Italia

Michele Matzeu

Dipartimento di Matematica

Universita di Roma Tor Vergata

Via della Ricerca Scientifica, 00133 Roma, Italia

March 1, 2005

Supported by MURST, Project Variational Methods and Nonlinear Diffe-rential Equations

Abstract

Semilinear equations with dependence on the gradient and on the hessian

are considered. The existence of a positive and a negative classical solution is

stated through an iterative scheme and a mountain-pass technique. Moreover

we show that these solutions are actually solutions of a quasilinear problem.

Keywords: Fully non-linear equations, mountain-pass theorem, iteration me-thods.

Subject Classification: 35J20, 35J25, 35J60

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1 Introduction

In this paper we are concerned with a class of the so called fully nonlinear dif-ferential equations, that is partial differential equations of the following kind

(F N L) P(x, u(x), u(x), D2u(x)) = 0

whereP := IR IRNSN, N3, SN denotes the set ofN Nmatrices,uis the gradient and D2u the Hessian matrix of the solution u = u(x) of (F N L).

As noted in a very general survey on partial differential equations by Brezisand Browder (see [1]) in the 2-dimensional case, a complete theory of a prioriestimates for fully nonlinear equations (F NL) was derived in 1953 by L.Nirenberg[12]using techniques developed earlier by C.Morrey [11]. In[1] the authors pointout that the general problem (F N L) was still open in the 3-dimensional case.Moreover they give an exaustive list of references about some particular problems

in this framework.A very important contribute to the study of problem (F N L) has been given

by the introduction of the so called viscosity solutions in three celebrated pa-pers by Crandall, Lions and Evans (see [4],[5],[6]). In the following an extensiveliterature was developed about this kind of solutions connected in particular tothe stochastic control theory (see[3],[2]for two very interesting surveys with anextensive list of references).

Actually the concept of viscosity solution is concerned with a weak solutionuwhich is only continuous, so the gradient and the Hessian ofu are to be intendedin a generalized sense which appears much appropriated in dealing with Hamilton-Jacobi equations associated with stochastic control problems.

In particular there are various existence results of viscosity solutions (see [4],[5], [6]for references).

However two other conditions onPare required in order to define the conceptof viscosity solution, that is a sort of monotonicity with respect to the u-variableand to the D2u-variable (in this case the order on SN is given saying that Y XiffY X is semidefinite positive).

The aim of the present paper is to state an existence result of a positive anda negative solution u of (F N L) in a classical sense, that is u is a C2 functionwhich verifies pointwise equation (F N L). Let us note that we dont assume anymonotonicity condition on P. Our problem is written in the following form

(P)

u= f(x, u(x), u(x), D2u(x)) in u= 0 on

where is a bounded open set of IR

N

, N 3, with smooth boundary . Oneassumes some standard superlinear growth conditions on the C1-functionf atzero and at infinity w.r.t. the u-variable (f0)-(f4). Actually the sharp assump-tion onf(f5) is given by requiring the constancy offw.r.t. the D2u-variable in a

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domainB of IR IRNSN whose diameter can be determined in terms of Sobolevconstant, the dimension Nof the space and the superlinear growth coefficients.

More precisely we require that our f in B coincides with a C1-function g thatdoesnt depend on the Hessian variable. In a rough way, we could say that thiscondition corresponds to assume that the diffusion term of the stochastic differ-ential equation associated with the control problem is actually a free diffusionon B, while the controls can give their effects on the diffusion only outside ofB.By our methods, one is able to localize a solution just living in B. Finally werequire that the Lipschitz constancy coefficients offw.r.t. the u, u-variables aresufficiently small in a suitable sense in connection with the first eigenvalue of theLaplacian.

In this framework of assumptions we obtain a classical positive solution u(and a respectively negative one) as a limit of a sequence {un} where un is aMountain-Pass type solution of a semilinear problem obtained by freezing theu and D2u-variables, more precisely we consider an iterative scheme such that,starting from some sufficiently smooth u0 H10 () H

2(), one defines un as asolution of Mountain-Pass type of the problem

(Pn)

un= f(x, un, un1, D

2un1) in un= 0 on .

At first, some a priori estimastes due to the Mountain-Pass nature ofun allow toclaim that

||un||C0 1, ||un||C0 2, ||D2un||C0 3,

where i, i= 1, 2, 3 are determined by Sobolev constant, the dimension Nof thespace and the superlinear growth coefficients off. On the other side, if one choisesi, i= 1, 2, 3 as the diameter ofB in each variable, one finds un as a solution ofthe quasilinear problem

(QLn)

un= g(x, un(x), un1(x)) in un= 0 on

A suitable boot strap argument gives un as a C3-solution of (QLn) such that

||D3un||C0 4.

This fact and the strong convergence of the whole sequence {un}in H10 () (which

is a consequence of the assumption of smallesness of the Lipschitz continuoslycoefficient offw.r.t. the u and u-variables in connection with 1) allow to passto the limit in (QLn) and get a classical solution of

(QL) u= g(x, u(x), u(x)) in u= 0 on

.

Actually, since u itself attains its values in the region B , one gets

g(x, u(x), u(x)) = f(x, u(x), u(x), D2u(x))

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Therefore one realizes that u solves the initial problem (P). One points out thatthis kind of method based on a priori estimates allows in some sense to localize

the solution of the problem in such a way that it lives in a region of the spacewhere one gives the suitable assumptions which enable to solve the problem in asimple case. On that point of view is based the argument in order to improve thehypothesis at infinity given on the function fgiven in [8]for the quasilinear case(see preprint [9]).

As a final remark, we point out that a technique of a similar type was deve-loped in[7],[8] for quasilinear equations and in [10]for integrodifferential equationsrelated to non symmetric kernels.

2 The problem

Let us consider the following fully non-linear elliptic equation:

(P)

u= f(x, u(x), u(x), D2u(x)) in u= 0 on

where is a bounded open set of IRN, N3, with smooth boundary . In thefollowing we will denote by ||the measure of the set . Let SN be the space ofN Nmatrices endowed with the norm

||X||= maxi,j=1,...,N

|xij | where X= (xij) SN.

Let fbe a real function satisfying the following assumptions:

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(f0) f : IR IRNSN is a locally Lipschitz continuous w.r.t.

the first three variables and continuous w.r.t the last one.

(f1) limt0f(x,t,,X)

t = 0 uniformly forx , IRN, X SN

(f2) a1 > 0, p

1, N+2N2

andr, s (0, 1) with r + s 0 and >2 : 0< F(x,t,,X) tf(x,t,,X)

x , |t| t0, IRN, X SN

where F(x,t,,X) =t0

f(x, , , X) d.

(f4) a2, a3 > 0 : F(x,t,,X) a2|t| a3

x , t IR, IRN, X SN.

As a standard example of a function f satisfying (f0)-(f4), one can consider

f(x,t,,X) =

a1t|t|p

1(1 + ||) |t| 1, || 2, ||X|| 3t|t|p1(1 + ||r)(1 + ||X||s) t|> 1, ||> 2, ||X||> 3

where p

1,N

+2N2

, > 1, r, s (0, 1) with r+ s < 1. Moreover j , j , (j =1, 2, 3) are fixed constants withj > j for anyj and a1has to be taken sufficientlysmall w.r.t. p and .

Remark 2.1 From (f2) and (f3) it follows that p + 1.

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Theorem 2.1 Suppose that the functionf satisfies(f0) (f4)and let us assumethat

(f5) there exist three positive numbers1, 2, 3 depend explicitely on

p, r, s, , a1, a2, a3, N, || given in the previous hyptotheses, such that

f(x,t,,X) = g(x,t,) (x,t,,X) B where

B = {(x,t,,X) : x , |t| [0, 1], || [0, 2], ||X|| [0, 3]}

andg is aC1 function on IR IRN such that (f1)-(f4) hold

withf replaced byg.

Then there exists a positive and a negative solutionu of problem(P)provided

the following relation holds:

1

2

1 L2+ 11 L1 0, the followingclosed convex set

CR:=

w H10 () H2() : w, D2w are

holder continuous on and ||w||C0 R, ||D2w||C0 R

and let us consider the class of the following problems

(Pw)

u= f(x, u(x), w(x), D2w(x)) in u= 0 on

where w CR. The following proposition holds:

Proposition 2.2 Suppose that the functionf satisfies(f0) (f4).

Then, for anyw CR, there exists a positive and a negative solutionuw of problem(Pw). Moreover there exist two positive constantsc1, c2 such that

c1 ||uw|| c2, w CR

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Proof

Let us recall that the energy functional Iw : H10 () IR, associated with problem

(Pw), is defined by

Iw(v) : 1

2

|v|2

F(x,v, w, D2w) dx.

We want to prove, by steps, that Iw has the geometry of the mountain-pass the-orem, that it satisfies the Palais-Smale condition and finally that the obtainedsolutions verify the uniform bounds stated in the theorem.

Step 1. Let w CR. Then there exists some , >0 depending on R such that

Iw(v) v: ||v||= .

Proof

By (f1), given any >0 there exists >0 such that

F(x,v, w, D2w) 0 such that

F(x,v, w, D2w)< K|v|p+1(1 + R)r+s x , |v| , w CR

hence,

F(x,v, w, D2w) dx 2

v2(x) dx + K(1 + R)r+s |v(x)|

p+1 dx

K 2 + K(1 + R)r+s||v||p1 ||v||2(2.2)

with a constant K >0 depending on Poincare and Sobolev inequalities. Choosing

||v||< ( 2K(1+R)r+s ) 1p1 , one gets

F(x,v, w, D2w) dx K||v||2

If one chooses < 12K and = (12 K

)2 the thesis easily follows.

Step 2. Let w CR. Fix v0 H10 () with ||v0|| = 1. Then there is a T > 0,

independent ofw and R, such that

Iw(tv0) 0 t T , (2.3)

hence there exists a TR, depending on R, such that (2.3) holds and ||v|| :=

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||TRv0||> with as in Step 1.ProofIt follows from (f4) that

Iw(tv0) 12 t

2 a2|t|

|v0| dx + a3||. (2.4)

By Sobolev embedding theorem ( p + 1 -see Remark 2.1) we obtain

Iw(tv0)1

2t2 a2|t|

(S) + a3||.

where S is the constant of the embedding ofH10 () into L

(). Since >2, weobtain T such that (2.3) holds. It is sufficient now, to choose TR > -where isfixed in Step 1- to obtain the thesis. Obviously, Tdepends on R.

Step 3. For any w CR, there exists some uwH10 () such that

a) Iw(uw) = 0

b) Iw(uw) = infmaxt[0,1] Iw((t)) where

=

C0([0, 1], H10 ()) : (0) = 0, (1) =v

where v is choosen as in Step 2.

c) uw >0

(2.5)

Proof

The existence of an element uw such that (2.5) holds is an immediate conse-quence of the mountain-pass theorem of Ambrosetti Rabinowitz (see [AR]). Indeed

Iw(0) = 0, Step 1 and 2 hold and hypotheses (f2) and (f3) imply, in a standardway, that Iw satisfies PS-condition.As for the positivity of uw, it derives from standard arguments. More preciselyone replaces f byfdefined as

f(x,t,,X) =

0 ift


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||uw|| c1 (2.6)

for all solution uw obtained in Step 3.Proof

By putting in the equation

uwv=

f(x, uw, w, D2w)v(x) dx

v= uw, one gets

|uw|2 =

f(x, uw, w, D2w)uw(x) dx. (2.7)

It follows from (f1) and (f2) that, given >0, there exists a positive constant c,independent ofw, such that

|f(x,t, w, D2w)| |t| + c|t|p.

Using this inequality, we estimate (2.7) and obtain

|uw|2

|uw|2 + c

|uw|p+1.

Again by Poincare inequality and the Sobolev embedding theorem, we obtain1

1

||uw||

2 c||uw||p+1

which implies(2.6) choosing < 1, since p + 1> 2.

Step 5. There exists a positive constant c2, independent ofw and R, such that

||uw|| c2. (2.8)

Proof

From the infmax characterization ofuw in Step 3, choosing the path in as thesegment line joining 0 and v , we obtain

Iw(uw) maxt0

Iw(tv).

We estimate Iw(tv) using (f4):

maxt0

Iw(tv) maxt0

t2

2

|v|2 a2|t|

|v| + a3||

.

Let us now remark that the function h(a2, a3, ||, v , )(t) defined as follows

t IR+ t2

2

|v|2 a2|t|

|v| + a3||,

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attains a maximum positive value independent ofR since, as one checks,

maxt h(a2, a3, ||, TRv0, )(t) = h(a2, a3, ||, v0, ).

Therefore we have obtained that

Iw(uw) const (2.9)

At this point, using the criticality ofuw for Iw, (2.9)and (f3), one has

1

2||uw||

2 const +1

f(x, uw, w, D2w)uw = const +

1

||uw||

2

so (2.8) follows as >2.

Remark 2.2(On the regularity of the solution of (Pw))

In Step 3 of the previous proposition we have obtained a weak solutionuw of (Pw)for each given w CR H

10 () H

2(). Since p < N+2N2 , a standard bootstrap

argument, using Lpregularity theory, shows that uw is, in fact, in C2,(), for

some (0, 1). As a consequence of the Sobolev embedding theorems and Step 5of Proposition2.2 we conclude that there exist three positive constants1, 2, 3,independent ofw, such that the solution uw satisfies

||uw||C0 K1 := 1(1 + R)r+s

||uw||C0 K2 := 2(1 + R)r+s

||D2uw||C0 K3 := 3(1 + R)r+s

(2.10)

Proposition 2.3 Suppose that the functionf satisfyies(f0) (f4).Then there exists a positive constantR such that

w CR uw CR (2.11)

whereuw is a solution of(Pw).

Proof

It easily follows by the estimates (2.10), and by the fact that r + s


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obtained by the mountain-pass theorem in Proposition 2.2. We start from anarbitraryu0 CR.

Now, using (Pn) and (Pn+1), we obtain

un+1(x)(un+1(x) un(x)) dx=

f(x, un+1(x), un, D2un)(un+1(x) un(x)) dx

un(x)(un+1(x) un(x)) dx=

f(x, un(x), un1, D2un1)(un+1(x) un(x)) dx

which gives

||un+1 un||2

|f(x, un+1(x), un, D2un) f(x, un(x), un1, D

2un1)|

|un+1(x) un(x)| dx

|f(x, un+1(x), un, D2un) f(x, un(x), un, D

2un1)||un+1(x) un(x)| dx

+

|f(x, un(x), un, D2un1) f(x, un(x), un1, D

2un1)||un+1(x) un(x)| dx

Let us observe now that, by the estimates (2.10),

(x, un+1(x), un, D2un), (x, un(x), un, D

2un1) (x, un(x), un1, D2un1) B

choosingi = R i= 1, 2, 3. By hyphotesis (f5) it results that

g(x, un+1(x), un) = f(x, un+1(x), un, D2un)

g(x, un(x), un) = f(x, un(x), un, D2un1)

g(x, un(x), un1) = f(x, un(x), un1, D2un1)

and the problem (Pn) is equivalent to the following quasi-linear problem

(QLn)

un= g(x, un(x), un1(x)) in un= 0 on

.

Therefore we can estimates the integral above using (f6) and (f7) in the followingway:

||un+1 un||2

|g(x, un+1(x), un) g(x, un(x), un)||un+1(x) un(x)| dx

+ |g(x, un(x), un) g(x, un(x), un1)||un+1(x) un(x)| dx

L1 ||un+1(x) un(x)||2L2()+ L2 ||un un1||L2()||un+1 un||L2().

(2.12)

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Next using Poincare inequality, we estimate further (2.12):

||un+1 un||2 L111 ||un+1 un||2 + L212

1 ||un un1| | | |un+1 un||

from which follows

||un+1 un|| L2

12

1

1 L111

||un un1||.

Since the coefficient L2

12

1

1L11

1

is less then 1, then it follows that the sequence{un}

strongly converges in H10() to some function u H10 (), as it easily follows from

the fact that {un} is a Cauchy sequence in H10 (). Since ||un|| c1 for all n, it

follows that u0. More precisely u0, as it follows from the positivity ofunandactually, u >0 by the maximum principle as in the proof of Step 3. Actually, by

Schauder theorem since g is a C1-function and un is C2,(), for some (0, 1)it follows that, actualy, un is C

3,(), for some (0, 1). Moreover there existsa positive constant 4 such that

||D3uw||C0 4(1 + R)r+s. (2.13)

By Ascoli-Arzela Theorem which can be applied by (2.10) and (2.13), we find asubsequence{unj} of{un}uniforme converging to u such that

unj u D2unj D

2u. (2.14)

Passing to the limit in (QLn) we find that u = u is a positive classical solution of

(QL)

u= g(x,u(x), u(x)) in u= 0 on .

By the estimates (2.10) and the convergence (2.14) it results that u B that isu= u is a classical positive solution of the initial problem (P).

Remark 2.3 We underline that we have find that the solution u of the problem(P) is, actually, a solution of a quasilinear problem (QL).

References

[1] H.Brezis, F.Browder, Partial Differential Equations in the 20th Century,Adv. in Math. 135, (1998), 76-144.

[2] X.Cabre, L.Caffarelli, Fully Nonlinear Differential Equations, Amer.Math. Society, Providence (1995).

[3] M.G.Crandall, Viscosity solutions: a primer, Lecture Notes in Mathematics1660.

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[4] M.G.Crandall, P.L.Lions, Condition dunicite pour les solutions gener-alisees des equations de Hamilton-Jacobi du premier ordre, C.R. Acad. Sci.

Paris, 292 (1981), 183-186.

[5] M.G.Crandall, P.L.Lions, Viscosity solutions of Hamilton-Jacobi equa-tions, Trans. Amer. Math. Soc. 277 (1983), 1-42.

[6] M.G.Crandall, P.L.Lions, L.C.Evans, Some properties of viscosity so-lutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984),487-502.

[7] D.De Figuereido, M.Girardi, M.Matzeu, Semilinear elliptic equationswith dependence on the gradient via mountain-pass techniques, Differentialand Integral Equations, 17, 1-2 (2004), 119-126.

[8] M.Girardi, M.Matzeu, Positive and negative solutions of a quasi-linearequation by a Mountain-Pass method and troncature techniques, NonlinearAnalysis,59 (2004), 199-210.

[9] S.Mataloni, Quasilinear elliptic problems with a non-linear term of arbitrarygrowth w.r.t. the gradient. Preprint

[10] S.Mataloni, M.Matzeu, Semilinear integrodifferential problems with non-symmetric kernels via mountain-pass techniques, Advanced Nonlinear Stud-ies, to appear

[11] C.B.Morrey, On the solutions of quasilinear elliptic partial differential equa-tions, Trans. Amer. Math. Soc., 43 (1938), 126-166.

[12] L.Nirenberg, On nonlinear elliptic partial differential equations and Holdercontinuity, Comm. Pure Appl. Math. 6 (1953) 103-156.

[13] P.H.Rabinowitz, Minimax methods in critical point theory with applica-tions to differential equations, CBMS Regional Conf. Series Mathy., 65 Amer.Math. Soc., Providence RI (1986).

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Girardi M., Mataloni S., Matzeu M. - Existence of classical solutions for fully non-linear elliptic equations via mountain-pass techniques(2005)(13).pdf