Perron’s method and the method of relaxed

limits for “unbounded” PDE in Hilbert

spaces

Djivede Kelome Andrzej Swiech∗

Abstract

We prove that Perron’s method and the method of half-relaxed limits of Barles-

Perthame works for the so called B-continuous viscosity solutions of a large class

of fully nonlinear unbounded partial differential equations in Hilbert spaces. Per-

ron’s method extends the existence of B-continuous viscosity solutions to many new

equations that are not of Bellman type. The method of half-relaxed limits allows

limiting operations with viscosity solutions without any a priori estimates. Possi-

ble applications of the method of half-relaxed limits to large deviations, singular

perturbation problems, and convergence of finite dimensional approximations are

discussed.

Key words: viscosity solutions, Hamilton-Jacobi-Bellman equations, Perron’s method,relaxed limits, Hilbert spaces.

2000 Mathematics Subject Classification: 49L25, 35R15, 35J60.

1 Introduction

In this paper we investigate the possibility of extending Perron’s method and the methodof half-relaxed limits of Barles-Perthame to a class of equations in infinite dimensionalHilbert spaces of the form

u+ 〈Ax,Du〉 + F (x,Du,D2u) = 0 (1.1)

and their time dependent versions

{

ut + 〈Ax,Du〉 + F (t, x,Du,D2u) = 0 (t, x) ∈ (0, T ) ×Hu(0, x) = g(x).

(1.2)

∗A. Swiech was supported by NSF grant DMS 0500270.

1

Above H is a real separable Hilbert space with the inner product 〈·, ·〉 and norm ‖ ·‖, and A is a linear, maximal monotone operator in H. The symbols Du,D2u denotethe Frechet derivatives of u. This is a large class of equations that includes Hamilton-Jacobi-Bellman (HJB) equations for optimal control of stochastic semilinear PDE (forinstance stochastically perturbed reaction diffusion equations) and delay equations, Isaacsequations, infinite dimensional Black-Scholes-Barenblatt equation for option pricing, andmany others.

There exists a good theory of such equations based on the notion of the so calledB-continuous viscosity solution [5, 6, 25]. The theory however still lacks several key com-ponents that are among the main tools of viscosity solutions in finite dimensions, namelyPerron’s method and the method of half-relaxed limits. Perron’s method is the main tech-nique for producing viscosity solutions of PDE in finite dimensional spaces (see [3]). It isbased on the principle that the supremum of the family of all viscosity subsolutions of anequation is a viscosity solution and so all we need to do to prove existence of a viscositysolution is to produce one sub- and one super-solution. Despite previous efforts it is stillnot known if a version of Perron’s method can be implemented for B-continuous viscositysolutions of (1.1) and (1.2), even if the equations are of first order. Perron’s method workswith Ishii’s definitions of solutions [17, 18] (see also [23]) however his notion of solution[18] does not seem applicable to stochastic optimal control problems and is not used.Half-relaxed limits of Barles-Perthame (see [3]) are perhaps an even more fundamentaltechnique in the theory of viscosity solutions that is widely used to pass to weak limitswithout any a priori estimates. A huge part of the success of viscosity solutions is basedon the fact that limiting operations are very easy in this framework. It is known that dueto the lack of local compactness in infinite dimensional spaces this procedure may notwork in general, even for simple equations with A = 0 (see [1, 26]).

In this paper we will show that both Perron’s method and the method of half-relaxedlimits can be adapted for equations (1.1) and (1.2) if the operator A satisfies a coercivitycondition (2.2). It has been noticed before [2, 8, 13, 14, 15, 16] that a condition of this typeleads to a stronger definition of viscosity solution and this stronger definition of solutionwill help us overcome technical difficulties needed to implement both methods.

Apart from providing an easy method to produce solutions another consequence ofPerron’s method will be new existence results for a large class of equations of the abovetypes. Currently there exist two methods for proving existence of B-continuous viscos-ity solutions of (1.1) and (1.2); by finite dimensional approximations [25], and by usingstochastic analysis to show that the value function of the associated stochastic optimalcontrol problem solves the PDE [13, 15, 16, 19]. The first is limited to the case of the oper-ator B (see Section 2) being compact and the second is limited to HJB equations relatedto optimal control problems. Perron’s method will allow to construct solutions for generaloperators B and equations that are not of Bellman type, for instance for Isaacs equationsrelated to stochastic differential games. The method of half-relaxed limits should havesignificant impact on the theory of PDE in Hilbert spaces, especially since passing tolimits in infinite dimensional spaces is very difficult even with good a priori estimates.Moreover we anticipate many other interesting applications, for instance in the theory oflarge deviations and risk sensitive control. In these problems one has to deal with small

2

noise limits that correspond to singular perturbation problems for the associated HJBequations. For instance for diffusions driven by stochastic PDE with additive noise theseHJB equations may have the form

(uε)t − εTr(QD2uε) + 〈Ax,Duε〉 + F (t, x,Duε) = 0

for some trace class operator Q = Q∗ ≥ 0, and one is interested in the behavior of theirsolutions uε as ε → 0. Recently Feng and Kurtz [12] proposed a very general frameworkfor large deviations based on viscosity solutions in abstract spaces. However they only useviscosity solutions of the limiting first-order equation and the rest of the method relieson convergence of nonlinear semigroups and stochastic analysis making it rather cumber-some to apply. Similar approach is used in [10, 11] for Hilbert space valued diffusions. In[11] Tataru’s definition of viscosity solution [27, 28, 7] is used for the limiting first orderHamilton-Jacobi equation and the passage to the limit is based on the convergence ofgenerators and the comparison principle for the limiting Hamilton-Jacobi equation. Wethink that the theory of second-order HJB equations is crucial to a good PDE approachto large deviations. The method of half-relaxed limits is a purely analytical technique thatmakes passing to singular limits almost trivial. It seems to be a perfect tool for large devi-ation arguments for Hilbert space valued diffusions, including exit time problems. We willpresent applications to large deviations in a future publication. Another possible applica-tions of the method of half-relaxed limits include convergence results for finite dimensionalapproximations that would give a “Galerkin” type procedure for (1.1) and (1.2) withoutany a priori estimates. Apart from its theoretical value such a method may for instancehelp produce numerical methods for solving infinite dimensional equations, and may helpdevelop procedures for constructing ε-optimal feedback controls. The possibilities seemwide open.

Finally we refer the reader to [9] for an overview of the established theory of PDE inHilbert spaces by methods other than viscosity solutions.

2 Notation and assumptions

We will always identify H with its dual space. With this identification we can interpretthe Frechet derivatives Du(x) and D2u(x) as respectively an element of H and a bounded,self-adjoint operator in H. We will denote the space of bounded, self-adjoint operators inH by S(H).

Throughout the paper B will be a fixed bounded, positive, self-adjoint operator suchthat A∗B is bounded and

〈(A∗B + C0B)x, x〉 ≥ 0, for some C0 > 0 and all x ∈ H. (2.1)

Such an operator always exists, for instance B = (A∗+I)−1(A+I)−1 or B = ((A+I)(A∗+I))−1/2. We refer the reader to [22] for the proof of the latter and to [5] for examples ofother possible B in some particular cases. Operator B defines spaces Hα. For α < 0we define Hα as the completion of H under the norm ‖x‖α = ‖Bα/2x‖, and for α > 0,Hα = R(Bα/2) equipped with the norm ‖x‖α = ‖B−α/2x‖. They are Hilbert spaces with

3

the inner product 〈x, y〉α = 〈B−α/2x,B−α/2x〉, Hα, and H−α are dual to each other, andBα/2 is an isometry between Hβ and Hβ+α.

We will require that A satisfy the coercivity condition

〈A∗x, x〉 ≥ λ‖x‖21, for x ∈ D(A∗) (2.2)

for some λ > 0.The above implies in particular that D(A∗) ⊂ H1. This assumption is satisfied for

instance for self-adjoint invertible operators A if B = A−1.We will always assume that F : [0, T ] × H × H × S(H) → IR is locally uniformly

continuous and is degenerate elliptic, i.e.

F (t, x, p,X) ≥ F (t, x, p, Y ) when X ≤ Y.

Let {e1, e2, . . . , en, . . .} be a basis of H−1 made of elements of H. Given N ≥ 1 letVN = span{e1, e2, . . . , eN}, and let PN denote the orthogonal projection of H−1 onto VN .Denote QN = I−PN where I is the identity in H−1 here. We will sometimes need severaladditional conditions on the Hamiltonian F . Let k ≥ 0.

(1)k There exists a radial function µ(x) = µ(‖x‖) satisfying Definition 3.1(iii) andK > 0 such that

lim‖x‖→∞

µ(x)

‖x‖k= +∞

and such that for every positive real number α, t ∈ [0, T ], x, p ∈ H,X ∈ S(H)

|F (t, x, p+ αDµ(x), X + αD2µ(x)) − F (t, x, p,X)| ≤ Kα(1 + µ(x)).

(2) For all t ∈ [0, T ], x, p ∈ H,R > 0

sup{|F (t, x, p,X + λBQN) − F (t, x, p,X)| : ‖X‖, |λ| ≤ R P ∗NXPN = X} → 0

as N → +∞.

(3) There exist moduli ωR such that

F (t, x,B(x− y)

ε,X) − F (t, y,

B(x− y)

ε,−Y ) ≥ −ωR(‖x− y‖−1(1 +

‖(x− y)‖−1

ε))

whenever ‖x‖, ‖y‖ ≤ R and X, Y satisfy the inequality

(

X 00 Y

)

≤2

ε

(

BPN −BPN

−BPN BPN

)

.

We point out that condition (3) can be weakened if A and B satisfy a stronger versionof (2.1), namely if

〈(A∗B + C0B)x, x〉 ≥ ‖x‖2, for some C0 > 0 and all x ∈ H. (2.3)

4

According to [22], (2.3) is always satisfied for B = ((A + I)(A∗ + I))−1/2 if (2.2)holds and [D(A), H] 1

2

= [D(A∗), H] 1

2

= H1, where [·, ·] 1

2

= is the interpolation space (see

[20]). Also it is easy to see that (2.3) holds if (2.2) holds and for instance if ‖(B1

2A∗B1

2 −A∗B)x‖ ≤ C1‖x‖−1. To see this it is enough to notice that

〈A∗Bx, x〉 ≥ 〈B1

2A∗B1

2x, x〉 − C2‖x‖−1‖x‖ ≥λ

2‖x‖2 − C2‖x‖

2−1.

In particular it holds if A is self-adjoint and invertible and B = A−1. However we will notstate any results for the stronger case (2.3).

We will say that a function u : [0, T ] × H → IR is B-upper semicontinuous (respec-tively, B-lower semicontinuous) if whenever tn → t and ‖xn − x‖−1 → 0 for a boundedsequence xn then lim supn→∞ u(tn, xn) ≤ u(t, x) (respectively, lim infn→∞ u(tn, xn) ≥u(t, x)). A function is B-continuous if it is both B-upper semicontinuous and B-lowersemicontinuous. A function is locally uniformly B-continuous on [0, T ] × H if it is uni-formly continuous in | · | × ‖ · ‖−1 norm on bounded subsets of [0, T ] ×H.

We will write u∗ and u∗ to denote the upper- and lower-semicontinuous envelopes ofu in the | · | × ‖ · ‖−1 norm, i.e.

u∗(t, x) = lim sup{u(s, y) : s→ t, ‖y − x‖−1 → 0},

u∗(t, x) = lim inf{u(s, y) : s→ t, ‖y − x‖−1 → 0}.

For a Hilbert space V we will be using the following function spaces.

C2(V ) = {u : V → IR : Du,D2u are continuous},

C1,2((0, T ) × V ) = {u : (0, T ) × V → IR : ut, Du,D2u are continuous}.

We will write L(V ) for the space of bounded, linear operators in V equipped withthe operator norm.

3 Viscosity solutions

In order to obtain Perron’s method we will have to deal with discontinuous solutions.Therefore we need two definitions of viscosity solutions. The more usual one that is astronger version of the definition from [25], and a discontinuous viscosity solution that isbased on the notion given for first order equations by Ishii [17].

Definition 3.1. A function ψ is a test function if ψ = ϕ± δ(t)h(‖x‖), where:

(i) ϕ ∈ C1,2 ((0, T ) ×H), and is such that ϕ is B-continuous, ϕt, A∗Dϕ, Dϕ, D2ϕ

are uniformly continuous on closed subsets of (0, T ) ×H.

(ii) δ ∈ C1 ((0, T )) is such that δ > 0 on (0, T ).

(iii) h ∈ C2([0,+∞)) and is such that h′(0) = 0, h′′(0) > 0, h′(r) > 0 for r ∈ (0,+∞).

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For stationary equations ϕ is independent of t and δ(t) ≡ 1.

We remark that even though ‖x‖ is not differentiable at 0, the function h(‖x‖) ∈C2(H) for a test function h as above. Notice also that if ‖x‖ → 0 then h′(‖x‖)/‖x‖ →h′′(0) > 0 so the term h′(‖x‖)/‖x‖ is bounded away from 0 on bounded sets.

Definition 3.2. A locally bounded B-upper semi-continuous function u is a viscositysubsolution of (1.1) if whenever u−ψ has a local maximum at a point x for a test functionψ = ϕ+ h(‖x‖) then

x ∈ H1

and

u(x) + λ‖x‖21

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(x)〉 + F (x,Dψ(x), D2ψ(x)) ≤ 0.

A locally bounded B-lower semi-continuous function u is a viscosity supersolution of (1.1)if whenever u− ψ has a local minimum at a point x for a test function ψ = ϕ − h(‖x‖)then

x ∈ H1

and

u(x) − λ‖x‖21

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(x)〉 + F (x,Dψ(x), D2ψ(x)) ≥ 0.

A viscosity solution of (1.1) is a function which is both a viscosity subsolution and aviscosity supersolution.

Definition 3.3. A locally bounded B-upper semi-continuous function u is a viscositysubsolution of (1.2) if whenever u−ψ has a local maximum at a point (t, x) ∈ (0, T )×Hfor a test function ψ(s, y) = ϕ(s, y) + δ(s)h(‖y‖) then

x ∈ H1

and

ψt(t, x) + λ‖x‖21δ(t)

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(t, x)〉 + F (t, x,Dψ(t, x), D2ψ(t, x)) ≤ 0.

A locally bounded B-lower semi-continuous function u is a viscosity supersolution of (1.2)if whenever u− ψ has a local minimum at a point (t, x) ∈ (0, T ) ×H for a test functionψ(s, y) = ϕ(s, y) − δ(s)h(‖y‖) then

x ∈ H1

and

ψt(t, x) − λ‖x‖21δ(t)

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(t, x)〉 + F (t, x,Dψ(t, x), D2ψ(t, x)) ≥ 0.

A viscosity solution of (1.2) is a function which is both a viscosity subsolution and aviscosity supersolution.

6

Definition 3.4. A locally bounded function u is a discontinuous viscosity subsolution of(1.1) if whenever (u− h(‖ · ‖))∗ − ϕ has a local maximum in the topology of ‖ · ‖−1 at apoint x for test functions ϕ, h(‖y‖) such that

u(y)− h(‖y‖) → −∞ as ‖y‖ → ∞ (3.1)

thenx ∈ H1

and

(u− h(‖ · ‖))∗(x) + h(‖x‖) + λ‖x‖21

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(x)〉

+F (x,Dψ(x), D2ψ(x)) ≤ 0,

where ψ = ϕ+ h(‖y‖).A locally bounded function u is a discontinuous viscosity supersolution of (1.1) if whenever(u + h(‖ · ‖))∗ − ϕ has a local minimum in the topology of ‖ · ‖−1 at a point x for testfunctions ϕ, h(‖y‖) such that

u(y) + h(‖y‖) → +∞ as ‖y‖ → ∞ (3.2)

thenx ∈ H1

and

(u+ h(‖ · ‖))∗(x) − h(‖x‖) − λ‖x‖21

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(x)〉

+F (x,Dψ(x), D2ψ(x)) ≥ 0,

where ψ = ϕ− h(‖y‖).A discontinuous viscosity solution of (1.1) is a function which is both a discontinuousviscosity subsolution and a discontinuous viscosity supersolution.

Definition 3.5. A locally bounded function u is a discontinuous viscosity subsolution of(1.2) if whenever (u− δ(·)h(‖ ·‖))∗−ϕ has a local maximum in the topology of | · |×‖ ·‖−1

at a point (t, x) for test functions ϕ, δ(s)h(‖y‖) such that

u(s, y) − δ(s)h(‖y‖) → −∞ as ‖y‖ → ∞ locally uniformly in s (3.3)

thenx ∈ H1

and

ψt(t, x) + λ‖x‖21δ(t)

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(t, x)〉

+F (t, x,Dψ(t, x), D2ψ(t, x)) ≤ 0,

where ψ(s, y) = ϕ(s, y) + δ(s)h(‖y‖).A locally bounded function u is a discontinuous viscosity supersolution of (1.2) if whenever

7

(u+ δ(·)h(‖ · ‖))∗ −ϕ has a local minimum in the topology of | · | × ‖ · ‖−1 at a point (t, x)for test functions ϕ, δ(s)h(‖y‖) such that

u(s, y) + δ(s)h(‖y‖) → +∞ as ‖y‖ → ∞ locally uniformly in s (3.4)

thenx ∈ H1

and

ψt(t, x) − λ‖x‖21δ(t)

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(t, x)〉

+F (t, x,Dψ(t, x), D2ψ(t, x)) ≥ 0,

where ψ(s, y) = ϕ(s, y) − δ(s)h(‖y‖).A discontinuous viscosity solution of (1.2) is a function which is both a discontinuousviscosity subsolution and a discontinuous viscosity supersolution.

If a subsolution (respectively, supersolution) in Definition 3.4 or 3.5 is B-upper semi-continuous (respectively, B-lower semi-continuous) then it is easy to see that Definitions3.4 and 3.5 reduce to Definitions 3.2 and 3.3 respectively, since if u is B-upper semi-continuous (respectively, B-lower semi-continuous) then (u−h(‖ ·‖))∗(x) = u(x)−h(‖x‖)(respectively, (u+ h(‖ · ‖))∗(x) = u(x) + h(‖x‖)).

Lemma 3.6. Without loss of generality the maxima and minima in Definitions 3.2 and3.3 can be assumed to be global and strict in the | · | × ‖ · ‖ norm and the maxima andminima in Definitions 3.4 and 3.5 can be assumed to be global and strict in the | · |×‖·‖−1

norm. However it is not clear if they can be strict in the | · | × ‖ · ‖ norm. Finally withoutloss of generality we can always assume that functions in Definitions 3.2 and 3.3 satisfy(3.1)-(3.4). Moreover we can also assume that functions in Definitions 3.4 and 3.5 satisfy

(u− δ(·)h(‖ · ‖))∗(t, x) − ϕ(t, x) → −∞, (u+ δ(·)h(‖ · ‖))∗(t, x) − ϕ(t, x) → +∞

as ‖x‖ → ∞ locally uniformly in t.

Proof. Let u be a B-upper semi-continuous function and let

(u− h− ϕ)(x) ≥ (u− h− ϕ)(y) for y ∈ BR(x)

for some R > 0, i.e. u− h− ϕ has a local maximum at x for test functions ϕ and h. Wewill show that there exist test functions ϕ and h such that Dϕ(x) = Dϕ(x), D2ϕ(x) =D2ϕ(x), Dh(x) = Dh(x), D2h(x) = D2h(x), and u− h− ϕ has a strict global maximumat x. Let g ∈ C2([0,∞)) be an increasing function such that

1 + r2 + sup‖y‖≤r

|u(y)| ≤ g(r).

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Let g1 ∈ C2([0,∞)) be a function such that

g1(r) =

0 r ≤ ‖x‖ + 1,increasing ‖x‖ + 1 < r < ‖x‖ + 2,g(r) r ≥ ‖x‖ + 2.

Let ϕ1 ∈ C2([0,∞)) be defined by

ϕ1(r) =

r4 r ≤ 1,increasing 1 < r < 2,2 r ≥ 2.

Now for n ≥ 1 consider the function

Φn(y) = u(y)− ϕ(y) − nϕ1(‖x− y‖−1) − h(‖y‖) − g1(‖y‖).

Obviously we haveΦn(x) = u(x) − ϕ(x) − h(‖x‖).

Suppose that for every n there exists yn such that Φn(yn) ≥ Φn(x). Then we must have‖x−yn‖−1 → o as n→ ∞ and ‖yn‖ ≤ C, i.e. yn ⇀ x. Since u is B-upper semi-continuous,ϕ is B-continuous, and h is strictly increasing, this implies that ‖yn‖ → ‖x‖, and thereforeyn → x in H. But then yn ∈ BR(x) for big n and so we get

Φn(yn) < u(y) − ϕ(y) − h(‖y‖) ≤ u(x) − ϕ(x) − h(‖x‖)

which is a contradiction. Therefore there must exist n0 such that Φn0(y) < Φn0

(x) fory 6= x. It then easily follows that Φn0+1 has a strict global maximum at x. Therefore theconclusion follows with ϕ(y) = ϕ(y)−(n0+1)ϕ1(‖x−y‖−1) and h(‖y‖) = h(‖y‖)−g1(‖y‖).

The fact that the maxima and minima in Definition 3.4 can be assumed to be globaland strict in the ‖ · ‖−1 norm is obvious and the final statements about the convergencesat ∞ follow from the construction of ϕ and h.

Remark 3.7. There are other possibilities for the choice of test functions that wouldgive good theory. For instance one can replace the functions ϕ in Definition 3.1 by thefunctions satisfying

• ϕ ∈ C1,2 ((0, T ) ×H−1), and is such that ϕt, Dϕ, D2ϕ are uniformly continuous

on closed subsets of (0, T ) ×H−1.

In this case one needs to assume that B1

2A∗B1

2 is bounded and that (2.2) is satisfied for

all x ∈ H1. (The term 〈A∗x, x〉 is now well defined for x ∈ H1.) Notice that if B1

2A∗B1

2

is bounded and ψ = ϕ± δ(s)h(‖y‖) then 〈x,A∗Dψ(x)〉 is well defined for x ∈ H1 and soDefinitions 3.2-3.5 can be simplified by replacing the terms

±λ‖x‖21δ(t)

h′(‖x‖)

‖x‖+ 〈x,A∗Dϕ(t, x)〉

wherever they appear in Definitions 3.2-3.5 by a single term 〈x,A∗Dψ(t, x)〉.

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Remark 3.8. It follows from the proof of Lemma 3.6 that if we know a priori that uhas certain growth at ∞ we can then obtain the same growth (at least quadratic) for h.For instance if u has a polynomial growth at ∞ we can have h which is a polynomialof some special form for big ‖x‖. This is important in applications to stochastic optimalcontrol where we may want to impose additional conditions on test functions to be ableto apply stochastic calculus. In these applications it may also be useful to assume thath′(r)/r is globally bounded away from 0 for the radial test functions h. To avoid technicaldifficulties it may then be more convenient to choose h belonging to one particular classof functions, say certain polynomials with growth depending on the growth of sub- andsuper-solutions we are dealing with. All results presented in this paper would hold with anappropriate version of such a definition. This approach has been successfully employed in[2, 13, 14, 15, 16]. However when using such narrow classes of radial test functions theglobal and local definitions of viscosity solutions may no longer be equivalent.

4 Comparison principles

In this section we prove comparison principles for discontinuous viscosity solutions. Webegin with the comparison result for the stationary case.

Theorem 4.1. Let (2.1) and (2.2) hold and let F satisfy (1)0, (2), (3). Let u be a viscositysubsolution and v be a viscosity supersolution of (1.1) in the sense of Definition 3.4, andlet u,−v be bounded from above. Then

limR↑∞

limr↓0

sup{u(x) − v(y) : ‖x− y‖−1 ≤ r, x, y ∈ BR} ≤ 0. (4.1)

In particular u ≤ v.

Remark 4.2. Theorem 4.1 shows that a bounded viscosity solution of (1.1) in the senseof Definition 3.4 is uniformly continuous in H−1 on bounded subsets of H and thereforeis a viscosity solution in the sense of Definition 3.2.

Proof. We argue by contradiction. Assume that (4.1) does not hold. Then there exists apositive real number η such that

limR↑∞

limr↓0

sup{u(x) − v(y) : ‖x− y‖−1 ≤ r, x, y ∈ BR} > 2η.

Let µ(x) be the function satisfying (1)0. For every positive real number α, we defineuα(x) = u(x) − αµ(x) and vα(y) = v(y) − αµ(y).

m := limR↑∞

limr↓0

sup{u(x) − v(y) : ‖x− y‖−1 < r, x, y ∈ BR},

mα := limr↓0

sup{(uα)∗(x) − (vα)∗(y) : , ‖x− y‖−1 < r}

= limr↓0

sup{uα(x) − vα(y) : , ‖x− y‖−1 < r}

10

mα,ε := sup{

(uα)∗(x) − (vα)∗(y) −1

2ε‖x− y‖2

−1

}

,

We have

m = limα↓0

mα, (4.2)

mα = limε↓0

mα,ε. (4.3)

Using perturbed optimization techniques [24] (see also [6]) we obtain sequences pn, qn ∈H such that ‖pn‖ + ‖qn‖ → 0 as n→ ∞, and such that

(uα)∗(x) − (vα)∗(y) −1

2ε‖x− y‖2

−1 + 〈Bpn, x〉 + 〈Bqn, y〉

achieves a strict global maximum at some point (x, y) ∈ H ×H. Convergences (4.2)-(4.3)yield

limε↓0

lim supn→∞

1

2ε‖x− y‖2

−1 = 0 for every α > 0, (4.4)

limα↓0

lim supε↓0

lim supn→∞

(αµ(x) + αµ(y)) = 0. (4.5)

We now have

‖x− y‖2−1 = ‖PN(x− y)‖2

−1 + ‖QN(x− y)‖2−1

and

‖QN (x− y)‖2−1 ≤ 2〈BQN(x− y), x− y〉 + 2‖QN(x− x)‖2

−1

+2‖QN(y − y)‖2−1 − ‖QN(x− y)‖2

−1

with equality at x, y. Therefore defining

u1(x) = (uα)∗(x) −〈BQN(x− y), x〉

ε−

‖QN(x− x)‖2−1

ε+

‖QN(x− y)‖2−1

2ε+ 〈Bpn, x〉

and

v1(y) = (vα)∗(y) −〈BQN (x− y), y〉

ε+

‖QN(y − y)‖2−1

ε− 〈Bqn, y〉.

we have that

u1(x) − v1(y) −1

2ε‖PN(x− y)‖2

−1

has a strict global maximum in H−1 at (x, y). At this step we need to produce appropriatetests functions to be able to use the definition of solution. This is done using partial sup-convolution techniques first introduced in [21] (see also [4]).

11

Lemma 4.3. Given N ≥ 1 there exist functions ϕk, ψk,∈ C2(H−1) with uniformly con-tinuous derivatives such that

u1(x) − ϕk(x)

has a global maximum at some point xk

v1(y) + ψk(y)

has a global minimum at some point yk, and

(

xk, u1(xk), Dϕk(xk), D2ϕk(xk)

)

→

(

x, u1(x),BPN(x− y)

ε,XN

)

, (4.6)(

yk, v1(yk), Dψk(yk), D2ψk(yk)

)

→

(

y, v1(y),BPN(y − x)

ε, YN

)

, (4.7)(

XN 00 YN

)

≤2

ε

(

B −B−B B

)

, (4.8)

with the convergences being in H × R ×H2 × L(H).

Proof of Lemma 4.3. Denote xN = PNx, x⊥N = QNx, yN = PNy, y

⊥N = QNy and define

u1(xN ) = supx⊥

N∈QNH

u1(xN + x⊥N ),

v1(yN) = infy⊥

N∈QNH

v1(yN + y⊥N),

the partial sup- and inf-convolutions of u1 and v1 respectively. Then

(u1)∗(xN ) − (v1)∗(yN) −

1

2ε‖xN − yN‖

2−1 (4.9)

has a strict global maximum over VN ×VN at (xN , yN), where (u1)∗ and (v1)∗ are the upper

and lower semi-continuous envelopes of u1 and v1 in VN . Moreover we have (u1)∗(xN) =

u1(x), (v1)∗(yN) = v1(y).We can now apply the finite dimensional maximum principle when we consider VN as a

space with the topology inherited from H−1 (which is equivalent to the topology inheritedfrom H). Denote VN with this topology by VN . Therefore there exist bounded functionsϕk, ψk,∈ C2(VN) with uniformly continuous derivatives such that (u1)

∗(xN ) − ϕk(xN )has a strict global maximum at some point (xk

N), (v1)∗(yN) + ψk(yN) has a strict global

12

minimum at some point (ykN), and such that as k → ∞

(

xkN , (u1)

∗(xkN ), DVN

ϕk(xkN ), D2

VN

ϕk(xkN)

)

→

(

xN , u1(x),xN − yN

ε, XN

)

, (4.10)(

ykN , (v1)∗(y

kN), DVN

ψk(ykN), D2

VN

ψk(ykN)

)

→

(

yN , v1(y),yN − xN

ε, YN

)

, (4.11)(

XN 0

0 YN

)

≤2

ε

(

I −I−I I

)

in H−1 ×H−1 (4.12)

for some N×N matrices XN and YN that as operators in L(H−1) satisfy XN = PNXNPN ,YN = PNYNPN and are symmetric. Furthermore we can set the above maximum andminimum to be equal to zero. (Above the symbols DVN

and D2VN

denote the Frechet

derivatives in VN .) Since in VN the topology of H−1 is equivalent to the topology of H theabove convergences hold in the topology of H × R × H × S(H). We now extend ϕk, ψk

to functions in C2(H−1) by setting ϕk(x) = ϕk(PNx), ψk(y) = ψk(PNy). Then, if DH−1ϕk

denotes the Frechet derivative of ϕk in H−1 we have

Dϕk(x) = BDH−1ϕk(x), D

2ϕk(x) = BD2H−1

ϕk(x) (4.13)

and the same also holds for ψk.Using a perturbed optimization result, we can find sequences pj, qj ∈ H, such that

‖pj‖ + ‖qj‖ → 0 as j → ∞, and

u1(x) − ϕk(x) − 〈Bpj, x〉

has a global maximum at some point xj, (4.14)

v1(y) + ψk(y) − 〈Bqj, y〉

has a global minimum at some point yj. (4.15)

Combining (4.14) and the fact that (u1)∗(xN ) − ϕk(xN) has a strict global maximum at

xkN , we can deduce that PNxj → xk

N and u1(PNxj) → (u1)∗(xk

N) as j → +∞. Similarly(4.15) and the fact that (v1)∗(yN)+ ψk(yN) has a strict global minimum at some point yk

N

implies that PNyj → ykN and v1(PNyj) → (v1)

∗(ykN) as j → +∞. We can then select an

appropriate subsequence jk such that PNxjk→ PN x and PNyjk

→ PN y, with the addi-tional requirements that u1(PNxjk

) → u1(x), v1(PNyjk) → v1(y), ϕk(PNxjk

)−ϕk(xN) → 0and ψk(PNyjk

) − ψk(yN) → 0, as k → ∞. Moreover we can choose the subsequence jkso that all the convergences in (4.10) hold when xk

N and ykN are replaced by PNxjk

andPNyjk

respectively. We may now repeat rather standard arguments of [4] (see also [13],pages 409-410) and use (4.14) and (4.15) to obtain that xjk

→ x and yjk→ y in H−1.

We now need to prove that xjk→ x and yjk

→ y in H. To obtain these convergences, itwill be enough to prove that xjk

and yjkare uniformly bounded in H1 (independently of

k). First we observe that xjkand yjk

are uniformly bounded in H (they remain in a ball

13

whose radius depends exclusively on α). Using (4.14) and the definition of subsolution weget

(uα)∗(xjk) + 2αδ‖xjk

‖21 + αµ(xjk

)

+〈xjk, A∗B(−pn + pjk

+QN(x− y)

ε+QN(x− xjk

)

ε) + A∗Dϕk(xjk

)〉

+F

(

xjk,−Bpn +Bpjk

+BQN (x− y)

ε+BQN (x− xjk

)

ε+ ϕk(xjk

) + αDµ(xjk),

α D2µ(xjk) +

BQN

ε+D2ϕk(xjk

)

)

≤ 0. (4.16)

Note that (uα)∗(xjk) → (uα)∗(x) so using the local boundedness of F we can deduce

from the above inequality that ‖xjk‖1 ≤ C for some positive constant C independent

of k. A similar argument can be used to prove that yjkis bounded in H1. Therefore

B1

2xjk→ B

1

2 x, B− 1

2xjk⇀ B− 1

2 x and so

‖xjk− x‖2 = 〈B− 1

2 (xjk− x), B− 1

2 (xjk− x)〉 → 0

as k → ∞. The same argument also shows that yjk→ y. Therefore the lemma holds with

ϕk(x) := ϕk(x) + 〈Bpjk, x〉 and ψk(y) := ψk(y) − 〈Bqjk

, y〉.

We now finish the proof of the theorem. Using (4.14), Definition 3.4 and taking lim infas k → +∞ we have

(uα)∗(x) + αµ(x) + 2αλ‖x‖21 + 〈x, A∗B(x−y)

ε〉 − 〈x, A∗Bpn〉

+F(

x, αDµ(x) − Bpn + B(x−y)ε

, αD2µ(x) +XN + BQN

ε

)

≤ 0.

Similarly (4.15) and Definition 3.4 yield

(vα)∗(y) − αµ(y) − 2λα‖y‖21 + 〈y, A∗B(x−y)

ε〉 + 〈y, A∗Bqn〉

+F(

y,−αDµ(y) +Bqn + B(x−y)ε

,−αD2µ(y) − YN − BQN

ε

)

≥ 0.

We then use (1)0 and (2) to obtain

(uα)∗(x) − (vα)∗(y) ≤ σ1(N) + σ2(n) + C0‖x−y‖2

−1

ε

+F (y, B(x−y)ε

,−YN)) − F (x, B(x−y)ε

, XN)) +Kα(2 + µ(x) + µ(y))

where σ1(N) → 0 for n, ε, α fixed and σ2(n) → ∞ for α, ε fixed. Finally (3) gives

(uα)∗(x) − (vα)∗(y) ≤ σ1(N) + σ2(n) + C0‖x−y‖2

−1

ε

+ωR(α)(‖x− y‖−1(1 + ‖(x−y)‖−1

ε)) +Kα(2 + µ(x) + µ(y)).

If m > 2η, then for ε, α sufficiently small and n large enough we have (uα)∗(x)−(vα)∗(y) >η. Therefore we get

η ≤ σ1(N) + σ2(n) + C0‖x−y‖2

−1

ε

+ωR(α)(‖x− y‖−1(1 + ‖(x−y)‖−1

ε)) +Kα(2 + µ(x) + µ(y)).

14

Letting now N → ∞, n → ∞, ε → 0 and α → 0 in that order we arrive at η ≤ 0 whichis a contradiction.

In the next theorem we prove a comparison result for the time dependent problem(1.2).

Theorem 4.4. Let (2.1) and (2.2) hold and let F satisfy (1)k, (2), (3) for some k ≥ 0.Let g be locally uniformly B-continuous and such that

|g(x)| ≤ C(1 + ‖x‖k) for some C > 0. (4.17)

Let u be a viscosity subsolution of (1.2), and v be a viscosity supersolution of (1.2) in thesense of Definition 3.5 such that

u,−v ≤ C(1 + ‖x‖k) for some C > 0 (4.18)

and

limt→0

{(u(t, x) − g(x))+ + (v(t, x) − g(x))−} = 0,

uniformly on bounded sets. (4.19)

Then for every 0 < T1 < T

limR↑∞

lim(r,η)↓(0,0)

sup{u(t, x) − v(s, y) : |t− s| < η, ‖x− y‖−1 < r

x, y ∈ BR, 0 ≤ t, s ≤ T1} ≤ 0. (4.20)

In particular u ≤ v.

Proof. We will outline the proof of this theorem as it is similar to the proof of Theorem4.1. We argue by contradiction and assume that (4.20) is not true. Then for a sufficientlysmall σ > 0 and some γ > 0 we have

γ < limR↑∞

lim(r,η)↓(0,0)

sup{uσ(t, x) − vσ(s, y) : |t− s| < η, ‖x− y‖−1 < r, x, y ∈ BR}

= limR↑∞

limr↓0

limη↓0

sup{uσ(t, x) − vσ(s, y) : |t− s| < η, ‖x− y‖−1 < r, x, y ∈ BR}, (4.21)

where we have set uσ(t, x) = u(t, x)− σT−t

and vσ(s, y) = v(s, y)− σT−s

. We define uα(t, x) =

uσ(t, x)− αeKtµ(x), vα(s, y) = vσ(s, y) + αeKsµ(y) where µ(x) satisfies (1)k and K is theconstant from (1)k. Let

m := limR↑∞

limr↓0

limη↓0

sup{uσ(t, x) − vσ(s, y) : ‖x− y‖−1 < r, |t− s| < η, x, y ∈ BR},

mα := limr↓0

limη↓0

sup{(uα)∗(t, x) − (vα)∗(s, y) : , ‖x− y‖−1 < r, |t− s| < η}

= limr↓0

limη↓0

sup{uα(t, x) − vα(s, y) : , ‖x− y‖−1 < r, |t− s| < η},

15

mα,ε := sup{

(uα)∗(t, x) − (vα)∗(s, y)−1

2ε‖x− y‖2

−1

}

,

mα,ε,β := sup{

(uα)∗(t, x) − (vα)∗(s, y) −1

2ε‖x− y‖2

−1 −(t− s)2

2β

}

.

We have

m = limα↓0

mα, (4.22)

mα = limε↓0

mα,ε, (4.23)

mα,ε = limβ↓0

mα,ε,β, (4.24)

however m can now be +∞. Using perturbed optimization results, we can find sequencesan, bn ∈ R, and pn, qn ∈ H such that |an| + |bn| + ‖pn‖ + ‖qn‖ ≤ 1

nsuch that

(uα)∗(t, x) − (vα)∗(s, y)−1

2ε‖x− y‖2

−1 −1

2β|t− s|2 + ant+ bns+ 〈Bpn, x〉 + 〈Bqn, y〉

achieves a strict global maximum at some point (t, s, x, y) ∈ (0, T ] × [0, T ] × H × H.Convergences (4.22)-(4.24) yield

limβ↓0

lim supn→∞

1

2β|t− s|2 = 0 for every α, ε > 0, (4.25)

limε↓0

lim supβ↓0

lim supn→∞

1

2ε‖x− y‖2

−1 = 0 for every α > 0. (4.26)

By the definition of uσ and vσ we have t < T and s < T . In light of (4.19), (4.21)-(4.26),and the uniform continuity of g in H−1 on bounded sets of H, we can conclude that t > 0and s > 0 for big n and small β, ε. We define as before

u1(t, x) = (uα)∗(t, x) −〈BQN(x− y), x〉

ε−

‖QN(x− x)‖2−1

ε

+‖QN(x− y)‖2

−1

2ε+ ant+ 〈Bpn, x〉

and

v1(s, y) = (vα)∗(s, y) −〈BQN (x− y), y〉

ε+

‖QN(y − y)‖2−1

ε+ αµ(y)− bns− 〈Bqn, y〉.

so that

u1(t, x) − v1(s, y) −1

2ε‖PN(x− y)‖2

−1 −1

2β|t− s|2

16

has a strict global maximum at (t, s, x, y). Arguing now as in the proof of Lemma 4.3 wecan claim the existence of functions ϕk, ψk,∈ C1,2((0, T )×H−1) with uniformly continuousderivatives such that

u1(t, x) − ϕk(t, x)

has a global maximum at some point (tk, xk),

v1(s, y) + ψk(s, y)

has a global minimum at some point (sk, yk), and

(

tk, xk, u1(tk, xk),∂ϕk

∂t(tk, xk), Dϕk(tk, xk), D

2ϕk(tk, xk))

→

(

t, x, u1(t, x),t− s

β,BPN (x− y)

ε,XN

)

, (4.27)

(

sk, yk, v1(sk, yk),∂ψk

∂t(sk, yk), Dψk(sk, yk), D

2ψk(sk, yk))

→

(

s, y, v1(s, y),s− t

β,BPN(y − x)

ε, YN

)

, (4.28)

(

XN 00 YN

)

≤2

ε

(

B −B−B B

)

, (4.29)

with the convergences being in R ×H × R × R ×H2 × L(H).Therefore, using Definition 3.5 we obtain as in the proof of Theorem 4.1 that

σ

(T − t)2− an +

t− s

β+ αKeKtµ(x) + αeKtλ

µ′(‖x‖)

‖x‖‖x‖2

1

+ 〈x,A∗B(x− y)

ε〉 − 〈x, A∗Bpn〉

+ F(

t, x, αeKtDµ(x) −Bpn +B(x− y)

ε, αeKtD2µ(x) +XN +

BQN

ε

)

≤ 0

(4.30)

and

bn −σ

(T − s)2+t− s

β− αKeKsµ(y) − αeKsλ

µ′(‖y‖)

‖y‖‖y‖2

1

+ 〈y,A∗B(x− y)

ε〉 + 〈y, A∗Bqn〉

+ F(

s, y,−αeKsDµ(y) +Bqn +B(x− y)

ε,−αeKsD2µ(y) − YN −

BQN

ε

)

≥ 0.

(4.31)

Combining (4.30) and (4.31) and using assumptions (1)k, (2), and the local uniformcontinuity of F , we have

17

2σ

T 2≤ σ1(N) + σ2(n) + C0

‖x− y‖2−1

ε− αK(eKtµ(x) + eKsµ(y))

+ F (s, y,B(x− y)

ε,−YN)) − F (t, x,

B(x− y)

ε,XN)) + αK(2 + eKtµ(x) + eKsµ(y)),

where σ1(N) → 0 for n, ε, α, β fixed and σ2(n) → ∞ for α, ε, β fixed. Using (3) and thelocal uniform continuity of F we then conclude that

σ

T 2≤ σ1(N) + σ2(n) + σ3(β) +C0

‖x− y‖2−1

ε+ωR(α)(‖x− y‖−1(1 +

‖(x− y)‖−1

ε)) + 2αK,

where σ3(β) → 0 for α, ε fixed. We can now let N → ∞, n→ ∞, β → 0, ε→ 0, and α → 0in this order to arrive at σ ≤ 0 which is a contradiction.

5 Perron’s method and existence of solutions

In this section we will show that viscosity solutions of (1.1) in the sense of Definition 3.4(respectively, viscosity solutions of (1.2) in the sense of Definition 3.5) can be obtained byPerron’s method, i.e. by taking the supremum of all such viscosity subsolutions of (1.1)(respectively, (1.2)) provided that a viscosity subsolution and a viscosity supersolutionexist. Therefore, if Theorem 4.1 (respectively, Theorem 4.4) holds, we will obtain thatthe solution produced by this method is B-continuous, and so it is a viscosity solution inthe sense of Definition 3.2 (respectively, Definition 3.3). A posteriori this will also showthat this solution is equal to the supremum of all viscosity subsolutions in the sense ofDefinition 3.2 (respectively, Definition 3.3).

It is convenient to state a simple lemma for future reference.

Lemma 5.1. Let (2.2) hold. Let ϕ ∈ C2 (H) be B-upper semi-continuous and be suchthat A∗Dϕ is continuous, and let h ∈ C2((−∞,+∞)) be even and such that h′(r) ≥ 0 forr ∈ (0,+∞). Let w(x) = ϕ(x) − h(‖x‖) (respectively, w(x) = −ϕ(x) + h(‖x‖)) satisfy

w(x) + 〈x,A∗Dw(x)〉 + F (x,Dw(x), D2w(x)) ≤ 0 for x ∈ D(A∗)

(respectively,

w(x) + 〈x,A∗Dw(x)〉 + F (x,Dw(x), D2w(x)) ≥ 0 for x ∈ D(A∗).)

Then the function w is a viscosity subsolution (respectively, supersolution) of (1.1).

Proof. We will only do the proof in the subsolution case. Since w is B-upper semi-continuous Definition 3.2 and Definition 3.4 are equivalent. Suppose that w(y)−h1(‖y‖)−ϕ1(y) has a local maximum at x for test functions ϕ1, h1. Then

Dϕ(x) −h′(‖x‖)

‖x‖x−Dϕ1(x) −

h′1(‖x‖)

‖x‖x = 0

18

andD2w(x) ≤ D2(ϕ1 + h(‖ · ‖))(x).

Therefore either x = 0 or(

h′(‖x‖)

‖x‖+h′1(‖x‖)

‖x‖

)

x = Dϕ(x) −Dϕ1(x) ∈ D(A∗) ⊂ H1.

Denote ψ = ϕ1 + h1(‖y‖). Now, using (2.2) and the degenerate ellipticity of F , we obtain

w(x) + λ‖x‖21

h′

1(‖x‖)

‖x‖+ 〈x,A∗Dϕ1(x)〉 + F (x,Dψ(x), D2ψ(x))

≤ w(x) +h′

1(‖x‖)

‖x‖〈x,A∗x〉 + 〈x,A∗Dϕ1(x)〉 + F (x,Dw(x), D2w(x))

= w(x) + 〈x,A∗Dw(x)〉 + F (x,Dw(x), D2w(x)) ≤ 0

and the claim is proved.

Proposition 5.2. Let (2.2) be satisfied. Let A be a family of viscosity subsolutions of(1.1) in the sense of Definition 3.4. Suppose that the function

u(x) = sup{w(x) : w ∈ A} (5.1)

is locally bounded. Then u is a viscosity subsolution of (1.1) in the sense of Definition 3.4.

Proof. Suppose that (u− h(‖ · ‖))∗ −ϕ has a strict in ‖ · ‖−1 norm global maximum at apoint x for test functions h and ϕ. (We can assume that (u(y)−h(‖·‖))∗(y)−ϕ(y) → −∞as ‖y‖ → ∞.) Perturbed optimization techniques and Definition 3.2 then give that thereexist viscosity subsolutions wn of (1.1), xn ∈ H1, and pn ∈ H, ‖pn‖ ≤ 1/n such that

B1

2xn → B1

2x, xn ⇀ x in H as n→ ∞, (5.2)

(wn(y) − h(‖y‖))∗ − ϕ(y)− < Bpn, y >

has a strict in ‖ · ‖−1 norm global maximum at xn, and

(wn − h(‖ · ‖))∗(xn) → (u− h(‖ · ‖))∗(x) as n→ ∞. (5.3)

Therefore, denoting ψ(y) = ϕ(y) − h(‖y‖),

(wn − h(‖ · ‖))∗(xn) − h(‖xn‖) + λ‖xn‖21

h′(‖xn‖)‖xn‖

+〈xn, A∗Dϕ(xn)〉 + 〈xn, A

∗Bpn〉 + F (xn, Dψ(xn) +Bpn, D2ψ(xn)) ≤ 0.

(5.4)

Since the xn are bounded, using the local boundedness of F we thus obtain that

‖xn‖21 ≤ C

for some constant C which, together with (5.2), implies that x ∈ H1, and B− 1

2xn ⇀ B− 1

2xas n→ ∞. Therefore, by (5.2),

‖xn − x‖2 = 〈B− 1

2 (xn − x), B1

2 (xn − x)〉 → 0 as n→ ∞,

19

i.e. xn → x in H. Using this, (5.3), the continuity of F , and the fact that ‖ · ‖1 is lowersemi-continuous in H, we can now pass to the lim inf as n→ ∞ in (5.4) to obtain

(u− h(‖ · ‖))∗(x) + h(‖x‖) + λ‖x‖21

h′(‖x‖)‖x‖

+ 〈x,A∗Dϕ(x)〉

+F (x,Dψ(x), D2ψ(x)) ≤ 0

which completes the proof.

Proposition 5.3. Let (2.2) be satisfied. Let u0, v0 be respectively a viscosity subsolutionand a viscosity supersolution of (1.1) in the sense of Definition 3.4 such that u0 ≤ v0.Then the function

u(x) = sup{w(x) : u0 ≤ w ≤ v0, w is a viscosity subsolution

of (1.1) in the sense of Definition 3.4} (5.5)

is a viscosity solution of (1.1) in the sense of Definition 3.4.

Proof. The fact that u is a subsolution follows from Proposition 5.2. Suppose now that(u+h(‖·‖))∗−ϕ has a strict in ‖·‖−1 norm global minimum at a point x for test functionsh(‖y‖) and ϕ. First we notice that if

(u+ h(‖ · ‖))∗(x) = (v0 + h(‖ · ‖))∗(x)

then (v0 + h(‖ · ‖))∗ − ϕ has a global minimum at x and so we are done since v0 is aviscosity supersolution. Therefore we only need to consider the case

(u+ h(‖ · ‖))∗(x) < (v0 + h(‖ · ‖))∗(x).

It then follows from the B-continuity of ϕ and the weak lower semi-continuity of ‖ · ‖ thatfor every R > 0

ε + (u+ h(‖ · ‖))∗(x) − ϕ(x) + ϕ(y) − h(‖y‖)< (v0 + h(‖ · ‖))∗(y) − h(‖y‖) ≤ v0(y)

(5.6)

for y ∈ B−1(x, r) ∩B(x,R) for some small r, ε > 0. Denote

w(y) = ε+ (u+ h(‖ · ‖))∗(x) − ϕ(x) + ϕ(y) − h(‖y‖).

If the condition for u being a viscosity supersolution of (1.1) is violated at x theneither

(i) x 6∈ H1

or(ii) x ∈ H1 but

(u+ h(‖ · ‖))∗(x) − h(‖x‖) − λ‖x‖21

h′(‖x‖)‖x‖

+ 〈x,A∗Dϕ(x)〉

+F (x,Dψ(x), D2ψ(x)) < −ν < 0,(5.7)

where ψ(y) = ϕ(y) − h(‖y‖), ν > 0.

20

If (i) happens, i.e. if x 6∈ H1 then

lim infy→x in H−1,y∈H1

‖y‖1 = +∞. (5.8)

Otherwise we would have a sequence yn such that B1

2 yn → B1

2x and ‖B− 1

2 yn‖ ≤ C. Then

for some subsequence (still denoted by yn) B− 1

2yn ⇀ z for some z ∈ H. But this implies

that x ∈ H1 and z = B− 1

2x. Using the local boundedness of F , condition (5.8) yields nowthat for every R > 0

w(y)− λ‖y‖21

h′(‖y‖)

‖y‖+ 〈y, A∗Dϕ(y)〉 + F (y,Dw(y), D2w(y)) < −

ν

2, (5.9)

for y ∈ B−1(x, r) ∩B(x,R) ∩H1 for some small r > 0.Suppose that (ii) is true. We will show that for every R > 0 (5.9) holds for y ∈

B−1(x, r) ∩B(x,R) ∩H1 for ε = µ/4 for some small r > 0. If not there exists a sequencexn → x in H−1, ‖xn‖ ≤ R such that

w(xn) − λ‖xn‖21

h′(‖xn‖)

‖xn‖+ 〈xn, A

∗Dϕ(xn)〉 + F (xn, Dw(xn), D2w(xn)) ≥ −

ν

2. (5.10)

Then of course ‖xn‖1 ≤ C for some constant C as otherwise (5.10) would be violated. But

then we must have B− 1

2xn ⇀ B− 1

2x in H and so arguing as in the proof of Proposition5.2 we obtain xn → x. However then (5.7) and the lower semi-continuity of ‖ · ‖1 imply

lim supn→∞

(

w(xn) − λ‖xn‖21

h′(‖xn‖)

‖xn‖+ 〈xn, A

∗Dϕ(xn)〉 + F (xn, Dw(xn), D2w(xn))

)

< −ν

2

which gives a contradiction.So we have obtained that in either case for every R > 0 (5.9) is satisfied for y ∈

B−1(x, r) ∩B(x,R) ∩H1 for some small r, ε > 0.It follows from the fact that (u + h(‖ · ‖))∗ − ϕ has a (strict in ‖ · ‖−1 norm) global

minimum at x that for every r > 0 if ε is small enough (depending on r) there exists aconstant µr > 0 such that

w(y) < (u+ h(‖ · ‖))∗(y) − h(‖y‖) − µr ≤ u(y)− µr

for y 6∈ B−1(x, r). Moreover it is clear by the growth condition on the test function in theDefinition 3.4 that if R is big enough then

w(y) ≤ u(y)− 1

for y 6∈ B(x,R).Using the last two inequalities and (5.6) we can therefore conclude that there exist

numbers R, r, ε, µ > 0 such that

w ≤ v0 in H, and w(y) < u(y)− µ for y 6∈ B−1(x, r) ∩B(x,R), (5.11)

21

and such that (5.9) is satisfied for y ∈ B−1(x, r) ∩B(x,R) ∩H1.Finally, if R is big enough there exist yn ∈ B(x,R) such that yn → x in H−1 such

thatu(yn) + h(‖yn‖) − ϕ(yn) → (u+ h(‖ · ‖))∗(x) − ϕ(x)

which means that there exist points y ∈ B−1(x, r) ∩ B(x,R) for which

u(y) < w(y). (5.12)

We now claim that the function w is a viscosity subsolution of (1.1) in B−1(x, r) ∩B(x,R). (Since w is B-upper semi-continuous Definition 3.2 and Definition 3.4 are equiva-lent, however we point out that being a viscosity subsolution of (1.1) in B−1(x, r)∩B(x,R)still requires that the maxima in Definition 3.4 be local in the ‖ · ‖−1 norm in the wholespace.) The claim follows from Lemma 5.1 upon noticing that by (5.9) and (2.2) we have

w(y) + 〈y, A∗Dw(y)〉+ F (y,Dw(y), D2w(y))

≤ w(y) − λ‖y‖21

h′(‖y‖)‖y‖

+ 〈y, A∗Dϕ(y)〉+ F (y,Dw(y), D2w(y)) < 0

for y ∈ B−1(x, r) ∩B(x,R) ∩D(A∗).It now remains to show that the function

u1 = max(w, u)

is a viscosity subsolution in the sense of Definition 3.4 and u0 ≤ u1 ≤ v0. But this is clearfrom Proposition 5.2 (more precisely from its proof) and the fact that w is a viscositysubsolution of (1.1) in B−1(x, r) ∩ B(x,R), and (5.11). This, together with (5.12), givesus a contradiction and the proof is complete.

Combining the above proposition with Theorem 4.1 we therefore obtain the followingresult.

Theorem 5.4. Let (2.1) and (2.2) hold and let F satisfy (1)0, (2), (3). Let u0, v0 berespectively a bounded viscosity subsolution and a bounded viscosity supersolution of (1.1)in the sense of Definition 3.4. Then the function

u(x) = sup{w(x) : u0 ≤ w ≤ v0, w is a viscosity subsolution

of (1.1) the sense of Definition 3.4}

is the unique bounded viscosity solution of (1.1) in the sense of Definition 3.2. Moreoveru is locally uniformly B-continuous.

The same technique applies to the time dependent problems (1.2). We just state herethe final existence result that can be proved in the same way as Theorem 5.4.

Theorem 5.5. Let (2.1) and (2.2) hold and let F satisfy (1)k, (2), (3) for some k ≥ 0.Let g be locally uniformly B-continuous and such that

|g(x)| ≤ C(1 + ‖x‖k) for some C > 0. (5.13)

22

Let u0 be a viscosity subsolution of (1.2), and v0 be a viscosity supersolution of (1.2) inthe sense of Definition 3.5 such that

u0,−v0 ≤ C(1 + ‖x‖k) for some C > 0 (5.14)

and

limt→0

{|u0(t, x) − g(x)| + |v0(t, x) − g(x)|} = 0

uniformly on bounded sets. (5.15)

Then the function

u(t, x) = sup{w(t, x) : u0 ≤ w ≤ v0, w is a viscosity subsolution

of (1.2) the sense of Definition 3.5}

is the unique viscosity solution of (1.2) in the sense of Definition 3.3 satisfying (5.14) and(5.15). Moreover u is locally uniformly B-continuous on [0, T1]×H for every 0 < T1 < T .

We will construct a subsolution and a supersolution so that we can apply Perron’smethod. We remark that if supx∈H |F (x, 0, 0)| = M < +∞, then the functions u(x) = −Mand v(x) = M are respectively a viscosity subsolution and a viscosity supersolution of(1.1) in the sense of Definition 3.2. In the proposition below, we will show how the con-struction of the supersolution can be done in the time-dependent case. The constructionof a subsolution is very similar.

Proposition 5.6. Let (2.2) hold and let g be locally uniformly B-continuous and suchthat |g(x)| ≤ µ(1+ ‖x‖) for x ∈ H for some positive constant µ. Then there is a viscositysupersolution V of equation (1.2) such that limt↓0 V (t, x) = g(x) uniformly on boundedsets of H.

Proof. We define C(r) = sup{|F (t, x, p,X)| : x ∈ H, t ∈ [0, T ], ‖p‖, ‖X‖ ≤ r}. Letv(t, x) = αt + 2µ

√

1 + ‖x‖2. By a time dependent version of Lemma 5.1 a condition forv to be a viscosity supersolution of (1.2) is

α+ F(

t, x,Dv(t, x), D2v(t, x))

≥ 0

for all (t, x) ∈ (0, T ) × H. Since Dv(t, x) and D2v(t, x) are bounded we can thereforeselect α depending only on µ such that the above condition is satisfied.

Let z ∈ H, ε > 0. We first choose a constant R = R(‖z‖) such that (‖x‖ − ‖z‖)4+ ≥

v(t, x) for ‖x‖ ≥ R. We then find M = M(‖z‖, ε) such that

wz,ε(x) = g(z) + ε+M‖x − z‖2−1 + (‖x‖ − ‖z‖)4

+ ≥ g(x)

for ‖x‖ ≤ R. Let now γ = sup{‖Dwz,ε(x)‖ + ‖D2wz,ε(x)‖ : ‖x‖ ≤ R}. In order forwz,ε(t, x) = βt+ wz,ε(x) to be a viscosity supersolution of (1.2) in (0, T )×B(0, R) we need

β + 〈x,A∗B(x− z)〉 + F (t, x,Dwz,ε(t, x), D2wz,ε(t, x)) ≥ 0.

This can be achieved by taking β = R(R + ‖z‖)‖A∗B‖ + C(γ).

23

It now follows that

ωz,ε(t, x) = min{wz,ε(t, x), v(t, x)}

is a lower B- semicontinuous viscosity supersolution of (1.2) in (0, T )×H. It is now clearfrom the construction of the ωz,ε and the time dependent version of Proposition 5.2 forsupersolutions that the function V (t, x) = infz,ε ωz,ε(t, x) is a viscosity supersolution of(1.2) in the sense of Definition 3.5 such that limt↓0 V (t, x) = g(x) uniformly on boundedsets of H.

6 Relaxed limits

In this section we will show how the method of half-relaxed limits of Barles-Perthame canbe generalized to infinite dimensional spaces. We will consider two separate cases. Thefirst will deal with limits of sub- and super-solutions of equations on the whole space withoperators A satisfying similar structure conditions. The second will deal with limits offinite dimensional approximations.

Let Fn : [0, T ]×H ×H ×S(H) → IR be continuous, locally bounded uniformly in n,and degenerate elliptic. Denote

F+(t, x, p,X) = limm→∞

sup{Fn(s, y, q, Y ) : n ≥ m, |t−s|+‖x−y‖+‖p−q‖+‖X−Y ‖ ≤1

m}

and

F−(t, x, p,X) = limm→∞

inf{Fn(s, y, q, Y ) : n ≥ m, |t−s|+‖x−y‖+‖p−q‖+‖X−Y ‖ ≤1

m}.

Let An be linear, maximal monotone operators in H such that D(A∗) ⊂ D(A∗n) and

〈A∗nx, x〉 ≥ λn‖x‖

21, for x ∈ D(A∗

n), (6.1)

where lim infn→∞ λn ≥ λ. Assume moreover that

If xn → x,A∗xn → A∗x then A∗nxn ⇀ A∗x, (6.2)

and that for every test function ϕ, the family A∗nDϕ is locally uniformly bounded. We

then have the following theorem.

Theorem 6.1. Let the assumptions of this section be satisfied and let B be compact. Letun be locally uniformly bounded viscosity subsolutions, (respectively, supersolutions) of

un + 〈Anx,Dun〉 + Fn(x,Dun, D2un) = 0 in H (6.3)

in the sense of Definition 3.2. Then the function

u+(x) = limm→∞

sup{un(y) : n ≥ m, ‖x− y‖ ≤1

m}

24

(respectively,

u−(x) = limm→∞

inf{un(y) : n ≥ m, ‖x− y‖ ≤1

m})

is a viscosity subsolution (respectively, supersolution) of

u+ + 〈Ax,Du+〉 + F−(x,Du+, D2u+) = 0 in H

(respectively,u− + 〈Ax,Du−〉 + F+(x,Du−, D

2u−) = 0 in H)

in the sense of Definition 3.4.

Please notice that the function u+ does not have to be B-upper semi-continuous.

Proof. Let (u+ − h(‖ · ‖))∗ −ϕ have a local maximum equal to 0 at x. In light of Lemma3.6 and local uniform boundedness of the un we can assume that the maximum is global,strict in the ‖ · ‖−1 norm, and such that

u+(y) − h(‖y‖), (u+ − h(‖ · ‖))∗(y) − ϕ(y) → −∞,

andun(y) − h(‖y‖) − ϕ(y) → −∞

as ‖y‖ → +∞, uniformly in n. Then there must exist a sequence xn such that ‖xn−x‖−1 →0, ‖xn‖ ≤ C, and

u+(xn) − h(‖xn‖) − ϕ(xn) ≥ −1

n.

Therefore there exist yn and mn such that

umn(yn) − h(‖yn‖) − ϕ(yn) ≥ −

2

n. (6.4)

Let zn be a global maximum of

umn(y) − h(‖y‖)− ϕ(y).

It exists because of the decay of this function at infinity and the fact that, because B iscompact, B-upper semi-continuity is equivalent to weak sequential upper-semi-continuity.Obviously ‖zn‖ ≤ C1 and we also have

umn(zn) + λmn

‖zn‖21

h′(‖zn‖)

‖zn‖+ 〈zn, A

∗mnDϕ(zn)〉 + Fmn

(zn, Dψ(zn), D2ψ(zn)) ≤ 0, (6.5)

where ψ(y) = h(‖y‖)−ϕ(y). Using the boundedness of zn, the fact that h′(‖y‖)/‖y‖ > c >0 for ‖y‖ ≤ C1, and the local uniform boundedness of the Fn and A∗

mnDϕ, we therefore

obtain that ‖zn‖1 ≤ C2 which implies that zn ⇀ z in H1 and zn → z in H for somez ∈ H1.

25

Therefore u+(z) ≥ lim supn→∞ umn(zn) and this then yields u+(z)−h(‖z‖)−ϕ(z) ≥ 0,

i.e. that z = x. Moreover it also follows that

limn→∞

umn(zn) = (u+ − h(‖ · ‖))∗(x) + h(‖x‖).

We can now pass to the lim inf in (6.5) to conclude the proof.

The following theorem is an immediate corollary of Theorems 6.1 and 4.1.

Theorem 6.2. Let A satisfy (2.1) and (2.2). Let the assumptions of this section on An

and Fn be satisfied, and let B be compact. Let F− = F+ =: F satisfy the assumptions ofTheorem 4.1. Let un be locally bounded uniformly in n viscosity solutions of (6.3) in thesense of Definition 3.2. Let u+ and −u− be bounded from above. Then u+ = u− =: u,u is locally uniformly B-continuous (i.e. u is weakly sequentially continuous), and u isthe unique bounded viscosity solution of (1.1) in the sense of Definition 3.2. Moreoverthe functions un converge to u pointwise as n → ∞ and the convergence is uniform onbounded subsets of Hα for every α > 0.

We point out that the limiting Hamiltonians F+ and F− may be of first order so theabove theorems can be applied to singular perturbation problems and small noise limits.

The time dependent version of Theorem 6.2 is the following. The functions u+ andu− below are now defined by taking the lim sup and lim inf in both variables s and y.

Theorem 6.3. Let A satisfy (2.1) and (2.2). Let the assumptions of this section on An

and Fn be satisfied, and let B be compact. Let F− = F+ =: F and let F, g satisfy theassumptions of Theorem 4.4. Let un be viscosity solutions of

(un)t + 〈Anx,Dun〉 + Fn(t, x,Dun, D2un) = 0 in (0, T ) ×H

in the sense of Definition 3.3 and such that

|un(x)| ≤ C(1 + ‖x‖k) for some C > 0 (6.6)

and

limt→0

|un(t, x) − g(x)| = 0 (6.7)

uniformly on bounded sets, uniformly in n.

Then u+ = u− =: u, u is weakly sequentially continuous, and u is the unique viscositysolution of (1.2) in the sense of Definition 3.3 satisfying (6.6) and (6.7). Moreover thefunctions un converge to u pointwise as n→ ∞ and the convergence is uniform on boundedsubsets of [0, T1] ×Hα for every α > 0 and 0 < T1 < T .

We want to close this section by showing how half-relaxed limits can be applied toshow convergence of finite dimensional approximations.

Denote by VN the space spanned by the eigenvectors of B corresponding to theeigenvalues that are greater than or equal to 1/N . Let PN be the orthogonal projectionin H onto VN . Define

AN = (PNA∗PN )∗, BN = BPN .

26

Then AN is bounded and monotone in H, AN and BN satisfy (2.1), and

〈A∗Nx, x〉 ≥ λ‖PNx‖

21, for x ∈ H. (6.8)

We now have the following result.

Theorem 6.4. Let A satisfy (2.1), (2.2), and let D(A∗) = R(B). Let F satisfy (1)0,(2), (3), let supx∈H |F (x, 0, 0)| = M < +∞, and let B be compact. Let u be the uniquebounded viscosity solution of (1.1), and let uN(x) = vN(PNx), where the vN are theviscosity solutions of

vN + 〈ANx,DvN〉 + F (x,DvN , D2vN) = 0 in HN . (6.9)

Then uN → u pointwise in H as N → ∞ and the convergence is uniform on boundedsubsets of Hα for every α > 0.

Proof. Under our assumptions equation (6.9) has a unique viscosity solution vN suchthat |vN | ≤M for every N ≥ 1. Also (see [25]) the functions uN are viscosity solutions of

uN + 〈ANx,DuN〉 + F (PNx, PNDuN , PND2uNPN) = 0 in H.

Since the above equations have only bounded terms the solutions can be interpreted inthe usual sense of [21] which in particular implies that the uN are solutions in the senseof Definition 3.2.

We first observe that the AN satisfy (6.2) with a strong convergence. This followsfrom the proof of Lemma 2.3 of [5] upon noticing that D(A∗) = R(B) guarantees thatthe operator Q = A∗B + cB has bounded inverse Q−1 = B−1(A∗ + cI)−1 for every c > 0.The last statement is a trivial consequence of the closed graph theorem.

We next claim that for every test function ϕ, the family A∗NDϕ is locally uniformly

bounded. This is true since

‖A∗NDϕ(x)‖ ≤ ‖A∗BPNB

−1(A∗ + I)−1(A∗ + I)Dϕ(x)‖

≤ ‖A∗B‖‖B−1(A∗ + I)−1‖‖(A∗ + I)Dϕ(x)‖ ≤ C‖(A∗ + I)Dϕ(x)‖.

However we cannot invoke Theorem 6.1 directly as the AN only satisfy (6.8). Insteadwe will follow its proof pointing out the main differences. Repeating its arguments wehave instead of (6.4) that

Therefore there exist yN and mN such that

umN(PmN

yN) − h(‖PmNyN‖) − ϕ(PmN

yn) ≥ −2

N.

We then take zN to be a global maximum of

vmN(y) − h(‖y‖) − ϕ(y) over HmN

.

27

As before obviously ‖zN‖ ≤ C1 and we have in (6.5)

vmN(zN ) + λ‖zN‖

21

h′(‖zN‖)

‖zN‖+ 〈zN , A

∗mNDϕ(zN)〉

+ Fmn(zN , PmN

Dψ(zN ), PmND2ψ(zN )PmN

) ≤ 0,

(6.10)

where ψ(y) = h(‖y‖) − ϕ(y). This implies that (notice that zN = PmNzN) ‖zN‖1 ≤ C2,

which gives zN ⇀ z in H1 and zN → z in H for some z ∈ H1.Therefore we obtain u+(z) ≥ lim supN→∞ umN

(zN ) = lim supN→∞ vmN(zN) and this

gives u+(z) − h(‖z‖) − ϕ(z) ≥ 0, i.e. that z = x. We also obtain

limN→∞

umN(zN ) = (u+ − h(‖ · ‖))∗(x) + h(‖x‖).

We can now pass to the lim inf in (6.10) using Lemma 2.8 of [25].In spite of the novelty of the method of half-relaxed limits in infinite dimensions,

Theorem 6.4 is not really new under our assumptions. Convergence of finite dimensionalapproximations was proved in [25] (following a similar method for first order equations of[5]) by first proving uniform continuity estimates for the uN and then showing their localuniform convergence. In [25] assumption (2.2) was not needed but here condition (2) isa little more general. However our new method may succeed in situations where we maynot be able to obtain uniform a priori estimates for the continuity of uN .

Similar results can also be obtained for time dependent problems and for problemswhere we do not assume that D(A∗) = R(B). We do not work out the details hereas they are technical and lengthy, and the final statements are similar to the results of[25]. However there is a significant difference between the approximations used in [25](and in [5] before) and the ones we would need here for the half-relaxed limits. In [25]the operator A was first approximated by its Yoshida approximation Aλ and then byAλ,N = PNAλPN . We do not know if this process would succeed here. It seems that weneed first to take Aλ = A + λB−1 and then use Aλ,N = PNAλPN for the above Aλ. Thiskind of approximation procedure was used in [6] and we refer the readers to this paperfor some ideas and hints on how the proof should proceed in our case.

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[27] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded non-linear terms, J. Math. Anal. Appl. 163 (1992), 345–392.

[28] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded non-linear term-a simplified approach, J. Differential Equations 111 (1994), 123–146.

Department of Mathematics and StatisticsUniversity of MassachusettsAmherst, MA 01003, U.S.A.kelome@math.umass.edu

School of MathematicsGeorgia Institute of TechnologyAtlanta, GA 30332, U.S.A.swiech@math.gatech.edu

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