V I S C O S I T Y S O L U T I O N S : A P R I M E R

by

Michael G. Crandall (t) Department of Mathematics

University of California, Santa Barbara Santa Barbara, CA 93106

0. I n t r o d u c t i o n

These lectures present the most basic theory of "viscosity solutions" of fully nonlinear scalar partial differential equations of first and second order. Other contributions to this volume develop some of the amazing range of applications in which viscosity solutions play an essential role and various refinements of this basic material.

In this introductory section we describe the class of equations which are treated within the theory and then our plan of presentation.

The theory applies to scalar second order partial differential equations

(PDE) F(x, u, Du, D2u) = 0

on open sets f~ C IR N. The unknown function u : f} --+ IR is real-valued, Du corresponds to the gradient (uxl , . . . ,uxN) of u and D2u corresponds to the Hessian matrix (ux~,zj) of second derivatives of u. Consistently, F is a mapping

F : f ~ x I R x I R N x S ( N ) ~ I R

where S(N) is the set of real symmetric N x N matrices. We say that Du (D2u) "corresponds" to the gradient (respectively, the Hessian) because, as we shall see, solutions u may not be differentiable, let alone twice differentiable, and still "solve" (PDE). We write F(x, r, p, X) to indicate the value of F at (x , r ,p ,X) C f~ x IR x IR N x S(N). (PDE) is said to be fully nonlinear to emphasize that F(x, r ,p, X) need not be linear in any argument, including the X in the second derivative slot.

F is called degenerate elliptic if it is nonincreasing in its matrix argmnent:

Y(x , r , p ,X ) <_F(x,r,p,Y) for Y<_X.

The usual ordering is used on S(N); that is Y <_ X means

(Xr 4) -< (Y~, 4) for ~ E ]R N

where (., .} is the Euclidean inner product. If F is degenerate elliptic, we say that it is proper if it is also nondecreasing in r. That is, F is proper if

F(x , s , p ,X )<_F(x , r , p ,Y ) for Y < X , s < r .

(t) Supported in part by NSF Grant DMS93-02995 and in part by the author's appointment as a Miller Professor at the University of California, Berkeley for Fall 1996.

As a first example, F might be of first order

F ( x , r , p , X ) = Lr(~:,r,p);

every first order F is obviously (very) degenerate elliptic, and then proper if it is nondecreasing in r. For an explicit example, the equation ut + (u~) 2 = 0 with (t, x) E IR 2 is a proper equation (we are thinking of (t, x) as (xl, x2) above). On the other hand, the Burger 's equation ut + uuz = 0 is not proper, for it is not monotone in u. We refer to proper first order equations H(x,u, Dr,) = 0 and ut + H(x, u, Du) = 0 as "Hamilton-Jacobi" equations.

Famous second order examples are given by F(x, r, p, X) = - T r a c e (X) and F(x, r,p, X) = - T r a c e (X) - f(x) where f is given; the pdes are then Laplace's equation and Poisson's equation:

N

F(D~u) =-E 'uz~x~ = - A u = 0 and - A u = f ( x ) i=1

The equations are degenerate elliptic since X + Trace (X) is monotone increas- ing on S ( N ) . We do not rule out the linear case! Incorporating t as an additional variable as above, the heat equation ut - Au = 0 provides another famous ex- ample. The convention used here, that Du, D2u stand for the spatial gradient and spatial Hessian, will be in force whenever we write "ut + F(x, u, Du, D2u) ''.

Note the preference implied by these examples; we prefer A to A. A reason is tha t (in various settings), - A has an order preserving inverse. This convention is not uniform; for example, Souganidis [35] does not follow it and reverses the inequality in the definition of degenerate ellipticity.

More generally, the linear equation

N N

- + + - = o

i , j = l i=1

may be writ ten in the form F = 0 by setting

(0.1) F(x,r ,p ,X) = -Trace(A(x)X) + {b(z),p) +c(x)r - f(x)

where A(x) is a symmetr ic matr ix with the elements ai,~(x) and b(x) = (bl(X), . . . ,bN(x)). This F is degenerate elliptic if 0 < A(x) and proper if also 0 < c(x).

In the text we will pose some exercises which are intended to help readers orient themselves (and to replace boring text with pleasant activities). We vi- olate all conventions by doing so even in this introduction. Some exercises are "starred" which means "please do it now" if the fact is not familiar.

E x e r c i s e 0.1.* Verify tha t F given in (0.1) is degenerate elliptic if and only if A(x) is nonnegative.

The second order examples given above are associated with the "maximum principle". Indeed, the calculus of the maxinmm principle is a fundamental idea in the entire theory.

E x e r c i s e 0.2.* Show tha t F is proper if and only if whenever ~, I/) C C 2 and p - r has a nonnegative maximum (equivalently, r - ~ has a nonpositive minimum) at 2, then

F(~, ~(~), D~(~?), D2~(97)) _< F(~, ~(37), D~(~) , D2~(37)).

So far, we have presented nonlinear first order examples and linear second order examples. However, the class of proper equations is very rich. Indeed, if F, G are both proper, then so is AF + pC for 0 < A, #. More interesting is the following simple fact: if F~,~ is proper for c~ E .4,/3 E 13 (some index sets), then so is

F = sup i n f F ~ aEA/3E g '

provided only it is finite. This generality is essential to applications of the theory in differential games (see Bardi [2]), while applications in control theory correspond to the case "F~" in which there is only one index (see Bardi [2] and Soner [34]) .

For example, max(ut + I D u l 2 - g ( z ) , - A u - f ( x ) ) = 0 is a proper equation. The other lecture series will present many examples of scientific significance. We have only a t t empted here to indicate that that class of proper equations is broad and interesting.

Here we aim at a clear and congenial presentation of the most basic ele- ments of the theory of viscosity solutions of proper equations F = 0. These are the notion of a viscosity solution, maximum principle type comparison results for viscosity solutions, and existence results for viscosity solutions via Perron's method. We do not aim at completeness or technical generality, which often distract from ideas.

The text is organized in sections, many of which are quite brief. The de- scriptions below contain remarks about the logic of the presentation. By the numbers, the topics are:

Section 1: An illustration of the need to be able to consider nondifferentiable functions as solutions of proper fully nonlinear equations is given using first order examples.

Section 2: The notions of viscosity subsolutions, supersolutions and solutions are presented. The convention tha t the modifier "viscosity" will be dropped thereafter in the text is introduced. It is essential to deal with semicontinuous functions in the theory, and this generality appears here.

Section 3: Striking general existence and uniqueness theorems are presented without proof to indicate the success of viscosity solutions in this arena. The contrast with the examples in Section 1 is dramatic.

Sections 4, 5, 6: A pr imary test of a notion of generalized solutions is whether or not appropriate uniqueness results can be obtained (when suitable side con- ditions - boundary conditions, growth conditions, initial conditions, etc. - are satisfied). Actually, one wants a bit more here, that is the sort of comparison theorems which follow from the maximum principle. Basic arguments needed

in proofs of comparison results for viscosity solutions of first order s tat ionary problems (those without "t") are presented here and typical results are deduced. Section 4 concerns the Dirichlet problem, Section 5 concerns bounded solutions of a problem in IR N, and Section 6 provides an example of treat ing unbounded solutions. The second order case is more complex and is not taken up until Sec- tion 10. However, nothing is wasted, and all the arguments presented in these sections are invoked in the second order setting.

Section 7: The notions of Section 2 are recast in a form convenient for use in the next section and in the comparison theory in Sections 8 and 9.

Section 8: Two related results, each an important tool, are established. One states roughly that the supremum of a family of subsolutions is again a sub- solution, and the other that the limit of a sequence of viscosity subsolutions (supersolutions, solutions) of a converging sequence of equations (meaning the F~'s converge) is a subsolution (respectively, a supersolution, solution) of the limiting equation. We call this last theme "stability" of the notion; it is one of the great tools of the theory in applications. The mathematics involved is e lementary with a "point-set" flavor.

Section 9: Existence is proved via Perron's Method using a result of the previous section. The existence theory presupposes "comparison". At this stage, com- parison has only been treated in the first order case, and is simply assumed for the second order case. This does not affect either clarity or the basic argument. At this juncture, the most basic ideas have been presented with the exception of comparison for second order equations.

Section 10: The pr imary difference between the first and second order cases is explained. Then the rather deep result which is used here to bridge the gap, called here "the Theorem on Sums" (an analytical result about semicontinuous functions), is s tated without proof. An example is given to show how this tool theorem renders the second order case as easy to treat as the first order case.

Section 11: The Theorem on Sums is proved.

Section 12: In the preceding sections comparison was only demonstrated for various equations of the form F(x, u, Du, DUu) = 0. Here the main additional points needed to t reat ut + F(x, u, Du, D2u) = 0 are sketched.

Regarding notation, we use s tandard expressions like "C2(~2) '' (the twice continuously differentiable functions on f/) and "IPI" (the Euclidean length of p) without further comment when it seems reasonable. With some exceptions, we minimize distracting notation.

Regarding the literature, it is too vast to t ry to summarize in a work like this, which aims at presenting basic ideas and not at technical generality or great precision. We will basically rely on the big brother to this work, the more intense (and reportedly less friendly) [12] for its extensive references, together with those in the other contributions to this volume. (We recommend the current work as preparat ion for reading [12], especially the topics therein not taken up here.) We do give some references corresponding to the original works initiating the themes t reated here. A few more recent papers are cited as appropriate. All

references appear at the ends of sections. In addition, we mention the books by Cabr6 and Caffarelli [7] and Dong [17] for recent expositions of regularity theory of solutions, which is not t reated here, as well as the classic text of Gilbarg and Trudinger [23]. Regularity theory is also one of the themes of Evans [20]. The recent book of Barles [3] presents a complete theory of the first order case (which itself fills a book tha t contains 154 references!). The book of Fleming and Soner [22, Chapters I I and V] also nicely covers the basic theory. There are alternative theories for first order equations; see, e.g., [9] and [36]. Of course, MathSciNet now allows one to become nearly current regarding the state of the literature relatively easily, and one can profitably search on any of the leads given above.

A significant l imitation of our presentation is that only the Dirichlet bound- ary condition is discussed at any length, and this in its usual form rather than the generalized version. Other boundary conditions appear in the contributions of Bardi [2] and Soner [34] in an essential way. In addition to the references they give, the reader may refer for example to [12, Section 7] for a discussion in the spirit of this work. Another limitation is tha t singular equations are not t reated at all. Equations with singularities appear in contributions of Evans [20] and Souganidis [35]. See also [12, Section 9]. Finally, only continuous solutions are discussed here, while within applications one meets the discontinuous solutions. The contribution of Bardi [2, Section V] treats this issue, and discontinuous functions appear quickly in the exposition of Souganidis [35].

1. O n t h e N e e d for N o n s m o o t h S o l u t i o n s

The fact is tha t it is difficult to give examples of solutions (in any sense) of equations F = 0 which are not classical solutions unless the equation is pre t ty "degenerate" (roughly, the monotonicity of X ---* F ( x , r , p , X ) is not strong enough) or "singular" ( that is, F may have discontinuities or other types of sin- gularities). (A "classical" solution of an equation F(x, u(x), Du(x), D2u(x)) = 0 is a twice continuously differentiable function which satisfies the equation point- wise; if the equation is first order classical solutions are once continuously differ- entiable; if the equation has the form ut + F(x, u, Du, D2u) = 0, then a classical solution will possess the derivatives ut, Du, D2u in the classical sense. Simi- lar remarks apply to subsolutions and supersolutions.) The reason is tha t the regularity theory of sufficiently nondegenerate and nonsingular equations is still unsettled. In particular, it may be that nondegenerate nonsingular equations F = 0 with smooth F admit only classical solutions, although some suspect tha t this is not so.

However, if the equation is first order (so very degenerate), then examples are easy. The next exercise gives a simple problem without classical solutions and for which there are solutions slightly less regular than "classical"; however allowing less regular solutions generates "nommiqueness".

E x e r c i s e 1.1.* Pu t N = 1, ~ = ( - 1 , 1 ) and F ( x , r , p , X ) = [pl 2 - 1. Verify tha t there is no classical (here this means C 1 ( - 1 , 1) A C( [ -1 , 1])) solution u of F(u') = (u') 2 - 1 = 0 on (0, 1) satisfying the Dirichlet conditions u ( - 1 ) = u(1) = 0. Verify tha t u(x) = 1 - Ixl and v(x) = Ixl - 1 are both "strong" solutions: in

th is case, t h e y are Lipschi tz cont inuous and the equa t ion is sat isf ied poin twise excep t a t x = 0 (so a lmos t everywhere) .

Of course, t he p rob lem in Exercise 1.1 has a unique solut ion wi th in our theory, as we will see l a te r (it is u(x) = 1 - Ixl).

To fu r the r es tab l i sh the des i rab i l i ty of al lowing nondif ferent iable solut ions, we recal l the c lass ical m e t h o d of charac te r i s t i cs as it appl ies to the Cauchy p r ob l em for a H a m i l t o n - J a c o b i equa t ion ut + H(Du) = O:

ut + H ( D u ) = 0 for x C ]R N, t > 0 (1.1) / ~t(0, X) = ffJ(X), for X C IR N.

Suppose t h a t H is smoo th and t h a t u is a smoo th solut ion of ut + H ( D u ) = 0 on t >_ 0, x E IR N. Define Z(t) E IR N to be the solut ion of the ini t ia l value p r ob l em

Z'(t) : ~Z(t) = DH(Du(t,Z(t))), Z(O) : 9

I

over the largest interval for which this solution exists. A computation yields

Ou d Du(< Z(t)) = D ~ ( t , Z(t)) + D2u(t, Z ( t ) )Z ' ( t )

Ou = D-~-[(t, Z(t)) + D2u(t, Z ( t ) ) D H ( D u ( t , Z( t ))

= 0

where the las t equa t ion arises from di f ferent ia t ing ut + H(Du) = 0 wi th respec t to x.

R e m a r k 1.1. In ca lcu la t ions such as the above, one has to decide whe the r the t he g rad i en t Dv of a scalar f lmct ion v is to be a co lumn vector or a row vector . T h e r e is no a m b i g u i t y a b o u t D2v, for it is to be square and s y m m e t r i c in any case. In t he i n t r o d u c t i o n we wro te the g rad ien t as a row vector, bu t above in t e rp re t it as a co lumn vector. Th is is cons is ten t wi th in t e rp re t ing po in ts of IR N as co lumn vec tors while wr i t ing row vectors , and wi th these s loppy convent ions the above is correct .

We conc lude t h a t Du is cons tan t on the curve t -+ (t, Z(t)) . I t then would follow t h a t Z(t ) = 5c + tDH(Dr However, the resul t ing equa t ion Du(t, 5c + tDH(Dr -- D~b(3c) yields con t rad ic t ions as soon as we have charac te r i s t i cs crossing, t h a t is y # z bu t t > 0 such t h a t y+tDH(D~b(y ) ) = z + t D H ( D r In th i s case, one says t h a t "shocks form" and there are no smoo th solut ions u def ined for all t _> 0 in general .

E x e r c i s e 1.2. (i) Con t inue the analys is above to find

u(t, Z ( t ) ) = r + t((Dga(a?), DH(D~b(5:))) - H(D~(Sc)))

where (., .) is t he Euc l idean inner p roduc t . (ii) If N = 1, t hen shocks will form unless r --+ H' (~b ' (x)) is monotone .

Under reasonable assumptions, as is shown in elementary courses, analysis by characteristics provides a smooth solution of (1.1) until shocks form. When classical solutions break down, in this area and others, one is led to think of the problem of finding a way to continue past the breakdown with a less regular solution. However, one can also immediately think of the problem of finding solutions in cases where the da ta does not allow the classical analysis. E.g., what does one do if H and/or ~/~ above is not smooth? The "breakdown" idea is not central in this view.

Just as in the case of Exercise 1.1, relaxing the regularity requirement for a solution just a t iny bit leads to nonuniqueness for (1.1). One does not expect uniqueness in general for s ta t ionary problems, but one does expect uniqueness for initial-value problems.

E x e r c i s e 1.3. Consider the equation ut + (u~) 2 = 0 for t > 0, x E IR coupled with the initial condition u(0, x) - 0. Verify that the function

v(t,x)=--O for 0 < t _ < Ixl,

v ( t , x ) = - t + l x l for Ixl_<t,

satisfies the initial condition, is continuous and has all the regularity one desires off the lines z = 0, t = Ixl, and satisfies the equation off these lines. Thus u = 0 and v are distinct nearly classical - even piecewise linear - solutions of the Cauchy problem.

We have not given second order examples. However, here is a model equa- tion which will be covered under the theory to be described and for which the issue of how smooth solutions are is unsettled. Let As E S (N) , i = 1, 2, 3 satisfy I _ < A i < 2 I for i = 1, 2, 3 , 4 a n d

F(X) = - max(Trace (A1X), min(Trace (A2X), Trace (AaX))).

This is a uniformly elliptic equation - here this means that there are constants 0 < A < A such tha t

F(X + P) <_ F(X) - ~Trace (P) and IF(X) - F(Y)I _< A I I X - YII

for X, Y, P E $ ( N ) , P > 0. Here Ilxll can be any reasonable matr ix norm of X; a good one is the sum of the absolute values of the eigenvatues of X, as it coincides with the trace on nonnegative matrices.

E x e r c i s e 1.4. Determine A, A which work above.

I t is known tha t solutions of uniformly elliptic equations typically have H61der continuous first derivatives, but it is not known if these solutions are necessarily C 2. If the equation is uniformly elliptic and convex in X, regularity is known. See Evans [20], Cabr~ and Caffarelli [7], Dong [17], the references therein, as well as Trudinger [39] and Switch [37] for a recent result concerning Sobolev rather than Hhlder regularity.

2. T h e N o t i o n o f V i s c o s i t y S o l u t i o n s

As we will see, the theory will require us to deal with semicontinuous func- tions, there is no escape. Therefore, let us recall the notions of the upper semi- continuous envelope u* and the lower semieontinuous envelope u . of a function u : ~t ~ IR:

u*(x) = l i m s u p { u ( y ) : y E ~ , ] y - x ] < r} r$0

(2.1) U,(X) = l i m i n f { u ( y ) : y C ~t, l y - x l -< r} . riO

Recall t ha t u is upper semicontinuous if u = u* and lower semicontinuous if u = u . ; equivalently, u is upper semicontinuous if Xk --* x implies u(x) > limsupk__. ~ u(xk), etc. Of course, u* is upper semicontinuous and u . is lower semicontinuous.

E x e r c i s e 2.1.* In the above definition ~ could be replaced by an arb i t rary metric space (9 if lY - x ] is replaced distance between x, y c (9. Show in this general i ty t ha t u is upper semicontinuous if and only if v = - u is lower semi- continuous if and only if {x E (9 : u(x) <_ r} is closed for each r E IR. Show tha t a funct ion which is bo th upper semicontinuous and lower semicontinous is continuous. Show tha t if (9 is compact and u is upper semicontinuous on (9, then u has a max imum point ~ such tha t u(x) <_ u(:~) for x E (9.

Mot ivat ion for the following definition is found in Exercise 0.2; see also Exercise 2.4 below. The semicontinui ty requirements in the definition are par t ly explained by the last par t of Exercise 2.1 and the fact t ha t we will want to produce the max ima associated with subsolutions, etc., in proofs.

D e f i n i t i o n 2.1. Let F be proper, ~ be open and u : ~ --~ IR. Then u is a viscosity subsolution of F = 0 in ~ if it is upper semicontinuous and for every ~ E C 2 ( ~ ) and local m a x i m u m point ~ E ~ of u - ~, we have F(:~,u(J:), D~(~ ) , D2~(:~)) < O. Similarly, u : ~ --* IR is a viscosity superso- lution o f f = 0 in ~ i f it is lower semieontinuous and for every ~ E C2(~ ) and local m i n i m u m point :~ E ~ of u - ~, we have F(~ , u(~), D~(~) , D2~(2) ) > O. Finally, u is a viscosity solution of F = 0 in ~ if is both a viscosity subsolution and a viscosity supersolution (hence continuous) o f F = O.

R e m a r k 2.2. Hereafter we use the following conventions: "supersolution", "subsolution" and "solution" mean "viscosity supersolut ion", "viscosity subso- lution" and "viscosity solution" - other notions will car ry the modifiers (e.g., classical solutions, etc.). Moreover, the phrases "subsolution of F = 0" and "solution of F < 0" mean the same (and similarly for supersolutions).

R e m a r k 2.3. Explicit subsolutions and supersolutions which are semicontin- uous and not continuous will not appear in these lectures. They intervene ab- s t rac t ly in proofs, however, in an essential way.

E x e r c i s e 2.2.* Reconcile Definition 2.1 with Exercise 0.2 in the following sense: Show tha t if F is proper, u c C2(~) and

F(x , u(x), Du(x) , D2u(x)) <_ 0

( F ( x , u ( x ) , D u ( x ) , D 2 u ( x ) ) > 0) for x e ~, then u is a solution of F < 0 (respectively F > 0) in the above sense.

E x e r c i s e 2.3.* W i t h F as in Exercise 1.1, verify tha t u(x) = 1 - I x l is a solution of F = 0 on (-1,1), but tha t u(x) = Ix[ - 1 is not. A t t empt to show tha t u(x) = 1 - Ixl is the only solution of F = 0 in (-1,1) which vanishes at x = - 1 , 1. Verify tha t u(x) = Ixl - 1 is a solution of - ( u ' ) 2 + 1 = 0. In general, verify tha t if F is proper then u is a solution of F < 0 if and only if v = - u is a solution of G > 0 where G ( x , r , p , X ) = - F ( x , - r , - p , - X ) and tha t G is proper. Thus any result about subsolutions provides a dual result about supersolutions.

E x e r c i s e 2.4. In general, if ~ is bounded and open in IR N, verify tha t u(x) = distance(x, c9~) is a solution of ]Du I = 1 in ft.

We ment ion tha t the idea of put t ing derivatives on test functions in this m a x i m u m principle context was first used to good effect in Evans [18, 19]. The full definitions above in all their semicontinuous glory, evolved after the unique- ness theory was init iated in [14], [15]. The definition in these works was equiva- lent to t ha t above, but was formulated differently and all functions were assumed continuous. The paper [16] comments on equivalences and writes proofs more similar to those given today. Ishii 's in t roduct ion of the Perron method in [24] was a key point in establishing the essential role of semicontinuous functions in the theory. Ishii in fact defines a "solution" to be a function u such tha t u* is a subsolut ion and u . is a supersolution. See Bardi ' s lectures [2] in this regard.

3. S t a t e m e n t s o f M o d e l E x i s t e n c e - U n i q u e n e s s T h e o r e m s

Recalling the discussion of classical solutions of the Cauchy problem (1.1) and Exercise 1.3, the following results are a striking affirmation tha t the solutions in t roduced in Definition 2.1 are appropriate.

For Hami l ton-Jacobi equat ions we have:

T h e o r e m 3.1. Let H : IR N --+ IR be continuous and ~b : ]R N ~ IR be uniformly continuous. Then there is a unique continuous function u : [0, oo) • ] R N ---+ ]R with the following properties: u is uniformly continuous in x uniformly in t, u is a solution of ut + H ( D u ) = 0 in (0, oo) • ] R N and u satisfies u(O, x) = ~b(x) for x E I R N .

Even more striking is the following even more unequivocal generalization to include second order equations:

T h e o r e m 3.2. Let F : IR N x $ ( N ) ---+ ]R be continuous and degenerate elliptic. Then the statement of Theorem 3.1 remains true with the equation u t+ H( Du) = 0 replaced by the equation ut + F(Du , D2u) = O.

]0

The analogue of 3.2 for the stationary problem (i.e., without "t") is

T h e o r e m 3.3. Let F : IR N • S ( N ) --~ IR be continuous and degenerate elliptic and f : I ~ N ~ I~ be uniformly continuous. Then there is a unique uniformly continuous u : IR g -~ IR which is a solution of u + F(Du , D2u) - f ( x ) = in IR N"

Moreover, the solutions whose unique existence is asserted above are the ones which are demanded by the theories developed in the other lectures in this volume. In Bardi [2] and Soner [34] formulas are given for potential solutions of various problems, in control theoretic and differential games settings, and it is a tr iumph of the theory that the functions given by the formulas can be shown to be the unique solutions given by the theory.

All of the heavy lifting needed to prove these results is done below. However, some of the details are left for the reader's pleasure. The proof of Theorem 3.1 is indicated at the end of Section 9, the proof of Theorem 3.3 is completed in Exercise 10.3 and the proof of Theorem 3.2 is completed in Exercise 12.1.

4. C o m p a r i s o n for H a m i l t o n - J a c o b i E q u a t i o n s : t he Di r i ch le t P r o b l e m

The technology of the proof of comparison in the second order case is more complex than in the first order case, so at this first stage we offer some sample first order comparison proofs. As a pedagogical device, we present a sequence of proofs illustrating various technical concerns. We begin with simplest case, that is the Dirichlet problem. The next two sections concern variants. Arguments are the main point, so we do not package the material as "theorems", etc. All of the arguments given are invoked later in the second order case so no time is wasted by passing through the first order case along the way.

Let ~t be a bounded open set in IR N. The Dirichlet problem is:

(DP) H ( x , u , Du) = 0 in Q, u = g on 0~.

Here H is continuous and proper on ~ • IR • ]R N and g ~ C(c9~). We say that u : ft --~ IR is a subsolution (supersolution) of (DP) if u is upper semicontinuous (respectively, lower semicontinuous), solves H _< 0 (respectively, H > 0) in gt and satisfies g _< u on cg~t (respectively, u > g on a~).

E x e r c i s e 4.1. One does not expect (DP) to have solutions in general. Show that if N = 1, ~ = (0, 1), the Dirichlet problem u + u ~ = 1, u(0) = u(1) = 0 does not have solutions (in the sense of Definition 2.1!).

We seek to show that if u is a subsolution of (DP) and v is a supersolution of (DP), then u _< v. We will not succeed without further conditions on H. Indeed, choose f~ to be the unit ball and let w(x) E C1(~) be any function which vanishes on 0f~ but does not vanish identically. Then w and - w are distinct classical solutions (and hence viscosity solutions, via Exercise 2.4) of (DP) with H(x , u, p) = Ipl 2 - I D w l 2, g = 0. We will discover sufficient conditions to guarantee the comparison theorem along the way.

11

T h e idea of compar i son proofs for viscosity solutions is this: we would like to consider an interior m a x i m u m 2 of u(x) - v(x) and use H(:~, u(2) , Du(d~)) <_ 0, H(~?,v(dc),Dv(c?)) > 0 to conclude t h a t u(37) < v(2) or u _< v. A p r imary difficulty is t h a t u and v need not be differentiable at such a m a x i m u m ~?. Thus ins tead one chooses smoo th "test functions" 9~(x, y) for which u ( x ) - v ( y ) - 9 ~ ( x , y) has a m a x i m u m (~, ~)). Assuming tha t 2, !) E f~, 97 is a m a x i m u m of x -* u (z ) - p(.~:, 9) and so, by the definit ion of subsolution, H(~?, u(a?), Dx~(2 , ~))) <_ 0. Similarly, H(9 , v ( ~ ) ) , - D y e ( 2 , ~))) > 0 and then

H(Yc, u(5:), D~7:(:?, ~)) ) - H@, v(~), -Dye(re , [1)) <_ O.

I t remains to conclude t h a t u < v by playing with the choice of qo and perhaps mak ing auxi l iary es t imates .

Pick e > 0 and small and let us maximize

1 (4.1) '~(x ,y) = u(x) - v(y) - ~ l x - yl 2

over f t x fL Since �9 is upper semicont inuous a m a x i m u m (xc, ~)~) exists. The tes t funct ion 7)(x, y) = I x - y12/(2e) is chosen to "penalize" large values of I x - y [ when E is sent to zero. I t fur ther has the desirable p rope r ty t ha t D x ~ = - D y e , the ut i l i ty of which is seen below.

We p repa re a useful l e m m a abou t penalized m a x i m u m s of semicont inuous funct ions for use now and later.

L e m m a 4.1 . Suppose (~ C ]1~ N . Let w, q~ : �9 ---, IR, 0 <_ q!, and w , - q ! be upper semicontinous. Let

(4.2) Af = {z E O : q/(z) = 0} r O,

and sup(w(z) - ~ ( z ) ) < oo. zE�9

Let M~ = s u p z E o ( w ( z ) - e~(z)/e) for e <_ 1. I f z~ E 0 is such that

(4.3)

then

(4.4 / 1 o.

Moreover, and i f ~ E (9 is a cluster point of z~ as ~ ~ O, then ~ E N" and w(z) < w(~) for z E Af.

Proof. M~ is clearly a decreasing funct ion Of 0 < ~ <_ 1. Since supAr w _< ~/I~ <_ M1 < oo where Af is the n o n e m p t y set of (4.2), Mo = lim~+o M~ exists and is

12

finite. Lett ing g(5) be the left-hand side of (4.3), for 0 < #, 5 we have

~ ( ~ ) - ~ , ( z~ ) _> M~ -g (5) .

Taking # = 25 we conclude from the extreme inequMities that

1 -~ (z~ ) <_ 2 (M~ - ~ + g(~)) 5

and the right-hand side tends to zero as 5 ~ 0. Assume now tha t z~ ~ ~ E O along a sequence of 5% tending to zero. Then

0 = limsup~l 0 ~(zE) _> q(~) by lower semicontinuity, and ~ E J~. Moreover, by the upper semicontinuity of w,

elO \ 5 / zEH

[]

Since ~ x ~ is compact, the maximum point (S:~, ~)~) of �9 of (4.1) has a limit point c $ 0. It follows from Lemma 4.1 that

(4.5) l I ~ - ~ l 2 -~ 0 c

and any limit point has the form (2,2). If ~ c c0~, then u(2) _< g(2) < v(S:) shows tha t

lim sup ~(2~, ~)~) < 0. el0

If no such limit 2 E c0~ , then 2 ~ , ~ must lie in ~ for small 5. In this case, as explained before, we have

When does this information imply u _< v? For a simple example, let us assume that G has the form H(x, r,p) = r + G(p) - f ( x ) where f is continuous

I

on ~. Then (4.6) rewrites to

u ( ~ ) - v(#~) < f ( ~ ) - f(~)~);

in view of (4.5) and the uniform continuity of f the right-hand side tends to zero as 5 ~ 0 and we conclude again tha t

l imsup r ~)~) _< l i m s u p ( u ( ~ ) - v(~)~)) _< O. ~10 elO

13

Since u(x) - v(x) = ~ ( x , x ) _< (I)(~, ~)~), we conclude u <_ v in the limit ~ ~ 0. The case in which H has the form H(x , r ,p ) = G(r,p) - f ( x ) and H is strictly increasing in r uniformly in p E IR N is essentially the same.

In the above examples, the x dependence is "separated". When it is not, the situation is more subtle and it convenient to use the full force of (4.5).

E x e r c i s e 4.2. Establish comparison for (DP) when H(x, r,p) satisfies

IH(x,r ,P) - H(y,r ,P)I <- c~ YI( 1 + Ipl))

for some function satisfying aJ(0+) = 0 and H is sufficiently increasing in r. Show that the "sufficiently increasing" (however you formulated it) assumption can be dropped when there is a c > 0 such that either u solves H _< - c or v solves H > c.

It was remarked at the beginning that solutions of (DP) for equations of the form IDul 2 = f ( x ) are not necessarily unique. However, they are unique if f ( x ) > 0 in f~ as shown by the next two exercises.

E x e r c i s e 4.3.* Let F(x , r, p) be proper, u be a solution of F _< 0 (respectively, F _> 0) and consider a change of unknown function according to u = K(w) . Here K is continuously differentiable, K'(r) > 0 for r in the domain of K and the range of K includes the range of u. Show that w is then a subsolution (respectively, su- persolution) of the resulting equation, G(x, w, Dw) = F(x , K(w) , K ' (w)D w) = 0. Note, however, that G may not be proper. Discuss the second order case.

Exe rc i s e 4.4.* Let f E C(-~), f ( x ) > 0 for x E ft. Find a change of unknown in the equation I D u l 2 - f(x) = o which - with a little massaging - produces a proper equation H(x , w, Dw) = 0 for which comparison in the Dirichlet problem holds.

Except for the semicontinuous generality, which plays a small role, com- parison results of these forms have been known since [15]. However, the proofs above are certainly clearer than the original ones.

5. Compar i son for Hami l ton-Jacobi Equat ions in IR N

The point of this section is to indicate how to handle unbounded domains. The reader may skip ahead now to Section 7 if desired.

We consider the model stationary Hamilton-Jacobi equation on IRN:

(SHJE) u + H(Du) = f ( x ) for x �9 ]~N.

Means to treat more general equations used in the previous section will also work here, and we focus only the modifications of arguments required by the unbounded domain IR N. Everywhere below, u, v : ]R N --~ IR, u is a subsolution and v is a supersolution of (SHJE). Moreover, H, f : IR N --+ IR, H is continuous and f is uniformly continuous. The goal is again to prove u < v.

14

We suppose t h a t u, v are bounded on IRN; this is relaxed in the next section. For 0 < e, d define the funct ion

1 Ix - yl 2 ~ ( x , y) = u ( z ) - v ( y ) - ~ - ~ (1< 2 + ly] 2)

on IR N x IR N. T h e t e r m $(Ixl 2 + lyl2)/2 is present to guarantee t ha t r has a m a x i m u m on its unbounded domain and will be removed by sending (5 ; 0. �9 is uppe r semicont inuous and, since u(x) - v(y) is bounded above by assumpt ion ,

tends to - o o as Ixl, lyl -~ oo. Thus �9 has a m a x i m u m point (a?, 1)) (it depends on e,6; however we no longer indicate this dependence) . Proceeding as above we have

From the assumed boundedness of u, v, it follows tha t

1 (5.2) s u p ( ~ ( z ) - ~(v) - ~ Ix - yl 2) < oo.

x , y

A slight modif icat ion of L e m m a 4.1 then yields

g (5.a)

where

(5.4) lim lira sup C~,e = O. riO 510

In addit ion, for e < 1, u(0) - v(0) < ~b(,, 1)) and (5.2) imply

c5 1 ^ (la?l 2 + 1912) < v(0) - ~ ( 0 ) + u(a?) - v ( 9 ) - ~ lx - ~ 1 2 <_ c1

I t follows f rom (5.3) and the above that :

(i) ta~ - 91/e = (la? - ~)12/e)~/2/v~ _< (c~,~/e) 1/2,

(5.5) (ii) I :~ -~ ) l -< (Ce,,hc) 1/2,

(iii) ~15:[ + ~l~)l = v/~((6la?12) 1/2 + (~l;)12) 1/2) -< 2x/~c~/2-

L e t / ) f be the modu lus of cont inui ty of f , t ha t is the least nondecreasing funct ion such t h a t

IS(x) - f (y ) l <- as ( Ix - yl);

p) is continuous. Likewise, the merely continuous function H is uni formly con- t inuous on compac t sets, so there is a least funct ion PH such t h a t

I H ( p + q ) - H ( p ) I _ < p H ( R , r ) for IPl _<R, IV[ _<r.

15

We have as well p/(O+) = pH(R, 0+) = 0 for R > 0. Return ing to (5.1), we may use (i)-(iii) above to conclude (with m~seemly precision) t ha t

U(:~) -- V(9 ) < pH((Ce,6/g) 1/2, 2v/~C~/2) -t- pf((cVe,6)l/2).

Thus

Therefore

limsup(u(:~) - v(~))) _< pf((elimsupC<5)~/2). 51o ~o

u ( x ) - v(y) I x - Yl2 - l im~(x , y ) < lim~(a?,~)) < (5.0) 2c 510 - 510 -

limsup(u(.~) - v(*))) < p/((sl imsupC~,5)l /2) . 510 510

Pu t t ing x = y and lett ing c --+ 0, we conclude tha t u(x) < v(x) for all x. Another use of es t imates like (5.6) is this: if u itself is a bounded solution

of (SHJE), we may take u = v in (5.6) to conclude tha t

u ( x ) - u ( y ) <- 0<s_<l ( i n f --Yl22~-- +P/((climsupCe'5)l/2)~'bio -j

which provides a modulus of continui ty for u determined by f . This me thod generalizes to allow H(x,p) to depend on x as well; in the current case there is a simpler way to obta in a modulus for a solution u.

E x e r c i s e 5.1. Generalize the above to show tha t if f , 9 : IRN --~ IR are uniformly continuous and u, - v are upper semicontinuous and bounded, u solves u+ H(Du) < f and v solves v+ H(Dv) > 9, then u ( x ) - v ( x ) <_ supzc~N( I ( z ) -- g(z)). If U is a solution of u + H(Du) = f and z E IR N, then v(x) = u(x + z) is a solution of v + H(Dv) = g with 9(x) =: f ( x + z). Consequently, u(x) - u(y) <_

- y l ) .

6. Hamil ton-Jacobi Equations in IRN: Unbounded Solutions

We treat some technical difficulties "at oo" caused by allowing unbounded u, v in the problem t rea ted in the Section 5. The devices used adapt to the second order case. The reader m a y skip ahead to Section 7 at this t ime wi thout disrupt ing the flow.

This time, we allow a linear growth at infinity, tha t is

(LG) u(x) - v(y) <_ L(Ix I + l y l + l ) for x, y E IR N

for some cons tant L (note t ha t this amounts to bounding u , - v separately from above). We show tha t then u < v. A review of the proof of Section 5 shows tha t bounds on u, v were not in fact needed except in so far as they guaranteed (5.2) (which itself guarantees the existence of the maxima used). If we verify (5.2) (or an close subst i tute) , we will be finished.

16

R e m a r k 6.1. The linear growth is "critical" in the class of powers of Ixl. The equation u - IDul ~ = 0 with 7 > 1 has the two distinct solutions u ~ 0 and u = ((7 - 1 ) / Y ) ~ / ( ' ~ - l ) l x l ~ / (~-1) . Choosing 7 large, the growth is as close to linear as we please.

The next exercise is used immediately.

E x e r c i s e 6.1.* Let f : IR N ~ IR be uniformly continuous and

of(r) = s u p { I f ( x ) - f(Y)l : x , Y ~ I R N , Ix-yl <r} be its modulus of continuity. Show that p f ( r + s) < p i ( r ) + fiX(S) (pf is subad- ditive) and p s ( r ) < p f ( 5 ) + ( p f ( f ) / ( 5 ) r for 0 _< r, s, 0 < s

Since f is uniformly continuous on IR N, by the above exercise it admits the estimate

(6.1)

w h e r e / ( ~ pf(1). We claim that

I f ( x ) - f ( y ) l - K l x - Yl ~ K

sup ( ~ ( x ) - v ( y ) - K I x - Yl) < oo 1R N x 1R N

(and then u ( x ) - v ( y ) - K I x - yl 2 is bounded as well). tn view of (LG), the upper semicontinuous function

�9 ( x , y ) = ~ ( x ) - v ( y ) - K(1 + Ix - y12) 1/2 - ~(IxP + lyl ~)

attains its maximum at some point (37,/)). (LG) and u(0) - v(0) < r 7)) imply that

~(1~:12 + I~)l 2) _< v(O) - ~ (o ) § u ( ~ ) - v(~)) <_ c + L(IS:I + I~)l)

which implies

(6.2) ~5(IJ:l + lYl) -< C

where the constants C are various. Using the equations

( + '-~:~ - t.)_ ~)t2) 1/2 _ @ ) _ u(5~) - v(?)) < H K(1

(6.3)

H K(1 + 5 ~ + f ( ~ ) - f ( y ) .

The key thing here is that

I:~ - DI < 1 (1 + I~ - ~1~) ~/~ -

17

while we also have (6.2). Thus the arguments of H above are bounded indepen- dent ly of 6. Invoking (6.1) as well, we conclude tha t

But then

u(k) - v(?)) - K(1 + 15 - I)12) 1 /2 _< % ( 5 ) - v (~) ) - KI5 - ~)l < C

and finally

�9 (x,y) < (I)(5, 7)) < u ( 5 ) - v(t)) - K ( 1 + 15 - 912) 1/2 _< c .

Passing to the limit as 6 1 0, we conclude tha t u ( x ) - v (y ) - K(1 + Ix - yl2) 1/2

is bounded.

E x e r c i s e 6 .2. Show tha t linearly bounded solutions of (SHJE) are uniformly continuous.

See the references given in [12, Section 5D].

7. D e f i n i t i o n s R e v i s i t e d - S e m i j e t s

In this section the notion of Definition 2.1 will be recast for later conve- nience. The definition itself involves ex t rema of differences u - ~ and then evaluat ion of the equat ion at da ta from the second order Taylor expansion of

at these extrema. It is only the information from the expansion of ~ which matters , and we now emphasize this.

If ~ C Cl ( f t ) and u - ~ has a local max imum relative to ft at 5 E t2, then

~(x) _<~(5) + ~ (x ) - ~ (5 ) = (7.1)

u ( 5 ) + ( p , x - 5 ) + o ( I x - 5 1 ) as f ~ x - ~ 5

a n d if ~ E C2(ft) , then

~(x) _< ~(5) + ~ (x ) - ~ (5)

(7.2) = ~(5) + (p, x - 5) +

l(X(x 5),x 5)+o(Ix-512 ) as a ~ x 5 2

where

(7.3) p = Dg)(5) and X = D2qo(~?).

Conversely, if p E IR g and (7.1) holds, then there exists ~ c Cl ( f / ) such tha t u - ~ has a s t r ic t maximum at a? and Dr)(5) = p. Here a max imum 5 of u - is str ict if there is a nondecreasing function h : (0, oc) --+ (0, oc) and r0 > 0 such tha t

(7.4) u ( x ) - ~ p ( x ) < u ( 5 ) - ~ ( c ? ) - h ( r ) for r < l x - 5 I < r 0 .

18

Let us call h the "strictness" in this situation. The proof goes like this: assume (7.1) and set

~(~) -- s . p { ( ~ ( x ) - ~ ( ~ ) - < p , x ~ > ) + : x �9 ~ , Ix - ~1 -< ~}.

By (7.1), g(r) - o(r) as r I 0 and is nondecreasing. Choose a continuous nondecreasing ma jo ran t 0 with the same properties; tha t is, we want g(r) < O(r), 0(r) = o(r) and ~ is nondecreasing. Now- put

and check tha t

G(r) 1 f2< : - 0 ( ~ ) & P

~(x) : c(Ix - ~I) + <>z - }> + Ix - ~I 4

has the desired properties (with h(r) -- r 4 as the strictness). Exercise 7.i.* Formulate and prove the corresponding statement for (7.9).

We focus on the second case. The quadratic appearing on the right hand side of (7.2) is defined by the "jet" (u(3c),p, X) and we write

(~(~),;, x ) c J~,+~(~)

when (7.2) holds. The quant i ty u(a~) on the left appears to be redundant , but is incorpora ted for technical reasons. For any function u : ~ --+ IR, J2,+u maps ~ into the set of subsets of ~ • 11:{ N • $(N) (and the empty set may well be a value).

E x e r c i s e 7.2.* Let u(x) = Ix] for x ~ 1R. Compute J2'+u(O) and J2'-u(O). Whenever J2,+u(x) is not empty, it is infinite, since whenever (7.2) holds for

p ,X it also holds for p ,Y whenever X _< Y. Likewise, we define (u(~),p,X) E J2 , -u(s to mean tha t (7.2) holds with the inequality reversed.

E x e r c i s e 7.3. Define the first order analogues j l , + , j , , - of the second order semijets. Observe tha t (u(x),p, X) E J2'+u(x) implies (u(x),p) E Jl'+u(x), but show by example tha t the converse fails: J2,+u(x) may be empty while Jl,+u(x) is nonempty.

E x e r c i s e 7.4. Let ft be open and u : f~ --+ IR be upper semicontinuous. Show tha t {x �9 f~ : J2'+u(x) r 1?} is dense in f~. Show tha t if Jl,+u(:~) n J1,-u(0?) is

nonempty, then there is a p E IR N such tha t

u(x)=u(:~)+<p,x- :~}+o( lx - :~ l ) for x C f t ;

in this case we say tha t u is differentiable at a? and p = Du(3:). Conclude tha t there are cont inuous functions u : (0, 1) -* I1% such tha t

g2'+u(x) N J 2 ' - u ( x ) = 0

for every x E (0, 1).

E x e r c i s e 7 .5, Let u : IR N -~ IR and lu(x)-u(y)l <_ L I x - y I for x, y �9 ]R N (i.e., u is Lipschitz cont inuous with constant L). Show tha t if (u(0~),p) E Jl '+u(a?),

19

then ]p] < L. Conversely, if (u(x) ,p,X) E J2'+u(x)implies t ha t [p[ < L, show t h a t u is Lipschitz with constant L.

According to Exercise 7.1, an upper semicontinuous function u : ~t --~ ]R is a subsolut ion of a proper equat ion G = 0 if and only if G(x,u(x) ,p ,X) <_ 0 for every x E ~ and (u(x) ,p,X) ~ J2'+u(x). If G is continuous (or even lower semicontinuous) , the relation

a(x, u(x), p, x ) <_ o

persists under taking limits, and this leads us to define the closure -]2'+u of

J2'+u. This goes as follows: (r ,p ,X) c 72 '+u(x) if there exists

xn x and �9 J '+u(xn)

such that (u(xn),pn, Xn) -~ (r,p, X).

We have then that an upper semicontinuous function u : ~ -~ IR is a

subsolution of a proper equation G = 0 if and only if G(x, r, p, X) <_ 0 for every

x E Q and (r,p,X) c J2'+u(x). Note that upper semicontinuity of u implies

t ha t r <_ u(x), so perhaps G(x, u(x), p, X) > O. One defines J 2 ' - u similarly and then u : f~ + IR is a subsolution of a

proper equat ion G = 0 if and only if G(x , r ,p ,X) >_ 0 for every x E ~2 and

( r , p ,X)

R e m a r k 7.1. The nota t ion in use here differs from tha t in [12] in tha t the values of j2 ,+ are taken here to include "u(x)" and this was not so in [12]. N o b o d y much likes this "jet" business and perhaps we should refer to "second order superdifferentials" or some such. There seems to be a law of conservation of pedant ic excess in a t t emp t s to resolve this issue. It is a bookkeeping question, and when we get to the Theorem on Sums, one needs to do the bookkeeping somehow.

The cons t ruc t ion of the test functions used to show tha t the "jet" formula- t ions are equivalent to the "maximum of u - ~" formulat ions appears in Evans [18].

8. S t a b i l i t y o f t h e N o t i o n s

We begin by considering two related issues. First, if ~ is a collection of solutions of F < 0, then ( supuc7 u)* (recall (2.1)) is another subsolution. Next, if u~ is a solution of F~ < 0 for n = 1, 2 , . . . , F~ = 0, and u~ --* u, F~ --* F in a suitable sense, then u is a subsolut ion of F = 0. These facts are linked in t ha t they bo th rely on the following result.

The reader will notice in the s ta tement below tha t there is a set O C ]R N which is "locally compac t" ; for example, bo th open sets and closed subsets of IR N contain a compac t (relative) ne ighborhood of each point and so are locally

20

compact, as are various other sets. In fact, the above considerations easily generalize to allow locally compact sets ft in the definition of the jets, etc. If ft is locally compact , we take p E C2(ft) to mean it is the restriction to ft of a twice continuously differentiable function defined on a neighborhood (in IR N) of ft. The relation (7.2) needs no modification, for we have already appended "as f~ ~ x --~ a?" to emphasize that ft may not be open, etc. The jet "operators"

~,+ 2,- J a and J~ when f~ is j2,+, j 2 , - are writ ten J~ , J~ and their closures --2,+ - 2 , - not necessarily open. At the moment, we pose this technical generality "because we can" and it doesn ' t affect the presentation; however, it is essential in various ways in other parts of the theory.

P r o p o s i t i o n 8.1. Let (9 C IR N be locally compact, U : (9 ~ IR be upper 2 , + Z semicontinuous, z E (9 and ( U ( z ) , p , X ) c J~ U ( ) . Suppose also that un is a

sequence of upper- semicontinuous functions on �9 such that

(i) there exists x~ ~ (9 such that (x,~,u,~(xn)) -~ (z ,U(z)) and

(ii) if z,~ ~ (9 and z,~ ---+ x E O, then l imsupun(z~) _< V(x). n- -+ OO

(8.1)

Then

(s.2) 2,+ there exists fc~ c (9, (Un(JC,~),p~,Xn) E Jo un( n)

such that (Jz~,un(3c~),p~,X~) --~ ( z , U ( z ) , p , X ) .

Before proving this result, we use it.

P r o p o s i t i o n 8.2. Let 5 be a nonempty collection of solutions of F <_ 0 on where F is proper and continuous. I f U(x) = s u p ~ 7 u(x) and U* is finite on ft, then U* is a solution of F <_ 0 on fL

Proof. Suppose z c f t and (U*(z),p, X ) c J2'+U*(z). It is clear that there exists a sequence u~ E 5 r and xn c ft such that (x,~,Un(Xn)) --+ (z,U*(z)) and that then the assumption of Proposition 8.1 are satisfied with U replaced by U*. Let (2n, Un (2n), p~, Xn) -+ (z, U* (z), p, X) be as in Proposition 8.1; by assumption F(2n,u,~(:~,~),pn,Xn) <_ 0 and we conclude F ( z , U * ( z ) , p , X ) <_ 0 in the limit. []

We use Proposit ion 8.1 again. For an arbi t rary sequence of functions u n on (9 we can form the smallest

function U such tha t if (9 ~ Xn -~ x E O, then limsupn_~oo u,~(Xn ) <_ U(x). U is given by

U(x) = lim sup Un(y)

= lim s u p ~ u n ( y ) : n > _ m , y E ( 9 , l y - x l < - 1 fo, ---~ OO / m )

21

we wr i t e U = l im sup~__.~ us . In the oppos i t e sense, we define

l im inf us = - lira sup* ( - u n ) . n - - * o o . n ___~ o o

Note t h a t for any x E (9 the re exists sequences n j --+ OO and O ~ xnj --~ x such t h a t u~j ( x ~ ) ~ U(x) .

E x e r c i s e 8 .1 . Show l im sup~__.oo u~ is upper semicont inuous and t h a t if u~ -= u for all n, t hen " * = u , h m s u p ~ o ~ un * the uppe r semicont inuous envelope of u.

T h e following s t a t e m e n t is remarkable , in t ha t it p roduces a subso lu t ion of a l imi t p rob l em from an arbitrary sequence of subsolu t ions of a p p r o x i m a t e problems. No cont ro l of der ivat ives of any kind is assumed.

T h e o r e m 8.3 . For n = 1, 2 , . . . , let u,~ be a subsolution of a proper equation Fn = 0 on f}. Let U = l im s u p * ~ o ~ u~ and F be proper and satisfy

F < lira inf Fn. n ---~ o c .

I f U is finite, then it is a solution of F <_ O. In particular, i f Un ---* U, Fn --~ F locally uniformly, then U is a solution of F < O.

Proof. Accord ing to t he above discussion, if (U(z) , p, X ) c J2 '+U(z) , t hen there is a subsequence of t he u,~ (which we again call un) such t h a t the hypo theses of P r o p o s i t i o n 8.1 are satisfied. If ( ~ , u(x,~),pn, Xn ) is as in the propos i t ion , our

a s sumpt ions imp ly

F ( z , U ( z ) , p , X ) <_ l im in f Fn(SCn,Un(Xn),pn,Xn) <<_ O. n ~ c ~

[]

R e m a r k 8 .4 . Recal l Exercise 2.3, whereby resul ts concerning subsolu t ions a u t o m a t i c a l l y i m p l y the co r respond ing resul t for supersolut ions .

E x e r c i s e 8 .2 . T h e equa t ion

N

- A p U = ~-~,(IDulP-2u~,)x{ : 0 i : 1

is cal led the "p-Laplace" equat ion; p is a number and we focus on large p. C a r r y i n g ou t t he di f ferent ia t ions , show t h a t th is is a degenera te el l ipt ic equa t ion (and t hen p rope r as it does not have a "u" dependence) . Suppose Up is a so lu t ion of th is equa t ion for large p and t h a t Up --~ u uni formly on ~t as p --* oc. Show t h a t u is t hen a so lu t ion of the "infini ty Laplace" equat ion ,

N

- - /~oeU : -- ~ Ux~UxjUx~,x j : O.

i , j = l

22

E x e r c i s e 8 .3. Suppose tha t {un} are uniformly bounded and un solves

Us + H ( D u s ) - l z ~ u n ~ - f ( x ) 7Z

on IR N where f is uniformly continuous. Prove via comparison for u + H ( D u ) = f t ha t then

l imsup*us < l iminf us. S ~ O O n - -+ ( x ) *

Show this implies t ha t u~ converges locally uniformly to a limit u, and u is the unique bounded solution of u + H ( D u ) = f .

Proof of Proposition 8 . / .Wi thout loss of generali ty we put z = 0. By the as- sumpt ions and Exercise 7.1, there is an r > 0 such tha t Nr = {x E 59 : lxl _< r} is compac t and a twice cont inuously differentiable function p defined in a neigh- borhood of O such tha t

(8.3) U(x) - ~(x) <_ U(0) - ~(0) for x c NT,

the m a x i m u m 0 is bo th strict and the only max imum of U - ~ in N~, p = Dg)(0), and X = D2~(0) . By the assumpt ion (8.1) (i), there exists x~ E (9 such tha t (x,~, Ur~(Xs)) -~ (0, U(0)). Let a?~ E Nr be a max imum point of us(x) - g)(x) over N~ so tha t

(8.4) us(x) - ~(x) <_ u~(3:,~) - ~(:cs) for x E Nr.

Suppose tha t (passing to a subsequence if necessary) ~n --+ Y as n --+ co. Pu t t ing x = x~ in (8.4) and taking the limit inferior as n --+ oo, we find

U(0) - ~(0) < lim inf un (a?n) - p(y) ;

on the o ther hand, by (8.1) (ii) lim inf un(a?,~) < U(y). Thus U ( 0 ) - ~ ( 0 ) < U ( y ) - ~(y) and we conclude tha t y = 0 (because 0 is the only maximum) and tha t the inequali ty lim inf un(5:n) <_ U(O) cannot be strict - tha t is, (:?s, us(0?~) --+ (0, U(0)). Since this holds no mat te r the subsequence, it holds wi thout passing to a subsequence. Final ly

J2'+Un(~On) 9 (Un(}n), D(p(~On), D2~(~,~))--* (U(O),p, X )

and we are done. [ ]

R e m a r k 8.5. Propos i t ion 8.1 could be restated in the form it is proved; a strict m a x i m u m of U - ~ per turbs to max ima of u~ - ~ which converge, etc. Which form you prefer is a ma t t e r of taste; this au thor "thinks" in terms of the da ta at a point while el iminat ing "jets" in the s ta tement does make it less unat t ract ive .

Propos i t ion 8.2 is due to Ishii [24]. Theorem 8.3 is of great uti l i ty in ap- plications and is due to Barles and Pe r thame [4], [5]. See the comments of [12, Section 6] and Barles [3], for fur ther orientation. In the case of uniform conver- gence, Evans [18], [19] already employed the essential idea. Sophist icated limit

23

questions are discussed in Souganidis [35] and many references are given. Barles and Souganidis [6] is typical significant contribution reflecting the utility of the ability to take limits easily. Exercise 8.3 reflects the origin of the term "viscos- ity"; if one can solve regularized equations by adding "artificial viscosity", then passage to the limit is an easy matter . Note that with the technology of this section, one does not even need to estimate the modulus of continuity of the un. Exercise 8.2 is light-hearted, and is from Jensen [29], which is quite fascinating. At the moment, a pure viscosity solution proof of the uniqueness theorem in [29] (comparison for the Dirichlet problem for - A o o u = 0) is not known.

9. E x i s t e n c e V i a P e r r o n ' s M e t h o d

We establish existence results for Dirichlet problems via Perron's method. Below (DP) means:

(DP) F ( x , u , Du, D 2 u ) = O in f~, u = g on 0fL

Subsolutions, etc., are defined exactly as in Section 4. F is to be proper and continuous, while g is continuous. We call the following

implementat ion of "Perron's Method" Ishii's Theorem, as it is a good example of the method he introduced into this subject. At this stage, we have not verified the hypotheses of the theorem for second order equations, but will take this up later. The assumptions have been verified in first order cases in Section 4.

T h e o r e m ( I sh i i ) . Let comparison hold for (DP); i.e., i f w is a subsolution of (DP) and v is a supersolution of (DR), then w << v. Suppose also that there is a subsolution u and a supersolution ~ of (I71)) which satisfy the boundary condition u_.(x) = g*(x) = g(x) for x r ~)f~. Then

(9.1) W ( x ) = sup{w(x) : u < w <_ ~ and w is a subsolution of (DP)}

is a solution of (DP).

The first step in the proof of Ishii's Theorem was given in Proposition 8.2. The second is a simple construction which we now describe. Roughly speaking, it says tha t if a subsolution of (DP) is not a solution, then it is not a maximal subsolution. Of course, if comparison holds for (DP) and it has a solution, then the solution is the largest subsolution. We have to take care of semicontinuity considerations. Suppose tha t ~ is open, u is a solution of F _< 0 and that u. is not a solution of F >_ 0; in particular, assume 0 E f~ and we have

(9.2) F(0, u.(0) , D~(0), D2~(0)) < 0

for some ~ C C 2 such tha t for some r0 > 0

u , (x) - ~(x) > u.(0) - ~(0) + h(r) for r _< Ixl _< r0

24

where h > 0 is the strictness of the minimum. Adjusting ~ by a constant, we assume u.(0) = ~(0). Then, by continuity,

F(z , 7:(x) + 6, Dp(x) , D2cr ) < 0

for 0 < 6 < 6o and Ix] _< ro provided 60 and ro are sufficiently small; that is, u~ = 7)(x) + 6 is a classical solution of F < 0 in Ix[ _< to. Since

u(x) >_ u . (x) >> ~(x) + h(r) for r<_lx l_<r0 ,

if we choose 6 < (1/2)h(ro/2), then u(x) > u~(z) for ro/2 <_ Izl _< r0 and then, by Proposition 8.2, the function

= if Ixl < r0, U(x) [ u(x) otherwise

is a solution of F < 0 in ft. The last observation is that in every neighborhood of 0 there are points such that U(x) > u(x); indeed, by definition, there is a sequence (x~, u(x~)) convergent to (0, u.(0)) and then

lim (U(x~) - u(x~)) us(0) - u~(0) = u.(0) + 6 - u.(0) > 0. f t ~ O O

We summarize what this "bump" construction provides in the following lemma, the proof of which consists only of choosing r0 sufficiently small.

L e m m a 9.1. Let ft be open, and u be solution o f F < 0 in ft. I f u. fails to be 2 - ^

a supersoIution at some point 3c, i.e., there exists (u.(Jc),p,X) E J~' u , (x ) for which F(fc, u.(:?),p, X ) < O, then for any small ~ > 0 there is a subsolution U~ of F <_ 0 in ft satisfying

{ U~(x) > u(x) and sup(U~ - u) > 0, (9.3)

= u ( x ) for x e a n d Ix - 51 _>

Proof of Ishii's Theorem.With the notation of the theorem observe tha t u. _< W. _< W <_ W* _< g* and, in particular, W. = W = W* = 0 on 0fL By Proposit ion 8.2 W* is a subsolution of (DP) and hence, by comparison, W* _< g. I t then follows from the definition of W tha t W = W* (so W is a subsolution). If W. fails to be a supersolution at some point 5: C ft, let W~ be provided by Lemma 9.1. Clearly _u _< W~ and W~ = 0 on Oft for sufficiently small n. By comparison, W~ < g and since W is the maximal subsolution between _u and g, we arrive at the contradiction W~ < W. Hence W. is a supersolution of (DP) and then, by comparison for (DP), W* = W <_ W., showing that W is continuous and is a solution. []

Ishii 's Theorem leaves open the question of when a subsolution _u and a supersolution g of (DP) which agree with g on 0f~ can be found. Some general

25

discussion can be found in [12, Section 4]. Here we only discuss simple i l lustrat ive cases to show the power.

First , according to Exercise 4.4, compar ison holds for the equat ion IDul 2 - f ( x ) = 0 if f > 0 on ft. P u t g = 0. Assuming t h a t f > 0, H(x,O) - f ( x ) <_ O, and we have a subsolut ion, name ly u ~ 0. To find a supersolut ion, we rely on Exercise 2.4. Tak ing u = Mdis t ance (x , 0~ ) , MID u I = M and if M Z supa f , u is a supersolut ion. We conclude t h a t a unique solution exists. No requi rements had to be laid on Oft.

Next we take up the cons t ruc t ion of a supersolut ion for a general class of un i formly elliptic opera tors . T h e cons t ruc t ion is s t andard (see e.g. Gi lbarg and Trudinger [23]); bu t the presenta t ion is made consistent wi th the t h e m e here. The implicat ions, v ia Ishii 's T h e o r e m and compar i son results to follow, is a quite general existence and uniqueness theorem. At this juncture , we r emark t h a t the appl ica t ions of the theory to degenera te equat ions is more significant, bu t we leave it to the reader to visit [12] or o ther works in this regard. Nonetheless, even in the uni formly elliptic case, we are outs ide the rea lm of classical solutions, regular i ty is not known for general uniformly elliptic equations.

Define the " t race" norm on S ( N ) :

( 9 . 4 ) IlXll = /~Ceig(X)

where eig(X) is the set of eigenvalues of X counted according to their multiplic- ity.

E x e r c i s e 9.1.* Verify t h a t the t race norm is indeed a norm on X.

Recall t h a t F is called uniformly elliptic if there exists constants 0 < )` _< A such t h a t

F ( x , r , p , X + Z) < F ( x , r , p , X ) - ) ,Trace (Z) and (9.5)

I F ( x , r , p , X ) - F ( z , r , p , Y ) ] <_ AIIX - YII

for x , Y , Z E 8 ( N ) , Z >_ O. For example , F = - T r a c e ( X ) satisfies (9.5) wi th )` = A = 1. T h e two inequalities in (9.5) say first t ha t F(x, r,p, X ) is decreasing wi th respect to X at an a t least linear ra te and then t h a t it is Lipschitz cont inuous in X as well. A bit less intuit ive is the condit ion

F ( x , r , p , X ) - h T r a c e (Z) _< F ( x , r , p , X + Z) <_ (9.6)

F ( x , r , p , X ) - ) `Trace (Z) for Z _> 0,

but it is easy to see the equivalence or (9.5) and (9.6). We are assuming t h a t F is proper . Rewri t ing F = 0 as

F(x , u, Du, D2u) - F(x , 0, 0, 0) = F(x, 0, 0, 0),

we m a y as well consider the equat ion F = f and assume t h a t

(9.7) F(x , 0, 0, 0) = 0.

26

Finally, we add a condition of Lipschitz continuity with respect to the gradient:

(9.8) I F ( x , u , p , X ) - F ( x , u , q , X ) [ <_ TIP - ql

for some constant %

E x e r c i s e 9.2.* Show tha t for b C IR N, A E $ ( N ) , c c IR,

F ( x , r, p, X) = - T r a c e ( A X ) + (b,p) + er

is proper and satisfies (9.6) if and only if eig(A) c [~, A], Ibt <_ "y and c >_ 0.

The goal is to construct supersolutions of (DP) with 9 = 0 for the equation F = f where f E C(~) . Concerning the region ~, it is assumed that there is an r0 such tha t every point Xb E O~ is on the boundary of a ball of radius r0 which does not otherwise meet ~. For each X b let Z b be such that ]Zb--Xbl = T 0 < ]X--Zbl

for x E f~, x ~ Xb. We seek a supersolution of F = f in the form U(x) = G(r) where r = lx - Zbl which will satisfy U(Xb) = 0 and U _> 0 in ~2.

A computat ion shows

= a " ( r ) - (x - zb) (x- zb)5 + 6, j

T 2 T '

Thus any vector orthogonal to x - Zb is an eigenvector of D2G with G~(r)/r as the eigenvalue, while x - Zb is itself an eigenvector with eigenvalue G"(r) . Letting P be the orthogonal projection along x - Zb we have

D2G(Ix - Zbl) = G " ( r ) P + 1 G ' ( r ) ( I - P) . r

(9.9)

Taking

(9.10) 1 1

G ( r ) - r ~ r cr

for cr > 0, G is nonnegative, concave and increasing on r0 _< r. The decomposi- tion (9.9) above then represents D2G as an orthogonal sum of a positive matr ix and a negative matrix. Using this above in conjunction with F being proper, (9.6), (9.7), and (9.8) we find

F ( x , G, DG, D2G) >_ F ( x , 0, DG, D2G)

> F ( x , O, 0,, D2G) - "~IDGI

> F ( x , 0, 0, G" ( r )P ) - AG' ( r )T race ( I - P ) - 7G' (r)

> F ( x , O, 0, 0) - AG"(r) A(N - 1) G'(r) - vG ' ( r ) ?-

= - AG"(r) A(N - 1) G'(r) - vG' ( r ) . r

27

Using (9.10), we find

(7 (9.11) F(x,G, DG, D2G) >_ r - T ~ ( ( a + I ) A - A ( N - 1 ) - T r ).

Taking cr sufficiently large, we can guarantee that F(x, G, DG, D2G) > e; > 0 on f~ and then if M = supn f+/~, we have

F(x, MG, MDG, MD2G) >_ f

o n 02.

We do not yet have our supersolution, since MG does not satisfy the bound- ary condition MG = 0 on 0fL However

inf MG(lx - Zbl) x ~ , E O f ~

does. Moreover, it is continuous since the family {MG([x - Zbl : XD C 0f~} is equicontinuous; it is then also a supersolution by Proposition 8.2.

E x e r c i s e 9.3. Construct a supersolution for a general continuous boundary function g on 0t2. (Hint: g(xb) + e + MG(lx - zbl). )

E x e r c i s e 9.4. Verify that if each of FA,~ satisfies (9.6) with fixed constants A,A, then so does supAinf~ FA,B. Observe that the supersolution just con- structed depends only on structure conditions and the implications of this.

As a last example, we take up the stationary Hamilton-Jacobi equation of Section 6 for which we established comparison (if you skipped that section, you can either return or skip this example). First of all, it is clear from the proof given for the Dirichlet problem that Perron's method applies to u + H(Du) = f in 1R N where f is uniformly continuous. Coupled with the comparison proved in Section 6, to prove the unique existence of a uniformly continuous solution, we need only produce linearly bounded sub and supersolutions. We. seek a supersolution in the form u(x) = A(1 + ]x12)l/2 + B for some constants A and B. Since f is uniformly continuous, f (x) <_ KIz[ + K for some K and it suffices to have

A(1 + + + H (A(I ) +lxl2)l /~ >_K[x I + K for x E I R N

We may take A = K and then B large enough to guarantee this inequality (and subsolutions are obtained in a similar way).

Ishii [24] introduced Perron's method into this arena. The construction carried out for uniformly elliptic equations can be modified (incorporating an angular variable) to the situation in which there are exterior cones rather than balls at each point of 0f~, see Miller [33]. The existence theory in the uniformly elliptic case is completed in Exercise 10.2. It is interesting that the Dirichlet problem for the equation F(u, Du, D2u) = f(x) where f is merely in LX(t2) can be shown to have unique "viscosity solutions" as well. Of course, this requires adapting the notions and other machinery. See Caffarelli, Crandall, Kocan, and Swigch [8].

28

10. T h e U n i q u e n e s s M a c h i n e r y for S e c o n d O r d e r E q u a t i o n s

Let us begin by indicating why uniqueness for second equations cannot be treated by simple extensions of the arguments for the first order case.

We consider a Dirichlet problem

} u + F(Du, D2u) = f ( x ) in ft (DP) { u(x):~(x) on 0~

and seek to prove a comparison result for subsolutions and supersolutions. For

a change, we package a comparison result as a formal theorem.

T h e o r e m 10.1. Let f~ be a bounded open subset of IR N, f �9 C(~) , and F(p, X ) be continuous and degenerate elliptic. Let, u, v : ft ---* IR be upper semicontinuous and lower semicontinuous respectively, u be a solution of u + F(Du, D2u) < f , v be a solution o f v + F ( D v , D2v) > f i n f ' , a n d u < v onOfL T h e n u < v in

The strategy of the first order proof already given suggests that we consider a maximum of u(x) - v(y) - Ix - yl2/(2c) over f t x f~ and let e ; 0. Following the corresponding proof in Section 3, we may assume (2, 9) lies in f~ x ft if e is small, and apply the definitions to find that

u ( Y c ) + F ( c ? - Y ~ , ~ I ) <f(~?),_ v ( 9 ) + F ( X - 9 c , ~ / ) >f(9),_

which turns out to be useless since [ >_ - I . We need refined information about this maximum (~?, 9), which turns out to be a substantial and interesting issue. The information we need corresponds to the fact that if ~(x, y) = u(x) - v(y) - ix - y12/(2r has a maximum at (~, 9) and u, v are twice differentiable, then the full Hessian of ~5 in (x, y) is nonpositive, or

o ) ,(: , 7) o - D % ( y ) <- -~

the failed a t tempt above did reflect the full second order test for a maximum in the doubled variables (x, y). (The notation "I" is used here and later to denote

the identity in any dimension.)Since the matrix on the right annihilates ( ~ )

for ~ �9 IR N, the above inequality implies that D2u(b) <_ D2v(9), which is the sort of thing we need. To get there, we require some preparations.

The basic fact concerns semijets of differences (equivalently, sums), as hinted above. We call it the "Theorem on Sums".

T h e o r e m o n Sums . Let 0 be a locally compact subset of lR N Let u, - v : (.9 ~ IR be upper semicontinuous and ~ be twice continuously differentiable in a neighborhood of (9 • 0 Set

w(x ,y ) = u ( x ) - v ( y ) for x , y � 9

29

and suppose (:?, Y) �9 O x O is a local maximum of w(x, y) - ~(x, y) relative to 0 • O. Then for each ~ > 0 with nD27)(~, Y) < I there exists X, Y �9 8(N) such that

- - 2 , ^

(u(~), D~c2(5:, Y), X) �9 7~+u(:?), (v(9), -Dy~(S:, 9), Y) �9 Jo v(y)

and the block diagonal matrix with entries X , - Y satisfies

(10.1) ~ ; I I < ( X - 0 -YO )<(i_t~D2(p(:~c,9))_lD2qo(fc,9)._

We use the Theorem on Sums to prove Theorem 10.1.

Proof of Theorem 10./.Set

, ~ ( x , y ) = u ( ~ ) - v ( y ) - 2 s - y12

and consider a maximum (5, 9) over f i x fL Let :1(' ;,)

= ^ ,Y=Y s - - I

and note that A < (2/s ; moreover, for 2~/e < 1

( I - ~ A ) - I A - 1 ( I - I ) s - 2~ - I I "

By Theorem on Sums, there exists X, Y E $(N) such that

(u(~), ~ - 9, x ) �9 7~'%(~), (v(9), ~ - 9, y ) �9 72,-v(9) s s

(10.2) 1 i < ( X 0 ) < _ _ 1 ( I - I ) - 0 - Y - e - 2 s - I I "

Since the right hand side

Y and then F((2 - 9)/e, Moreover,

u(2) + F (a? \ s

SO

annihilates ( ~ ) f o r { c I R N , w e c o n c l u d e t h a t X _ <

Y) _< F((a5 - 9)/e, X) since F is degenerate elliptic.

Y,X <_f(s v ( 9 ) + F ( ,Y)>_f (9 )

u(~) - v@) < f(~) - f(Y)

and we conclude (using Lemma 4.1) that u(~) - v ( 9 ) - ~ 0 a s c ~ 0. Since

we conclude. []

3(1

Note that the choice ~ = c/3 in (10.2) yields

c - 0 - Y - E - I

E x e r c i s e 10.1. Extend the above comparison proof to the equation

P(x , u, Du, D2u) = 0

in place of u + F(Du , D2u) = f ( x ) under the conditions that there exists ~/> 0 such that

(10.4) "),(r - s) <_ F ( x , r , p , X ) - F ( x , s , p , X )

for r >_ s, ( x , p , X ) ~ ~ x IR N x S ( N ) and there is a function w: [0, ce] -4 [0, oc] which satisfies a3(0+) = 0 such that

( x - Y - F ( x - Y X ) , (10.5) F y,r, e ' e -

whenever x, y E ~, r E IR, X, Y E S ( N ) and (10.3) holds.

See [12, Section 3] regarding verifying the assumption (10.5).

E x e r c i s e 10.2.* Show, as in Exercise 4.2 that the "strictly increasing in r" assumption of the previous exercise can be dropped under the assumption that for c > 0 such that either u solves F < - c or v solves F > c. Show that if F satisfies (9.6) and v solves F >_ 0, then there are arbitrarily small radial perturbations r such that v + r solves F >_ c > 0.

R e m a r k 10.2. At this point, via Perron's Method, the supersolutions con- structed in Section 9, and the preceding exercise, we have uniquely solved (for example) the Dirichlet problem for F ( u, Du, D2u ) = f ( x ).

E x e r c i s e 10.3. Adapt the above comparison proof to handle linear growth of the sub and supersolutions in the problem u + F(Du, D2u) = f ( x ) on IR N. Use Perron's method to complete the proof of Theorem 3.3. (Hint: t ry supersolutions of the form employed for the first order case at the end of Section 9.)

The second order uniqueness theory has a long history with important con- tributions by R. Jensen ([27] contains the first proof for second order equations without convexity conditions and introduced new ideas), H. Ishii, P. L. Lions and P. E. Souganidis (e.g., [31], [30], [25], [26]). See the comments ending [12, Section 3]. The result called here the "Theorem on Sums", which makes life so easy, is a mild refinement of the result called the "maximum principle for semi- continuous functions" in Crandall and Ishii [11] and was preceded by Crandall [10]. Note also "Ishii's Lemma" in [22, Chapter V]. The proof below is slicker too.

31

11. P r o o f o f t h e T h e o r e m on S u m s

In this section we sketch the proof of Theorem on Sums. By now, you know that it can be used to good effect. It was stated before in the form used in the theory, but for notational simplicity we change the statement a trifle, so it is, at last, about sums.

T h e o r e m on Sums . Let (9 be a locally compact subset of IR N. Let u, v : (9 -~ IR and ~ be twice continuously differentiable in a neighborhood of (9 • (9. Set

w(x ,y ) = u ( x ) + v ( y ) for x, y 6 ( 9

and suppose (J:, ~)) E 0 x 0 is a local maximum of w(x, y) - ~(x, y) relative to (9 • (9. Then for each ~ > 0 with ~D2~(~,y) < I, there exists X , Y E S ( N ) such that

(u(:~), Dx~(2,, ~l), X ) e 7~+u(~), (v(t)), Dye(2, Y), Y) �9 7~4+v(fi)

and the block diagonal matrix with entries X , Y satisfies

(11.1) --~II < ( X y ) <(I-~D2~(~,?)))- lD2qo(~,~)) .

[1] Consider the function u(x) = xsinx, u(0) = 0 on JR. Show that J2'+u(O)

is empty while J2'+u(0) is quite large.

In order to prove this result we will need to "regularize" merely semicon- tinuous functions. The method we will use is called "sup convolution". This operation is introduced next, the relevant properties are established, and these results are then used in the proof.

Let (_9 be a closed subset of IR M, ~ > 0, ~b : (9 --+ IR and ( �9 IR M, and put

( 1 ) (11.2) = s u p - - r

zEO

Obviously the supremum is assumed and finite if ~b ~ - c o and

(11.3) lim su- ~(z) 1 zcO,lzl~oo Izt <

which we always assume, often without comment. If we extend ~b to all of IR M by

~ b ( z ) = - o o for zr

the formula becomes

(11.4) 1 Iz - 41=). ~ ( 4 ) ---- m a x ( g , ' ( z ) - z E I R M

We make this extension without further comment, and treat upper semicontinous ~b : IR M --~ [-oo, oo) where we now allow -oo as a value, but insist that ~b ~ -oc .

32

Obviously ~ depends on ~ which is not indicated in our notation; ~ will be obvious from the context. We note some obvious properties of sup convolution:

(i) If ~ < ~, then ~ < ~,

(11.5) (ii) ~ < ~.

(iii) ~(C) + (1 /2~) f l 2 is convex.

Proper ty (i) needs no comment. Property (ii) is seen by putting z = ~ in the defining supremum. Proper ty (iii) holds because 9 (z) - ( t / 2 ~ ) l z - 4l 2 + ( 1/2~)1412 is affine in <, and the supremum of a family convex functions is convex.

The property (iii) is called "semiconvexity"; ~ is semiconvex (with constant 1/(2~)). Less obvious properties include that sup convolution is an approxima- tion of the identity, that is

lim ~(C) ~)(r ~10

tbr r E IR N when the left-hand side of (11.3) is finite.

E x e r c i s e 11.1. Prove this claim.

Hence sup convolution provides an approximation of ~ which is pret ty reg- ular, that is semiconvex. The function ~ enjoys all the regularity of convex functions.

We later use the following special case:

E x e r c i s e 11.2.* If B ~ S (M) and 2~B < I, prove that if 9(z) <Bz, z), then

~(r = <B(Z - 2t~m)-l<,r .

Finally, we note the "magic" properties of sup convolution - it respects j2,+

in the following sense:

Theorem On Magic^Properties. Let J : ]R N --+ ]]% be upper semicontinuous

and satisfy (11.3). I r e C IR M and

then for 2 = < + ~ p

and for every real M x M matrix T,

+ np),p, 1 (1 - T*) ( I - T) + T * X T ) e j2,+r

where T* is the adjoint of T. Moreover, ~ is the unique point for which

1 (11.6) .~(~) ~(~) - ~1~ - ~l <

33

In particular, choosing T = I,

Proof. We m a y assume tha t there exists ~ E C2(IR M) such tha t

~(4.) - ~(4.) < ~ ( 4 ) - ~ ( 4 )

for ; e IR M and D ~ ( 4 ) = p, D2p(~) = X. From the definition of ~ this implies there exists 2 such t h a t

l l z 1 i~ _ (.12 _ ~ ( 4 ) ~(z) - 2 ,~ - 412 - ~;(() -< ~(~) -

for all z E IR M. Pu t t i ng first z = ~, we find t ha t 1 ~ - ( 1 2 / ( 2 ~ ) + ~ ( 4 . ) has a

m i n i m u m a t 4, and thus by the first and second order tes t for a m a x i m u m ,

(11.7) z = ~ + ~D9~(4) : 4 + ~P, D2~(~) : X > _ 1 i .

Next, leave z free bu t put ~ = T(z - ~) + 4. This leads to

~ ( z ) - ~ ( z ) < r - ~ ( ~ )

where

�9 (z) --- ~ l ( Z - T ) z + T ~ - ~12 + ~(Tz - T ~ + 4). We conclude t h a t

(~(2) , D ~ ( ~ ) , D2~5(~)) E J~ '+ r

Finally, by (11.7) and direct compu ta t i on

D ~ ( ~ ) = 1 ( ~ _ 4) = P,

= 1 - ( I - T * ) ( I - T) + T*D2~(4)T = 1 ( I - T * ) ( I - T) + T*XT.

[]

R e m a r k 11.1 . I t is na tu ra l to ask wha t is the smallest ma t r i x Y which can be wr i t t en in the form

Y = 1 - ( I - T * ) ( I - T) +T*XT,

where we recall t h a t X > - 1 / ~ . There is no op t imal choice if - 1 / ~ is an eigenvalue of X - this can be seen from the scalar case: pu t t ing X = - 1 / ~ , T = t leads to Y = (1 - 2t)/n, which is not bounded below. If X > - l / n , t hen the answer is (exercise)

T = ( I + n X ) -1

34

which yields, with a little algebra,

(11.8) 1 ( I - ( I + t~X)- l ) 2 + (I + ecX)-2X = (I + t ~ X ) - I x .

E x e r c i s e 11.3. Com pu t e ~ if Ib(0) = 1 and ~(x) = 0 for x • 0.

E x e r c i s e 11.4.* Show tha t if u is a subsolution of a proper equation

F(u, Du, D2u) = O,

on ]R N, then g is as well. Using (11.6), show

I~- ( I = _< 2~(r - r

Conclude tha t if u is a bounded subsolut ion of an equat ion F(u, Du, D2u) = f ( x ) where f is uniformly continuous, then ~ is a solution of F(~ , Dit, D2~t) <_ f ( x ) + 5~ for some constant f~ --* 0 as n ~ 0. Discuss the general case, F(x, Du, D2u) <_ 0.

It is always an opt ion to use the method (direct use of the approximat ions in the equations) as indicated by Exercise E x e r c i s e 11.5.* in place of the Theorem on Sums while working on problems in this area.

We will also employ two nontrivial facts about semiconvex functions. The first assert ion is a classical result of Aleksandrov:

T h e o r e m ( A l e k s a n d r o v ) . Let g : IR M -~ IR be semiconvex. Then g is twice differentiable almost everywhere on IR N.

Here g is "twice differentiable" at g means tha t

1 ( X ( z - ~ ) , , - - ~ } + o ( I z - ~ l 2) g(~) = g(S) + (p, ~ - S) +

for some p �9 IR M, X E S ( M ) , and then we say Dg(~.) = p, D2g(~) = X. We will not prove this classical result.

The next result we will need concerning semiconvex functions follows. In the s ta tement , B~(z) is the ball of radius r about z. We will prove this result after complet ing the proof of the Theorem on Sums. I t is a variant of an a rgument of Aleksandrov.

L e m m a 11.2. Let U : IRM ---+ IR be semiconvex and fc be a strict local maximum point of U. For p E ]R N, set V)p(X) = U(x) + (p, x}. Then for 0 < r sufficiently small and all 5 > O,

{y �9 B~(5:) : 3p �9 B6(O) such that Up(x) <_ Up(Y) for x �9 B~(~)}

has positive measure.

Proof of the Theorem on Sums.In the proof, we may as well assume tha t (9 = ]R N. Indeed, if not, we first restrict u, v to compact neighborhoods E l , N2 of b, 9 in (.9 and then extend the restrictions to ]R y by u(x) = v(y) = - o o for x • N1

35

--2,+ ^ and y r N2. One then checks t h a t Yo u(x) = J 2 ' + u ( 2 ) (provided - ~ < u(2)) , etc. We m a y assume t h a t ~ is C 2 in a ne ighborhood of N1 • N2, and then on IR N by modif ica t ion off N1 • N2. I t is clear t h a t (5, ~)) is still a local m a x i m u m of w - ~ relat ive t o ]R N • IR N.

Fur the r we m a y as well assume t h a t

and t h a t

2 = ~ = o, ~ ( o ) = v ( o ) = o

1((x) (x)) ~ ( x , y ) : ~ A Y , y

for some A E 8 ( 2 N ) is a pure quadrat ic , and 0 is a global m a x i m u m o f w - ~ . In- deed, a t r ans la t ion pu ts 5, ~) a t the origin and then by replacing ~(x , y), u(x), v(y) by ~(x , y) - (~(0) + ( D ~ ( 0 ) , x) + (Dye(0) , y)) and u(x) - (u(0) + ( D ~ ( 0 ) , x/) , v(y) - (v(O) + (Dy~a(O),y)) we reduce to the s i tua t ion 2 = ~) = Dx~(O) = D y e ( 0 ) = 0, and ~(0) = u(0) = v(0) = 0. Then, since

l l A ( X y ) , ( X y ) } + o ( ] x ] 2 + ] y ] 2 )

where A = D2~a(0), and w(x ,y ) - r < w(0) - ~(0) = 0 for small x ,y , if . > 0, we will have

(11.9) < x , y ) - \ (A 7I) ,

for small x, y r 0. Global i ty of the (strict) m a x i m u m at 0 m a y be achieved by localizing fur ther as above. I f the result holds in this case, we then pass to the limit a s , I 0 to ob ta in the general result.

F rom

w(x ,y ) = u(x) + v(y) < ~ m _ ~ y

and Exercise 11.2 we have

1 ( d ( I - nd ) - l~ , r (11.10) ~b(~) = ~(~) + ~)(,) <_

where we are wri t ing

Since u _< g, etc., 0 = u(O) <_ g(O); s imilarly 0 _< ~(0). On the o ther hand, by (11.10) we have g(O) + ~(0) _< O, and then g(O) = �9 = 0 and

1 (A( I - n A ) - I ~ , ~ ) ~(~) + ~ ( . ) -

has a m a x i m u m a t the origin, which is in fact s t r ic t (or increase A a litt le if you don ' t want to check this). B y Jensen ' s L e m m a and Aleksandrov ' s Theorem, for

36

each 0 < 8 we have pe,qe c Be(O) such that

1 ( A ( I - hA)-1 ( , (> + (Pe,~) + <qe, r]} ~(~) + ~(v) -

has a maximum (~e,r)e) c Be(0) x Be(0) and z~, ~3 are twice differentiable at (e, ~e. We now apply the magic properties, and let ~ ~ 0 to reach the conclusion. The second order test at a maximum and semiconvexity imply

1 i < D 2 (~e) 0 <(1 gA)-IA; - D 2 ~ ( O e ) -

therefore we can assume

along a sequence 8 ~ 0 and

_ 1 i < ( X ~ _ Y0) < ( I _ ~ A ) _ I A ' -

Let P project IR N x IR N on its first coordinates. By the magic properties,

(u(~e + ec(fe + PA(e) ) ,pe + P A ( e , X e ) E J2'+u((e + ec(pe + PA(e)) .

By the definitions and magic properties,

t~ u(~e + ~(Pe + PA(e) ) - ~lPe + PA(e[ 2 = u(~e);

since ~2 is continuous, we conclude that

u(~e + e;(pe + RAZe)) ~ ft(O) = u(O) = 0

and then (~(o), 0, x ) e 7~ '%(o)

follows upon letting 6 ~ 0. The analogous comments hold for v. [ ]

Proof of Jensen's Lemma.We assume that r is so small that ~ has :? as a unique maximum point in B(97, r) and assume for the moment that ~ is C 2. It follows from this that if 8 is sufficiently small and p ~ Be(0), then every maximum of ~ ; with respect to B(~?, r) lies in the interior of B(~, r). Since D~ + p = 0 holds at maximum points of ~p, D ~ ( K ) D Be(0). Let A _> 0 and ~(x) + (A/2)lxl 2 be convex; we then have - A I _< D2~; moreover, on K, D2~ < 0 and then

- A I < D 2 ~ ( x ) _ < 0 for x E K .

In particular, ]de tD~(x ) [ < A N for x E K. Thus

meas(Be(0)) _< meas(D~(K)) _< L [detD2~(x)ldx < meas(K)lA[~

and we have a lower bound on the measure of K depending only on A.

37

In the general case, in which p need not be smooth, we approximate it via mollification with smooth functions ~ which have the same semiconvexity constant A and which converge uniformly to ~ on B(~?,r). The corresponding sets K~ obey the above estimates for small ~ and

oo oo / r ( D ( ' I n = 1 U r n = n K 1

is evident. The result now follows. []

E x e r c i s e 11.6. Extend the Theorem on Sums to an arbitrary number of summands.

Concerning the "magic properties", perhaps they are not so magic if viewed through the lens explained by Evans [20] under the subheading "Jensen's regu- larizations"; in this regard see also Lasry and Lions [32] and Jensen, Lions and Souganidis [30]. The precise formulation and proof of the magic properties given above is from Crandall, Kocan, Soravia, and Switch [13]. It is interesting that the Theorem on Sums does not refer to regularizations; the information is stored as a general fact about semicontinuous functions. Of course, as noted before, using regularizations themselves in pdes is an important tool. Aleksandrov's theorem is from [1]; see also [21]. Jensen's lemma is proved in [27].

12. B r i e f l y P a r a b o l i c

We briefly indicate how to extend the results of the preceding sections to problems involving the "parabolic" equation

(PE) ut + F(t , x, u, Du, D2u) = 0

where now u is to be a function of ( t ,x ) and Du, D2u mean D x u ( t , x ) and D~u(t, x). We do this by discussing comparison for the Cauchy-Dirichlet prob- lem on a bounded domain; it will then be clear how to modify other proofs as well. Let O be a locally compact subset o f lR N, T > 0, and 0 T : ( 0 , T) x O. We

2 , + 2 -- , ~ . . . 2 + 2 -- denote by P~ , Po the parabolic variants of the semljets J[9' , Jc)' " For example, if u: OT --~ IR, P~'+u is defined by (u(s, z), a,p, X ) E IR • IR N x S ( N )

lies in P~'+u(s, z) if (s, z) E O r and

I - -

u ( t , x ) <_ u ( s , z ) + a(t - s) + (p ,x - z) + -~ (12.1)

+ o ( [ t - s} + Ix - zt 2) a s OT ~ (t, z ) ~ (s, z) ;

2 - 2 , + - - 2 + - - 2 , - s i m i l a r l y , P$' u = -P~o ( -u ) . The corresponding definitions of P ~ , P o are then clear. This definition reflects that (PE) is first order in t.

The notions of a subsolution, etc., of (PE) on an open set are contained in the previous discussion. As in Section 7, they may be reformulated, and with a little work one sees that we have: a subsolution of (PE) on OT is an upper semicontinuous function u : O r --~ IR such that

(12.2) a + F ( t , x , r , p , X ) <_ 0 for ( t , x ) E OT

3 8

whenever (r, a , p , X ) ~ P~'+u(t,z); likewise, a supersolution is a lower semicon- tinuous function v such that

(12.3) a + F ( t , x , r , p , X ) >_ 0 for ( t ,x) �9 50T

- - 2 -

whenever (r, a , p , X ) �9 P o v( t ,x) . We show how to treat the Cauchy-Dirichlet problem for (PE), exhibiting the

considerations which do not occur in the stationary case. Consider the problem

(E) ut + F(t , x, u, Du, D2u) = 0 in (0, T) x ft

(12.4) (BC) u ( t , x ) = O f o r O _ < t < T a n d x � 9

(IC) u(0, x) = ~ ( x ) f o r x � 9

where ft C IR N is open and T > 0 and r �9 C(~) are given. By a subsolution of (12.4) on [0, T) x ~ we mean an upper semicontinuous function u : [0, T) x ~ ~ IR such that u is a subsolution of (PE) in (0, T) x ft, u( t ,x ) < 0 for 0 _< t < T and x �9 0ft and u(0, x) _< r for x �9 ft and so on for supersolutions and solutions.

T h e o r e m 12.1. Let ft c lit N be open and bounded. Let F �9 C([0, T] x ~ x l R x IR N x $ ( N ) ) be continuous, proper and satisfy (10.5) for each fixed t �9 [0, T), with the same function w. I f u is a subsolution of (12.4) and v is a supersolution of (12.4), then u <<_ v on [0, T) x ft.

To continue, we require the parabolic analogue of The Theorem on Sums. It takes the form:

T h e o r e m 12.2. Let (9 be a locally compact subset ofIR N and u, v : (0, T) x 50 -~ IR be upper semicontinuous. Let ~ be defined on an open neighborhood of (0, T) x 50 z 50 and such that (t, x, y) -~ ~(t, x, y) is once continuously differentiable in t and twice continuously differentiable in (x, y). Let (t, :~, Y) c (0, T) x 50 x 50 and

w(t , x, y) =_ u(t, x) + v( t , y) - ~(t, x, y) < w(~, :~, 9)

for 0 < t < T and x, y E 50. Assume, moreover, that there is an r > 0 such that for every M > 0 there is a C such that

(12.5) b <_ C whenever ( u ( t , x ) , b , q , X ) �9 P~'+u(t ,x)

and

Ix-21+lt-il<r and l u ( t , x ) i + l q i + i X i < M.

Assume the same condition on v (with c? replaced by ~ just above). Then for each ~ > 0 with ~D2~(t, 3c, ~1) < I there are X , Y �9 $ (N) , bl, b2 �9 IR such that

- - 2 + ^ (u(i, :~), bl, Dx~(t , 3:, ~l), X ) �9 P~ u(t, x),

- - 2 + ^ (v([, ~t), b2, D~(p(i, 5, ~)), Y ) �9 P ~ v(t, Y)

39

and

and

1 i < ( X - 0 O ) ( I - gD29z(i, 2 ,~ ) ) ) - lD2~([ ,2 ,~ ) <

bl + b2 = ~t( t , 2, 9).

Observe that the condition (12.5) is guaranteed if u is a subsolution of a parabolic equation (and likewise for v). P r o o f o f T h e o r e m 12.1. First observe that (decreasing T if necessary), we can assume that u is bounded above and v is bounded below. Next, let c, c > 0, and notice that (t = u - ct - ~ / ( T - t) is also a subsolution of (12.4) and in fact satisfies (PE) with a strict inequality:

(tt + F( t , x, (t, D(t, D2(t) <_ - c (T - t) 2'

Since u <_ v follows from ~ _< v in the limit c, r $ 0, it will simply suffice to prove the comparison under the additional assumptions

(i) ut + F( t , x, u, Du, D2u) _< - c < 0 and

(12.6) (ii) limu(t, x) = -oo uniformly on ~. tTT

Let (t, 2, ~)) be a maximum point of u ( t , x ) - v(t , p) - Ix - yl~/(2r over [0, T) x ~ x ~ where a > 0; such a maximum exists in view of the assumed bound above on u, -v , the compactness of ~ and (12.6) (ii). By Lemma 4.1,

(12.7) Is}- ~12 § 0 c

as c $ 0. Let (t', 2, 2) be a limit point of (t, ~, ~)) as ~ $ 0. If

2, 2) {0} x u [0, T) x

upper semicontinuity and the side conditions imply that

l imin f (u ({ ,2 ) v ( t , ~ ) ) [ : ~ 2 / [ 2 ) - _< ~(t, 2 ) - v ( t , 2 ) < 0 . el0

Hence

(12.8) u ( t , x ) v(t,x) < lim (-u(t, 2) v(t,~)) I~ ~-c ~)t2) - - - - - - z O

E l 0

and we are done. If t > 0 and 2 r c9~, then we may assume that ~, ~ c ~ and use Theorem

12.2 at (t, 2, ~)) to learn that there are numbers a, b and X , Y c 8 ( N ) such that

- - 2 + ^ ^

40

and

such that

- - 2 m ^

(v(i, ~), b, ~(~ - ~), Y) c p-~ v(t, 9)

,129, a b 0 nd 0) 3 ( , , , ) E - 0 Y - c - I

We may assume that v([,:?) < "u([,d:) for otherwise we have (12.8). Then the relations

a + F([,:;','u(i, 3c),(s 9 ) / c , X ) <_ -c , b+ F([,~),v([,[l),(Jc- ) ) / e , Y ) >0,

and (12.9) imply

~' + 1 ~ - 9 1 g

which leads to a contradiction via (12.7).

E x e r c i s e 12.1.* Work out the full proof of Theorem 3.2. The primary steps are to adapt the above proof to the pure initial-value problem via the devices used in Sections 5 and 6, to note that Perron's Method applies, and to produce subsolutions and supersolutions. All is routine, except perhaps the last step. Here is a brute force way to go about this. First, via comparison and stability, it suffices to discuss Lipschitz continuous ~ (as uniformly continuous functions on IRN are precisely those which are uniform limits of Lipschitz continuous functions). If ~ has Lipschitz constant L, then for every e > 0, z ~ IR N,

_< ~z,~ where

~ , ~ ( z ) = ~,(z) + L ( I z - zl 2 + e) 1/2

Show that there is an A~ such that U~,z = A~t + ~,~ is a supersolution for the initial value problem, that inf~,~ u~,z is continuous on [0, oc) • IR N, and agrees w i t h C a t t = 0 .

Regarding the parabolic theorem on sums, see [11] and note again "Ishii's Lemma" in [22, Chapter V].

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