A Baire category approach in existence theory of differential equations

GIULIO PIANIGIANI Dipartimento di Matematica per le Decisioni Universit~ di Firenze, Via Lombroso, 6/17 50134 Firenze Italy gi ul i oOds. uni f i . i t

A BAIRE CATEGORY APPROACH IN EXISTENCE THEORY OF DIFFERENTIAL EQUATIONS Conferenza tenuta il giorno 4 Maggio 1998

1

Introduction

The idea of using Baire category in differential equations has a long history dating back to Orlicz, Peixoto, Smale to m e n t i o n only a few names. Usually one is given a complete metric space M of maps and for f E M one considers the corresponding differential equation x'(t) = F(t,x(t)).

(1.1)

In Banach spaces the question whether (1.1) has properties like existence, uniqueness, continuous dependence, existence of periodic solutions etc. is, in general, a difficult one. This has led mathematicians to rise, as a substitute, the question w h e t h e r the property u n d e r consideration is perhaps generically satisfied in M. Here Baire category enters naturally if we agree to say that a p r o p e r t y is generic

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in M provided that it is satisfied by all f in a residual subset Mo of M, i.e. with M \ M0 of the first Baire category. It is worthwhile to point out, that knowing that a p r o p e r t y of (1.1) is generic in M does not furnish, in general, any information whether a specific f E M belongs or not to the generic set. In this note we discuss a method, based on the Baire category, by which the existence of solutions can be established. 2

E x i s t e n c e r e s u l t s f o r differential inclusions

This area of research was started m o r e recently by De Blasi and Pianigiani [9, 10] in connection with existence problems for n o n convex valued differential inclusions of the f o r m x'(t) ~ F(t,x(t)),

x(O) = xo.

In this development a stimulus was offered by a paper by Cellina [3] in which it was shown that the solution set to the Cauchy p r o b l e m

x'(t) E {-1, 1},

x(0) = 0

is a residual subset of the solution set of

x ' ( t ) ~ [ - 1 , 1],

x ( o ) = o,

if the latter is e n d o w e d with the metric of u n i f o r m convergence. To give an idea of the m e t h o d we consider two simple situations. If E is a Banach space, we denote by K ( E ) ) (resp. K c ( E ) ) the space of all nonempty, bounded, closed (resp. nonempty, bounded, closed, convex) subsets of E and let I = [0, 1]. Consider the Cauchy problem x'(t) ~ aF(t,x(t)),

x ( O ) = xo

(2.1)

x'(t) ~ F(t,x(t)),

x ( 0 ) = x0,

(2.2)

and

where aF(t, x ( t ) ) denotes the b o u n d a r y of F(t, x ( t ) ) . By a solution of (2.1) or (2.2) we m e a n a Lipschitzean function x satisfying (2.1) or (2.2) a.e. on I.

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In E, an o p e n ball with center x0 and r a d i u s r is d e n o t e d by B(xo, r ) . For A c E we d e n o t e by intA, -d-dA respectively the interior of A a n d the closed convex hull of A. THEOREM 2.1 [9] Let E be a reflexive Banach space, F : I x B(xo, r) -Kc (E) be Hausdorff continuous and bounded by a constant, say M. Assume, moreover, that for every ( t , x ) E I x B(xo,r) the set F ( t , x ) has nonempty interior. Then the Cauchy problems (2.1) and (2.2) have solutions and the solution set MaF of (2.1) is a residual subset of the solution set ME of (2.2). PROOF. (Sketch) Set M F = {X E C(I,E) : x is a solution of(2.2)}. Mt: is n o n e m p t y and, u n d e r the m e t r i c of C(I,E), is a c o m p l e t e metric space. For any n ~ N set M n = {x E M E " I i d ( x ' ( t ) , 3 F ( t , x ( t ) ) ) d t < i / n } , where d ( x ' ( t ) , OF(t,x(t))) = inf{llx'(t) - z l l : z ~ F ( t , x ( t ) ) } . The m a i n step is to prove that each M n is an o p e n a n d d e n s e subset of MF. Then, by the Baire category theorem, it follows that the set N M n is a residual s u b s e t of Mp. Since Mat: = f'l M n the s t a t e m e n t follows. [] THEOREM 2.2 [l O]Let E be a reflexive, separable Banach space and let F : I x B(xo, r ) ~ K c (E) be Hausdorff continuous and bounded by M. Moreover, assume that for every ( t , x ) E I • B(xo, r) the set F ( t , x ) has nonempty interior. Then the solution set MextF of the Cauchy problem x'(t) ~ extF(t,x(t)), x(O) = xo

is nonempty. PROOF. (Sketch) Let M F be the solution set of the Cauchy p r o b l e m (2.2) and set M ' = cL{x ~ MF : x ' ( t ) ~ i n t F ( t , x ( t ) )

a.e.}.

Here cl d e n o t e s the closure in the t o p o l o g y of C(I,E). M ' is n o n empty, a n d e q u i p p e d with the metric of C(I, E) is a c o m p l e t e metric

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space. Now introduce the Choquet function[5, 10, 15] dF(t,x,z), which "measures" the distance of a point z E F(t, x) f r o m the set of the extreme points of the convex set F(t,x). We s u m m a r i z e below the m a i n properties of dF :

i. dF is non negative and b o u n d e d on graph F; ii. d F ( t , x , z ) = 0 if and only if z ~ extF(t,x); iii. if {Xn } c M ' converges to x, then limsup fI dF(t, xn(t), Xn(t))dt < h dF(t, x ( t ) , x ' ( t ) ) d t . Then we set M ~ = {x ~ M " f l d F ( t , x ( t ) , x ' ( t ) ) d t

< l/n}

and we prove that each M n is an o p e n and dense subset of M ' . Consequently A M ~ is a residual subset of M ' and hence nonempty. Finally we show that

[] We point out that the right choice of the underlying metric space, in our case M ' , is fundamental. For example the above a r g u m e n t certainly fails if we replace M ' by MF for, if valid, it would imply that MextF is dense in ME which, in general, is not true by a well k n o w n counterexample due to Plis.

3

Existence results for Hamflton-Jacobi

equations

In a recent paper Dacorogna and Marcellini [7], see also Bressan and Flores [2], have u s e d the Baire m e t h o d to establish existence of solutions to the Dirichlet p r o b l e m for Hamilton-Jacobi equation of the form "H(~Tu(x)) = 0 x E f2 a.e.

u ( x ) = ep(x)

x ~ af2,

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where D c ~n is open. They consider both, the scalar and the vectorial case. After rewriting the above p r o b l e m in the equivalent f o r m Vu(x) e C

x ~ D a.e.

u(x)=q~(x)

x~a~,

where C = {z E Nrt I H ( z ) = 0}, Dacorogna and Marcellini prove an existence t h e o r e m by using a Baire category approach very m u c h in the spirit of T h e o r e m s 1 and 2 above. Their basic a s s u m p t i o n s are that Vqg(x) is compactly contained in the interior of -~-6F(x, q)(x)), x ~ f2 and that qo E C 1(['2). In the sequel we discuss these results and show that, in the scalar case b o t h a s s u m p t i o n s can be relaxed. Details and generalizations can be f o u n d in [11]. Consider the following p r o b l e m H(x,u(x),Vu(x))

=0

u ( x ) = ep(x)

x~Da.e. x ~ aD.

(3.1)

We associate with (3.1) the partial differential inclusion Vu(x)) E F(x,u(x))

x E s a.e.

u ( x ) = q~(x)

x e aD,

(3.2)

where F ( x , y ) = {z E ~n . H ( x , y , z ) = 0} . Under suitable assumptions on H, the map UfF turns out to be compact valued and continuous. THEOREM 3.1 [11] Let F 9 D x ~ -. K ( ~ n) be b o u n d e d b y a c o n s t a n t M > 0 a n d a s s u m e that-d-fF be c o n t i n u o u s on fl x ~, w h e r e s c ~ n is open a n d bounded. Let q) E C(-D) n WI,~176 be s u c h t h a t V c p ( x ) E int~-BF(x, qv(x)), x ~ s a.e.. Then the Dirichlet p r o b l e m V u ( x ) ~ ext-C6F(x, u ( x ) )

x E D a.e.

u(x) = ~(x)

x~af~

has a solution u E C(-D) n Wl,~176

(3.3)

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PROOF. (Sketch) We introduce an appropriate space M, given by M

= C/{uEC(~)

C)wl'~176

" VU(X) E

i n t ~ - 6 F ( x , u ( x ) ) , x E ~ a.e.

and u ( x ) = ~p(x),x E as Here, the closure is understood in the metric of C(~). Under this metric M turns out to be a nonempty complete metric space. For n e ~, set Mn

{u

M"

fndF(x,u(x), Vu(x))dx < I / n } .

Then by using the properties of the Choquet function dF we show that M n is open and dense in M. A straightforward application of the Baire category theorem yields that the set M0 = A Mn is non empty. Further each u ~ Mo is actually a solution of the Dirichiet problem (3.3). As F is assumed to have compact values it follows that each solution of (3.3), is also a solution of the initial Dirichiet problem (3.2) and hence of (3.1). [] The vectorial case is by far more difficult. A "particularly simple" vectorial problem, yet unsolved in its full generality, is the prescribed singular values problem. It has been recently studied by Cellina and Perrotta [4] in terms of potential wells and by Dacorogna and Marcellini [7, 8] by means of the Baire method which has proven to be effective, beside this, also in more general vectorial problems.

References [11 J.P. AUBIN and A. CELLINA, Differential Inclusions, SpringerVerlag, Berlin, 1984.

[21 A. BRESSANand F. FLORES, On total differential inclusions, Rend. Sem. Mat. Univ. Padova 92 (1994) 9-16

[3] A. CELLINA,On the differential inclusion x ' E [ - 1, 1 ], Atti Accad. Naz. Lincei, Rend. Sc. Fis. Mat. Nat. 69 (1980) 1-6. [41 A. CELLINA and S. PERROTTA,On a problem of potential wells, J. Convex Anal. 2 (1995) 103-115.

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[5] G. CHOQUET, Lectures on Analysis, Mathematics Lecture Notes Series, Benjamin, Reading, Massachussetts, 1969.

[6] B. DACOROGNA, Direct methods in the calculus of variations, Springer-Verlag, Berlin, 1989.

[7] B. DACOROGNA and P. MARCELLINI, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997) 1-37.

[8] B. DACOROGNAand P. MARCELLINI, Cauchy-Dirichlet problem for first order nonlinear systems, J. Funct. Anal. 152 (1998) 404446.

[9] F.S. DE BLASI and G. PIANIGIANI, A Baire category approach to the existence of solutions of multivalued differential equations in Banach spaces, Funkcial. Ekvac. 25 (1982) 153-162.

[io] F.S. DE BLASI and G. PIANIGIANI, Non convex valued differential inclusions in Banach spaces, J. Math. Anal. Appl. 157 (1991) 469494.

[11] F.S. DE BIAS] and G. PIANIGIANI, On the Dirichlet problem for first order partial differential equations. A Baire category approach, NoDEA Nonlinear Diff. Eq. Appl. 6 (1999) 13-34.

[12] I. EKELANDand R. TEMAM, Analyse convexe et problemes variationnels, Gauthier-Villars, Paris, 1974. [13l D. KINDERLEHRERand G. STAMPACCHIA,An introduction to variational inequalities and their applications, Academic Press, New York, 1980.

[141 P.L LIONS, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Math. 69 Pitman, London, 1982.

[lSl G. PIANIGIANI, Differential inclusions. The Baire category method, in Methods of nonconvex analysis, edited by A. Cellina, Lecture Notes in Math. 1446, Springer-Verlag, Berlin, 1990, 104136.