Nonlinear Operators and the Calculus and Variations

Lecture Notes in Mathematics Edited by ~ Dold and 13. Eckmann 543 Nonlinear Operators and the Calculus of Variations Su...

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Lecture Notes in Mathematics Edited by ~ Dold and 13. Eckmann

543 Nonlinear Operators and the Calculus of Variations Summer School Held in Bruxelles 8-19 September 1975

Edited by J. P. Gossez, E. J. Lami Dozo, J. Mawhin, L. Waelbroeck H9

ETHICS ETH-BIB

I,m lll)lUIIilll HiU 00100000346926

Springer'Verlag Berlin. Heidelberg. New York 1976

Editors Jean Pierre G o s s e z Enrique Jos~ Lami Doze Lucien Waelbroeck Universite Libre de Bruxelles D6partement de Mathematique C P 214, 1050 Bruxelles/Belgium Jean Mawhin Universit~ Catholique de Louvain Institut de Math~matique 1348 Louvain-la-Neuve/Belgiu m

Library of Congress Cataloging in PubncatJen Data

Main entry under title: Non//near operators and the calculus of vaz~atlons. (Lectu1~ notes ~n mathematics ; 5~3) "Lecture notes for the five series of lectlu~s at the Stm~er School on Nonlinear Operators and the Calculus of Variations, held at the Un/versit~ Libre de Bz%~elles, September 8 to 19, 197Y' Sponsored by the NATO Science Commlttee. 1. Nonl/near operators--Addresses, essays, lectumes. 2. Calculus of vaz~ations--Addresses, essays, lectures. I. ~ossez, J. P., 194577. North Atlantic T~eaty Organization. Solenee Committee. III. Sezdes: Lecture notes in marematics (Ber/_in) ; 5~,3. 0/~.L28 no. 543 [ ~ 2 9 . 8 ] 510'.8s [515'.6~] 76-~0~76

AMS Subject Classifications (1970): 35B45, 47H05, 47H15, 4 9 B 3 0 ISBN 3-540-07867-3 Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07867-3 Springer-Verlag New York 9 Heidelberg 9 Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, HemsbachlBergstr.

PREFACE Thts volume c o n t a i n s

lecture

notes f o r

the f t v e

s e r t e s of l e c -

t u r e s at the Summer School on N o n l i n e a r Operators and the C a l c u l u s o f V a r i a t i o n s , held at the U n t v e r s t t 6 L i b r e de B r u x e l l e s , September 8 to 19 1975. A Semtnar program was o r g a n i z e d c o n c u r r e n t l y w t t h the School. We assume t h a t the Seminar Speakers w i l l s u l t s on t h e t r own I n i t i a t i v e . Let all

those who helped make t h t s

publish

their

re-

Summer School a suCCess f t n #

here an e x p r e s s i o n o f our g r a t i t u d e , the t n v t t e d l e c t u r e r s , t h e partic i p a n t s , the s e c r e t a r i e s o f the B r u s s e l s Mathematics Department, the Fonds N a t i o n a l de ]a Recherche S c t e n t t f t q u e , the Solvay F o u n d a t i o n , and foremost the NATO Sc|ence Committee who run a v e r y e f f e c t i v e Summer School program and f t n a n c e d most o f the expenses o f t h i s s p e c i f i c meeting.

J e a n - P i e r r e GOSSEZ

E n r l q u e LAMI DOZO

Jean MAWHIN

Lucten WAELBROECK

CONTRIBUTORS Herbert AMANN. I n s t i t u t fur Mathematlk.

Ruhr-Untverstt~t

Bochum.

463 Bochum, German~. Harm BREZIS. Unlverslt~ Pierre et Marie Curie.

4~ Place Jus,sleu

75230 Paris C~dex 05.

Umberto MOSCO. I s t i t u t o Matematlco. Universitl di Roma.

00100 Roma.

Italy.

R . T y r r e l l ROCKAFELLAR. Department of Mathematics. U n i v e r s i t y of Washington. S e a t t l e . WA 98195. U.S.A. Roger TEMAN. D~partemeflt Math6mattque$. 91405 Orsa~. France.

Unlversit~ de Paris-Sud.

TABLE OF CONTENTS

Herbert AMANN : " N o n l i n e a r operators in ordered Banach spaces and some a p p l i c a t i o n s to n o n l i n e a r boundary value problems" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HaTm BREZIS

:

"Quelques p r o p r i 6 t 6 s des op6rateurs monotones et des semi-groupes pon l i n ~ a i r e s " . . . . . . . . . . . . . . . . . . .

Umberto MOSCO : " I m p l i c i t v a r i a t i o n a l problems and quasivariational inequalities" .......................

R T .o y r r e l l

ROCKAFELLAR : " I n t e g r a l f u n c t i o n a l s , normal integrands and measurable s e l e c t i o n s " . . . . . . . . . . . . . .

R_oger TEMAM : " A p p l i c a t i o n s de 1'analyse convexe au c a l c u l des variations" , ......................................

56

83

157

208

NONLINEAR OPERATORS IN ORDEREDBANACH SPACES AND SOMEAPPLICATIONS TO NONLINEAR BOUNDARYVALUE PROBLEMS

by

Herbert Amann

Introduction

In recent years much research has been done in the f i e l d of nonlinear functional analysis and many of the obtained results were motivated by and have applications to the theory of nonlinear d i f f e r e n t i a l equations. I t is well-known that many d i f f e r e n t i a l equations s a t i s f y so-called maximum principles. Abstractly speaking,this means that the d i f f e r e n t i a l operators are in some sense compatible with the natural order structure of the underlying function spaces. Since i t is only reasonable to use this additional information one is led in a natural way to the study of nonlinear equations in ordered Banach spaces (OBSs). I t is the purpose of this paper to present some recent results about fixed point equations in OBSs together with some of its applications to nonlinear boundary value problems. I t is shown that the abstract results lead to very general existence and m u l t i p l i c i t y theorems for differential equations of the second order.

In the f i r s t paragraph we present some fundamental results about OBSs and positive linear operators which are the basis for the nonlinear theory. In order to demonstrate the importance of OBSs we include a r e l a t i v e l y long l i s t of OBSs occuring in analysis. Furthermore, we discuss some applications to linear e l l i p t i c eigenvalue problems. I t is to be noted that we consider also the case where the eigenvalue parameter occurs "on the boundary". In the second paragraph we study fixed point equations involving increasing maps, that i s , maps which are compatible with the ordering. In this case i t is r e l a t i v e l y easy to prove constructive existence theorems. By combining these results with topological methods we then deduce nont r i v i a l m u l t i p l i c i t y theorems. In Paragraph 3 the foregoing abstract results are applied to mildly nonlinear e l l i p t i c and parabolic boundary value problems. Again, in the case of e l l i p t i c equations, i t should be noted that we admit nonlinear boundary conditions also. In the last paragraph we study nonlinear eigenvalue problems for nonlinear operators mapping the positive cone into i t s e l f . In particular we prove some global results concerning the bifurcation of nontrivial solutions from the "line of t r i v i a l solutions". At the end of the Paragraphs 2-4 we give a few bibliographical remarks which are far from being complete. For more complete notes and remarks cf.

[6].

1. Ordered Banach Spaces and Positive Li near Operators

Let

P+PcP where

P is called a cone i f

E be a real Banach space. A subset

~R+ := [o,|

, ~+Pcp

, p n ( - p ) = {o} , l ~ = P ,

and l ~ denotes the closure of

induces an ordering

o . Then

is a nontrivial vector subspace of

E and i t is easy

to see that Ilxll e i s a norm on t h i s

:= i n f

{ x > o I - x e < x < xe}

subspace, t h e o r d e r

u~it

norm or

tl~

e-norm

. It

can

be shown (cf.[ 16,18,27] ) that Ee := ( u { x [ - e , e ]

is a Banach space such that (Ee,Pe)

I~, ~ JR+}, I I . l l e )

Ee~ E . Hence by the preceding example,

is an OBS where Pe := i - l ( P )

induces the natural ordering in

Ee . Since the e-norm is monotone, Pe is normal, and since

[,e,e]

is

0

the (closed) unit ball of

Ee , i t is easily seen that

Pe* ~ " In fact,

x ~ ~e i f f there exist positive numbers ~,B

such that

~e ~ x ~ Be

(1.8) Exa~ole (b~aces of Oontinuousl~ Differentiable I~nctions}: Let (E,P)

be an OBS and l e t

X be a nonempty compact m-dimensional differen-

tiable (that is, C|

in

~N

with or without boundary, where

1 ~m ~ N . For each k E B := {o,1,2 . . . . . }

we denote by ck(x,E)

the

Banach space of a l l k-times continuously differentiable maps f : X § E with the norm l l f l l k :=

~ llDOfll lel~k C(X,E)

'

where we employ the standard multi-index notation. Then ck(x,E)~-+ C~ ck(x,E)

= C(X,E) . Hence (cf. the examples (1.4) and (1.6)),

is an OBS with the natural ordering.

For every

k~ ~

vector subspace of

and every ck(x,E)

~ E (o,1) , we denote by

ck+u(X,E) the

consisting of a l l maps whose k-th partial

derivatives are u-H~Ider continuous, that i s , It f ( x ) - f ( y ) I I E H (D~f) :=

for every

~ E ~N

with

sup x,y E X x r y

< Ix-yl u

lel : k . I t is well-known that

ck+u(X,E)

is a

Banach space with the norm

Ilfilk+ p := i l f l l k +

E H (D~f) Iel=k

Since ck+~(X,E)--* ck(x,E) , each one of the spaces C~

, o ~ o < |

is an OBS with respect to the natural ordering. (Observe that for each TE{O,o) , the space CT(X,E) induces the natural ordering in

C~

.

In fact, a map f iff

belongs to the positive cone C+(X,E) of

Ca(X,E)

f(X) c P .) 0

I t is easy to see that

int C~(X,E) ~ # i f f

P ~ # . (Observe that

the constant map f(x) = e , x E X , is an interior point of 0

e ~ P .) I t can be shown (cf.{ 6 ] ) that o > o (provided, of course, that

C+(X,E) is never normal i f

E ~ {o}).

Occassionally we shall also use the spaces ck-(x,E) all maps in

ck'I(x,E)

C+(X,E) i f

consisting of

whose (k-1)-th partial derivatives are 1-H~Ider

continuous (that is, Lipschitz continuous). Hence ck-(x,E) c-~ CO(X,E) for every c < k .

(1.9) Example (Sobolev ~o,aces): Let ~ c ~RN be a nonempty bounded domain. For every k E IW~ := I~ "-{o}

and every p E [I,|

by W~(~) the usual Sobolev spaces consisting of all

we denote

f E Lp(~) such I

that all the distributional derivatives up to the order

k belong to I

Lp(~) . Then Wkp(R)'--* Lp(R) and, consequently, Wkp(~) is an OBS with the natural ordering. In general, the positive cone, Wkp,+(~) , has empty interior. However, i f

kp > N (and B~ is sufficiently regular), then,

by the Sobolev imbedding theorem, Wkp(~)"-~ C(~) and, consequently, ~ d 9 Furthermore i t can be shown (cf. [ 6 ] ) that int Wk,+(~) p

W~,+(~)

is not normal.

(1.1o) Example (3paces of Continuous Linear Operators): Let (F,Q)

be OBSs such tl~t

(E,P)

and

P is total. We denote by L(E,F) the Banach

space of a11 continuous linear operators T : E ~ F with the usual norm. Then i t is easy to see that L+(E,F) := {TE L(E,F) I T(P) c Q} is a cone in

L(E,F) which is said to induce the natural ordering. (Hence

a continuous linear operator Tx m o for every

T : E~ F is positive i f f

x m o .) In the special case that

L(E,R ) = E~ , we write

P~ := L+(E,~ )

and call

T * o and

F = ~ , that i s ,

P~ t l ~ dual cone of

P. We denote by

K(E,F) the closed vector subspace of

L(E,F)

of all c o ~ a o t linear operators. Then (cf. Example (1.6))

consisting

K(E,F) is an

OBS with the natural ordering whose positive cone is denoted by

K+(E,F) .

Finally, in the special case that

E = F we suppress the second argument

in these notations (e.g.

L(E,E) ).

L ( E ) :=

The basic result in the theory of compact positive linear operators is the famous Krein-Rutman theorem. For proofs and for some of i t s generalizations we refer to[16,18,1g,27].(Recall that for every T E L(E) the l i m i t I/k r(T) := lim llTkll exists and is called the spectral radius of T .) k-~==

(1.11) T~orem (Krein-Rutman):

and in

T

(E,P)

T E K+(E) such that

cone. Suppose that eigenvalue of

Let

be an OBS with total positive

r(T) > o . Then

and of the dual operator

r(T)

is an

T ~ with eigenvector8 in

P

P~ , respectively.

I t is well-known that much more precise results can be obtained i f the class of positive endomorphisms is further restricted. In applications to problems in analysis i t turns out that an important and useful subclass is given by the class of strongly positive and almost strongly positive linear operators. Let

(E,P)

and

(F,Q) be OBSs such that

T : E -* F is called strongly positive i f positive (a.s.p.) i f

o

Q* ~ . A linear operator o T(l5) c Q and almost strongly o

P\ ker T # # and T(P\ker T) c Q .

10

O

Suppose that

P ~ ~ and l e t

eigenvector of

T ~ L(E)

be a.s.p.. Then every positive

T tO a positive eigenvalue belongs to

~ . The following

lemma contains a related, somewhat weaker property for positive eigenvectors of

T~ .

(1.12) Len~na= Suppose that exist is,

p > 0

and

T E L(E)

is a.s.p, and suppose that there

T~@ = pC . Then @(P',ker T) > o

@E p~( w i t h

(that

@(P\ker T) c ~+ ).

O

Proof: Let

o . Hence

r(T) , and no solution i n If

X = r(T)

and

P\ker T i f

x ~ P if

~

o .

an a r b i t r a r y p o s i t i v e eigenvector o f

(cf. Theorem ( 1 . 1 1 ) ) . Then

r ( P \ k e r T) > o

by Lemma (1.12). (ii)

Suppose t h a t

xE P\ker T

and

x o , then there e x i s t s an element y E p

r(S)y = Sy ~ Ty . Hence r ( S ) < r Consequently,

= < l ~ , y >

such t h a t

= r(T)

r(S) ~ r(T) . The remaining p a r t o f the assertion f o l l o w s by

a s i m i l a r consideration

9

12 I t should be noted that under the hypotheses of Theorem (1.13) i t can be shown that

r(T)

is a simple eigenvalue and that

only eigenvalue of the complexification of radius

r(T)

r(T)

is the

T lying on the c i r c l e with

(cf.[7,18,19]).

In the remainder of this paragraph we indicate some

Applications to elliptic boundary value problems: Let ~ be a nonempty bounded domain in C%)manifoldsuch that words, ~

~N

whose boundary,

r , is a smooth (that i s ,

~ lies l o c a l l y on one side of

r .

(In other

is a compact connected N-dimensional differentiable manifold

with boundary r .) We denote by A a d i f f e r e n t i a l operator of the form N

Au := -

N

z aikDiDku + .Diu + aoU i,k=l i~1 al '

with smooth coefficients and a uniformly positive d e f i n i t matrix

coefficient

(aik) . (For much weaker regularity hypotheses c f . [ 6 , 7 ] ).

(1.14) Example (T~ Dirichlet Problem): We consider the linear BVP Au = f

(2)

8oU where BoU := u l r g E C2+U(s

= g

in

R ,

on

r

,

denotes the D i r i c h l e t boundary operator and f E C~(~) ,

for some u E (o,1) . By a solution we mean a classical

solution. Suppose that unique solution

ao ~ o . Then i t is well-known that the BVP (2) has a u = So(f,g ) E C2+U(~) . Moreover, the maximum principle

and the Schauder a p r i o r i estimates imply that the operator to

L+(CU~) x C2+P(r),C2+V(~)) .

SO belongs

13 We now define

K ~ L+ (Cu(~),C2+~(~)) Kf := So(f,o )

and

and R E L+(C2+U(r),C2+~(~))

by

Rg := So(o,g ) ,

respectively. Then So(f,g) = Kf + Rg . By using appropriate L -estimates, the theory of generalized solutions, P and Sobolev type imbedding theorems, i t can be shown that K (and hence SO )

has a unique extension, denoted again by

K , such that

K ~ K+(C~),CI~)) . Moreover, by means of the strong maximum principle i t can be shown that K E K+(C(~),Ce(~))where e := K ~

and

1(x)

= x

solution of the BVP Ae =:~. in "solution operator"

K is

for R ,

xE~

(that i s ,

Boe = o on

strongly positive

e

is the unique

r ). Lastly, the

as a map from Ca)

into

Ce~ ) . (Recall (cf. Example (1.7)) that this means that, for every u ~ C+~) , there exist positive constants

a,B

such that

ce ~ Ku ~ Be .)

For detailed proofs of these results we refer to [ 1,2,3,6,7 ] .

(1.15) Exan~le (The Neu~nann and the Regular Oblique Derivative BVP)= Let

8 be an outward pointing, nowhere tangent, smooth vector f i e l d on r ,

and l e t

BI

Bo be a smooth function on r . We define the boundary operator

by B1u :=~-~ @u + BOu

and we consider the BVP Au = f

(3)

in

R

BlU = g on r where (f,g) E C ~ ) Suppose that

, ,

x cl"(F) .

(f,g) E CU(~) x C1+U(r) and that

well-known that the BVP (3) has a unique solution

(ao,Bo) > o . Then i t is u : S1(f,g ) E C2+~(~) .

14

Moreover, the maximump r i n c i p l e and Schauder type a p r i o r i estimates imply that

S1 E L+(CU(~) x cl+~(r),C2+U(~)) . By similar (though more complica-

ted) arguments as above i t can be shown that denoted again by

S1 has a unique extension,

S1 , such that SI E K+(C(~) x C(r),C(~))

and such that

S1 i s s t r o n g l y p o s i t i v e . A detailed proof for these results

is given in [ 7] . In order to t r e a t the d i f f e r e n t boundary conditions simultaneously, we denote by

6

a variable which assumes the values

o

and

1 only. Then we

consider the l i n e a r eigenvalue problem (EVP) Au = ~au in

(4)

B~u = ~bu where (a,b) E Cu(~) x c l - ( r )

~ ,

on l" ,

such that

(a,~b) > o .

(1.16) Theorem: Suppose that there exists a number aa+ ao ~ o i f

6 = o

and

(aa + ao,ab + Bo) > o

linear EVP (4) posses~s a ~nallest eigenvalue, eigenvalue, o~d

a >_ o

if

s~ch that

a = 1 . Then the

~o(a,6b) , the

Lo(a,6b) > -~

T~ere exists exactly one linearly independent positive eigenfunction

uO E C2(R) n C1(~)

and

Uo(X ) > o for every Ix~stly,

~o(a,~b)

uO belong8 to the principal eigenvalue.~oreover,

x E ~ , and

(a,6b) < (a1,&bl) , then

P r o o f : Suppose that -

~a,Bo - ~b) > o

~ = I .

is a strictly decreasing function of its arguments.

M o r e preoisely:, if the pair

(ao

uO >> o if

(al,bl) E C~(~) x C1-(r)

satis~es

Xo(a,6b ) > ~o(a1,6b1) > o .

~ _< -~ . Then ao - ha _> o if

if

6 : o

and

~ = I , and, by the maximump r i n c i p l e , (4) has

15

the t r i v i a l

solution only.

Suppose that

~ > -~ . Then the EVP (4) is equivalent to (A + aa)u = (L + e)au

in

R ,

(B6 + ~6b)u = (~ + ~)6 bu on r . Hence we can assume without loss of generality t h a t (ao,8o) > o Let

if

~ = 1 , and t h a t

Eo := Ce('~)

and

T(u) := u l r . Define

if

6 = o

and

~ > o .

E1 := C('~)

i n j e c t i o n , and denote by

ao ~ o

, let

z : C(~) § C(r)

i 6 : E6 + C(~)

be the natural

the trace operator defined by

T6 E K+(E6) by

T6u := S6(ai6(u),b T o i 6 ( u ) ) . 0

Then i t is e a s i l y seen t h a t

T6

is a.s.p, and t h a t

P6 denotes the p o s i t i v e cone of to the EVP T6u = ~ ' l u o f Theorem (1.13)

in

p~ n ker T6 = r , where

E~ . Furthermore, the EVP (4) is equivalent

E6 . Hence the assertion is an easy consequence

9

(1.17) Theorem: Let the h~potheses of Theorem (1.16) be satisfied and suppose that

(f,g) E CU(~) x cl-(F) . Then the BVP Au

(s)

-

Xau = f

in

~ ,

B6u - ~bu = 6g on

l~s for every

X < Xo(a,6b ) exactly one solution (in C2(fl) n CI(~) ) which

ie positive (everB~here in ~ ) if

(f,6g) > o . If

(f,g) - o and

X > Xo(a,6b) j then (S) has no positive solution. I~nally~ (5) has no solution at a l l i f

X : ~o(a,~b)

and

_+(f,6g) > o .

ProOf: Since the BVP (5) i s equivalent to the BVP (A +aa)u - (X + ~)au = f

in

(B6 + a6b)u - (x + a) 6bu = g on

R

,

r

,

18

we can assume t h a t (ao,Bo) > o

if

~ > o

ao ~ o

if

6 : o , and

a = I . Hence (cf. the preceding p r o o f ) , (5) is equiva-

l e n t t o the equation v := ~-1S6(f,6g ) >> o Theorem (1.13)

and t h a t

~

~-lu - T6u = v if

in

Ea

, where

( f , a g ) > o . Now the assertion follows from

17 2. Fixed Points of Increasing Maps

Let

X be a nonempty subset of some Banach space and l e t

is compact. The map f

is called eo~letely

is compact on bounded subsets of

X . (Observe that in

is continuous and ~

continuous i f

f

be a

is called compact i f

map from X into a second Banach space. Then f f

f

the case of a linear map the l a t t e r property is being used for the definition of a compact operator. However, since the only linear operator which maps i t s domain into a compact set is the zero operator, no confusion seems possible.) Let

(E,P) and

(F,Q) be OBSs, and l e t

X be a nonempty subset of

E . A map f : X ~ F is called inorea~ng i f

x ~y

implies

f(x) ~ f(y) ,

s t r i c t l y increasing i f

x o

x + P c D (or

P-open (or (-P)-open) i f

D is called

P := { x ~ P I l l x l l < p} . p

z~ght (or left) neighborhood of x i f there

exists a positive number ~ such that The set

let

x - P c D ). E

D is a right (or a l e f t )

neighborhood of each of i t s points. For example, P and every open set are P-open. Every order interval neighborhood of Let

x

{x,y]

with nonempty i n t e r i o r is a right

and a l e f t neighborhood of

D c E be a right neighborhood of

y .

x E E and l e t

arbitrary Banach space. A map f : D ~ F is said to be

in

x

i f there exists

lim h~o hEP

F be an

right diffe~nticu~le

T E L(l~Zl~,F) such that

f(x+h)-f(x)-Th llhll

=

0

The operator T is uniquely determined, denoted by f~(x) , and called the

right derivative of the obvious way.

f

at

X

.

The

left derivative

f~(x)

is defined in

2O O

(2.3) Le,vna: L~t

(E,P)

a right neighborhood of

be an 0BS such t h a t

x E E

~ontinuou8 mo~. Suppose that

and let

f(x) =

x

P * # . Let

f : D ~ E

be a co,~letely

f~(x)

and that

D c E be

exists and i8

0

a. 8.p.. Then there e~st.8 a point

(c,h) E ~ + x p

such that, for every

T e (o,c) , f ( x + Th) > x + Th

if

r(f+(x)) > 1 .

and

Similarly, if the above assunptions hold for

i f ( X ) , thsn

f ( x - Th) >> x - Th

if

r(f'(x)) < 1

f ( x - Th) > y + Th f o r every

T E (o,e) . Since

a number T e (o,e)

~ < < y := ~ + Th 1 .

x

of a map f

is called

r ( f ' ( x ) ) -< 1 , and unstable i f

23 In the very special case that

E = ~ i t is an easy consequence of the

intermediate value theorem that each pair of stable fixed point is separated by an unstable fixed point. I t is the purpose of the following considerations to generalize this result to a more general (and more useful) setting. Since, by the above considerations, unstable fixed points cannot be constructed by monotone iterations we have to take recourse to topological methods. The appropriate topological tool for the proof of the existence of fixed points of compact maps in OBSs is the fixed point index. In the following we collect the basic properties of the fixed point index without proofs. For an elementary derivation from the better-known Leray-Schauder degree we refer to [ 6 ] . Let

X be a nonempty closed convex subset of some Banach space. For U of

every open subset

X and every compact map f : U ~ X which has

no fixed points on

BU , there exists an integer

index (of

U with respect to

f

over

i(f,U,X) , the fixed point

X ), satisfying the following con-

ditions:

(i)

(Normalization): FOr every constant map i(f,U,X)

(ii)

= I

f with

f(U) c U ,

.

(Additivity): For every pair of disjoint open subsets of

U

such that

f

has no ~ x e d points in

UI, U2

"U\ (U I u U2) ,

i(f,U,X) = i ( f , U i , X ) + i(f,U2,X) 9 (iii)

(general Homotopy Invariance) : Let interval of

~, let

U

be an open subset of

U~ := {x E X I (~,x) E U} h : l~-* X

(~,x)

e

~U

A be a nonen~ty co~pact

for every

is a co~oact m~p such that .

Then

i(h(~,.),UL,X)

,

~e A ,

A x X , and let

~ E A . 5~ppoee that

h(~,x) ~ x for every

24

is well-defined and independent of

(iv)

(Permanence): If

Y

f(U) c Y , then

~ 9 A

is a convex closed subset of

i(f,U,X)

X

and

: i ( f , U n y,y) .

As an easy consequence of the above properties which, by the way, characterize the fixed point index, we obtain the following additional properties:

(V)

(Excision):

For every open subset

has no fixed point in (vi)

such that

f

UkV , i(f,U,X) = i(f,V,X) .

(SolutionProperty): I f fized point in

V c U

i (f,U,X) # o , then f

has a

U .

By means of the fixed point index i t is now easy to generalize the one-dimensional situation described above in order to obtain the following "mintermedlate value"

theorem. 0

(2.6) Theorem: Let there are four points

(E,P)

be an OBS such that

P % ~ . Suppose that

yj , ~j 6 E , j = 1,2 , with

Yl < h < ?2 < Y2 and a co~pact increasing map

f : [y'1,92 ] -~ E such that

Yl -< f(-Yl ) ' f(Yl) < 91 ' ~2 < f(~2 ) ' f(Y2) -< 92 Then

f possesses a maximal fixed point

fixed point ~2 i n T~n

f

I'Y2,92|.

to

Suppose that

has a third fixed point

Proof: The existence of

R 1 in [~1,91 ] and a minimal

x*

such that

~1 and 72

XI := { y l , P l ] and X2 := [~2,92]

Xl - B~ , f ( ' , ~ ) -< ao~ , and

g(.,~) < Bo~

Then the BVP (1) has at least one solution in the order interval This follows from the fact that ~II

is a subsolution and ~II

i s a super-

solution for (1). Consequently, the BVP -au = 2 cos u - eu

in

@u ~6 =_ 9 sin u + eu has at least one solution

Corollary (3.2) and l e t

Xo :=

,

on I"

u such that

(3.41 Exc~nple: Suppose that

~

o < u(x) _< ~/2

a and b

for a l l

XE~"

satisfy the hypotheses of

Xo(a,6b) . Then we consider nonlinear BVPs

"in resonance", that is, Au - XoaU = f ( x , u )

(B)

in

B6u - Xo~bU = 6 g ( x , u ) I t f o l l o w s from the above c o r o l l a r y exists a

negatiue

E g ( . , E ) _< xb62

constant

for all

x

~ E l~

9

on

, s

t h a t the BVP (5) i s s o l v a b l e i f

such t h a t with

6f(.,6)

< xa62

there

and

161 >- 6o 9 Consequently, the BVP

-Au - Xo U = e c o s u - 8u2k+1 u=o has a t l e a s t one s o l u t i o n f o r every

R

in on

~E R

T

, B E JR+ , and

(Observe t h a t the Example ( 3 . 3 ) i s a l s o " i n r e s o n a n c e " , )

9

kE

]~ .

32 As an application of Theorem (2.4) we prove the existence of a positive solution of the BVP (1) in the case that ( I ) possesses a t r i v i a l solution.

(3.5) Tl~orem- Suppose that partlal derivatives D2f ~

f ( ' , O ) = 0 , g(',O) = 0 , ~

that tl~

D2g exist a~d are continuous i n a right

neighborhood o f zero. Yoreover, e~pose that b := D2g( ' , o ) E cl-(F) , such that a number ~>- o such that

a := D2f ( . , o ) E C+U(~) ,

(a,ab) > o , and that there exists

ea + a0 _> o i f

a =o

)

(aa + ao,=b + Bo) > o

T~n, if

if

a = I

.

9 > o is a supersolution for the BVP (1), there exiets a

mammal ~ o ~ tive solution in the order interval

[o,~] provided

~o(a,6b) < I .

Proof: We can assume that _> o

I : [ o,max 9] . Hence there exists a constant

such that the i n e q u a l i t i e s (2) and (3) are s a t i s f i e d . Let

Fu(u ) := F(u) + ~u and G (u) := G(u) + ~u , and denote by S~ the solution operator for the pair

(A + m,B6 + am) . Then the BVP (4) - and

hence the BVP (1) - is equivalent to the fixed point equation u = ~(u)

in

E6 , where ~(u) := S~(F (u), aG oT(u)) . Let

~ := ~(9) E Ea

. Then I

into

E6 such that

~(o) = o

maps [ o , ~ ] c Ea

increasingly and compactly

and I ( ~ ) _< ~ . Moreover, ~

has a r i g h t

d e r i v a t i v e at zero, namely ~(o)h = S~(F' +(o)i~(h),~G',+(o)~ o i~(h)) for every

h E Ea . Hence ~'(o)

is an a.s.p, compact endomorphismof

E~

33 Suppose that

x := r ( v ' ( o ) ) ~ I . By Theorem (1.13) there exists a

positive eigenvector u of

V'(o)

to the eigenvalue

~ . Hence u is

a positive solution to the linear BVP

Since

(A + ~)u = x-1(a + ~)u

in

~

(B6 + 6~)u = x-16(b + ~)u

on

r

x- I ~ 1

,

i t follows that

Au - au ~ o

in

~

B6u- 6hugo

on

r

,

But these inequalities contradict Theorem (1.17) since

1 > Xo(a,ab) .

Hence r ( I ' ( o ) ) > I . Clearly,

~ >> o . Hence Theorem (2.4) implies the existence of a

maximal positive solution

u

in the order interval

is a supersolution i t follows that

[o,~] follows from

~

(3.6) Example: For every pair

2 , the BVP (~,y) E ~ +

-au = a sin u

in

@~ = y sin

on F

has at least one solution Indeed, ~

. Since

o < u ~ p ~ ~ . Hence the existence

of a maximal positive so]ution in the order interval Theorem (3.1)

[0,9]r E

u such that

~

,

o < u(x) ~ ~ for every

is a supersolution and Xo(~,3y/2 ) = o

Hence the assertion follows from Theorem (3.5)

xE ~ .

(cf. Theorem (1.16)).

9

The transformation of the BVP (1) into an equivalent fixed point equation in the OBS E6 , which has been used in the above proofs, makes i t also possible to apply the abstract m u l t i p l i c i t y results to the nonlinear BVP (1). The following theorem, whose proof is l e f t to the reader (cf. also { 2,6 ] ), is an easy consequence of Theorem (2.6).

34 (3.7) Theorem= S~opose that there exist a subsolution

supersolution

Vl

, a strict subsolution

f o r the BVP ( I ) such that

~I

, a strict

~2 j and a s~persolution

V2

~ i < ~1 N and D(L) : : {u E W~(R) I Bu = o}

operator

L

in

Lp(R) by Lu := Au for every

, and define a linear u E D(L) . Then the BVP

(6) can be identified with the evolution equation u' + Lu = F(u)

(7)

u(o)

in

for

o < t < T ,

= uo

Lp(~) , where F denotes the Nemytskii operator induced by f . Since

we can add on both sides of the f i r s t equation in (7) the term au , where is an a r b i t r a r i l y large positive number, we can assume without loss of generality that operator t ~ o , in

o belongs to the resolvent set of the closed linear

L and that

L

-tL generates a holomorphic semi-group U(t) := e

L p ( ~ ) (cf. [131 ).

By using the regularity theory for parabolic BVPs (cf. {12,20] ) and the regularity theory for evolution equations (cf.]13, 21 ]

), i t can

be shown that the parabolic BVP (6) is equivalent to the nonlinear integral equation

in

t u(t) = U(t)u o + J U(t-~)F(u(T))dT , o ~ t ~ T o E := C([o,T],Ca)) , provided uoE D(L) .

I t is an easy consequence of the maximum principle for e l l i p t i c equations that, for every

~ ~ o , the linear operator

endomorphism of

Lp(~)

(L + L) -1

is a positive

(cf. Paragraph i ) . Hence the well-known exponential

formula ( c f . [ 1 5 , 2 1 ] ) . U(t) = s - lim n ~

(1 + t L)-n

3B

implies that

U(t) E L+(Lp(n))

for every

t ~ o .

I t is well-known (e.g. [13,15,21] ) that every D(L)

U(t) e L(Lp(R),D(L))

for

t > o , where D(L) is given the graph norm. Moreover (e.g. [ 13] ) is compactly imbedded in

tinuous imbedding of

Ca) 9 By using these facts and the con-

C(-~) in

Lp(~) , i t is easy to prove the following

lemma (cf. also [21, Chapter 4, Theorem 4.3] ). (3.9)

Lerrena:

Let

u o 9 D(L)

.

Pot every

u 9 E := C ( [ o , ~

,C(ff))

let

t

T~n

"}r (t) := U(t)u o + S U(t-T)u(T)dT o 36 9 K+(E) .

For every ~i~:=~(,o~"

,

o~ t ~ T .

u 9 E l e t Y ( u ) ( t ) := F(u(t)) , o s t ~ T , and l e t . Then, by the above considerations, the parabolic BVP (6)

is equivalent to the fixed point equation

u =~(u) in

E , and i t is easy to see that "J~ is a completely continuous map in

E . Moreover, the map ~I~ is increasing provided

f(x,')

is increasing

for every x 9 ~ . Hence we are in a position to apply the abstract results of the preceding paragraph. A function

u9

C2'I(QT) n C~'~ T u

ZT) n C(QT) is said to be a sub-

eolu~on for the parabolic BVP (6) i f @u - - ~ + Au ~ f ( x , u ) Bu ~ o u ~ uo

in

QT '

on

I:T ,

on

no

Supersolutions are defined by reversing the above inequalities. I t should be observed that every sub-(super-)solution for the e l l i p t i c BVP

37 Au = f(x,u)

in

~

Bu = o

on

r

,

can be i d e n t i f i e d w i t h a s u b - ( s u p e r - ) s o l u t i o n o f the p a r a b o l i c BVP (6).

(3.1o) Theorem: Suppose that solution for the parabolic

r

is a subsolution and

BVP (6) such that

possesses a unique solution

u

and

~

is a super-

@ < ~ . Then the

BVP (6)

u E [r

Proof= Without loss of generality we can assume that

I = [min r ,max r

Hence, by adding an appropriate term of the form ~u ,

m E ~ + , to both

sides of the f i r s t equation in (6), we can assume that

f(x,.)

creasing for every x E ~ . Consequently, ~ from the order interval Let

v

[r162 c E into

is in-

is a compact increasing map

E .

be the unique solution of the BVP ~v

~--~+ Av = f ( x , r Bv :

V :

Then v =~(~)

in

QT '

o

on

sT

,

U0

on

~0

"

and

@~(C-v) + A(r @t

< o

in

QT '

B(r

~ o

on

T "

r

~ o

on ~o "

Hence, by the maximumprinciple for parabolic equations (cf. [21] ) i t follows that

r s v , that is,

r s~(r

. Similarly,

r ~ ~(r

. Hence

the assertion follows from Theorem (2.1) and the well-known fact that the Lipschitz continuity of

f

guaranteesthe uniqueness 9

38

Suppose that ~

is a subsolution and ~ is a supersolution for the

e l l i p t i c BVP Au = f(x,u)

in ~ ,

Bu = o

on r

(8) ,

such that ~,9 E D(L) and ~ ~ ~ . By identifying ~ corresponding constant functions in

and ~ with the

E , i t follows that ~

and ~ are

subsolutions and supersolutions, respectively, for the parabolic BVP (6). Hence9 by the preceding theorem, there exists exactly one solution ~

of

the parabolic.BVP (6) with i n i t i a l condition uo = T . Since this is true for every

T > o , i t follows that the BVP BU + Au = f(x,u) Bt

in

~ • ~+ ,

Bu = 0

on

r x ~+

i

(9)

u

=~

,

on ~ x {o}

has a unique solution ~ . Moreover, by the proof of the preceding theorem,

: ~t(~) > ~ V ) _>V Suppose that T E ~+

f

possesses a continuous partial derivative D2f . Let

be arbitrary and l e t

w(t) "= -u(t+T) - - u ( t )

@w+ Aw = c(x,t)w Bt

in

n x ~+ ,

Bw=o

on

r

w-> o

on

~ x{o}

x

+

for

t E ~R+ . Then

9

,

I

where c(x,t) := f D2f(x,~(x,t ) + ~ w(x,t))do . Hence the maximumprinciple o for parabolic equations (cf. {23] ) implies that w ~ o , that is, the map :

~(x)

I~+ § C(~) ::

lira

is increasing. Since u E {~,~] u(t,x)

exists for every

, i t follows that

x E ~ . This fact implies that

t§174

F~(t)) § F(~)

in

Lp(n) as

t § =

. Consequently, by well-known results

about the asymptotic behaviour of solutions of parabolic BVPs (cf. [ 12,13) ).

3g i t follows that "u ~

is a solution of the e l l i p t i c BVP (8). Hence

C(~) and ~(t) ~ - u

in

C(~) by Dini's theorem. This proves

the f i r s t part of the following sta/~lity

(3.11) Tl~orem: Suppose that D2f

on

~ x I . Let

~

f

~ s a continuous partial derivative

be a subsolution and let

for the elliptic BVP (8) such that of the parabolic BVP (6) and let

t ~ ~

~ be a supersolution

-v o

and

x E ~ . Consider the BVP

in

~

,

on

r

,

~ is a nonnegative real parameter. Then (cf. Theorem (4.9)) i t can

be shown that there e x i s t s a

~> o

such that

> ~* and a minimai p o s i t i v e s o l u t i o n

~(~)

e (o,~ ~) , Then, by using the convexity of can be shown that there e x i s t a s t r i c t solution

~

of (I0~) such t h a t

only s o l u t i o n of

~
0

est surjectif

de

D(A)

de point fixe de Schauder,

il existe

u

on

solution de

e

~u2~ + AuE + Bu e = f ,

(30)

(on utilise H

ici le fair que

faible pour aboutir

d'autre

;~

stable

Supposons e

+

par que

B

est monotone

~

(eP 2 + A)-IB

iv lim I + =

part ia propri~t6

est laiss~e

quand

f e H

N(A).

volt que pour tout

dans

R(B).

IBvk Iv--~ = 0

l'application

h6micontinu continu implique

, done continu de de

H fort

qu'une

grande boule

u ~-~ (eP2 + A)-](f-Bu))"

f 9 R(A) + e o n v

R(B)

et montrons

que

; appliquant

la monotonie

de

elu2e I

f = Av + E t. Bw. i

+

B , il vient

i

(Buc - Bw.l , u~ - w i) ~ 0 et par suite (Bue,ue) Co i

- (BuE,w)

- ( ZtiBwi,u e) > - C l

sont ind~pendants

de

e

et o7

w = E t.w. ii

.

fort

H fort

0 .

En effet

o7 t o u s l e s

dans

H

0

;

68

Donc

(f-~'u2~ - Au ,uE-w) - ( f - A v , u ) ~ - C] i.e.

Elu2J 2 < ~lu2sl Iwl * [Aucl(lulr

lUll * c 2

Or i l r ~ s u l t e de (30) que

(3,)

If][ * IBl%[ < Ifl §

0

~C~

tel que

(Bu-Bv,u-v)

Dfimonstration.

Comme

(Bw-Bu,v-w) Prenant

w

~

- v + %~

comme

IBw I

Vu , v e H

T

- ~Jvl 2 - c~

B = ~

avec

on a par monotonie

l~I = l

et

JB(v+X~)[+

lim JwI§

~

i~l

cyclique

~ (Bu-Bv,u-v)

%JBuJ < ~ Sup

D'autre part,

suivant

!B_v~. = 0 .

Ivl~ Alors

du Lemme

% ~ 0

on a

(Bu-Bv,u-v).

._w.__ = 0 , alors

V~ > 0

_~M 6

lWl

~lwl + M~

Vw

e

H

Par suite + M 6] + (Bu-Bv,u-v) Choisissant

~ =I__

]Bu[

(Bu-Bv,u-v) I/2

~I

le Lemme

15 il vient

IBu12,, - ~ lwi 12 - C~

tel que

70

et par suite

1'

2

,

(~u c , u c) - (~Us,W) - ( ~ tiBwi,u ~1 ~ ~ lBucl o~

w = E t.w.11 Suivant

et

CI

est ind~pendant

la d~monstration

~lu2c 12 + T1 IBuc]2 o~

C

est ind~pandant

de

~

]BuE], ]u]c[ et slu2cl2 2~

Supposons

reste borne.

n'~tant

que

Le passage

E 9

du Th~or~me

13 on arrive

(f,u)

Vu e N(A)

=~

(36)

JB(U)

Vu e N(A),

u @ 0

Prouvons u e H

> (f,u)

d'abord

et

~ e IR

D'o~ l'on d~duit (u,B(tu)) Prouvons

que

tels que que

(36)

r > 0

tel que

existe

= N(A)

, u ~ 0

Hahn Banach

Va , Vb .

JB(U)

dim N(A)

- (f,u) > plul

tel que

R(B)]

, il existe d'apras

< (u,f)

ast une fonction

Comme

Raisonnons

+conv

et d'autre part

JB(u)

homog~ne.

f2 ~ P2 [ eonv R(B)] . Appliquant u e N(A)

R(B)

R . En particuller

JB(u)

R(B).

f e Int [R(A)

(u,Aa + Bb) < ~ < (u,f)

; notons d'abord que

et positivement

f e R(A) + c o n v

alors

Vte

~

f 4 R(A) + c o n v

u e N(A*)

< e < (u,f)

continues) existe

(35) ; si

f e R(A) + conv B(B)

< ~

Vu e N(A)

par l'absurde

(u,Bb) < (u,f)

s.c.i.

(sup de fonctions

on d~duit

de (36) qu'il

. Ii suffit done de montrer

. Sl

Hahn Banach dans

, ce qui est absurde.

f 4 R(A) + e o n v N(A)

Vb e H

R(B),

on volt qu'il

et l'on en d~duit

une contradiction.

Un exemple. (29).

Soit

On prend g : R § R

H - L2(n) une fonction

avac

I~[ < ~

croissante

et on suppose

continue

telle que

qua

A

v~rifie

71

I~(~>I

~m

On p o s e

g+

.

=

o

lim

g(]:)

.

Soit

f e L2(fl) .

Alors

f

g+u

fl g+u

§

- g_u

_

- g_u

~

N(A)

f fl fu

,

Vu

fu

,

Yu e N(A)

9

e

e

R(A~'-~)

,

~,

f

, u # 0

~

f 9 Int R(A+B)

fl E n effet o n peut applique]: le T h ~ o r ~ m e

14 avec

Bv = g(v)

et note]: que

JB(U) = ~ g + u + - g_u- 9 Un ] : ~ s u l t a t

c o m p a ] : a b l e , p o u r l e c a s o~

L a n d e s m a n - L a z e r [~'~ ] ( c s

a u s s i [ ~ ] [~,~] ) .

g

e s t b o r n ~ a ~t~ d~mont]:~ p a r

72

IS - SEMI GROUPES NON LINEAIRES

II.I

Rappel de quelques r~sultats usuels ; effet r~$ula~iSant Enon~ons d'abord un r~sulhat de bas~ dfl ~ K o m ~ a ,

(Cf. par exemple

Th~or~me

16.

[7

Soit A maximal monotone. Alors pour tout u ~ e D(A), il existe

u e C([O,+=);H)

unique t e l l e

u(t) e D(A)

u t > 0

que

lu 0

d~fini

s u r un r

c e s 3 p r o p r i ~ t ~ s o n prouve ( o f . [ 7 1 p.114) q u ' i l

ferm~ e t v ~ r i -

e x i s t e un A maximal mono-

t o n e unique tel que S(t) coincide avec 18 semf-gro~pe engendr~ par -A.

Lorsque

A = 3r

Tl~or~me 17.

Soit

unique telle que uest du --+ dt De

d~rivable Au ~ 0

plus on

(37)

on a u n

A

-

~.

r~sultat plus precis

u(t) ~ D(A)

~t

> 0.

p.p. et d~rivable ~ droite p.p.

,

i l e x i s ~ e u r CG O,+=);H)

Alors pour tout u o r D - ~

u(O) = u

~t > 0

o

a

d+u ( t ) l - IAoa(t)l g d-q-

1 I% - t t ( t ) l ~.

~

> 0 9

73

D@monstration. ~

ProUVOns

lipschitzien

(37) dans ie tag o~ @ est convexe de classe C 1 ave=

(le cas g~n@ral s'en d~duit en r~gularisant A par A l puis en

passant g la limite).

Soit

v s H ; on a du ~(v) - ~ ( u ) > (Au,v-~) - -(~-f , v-~)

et par suite (38)

I

T

T~(V) -

I [u(T)_vI2 I 2 ~(u) > ~ - ~[Uo-V[

0

D'autre part on a d I~-~I2 + ~-f ~(u) = o d'o~ l'on d~duit que I T t ~d u 2 o

(39)

Combinant

+ T ~(u(T)) - J T ~(U) ~ 0

(38) et (39), il vient avec v ~ u(T)

IT t ~au2< 89 luo_~(T)12 .

(40)

O

du I~-{(t)[ est a ~ c ~ o i s ~ n t e

Notons enfin que la fone~ion

I Tz Remarque. S(t) D--~) C

du IT6 (T)12 0 ; auCrement dit S(t) ~ Bn offer r@gularisa~t.

Indiquons

2 probl~mes ouverts I) Si A est seulement cycliquement mono~olle K l'ordre 3 est ce que S(t) a un offer r@gularisant ?

La r@ponse est positive dans le cgs lingaire d'apr~s Bn

de Kato affirmant que los op@rate~rs analytiques

(Cf. [~6 ] et aussi [ ~

2) On suppose que A = ~@ , H = L2(~) u ~ , 6 o , Vt.

Est ce que

~sultgt

m-sectoriels eng~ndrent des semi groupes ]pour une 4@monstration avec

I~I < ~

u ~ ~ D--~) ~ L P ~ A s ( t ) u

~r~s simple).

et ~S(t)u o - S(t)~ollL~ o ~ Lp

pour 2 < p < ~ ?

Lorsque A est lin~ai~@ la r@ponse est affirmative d'apr~s un r~sul~at de E. Stein

[~+]. 11.2

Fgrmulationde

eertaines ~quatiOns d'~volu~ion comae problames de minimisa-

tion convexe. Indiquons d'abord quelques compl~ments ~u Th@or~me

17.

74

1~

S l u o e D--~, alo~a

e t comme p . p . 2~

e D(r du ec donc r O

eat abcOlument oonts sur IO,TI , r (u) ~ L I (O,Z) du du - u) on a t~*(- ~ ) e L'(O,T).

(.~-~.~,

~du)

+ r

r

Siu

r

alore r ~ L 2 (O,T).

est absolument eos~inue

sur

dtt [O,T] , ~-~e L2(O,T;H)

Lee r ~ s u L t a t s qui e u i v e n t s o n t dGs g Br~gie-EkeLand (Of. [ i O ] ) . On d~finit le convexe

dV~ L ~ (O,T;H) , r ; ~-~

K - {v e C ( [ O , T ] ; H )

r L~(0,T)

, r

dd~) e LI(O,T) e t

v(O5 - % et la fonctionnelle

J(v) "

[r

+ r

+

[v(T) I z.

0 Ii eet clair que J e s t Th~or~me 18.

convege sUr g.

On suppose qua u ~ # D(r

Alors I~ solu~s

de l'~q~atlon

d' ~volution du ~-~ + ~ ( u ) 3 0

(41)

p.p. sur [O,T]

, u(0) - u O

est l'unique solution du probl~me varla~ion~el (42)

Minimiser J(v) pour

v e g.

D~monstration. On salt que

sou)

(435

9 r

u e get

du

~5

que

du

" -(~,

a)

Pour v quelconque darts g on a l ' i n ~ g a l i b ~ r

(445

+ r

dv ~'~-C5~ -

dv (~-'~, v)

d'o~

(4s5

89

J(u).

D'autre pert, on a l'~ga~it~ dane (45) el at setilement si on a l'~gaiit~ p.p. dana (44). Donc v r~allse le minimum de J sur K si et settlement s i v v~rlfie (41), i.e. V

=

U.

Remarque. l) I I n e semble pae ais~ de d~montrer direc~ement, c'est ~ dire sans passer par (41) l'exlstence et l'uniclt~ d'une solution de (42). 2) Lorsque u ~ e D--~', 1'Squatlon (415 admet ~ot~joure ttne eoltltion, male tot~tefoia

75 du ~*(-~)

n'est pas n~cessairement

~*(- ~du)

e L~(~,T) , u q >u 0 e et

int~grable sur [O,T]; on sait seulement qua

I T ~*(- W-z)dt ~du converge vers use limite quand

§ O. Le Th~or~me 18 rests a~O~S velaSle ~ ~o~di~ion de modifier le convexe K ec d'entendre l'int6grale Un ex,emple.

Cor~sid~rons l ' f i q u a t i o n de l a ch~l, eur

~-

I

(46)

Au - 0

sur

q = 0. • [0,TI

u = 0

sur

r x [0,T]

uo(x) su~

u(z,O)

avec u ~

19.

La solution de (46) est l'unique solution du probl~me

Minimiser

avec

v 9 Hi(Q)

= [

~(n)

l[rr=tt)U _ ~

r • [0,T]

sUr

~

] dt +

~(~)

, v(x,0)

I

= Uo(Z) Sur ft.

On applique le Th~or~me 18 avec H = L2(fl)

I

~.

~(v)

- ~llv'~l

~ i

On a doric

f

JO

~

, v = 0

Dfimonstration.

~(v)

fi

e HIo(Q).

Corollaire (47)

I T ~ * ( - -~t)dC au sans ~e la semi c o n v e r g e n c e . J0

,i

,

,

~*(g)

st

.

2 z = "~ g NH_

0u

Igll

z

~S~ I~ norms duals

o de Uv,I H1 o Plus precis&sent

~*(g) - ~

du probl~me de Dirichlet homog~ne

~*(g)

- Supl

yaH

[

g.v

oil

~ . ~

-- g

gut

t w - 0

sur

r,

En effet, on a

-

o

=~- ~ i II.3

w = ~-ig d~signe la solution

Comportement

i

au v?isina~e de t = O.

On se propose de comparer au voisinage de t = 0 le compor~ement de S(t) et de

J t = ( I + r.~) - i . Lorsqua A est un opfirateur lin~aire born~ lee d~veloppements

76

J~ - (I + tA) -I - I - ~ A

§

S(t) 9 e -r-~ = I - tA + montrent

de

que I - Jt

et

...

~A ~ +

l

I - S(t)

AZ§

tx

...

sont des quantit~s

du m~me ordre au volsinage

- 0.

t

En fairs

(48)

dans Is car g~n~r~l

Vx

D(A)

e

(non lin~airs)

on a toujou~s

lim Ix-s(t)x[. = I

t~o Ix-Jt~l (noter que x - Jtx ~ 0 sat~f si x-Jtx En effet on a lim - = A~ t t~O Lorsque

x 9 ~

consid~rons

,

x 4 D(A)

l'exempls

Ax 9 0 ,

auquei car x - S(t)x s 0).

lim t~O

st

x-S(t)x t

, la propri~t~

suiva~t

(construit --~

=

AOx.

448) n'est plus v~rifi~e.

d'apr~s une suggestion

,

u>O

,

u < 0

En effet,

de L. V~ron)

U

Dans

H m R

on pose

Au ~

[

avsc

~

II eat alors

facile de v~rlfier

qua I

1

Jt 0

~+'I

S(t)O -

et

=

~ > 0 .

1

(~+I) ~+!

t~-~T 9

1 DoRC

da~s

ce

car

I ~ - s(~)x] ~ (~,1)~*"~ 1. Ix - Jt~l Ix - s(oxl

N~anmoins

on peut ~tablir

pour x 9 D - ~ u n

encadrement

de

; c'est

I~ - Jtxl l'objet

des r~sultats

Proposition

20.

Ix

suivants.

Pour tout x 9 D-'~,

- s(t)xl ~

D~moastratlon.

31~

- Jt~l

Soit y e D(A)

Ix - s(=)~l < I~-yl * ly - s(t)yl Cholsissant

y ~ Jtx

on a

vt ~ o

; on a

9 Is(t)y

st sn rsmarquant

que

- s(t)~l [A~

le r~sultat. Th~or~me

(4 9)

21.

Pour

Ix-Jtxl 0)

et on ~tudie la r a p i d i ~

de la co,verge~ce de ]ue-Uol

vers O. b) u ~ ~tant donn~, on consid~ve l'~quation d'~volution et on ~tudie la r a p l d i ~

~du +

Au = 0 , u(O) - u ~

de la convergence de lu~)-uol vers O.

78

Ces considerations sont d~velopp~es darts [ ~ d'interpolation" interm~diaires entre

II.4

COMPORTEMENT AU VOISINAGE DE

D(A)

] par l'introduction de "classes et

D-~).

t = ~.

L'~tude du comportement d'un semi-groupe non lin~aire au voisinage de t = = e s t

assez

d~licate, except~ pour le (as trivial oO (Au-Av,u-v) > ~lu-v[ 2 Vu,v (ave( a 9 0) pour lequel on a convergence sxponentielle de

S(t)u ~ vers i'unique solution de Au 9 0 .

Commen~ons par un r~sultat r~cent de Baillon-Brezis Th~or~me 23.

[3

].

Soit S(t) un semi-groupe de contractions sur ~n r

ferm~

C. On suppose que S(t) admet a~ moins ~n point d'~quilibre (i.e. S(t)u = u Alors

It S(r)x dT converge faib~men~ q~and ~x e C , ~(t) = ~! -o

~t~O).

~ § ~ vers l'un des

points d'~qu~libre de S. D~monstration.

Soit F l'ensemble (convexe fermi) des points d'~quilibre de S.

On d~signe par P la projection sur F. On pose u(t) = S(~)x e~ v(t) = Pu(t). Appliquant l'~galit~ du paraLl~log=~mme a = v(t+h) - u(t+h) (sl)

Iv(t*h)

et

- v(t)[

la-bl 2 * la§

b = v(t) - u(t§ 2 9

Iv(t*h)

2 = 21al 2 , 21bl 2

ave=

on a

§ v(t)

-

2u(t*h)]

2 = /

= 2lvCt.h)

+ v(t)) e F

- ~(t,~)

[ ~ 9 2lv 4]v(t§

- ~(t,~)

]z

on a - u(C*h)[ 2

D'autre part on a

Iv(t) - u(t*h)l ~ = Is(h) v(t) - s(~) u(t)l ~ < [v(~) - ~(~)I ~

(s3)

Comblnant (50), (51) et (52) on est conduit ~

(s~) Iien

Iv(~*~) - v(t)l 2 < 21v(~) - u(t)l ~ - 21v(c,u) - ~(~*h)l ~ r~sulte que la fonctio~

quent v(t) est Cauchy quan~1

t § Iv(t) - ~(t)[ ~

est d~croissante eC par conse-

t ~ +~ .

On pose

s = lim v(t). t-w,~ Par ailleurs reprenant (50) on obtient ! ! Soit

~(tn),----% s

Wv e D(A)

pour

et par suite

Enfin notons que

My ~ F

tn § ~ ; 0 e As on a

(u(c) - v(t), y-v(t)) < o.

i.e.

on d~d~i~ de (SS) s

~ F.

que ~%v, v-~,') > 0

79

Donc

(u(t)-v(t),y-s

iu(t)-v(t)[ 1~-v(t)i < Ix-Pxl I~-v(t)1

uco(.) fortement clans L2(~) et uo= v~rifie l'~quation - Au~ = 0 sur ~ ; - ~-~7 = 8(u~) surr. Carte ~quation admet pour solutions routes lee fonctions constantes u = k avec B(k)= O. Ii serait int~ressant de connaitre u== en fonction de u facile de voir qua

u

=

l

I

Uo ).

(lorsque 6 -- 0, il est

o

81

BIBLIOGRAPHIE.

[11

J.B. BAILLON. Un th~or~me de type ergodique pour les contractions non lin~aires dans un espace de Hilbert, C.R. Acad. Sc. 280 (1975) p. 1511-1514.

[2]

J.B. BAILLON. Quelques propri~t~s de convergence asymptotique pour les semi-groupes de contractions impaires, C.R. Acad. Sc. (1976).

[3]

J.B. BAILLON. -

H~ BREZIS.

Une remarque sur le comportement asymptotique des semi groupes non lin~aires, Houston J. Math.

[4]

J.B. BAILLON. -

[S]

Quelques propri~t~s des op~rateurs angles horn, s, Israel,J.Math. D. BREZIS (~ para~tre).

G. HADDAD.

Classes d'interpolation associ~es g u n (~ paraitre).

[6]

op~rateur maximal monotone

H. BREZIS. Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, Contributions to Nonlinear Functional Analysis, ZarantDnello ed. (1971), Acad. Press.

[7]

H. BREZIS Op~rateurs maximaux monotones, Lectures Notes n=5 , North Holland (1973).

[8]

H. BREZIS. - F. BROWDER Nonlinear integral equations and systems of Ha~merstein type, Advances in Math. 18 (1975) p. I15-147.

[9]

H. BREZIS -

M. CRANDALL. - A. PAZY

Perturbations of n o n ~ e a r maximal monotone sets, Camm. Pure Appl. Math. 23 (]970) p. 123-144.

[i0]

H. BREZIS. -

I. EKELAND

Un principe variationnel associ~ ~ certaines ~quations paraboliques C.R. Acad. Sci. (1976).

[11]

H. BREZIS.- A. HARAUX. Image d'une somme d'op~rateurs monotones et applications, Israel J, Math. (1976).

[]2]

H.

BREZIS

-

L. NIRENBERG

On some nonlinear operators and their ranges, Ann. Sc. Norm. Sup.Pisa.

82

[]3]

R. BRUCK.

Asymptotic convergence of nonlinear contraction semi groups in Hilbert space, J. Funct. Anal. 18 (1975) p. 15-26.

[]4]

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[]5]

P. HESS. On a theorem by Landesman and Lazer, Indiana Univ. Math. J. 2-3 (1974) p. 827-830.

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T. KATO. Perturbation theory for linear operators, Springer

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E.M. LANDESMAN - A.C. LAEER Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech. 19 (1970) - p. 609-623.

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J.L. LIONS. - G. STAMPACCHIA. Variational inequalities, C o m .

[19

Pure

Appl. Math. 2 0 (1967).p.493-519.

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[ 20

L. NIRE~LBERG. Generalized degree and nonlinear problems, Contributions to Nonlinear Functional Analysis, Zarantonello ed. (197]). Acad. Press.

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R.T. ROCKAFELLAR. On the maximality of the sum of nonlinear monotone operators, Trans. A.M.S. 149 (1970) p, 75-88. j

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M. SCHATZMAN. Probl~mes aux limites, non lin~aires, non coercifs, Ann. Scuola Norm. Sup. Piss 27 (]973) p. 641-686

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E. STEIN Topics in harmonic analysis (1970) - Princeton Univ. Press.

IMPLICIT V A R I A T I O N A L P R O B L E M S AND QUASI

V A R I A T I O N A L INEQUALITIES U. M o s c o

The object of these lectures is to describe,

in a general fra-

mework, both the topological and the order methods that have been re cently p r o v e d to be useful in dealing w i t h v a r i a t i o n a l problems and inequalities i n v o l v i n g

implicit

c o n s t r a i n t s , t h a t is,constraints that

depend on the s o l u t i o n itself. The m a i n a p p l i c a t i o n s we have in m i n d are to s o - c a l l e d

riational inequalities,

quasi-v~

that have been r e c e n t l y i n t r o d u c e d by A. Ben

soussan and J . L . L i o n s in c o n n e c t i o n w i t h some s t o c h a s t i c impulse con trol problems,

see for instance ref. [ 2 ] [ 5 ] .

The t o p o l o g i c a l results d e s c r i b e d in Chapter I are based on JolyM o s c o [28]. The basic p r o p e r t i e s of v a r i a t i o n a l and quasi v a r i a t i o n a l inequalities in o r d e r e d Banach spaces are given in C h a p t e r 2. Some examples,

and a p p l i c a t i o n s are c o n s i d e r e d in C h a p t e r 3.

Lavoro eseguito nell'ambito del GNAFA, Comitato per la Matematica del C.N.R.

84

TABLE OF CONTENTS

CHAPTER I. Implicit vgriational problems by topolo@ical methods. I. A general framework. 2. Variational problems:

a) Ky Fan's inequality and variational ine-

qualities for bilinear forms. 3. Variational problems:

b) The monotone case

4. Existence results for the general implicit problem of section 1. 5. Nash equilibria under constraints. 6. Implicit Ky Fan's inequality for monotone functions.

Hartman-Stam-

pacchia theorem for "monotone plus compact" operators. 7. Selection of fixed-points by monotone functions. 8. Quasi variational inequalities for monotone operators. CHAPTER 2. Variational and quasi variational inequalities for monotone operators ~9 ordered Banach spaces. I. Ordered Banach space. 2. T-monotone operators. 3. Comparison theorems. 4. Dual estimates for solutions of variational inequalities. 5. Birkhoff-Tartar theorem. CHARTER 3. Some applications. I. A quasi-varlational

inequality with implicit obstacle on the bounda

ry. 2. A quasi variational inequality connected to a stochastic impulse control problem. 3. Regular solutions 4. Final Remarks.

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: C.R.Acad.Sc.Paris,

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t.278 s~rie A

C.R.Acad.Sc.Paris,

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1411-1415,

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ContrSle

In~quations

variationnelles

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[ 6]

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(D~cembre 1973).

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at the "Scuola di

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G.TROIANIELLO:

On the regular solution of a

quasi v a r i a t i o n a l i n e q u a l i t y c o n n e c t e d to a prob l e m of s t o c h a s t i c impulse control, 18(1975),

to appear.

[31] T . L A E T S C H

: J.Funct.Anal.

286-287.

[32] J . L . L I O N S

: Q u e l q u e s m ~ t h o d e s de r ~ s o l u t i o n des p r o b l m ~ m e s aux limites non lin~aires, V i l l a r s ed., Paris,

[33] J . L . L I O N S

: Sur la theories du contr61e, Vancouver,

[34]

J.L.LIONS

D u n o d and G a u t h i e r s -

1969. Int.Math.

Congress,

1974.

: On the n u m e r i c a l a p p r o x i m a t i o n of problems of impulse control,

I.F.I.P. Meeting, N o v o s i b i r s k , J u l y

1974. [35] J.L.LIONS,

E.MAGENES:

P r o b l ~ m e s aux limites non-homog~nes,

t.1

Dunod Paris, (I 968) . [36] J.L.LIONS,

G.STAMPACCHIA:

Comm. Pure AppI. Math.

20(1967),

493-519.

[37] W.LITTMAN, G.STAMPACCHIA, H.WEINBERGER: Regular points for

el-

liptic e q u a t i o n s w i t h d i s c o n t i n u o u s coefficients, Ann. [38]

KY FAN

Scuola Norm. Sup. Pisa,

: Math.Annales

142(1961);

ed., A c a d . P r e s s

1972.

I__7 (1963), 43-77.

Inequalities III,Shisha

88 [39] F.MIGNOT,

J.P.PUEL:

Solution maximum de certaines in~quations d'~volution paraboliques et in~quations quasi-variationnelles Acad.Sc.

[40] U.MOSCO

Paris,

paraboliques.

C.R.

s.A(1975).

Some quasi-variational

inequalities arising

in stochastic impulse control theory, Int. Summer School on "Theory on Nonlinear Operators. Constructive aspects",

Berlin GDR

Sept. 1976. [41] H.SCHAEFFER

Topological vector spaces, Mac-Millan ed., New York,

[42 ] G.STAMPACCHIA

1966.

Formes bilineaires coercitives sur les ensem bles convexes, C . R . A . S . t .

258

(1964),

4413-4416. [43 ] L.TARTAR

In~quatlons quasi variationnelles abstraites, C.R.A.S.,

[44] L.TARTAR

Paris,

t. 278

(1974), 1193-1196.

In~quations quasi variationnelles, Convessa e Applicazioni,

in Analisi

Univ. di Roma apri-

le 1974, Quaderno dei gruppi di ricerca matematica del CNR.

89

CHAPTER IMPLICIT

I.

VARIATIONAL

I

PROBLEMS

BY TOPOLOGICAL

consider

in o u r

METHODS

A GENERAL F R A M E W O R K The problems

into the

we

following

We are given

shall

general

a s e t C in a r e a l v e c t o r

(1.1)

CI

and two

lectures

can all be put

s p a c e E,

a subset

framework:

C C

C,

o f C,

,

functions,

(1.2)

: C, x C

~ ]- - , + ~ ], w i t h

~(u,.)~+~

for every

u 9 C,

,

and f : C, x C x C (I .3)

for every The problem

satisfy

the

(1.4)

~ ]-~,+~[,

u E C,

consistsin

following

with

f(u,v,v)

< 0

a n d a l l v E C. finding

system

all vectors

of s c a l a r

u of the

s e t C]

that

inequalities

- u e C0 ~(u,u) + f ( u , u , w )

where

Co

we consider,

Vw6

C

given s u b s e t o f C~.

is s o m e

We shall

! ~(u,w)

deal with

problem

for any flzed

(1.4)

in t w o

steps.

In t h e

first

step

vector u 6 C,

the

functions

(1.5)

~ and g defined

~(w)

"(1.6)

g(v,w)

and we

,

i

(I .7)

The

setting

w 9 C

= f(u,v,w)

look for all vectors

"variational"

where

= ~(u,w)

by

,

v,w 9 C

v of the

space

E which

solve

the

following

problem:

vqC

~(v)

set

+ g(v,w)

(possibly

% and g are given

< + (w)

empty) by

u w 6 C

of all

(1.5)

and

solutions (1.6),

v of problem

will

be denoted

(1.7), by

90

s (u) and the m a p

(1.8)

S

thus d e f i n e d

:

Ci

~

2C

will be c a l l e d

the 8 e l e c t i o n

map of our initial

problem

(1 .4). What must be p r o v e d

in this

first

assumptions

on the data E, C, CI,

some

properties,

"hice"

step

is that,

under

~ and f, the s e l e c t i o n

in p a r t i c u l a r ,

that the set S(u)

map

suitable (1.8)

has

is non-empty

for every u 9 CI. The second (1.4)

consists

belonging

step that m u s t be carry out in order then

to solve p r o b l e m

all fixed-point8 of the s e l e c t i o n

in finding

to the given subset Co of C,,

i.e.,

all vectors

S

u such

that

~

u E C o

(1.9)

u 9

It is indeed

s (u)

just a m a t t e r

u of E is a s o l u t i o n

lution of the f i x e d - p o i n t We should p e r h a p s data of the initial ned by the g i v e n

of d e f i n i t i o n

of i n e q u a l i t i e s problem

notice

problem

(1.4)

(1.9)

since

(u,w)

= 6(Q(u),w)

,

maps

it is only i m p l i c i t e l y

Remark 1.1. An i m p o r t a n t special case of p r o b l e m

(1.10)

if it is a soS.

S is not itself among the

~ and f via the s o l u t i o n

functions

that a v e c t o r

for the s e l e c t i o n

that the maps

(1.4),

to check

if and only

of p r o b l e m

(1.4)

defi(1.7).

is w h e n

u 6 Ci, w 6 C,

w here

(1.11)

Q

is a m a p that a s s o c i a t e s and

6 (T, 9 ) is defined,

6 (T,w)

When problem

: CI -~ 2 C

a non-empty

s u b s e t Q(u)

for an a r b i t r a r y

of C w i t h any u 6 CI

subset T of E, by setting

= 0 if w E T and = + = if w E E - T .

~ is given by

(1.10),

problem

(1.4)

reduces

to the f o l l o w i n g

91

~

u 6 Co

(1.12)

,

u 6 Q(u)

f ( u , u , w ) 0

N F(w) weT 2.2.

This

lemma

is the

~ ~

infinite-dimensional

9

generalization

of a

94 classical authors

lemma,

in t h e i r

proof we We

refer

due to Knaster-Kuratowski-Mazurkiewicz,

used by these

proof

~

shall mention

COROLLARY

of Brouwer's

to Ky Fan

two corollaries

I OF THEOREM

2.1

(Ky F a n

L e t C be a n o n - e m p t y E and g a real-valued diagonal, riable.

concave

Then,

(2.12)

0 sufficiently coercive,

hypothesis vector

: ,.v.,3 ~ + ~ as

Dvll ~ The u n i q u e n e s s

w h i c h we

omit

Remark 2.4.

of the

here

(2.13)

is the first,

variational inequalities. qualities

solution

argument

by now classical,

inequality

between

9 example

of so called

variational

ine-

see also B r e z i s - N i r e n b e r g -

[10].

3.VARIATIONAL

9

PROBLEMS:

The a s s u m p t i o n too strong

by a s t a n d a r d

3.1).

On the c o n n e c t i o n

and Ky Fan's m i n i m a x

Stampacchia

u follows

(see also P r o p o s i t i o n

that the f u n c t i o n

for many

applications.

non-linear v a r i a t i o n a l (3.1)

b) The monotone

g(v,w)

A being a non-linear

w E C, be l.s.c,

is the case,

when

=

operator

g(.,w),

This

inequalities,

case on E is

for instance,

for

g is of type

,

v,wqC

from the c o n v e x

subset C of the space E to

its dual E'. Assumption suitable chia,

(2.5) (j) of T h e o r e m

pse~domonotonicity

loc. cit.

and J o l y - M o s c o

A is a pseudomonotone

operator

For sake of simplicity,

2.1

property

can

indeed be r e p l a c e d

[28]),

that covers s

in the sense

we shall

case

of Brezis

confine

here

(3.1)

to functions

have the monotonicity

property

by any g such as (3.1)~hen A is a monotone

tor in the usual

sense,

(Av-

With

this b a s i c

(see Joly-Mosco,

Aw,

example

introduced

when

[~9] 9

property

is s a t i s f i e d

by a

(see B r e z i s - N i r e n b e r g - S t a m p a c -

in D e f i n i t i o n

g that

2.1 b e l o w . T h i s opera-

i.e.

v-w}

> 0 ,

in mind,

loc.cit.)~

v,wqE.

we give

the f o l l o w i n g

two d e f i n i t i o n s

96

Definition 3.1. W e s a y t h a t a f u n c t i o n g is monotone if it is a f u n ction

as

(2.2)

(3.2)

and

its

symmetric

g(v,w) + g ( w , v )

Notice every

that

> 0

,

part

Yv

for a monotone

is n o n - n e g a t i v e

,wEC

o n C • C,

i.e.,

m

.

function

(2.2)

we

h a v e g(v,v)

= 0 for

v E C.

Definition 3.2. W e s a y t h a t a f u n c t i o n g is hemicontinuous if it is a function

as

(2.2)

with

C a convex

g(v +t(w-

of t h e

real variable

vectors

t610,1]

s e t a n d the

function

v) , w)

is l.s.c,

as to ~ 0 + for a r b i t r a r y

given

v , w o f C.

9

Remark 3.1. W e r e c a l l t h a t a m a p A : C ~ E', w h e r e C is a c o n v e x s u b s e t o f E, segment

of C to the weak

Clearly, ven by

(3.1)

the

prove

the h y p o t h e s i s

(2.5)

For

each v6C,

(ii)

g is m o n o t o n e

Notice

and

that we

However,

is n o t

stance,variational such that

g(v,-)

g(v,.)

function

is s t i l l

true

g with

is c o n v e x

the

(and,

following

g gi-

above

is convex), p r o v i d e d

9

furthermowe replace

one:

and u.s.c.

and hemicontinuous,

but also

according

to Definitions

that

g(v,.)

not only

is c o n c a v e ,

u.s.c..

restrictive

inequalities

for many : the

applications,

function

is a n affine f u n c t i o n

~ and g be given

respectively.

tisfied. empty

function

then the

t o the d e f i n i t i o n

(3.1)

for every

as,

for in-

involved

is t h e n

v 6 C.

3.1.

L e t C, (3.3),

2.1

(2.7)

are now assuming

2.1,

this

of

line

3.2.

as in T h e o r e m

THEOREM

on the

(i)

3.1

according

now that Theorem solutions

from the

o f E'

f r o m C to E',

is h e m i c o n t i n u o u s

set of all

(3.3)

topology

if A is h e m i c o n t i n u o u s

We will re,

hemicontinuou8 if it is c o n t i n u o u s

is s a i d

Then,

convex

the

Let

that

set of all

compact

satisfy

the coercivess

subset

solutions

of B A C

(2.3),

(2.1) (2.4)

condition

(2.6)

v of p r o b l e m

, B being

any

and

be also

(2.7)

(2.2) sa-

is a n o n -

set v e r i f y i n g

(2.6). The proof fact,

since

the

of T h e o r e m functions

2.1

cannot

g(.,w)

be repeated

are no more

unchanged

assumed

to be

here. l.s.c,

In on

97

E, the fore,

set G(w), we c a n n o t

W h a t we lemma

w E C, g i v e n apply

shall

by

Ky F a n ' s

do is t h e n

can be a p p l i e d

(2.9)

lemma the

directely

following:

to the f a m i l y

(3.4)

may well

n o t be closed. to the

We

show

family that

There(2.8).

Ky F a n ' s

of sets

F (w)

,

w 6 C

where

(3.5)

F(w)

This

allows

is the c l o s u r e

us to c o n c l u d e

of G(w)

that

(3.6)

the

in the

space

E.

set

F (w) weC

is not-empty. sumptions

At

this

point

on the

function

Instrumental

to t h a t

we

show

g, t h e

that

set

as a c o n s e q u e n c e

(3.6)

of our

as-

coincides w i t h the set

(2.10).

zation

of a w e l l

and F . E . B r o w d e r g satisfies

know

lemma

[12].

(3.3),

is L e m m a

below.

on m o n o t o n e

Basically,

then

3.1

this

a vector

This

lemma

operators,

lemma

says

due

is a g e n e r a l l

to M i n t y

that

if the

function

v of C is a s o l u t i o n

of the

inequa-

lities

(3.7)

~(v)

if and o n l y

~(v)

the

family

,

of the

+ g(w,v)

H(w)

of

,

u

,

inequalities

,

VwEC.

sets

w 6 C

for e a c h w 6 C w e d e f i n e

(3.10)

then

This

Jlv AX0

, vES(D0)

is t h e c a s e

if A ( u , v )

A(u,w)

-- A v

+ Bu

is of t h e

,

:

pro-

family of

s p a c e Xo. holds

:

such that

+

, uniformly

C, -+

with

X'

respect

form

u 6 CI ,

where A

condition

set

|Xo

Iv

as

to t h e

that of the

A Q (w), w 6Do

image

coerciveness

respect

following

the

v E C

to uqD0

,

115

satisfies

the

following

There

exists

coerciveness

wo 6

condition

~ Q(w) w ED0

o n Do

, such that

L-< Av,v-w0> /Uvilx03 ~ + |

(8.13) as

llvllx0 ~ =

,

v6S(Do),

while

B

is s u c h

that

B(D0)

: CI -+ X'

is b o u n d e d

m

in X0

8.2.

Remark

In o r d e r according (8.14)

and

that the map Q be weakly

to D e f i n i t i o n (8.15)

8.1

below

The map

above,

hold

it s u f f i c e s

o n Do

,

that both properties

:

Q is c o n t i n u o u s

of convex

(A,v')-continuous

sets C(X),

f r o m Do w e a k

to t h e

topology

i.e.,

(i) Q is u . 8 . c . : If u k c o n v e r g e s (8.14)

v k E Q (Uk) in X (ii)Q is

weakly

, then v6Q(u)

If u k c o n v e r g e s then

T h e m a p u -~ A ( u , v ) , f r o m Do w e a k

COROLLARY Under

weakly

more,

that

injection with

Do

to X' to X',

strong, i.e.

of X0 • X0

I of T H E O R E M

8.1.

assumptions

there

exists

C_+ i n t o V,

Do C_+ C,

bounded

into bounded

sets

s p a c e V0,

both

A(.,.)

which

below

of the dual

has

closed hold

isbounded

s e t of Do • Do

(8.18) , s u p p o s e

convex

conditions

.

is c o n t i n u o u s

furthermore,

and

such

to w in X

it c a r r i e s

and a non-empty

such that

and

and

w k6Q(uk)

fixed vEC,

(8.16) , (8.17)

a Hilbert

to u in Do

exist

strongly

for each

for t h e n o r m

the

to v

,

there

that w k converges

f r o m D0•

weakly

~.~. ~. :

w6Q(u),

(8.15)

t o u in Do,

and v k converges

9

further-

a continuous

subset :

X'.

Do

of V0

,

116

Do is stable under S and the image S(De) (8.21)

is bounded in V0

The map Q is w e a k l y (8.22)

(a,v')-continuous

on De a c c o r d i n g to D e f i n i t i o n 8.

Then, p r o b l e m

(8.20) admits a s o l u t i o n u .

Let us r e m a r k that, as a special case of D e f i n i t i o n 8.1,we have the f o l l o w i n g

Definition 8.2. W i t h the data of T h e o r e m 8.1, we say that the m a p Q is weakly

(a,v')-contlnuou8

on Do if :

For every sequence

(Uk,V k) c o n v e r g i n g w e a k l y to

(u,v)

in Do • Do and s a t i s f y i n g v k 6 S(Uk) , we have in the limit both (8.23)

(i)

veQ(u)

(ii)

u

, ~ W k e Q ( u k)

such that

limksu p a ( v k , w - w k) >_ 0

Remark 8.3. A c c o r d i n g to Remark 8.3, a s u f f i c i e n t c o n d i t i o n in order that the map Q be weakl U

(a,v')-condition on Do, is that the m a p Q be con-

tinuous on Do, that is,

(8.14) hold.

In fact, the a d d i t i o n a l c o n d i t i o n A(u,v)

(8.15)

is t r i v i a l l y s a t i s f i e d w h e n

~ L, L b e i n g the b o u n d e d linear o p e r a t o r a s s o c i a t e d w i t h the

b i l i n e a r form a, i.e.

(8.24)

a(v,w) = 0

due

if w e h a v e

121

The pairing between

in

(2.2),

as all p a r i n g s

V a n d its d u a l

the

is the d u a l i t y

sign = holds

pairing

V'.

strictly T-monotone if it is T - m o n o t o n e

A is s a i d whenever

space

below,

in

and

if

(u-v)+=0

(2.1).

Lemma 2.1. If the o p e r a t o r its

restriction

V to V'

(2.1)

is T - m o n o t o n e

to V is a m o n o t o n e

in the u s u a l

[strictly

[strictly

T-monotone],

monotone]

operator

then from

sense.

9

Proof. For

arbitrary

u-v=

(u-v)

both

with

+-

(u-v)-=

( u - v) + a n d

(Au ~Av,u-

and

u a n d v in V, we h a v e

if the

(u-v)

( v - u ) + in V

v> = ( A u - A v , ( u -

sign = holds

(v-u)

+

; therefore,

v)+> +< A v - A u ,

in t h i s

since

(v- u)+>

A is T - m o n o t o n e

~ 0

t h e n we h a v e

separately

= 0

and

( A v - Au, ( v - u ) +)

what

implies,

if A is s t r i c t l y

(u - v) + =

which

is to say u = v

0

= 0

T-monotone,

,

that

(v - u) + =

0

,

.

Example 2.1. open

Second

order

linear

subset

of ~ N

and

PDO

in d i v e r g e n c e

! N (2.4)

a(u,v)

=

( ~ i,j=1

with

(2.5)

aij,bj,c 6L~(~) ,

form.

Let

~ be a b o u n d e d

n aij(x)u

v

+ ~ bj(x)u v+c(x)uv)dx x i xj 3'= I x3

122

(2.6)

c(x)

> 0

a.e.

in

> ~01~I = i~j _

a.e.

x 6 ~

and N

(2.7)

[ i,J =I

aij(x)~

L e t V be any (1.12),

closed

subspace

of the

u ~ 6 ~N

,

Sobolev

space

, 70 > 0 .

HI ( ~ ) s a t i s f y i n g

such that

(2.8)

a(v,w)

If V = H~ (~), coefficients

(2.8)

of the

>_ ~LLVLIH, (e)

follows

form.

from

YvqV,

the

If V = H* (~),

7 > 0 .

above

then

assumptions

(2.8)

still

on the

holds

provi-

ded

(2.9)

c(x)

The

This

a.e.

x6 ~ .

identity

(2.10)

(Au,

defines

> co > 0

a

(linear)

follows

v) = a(u,

following

a ( w +, w-)

In fact,

uEHI

(n), v e V

= 0

property

for e v e r y

f r o m X = H* (~) to V'

of the

weH'

form

(2.4)

:

(~)

we n o w h a v e

y H (u-v)

whenever

( u - v) + 6 V. T h e

is s a t i s f i e d F of ~

latter

u,vEH

I (~),

condition, u-

for

v < 0

instance,

a.e.

IIv

if V = H~ (~),

on the b o u n d a r y

.

Let ction

provided

Jc 2

us a l s o

remark

of A to V is,

f r o m V to V'.

by

that, (2.8)

under

the a b o v e

assumptions,

a eoeroive c o n t i n u o u s

linear

the r e s t r i operator 9

123

Example 2.2. Non

linear

second

pseudo-laplacians, N

(2.12)

order

Au=-

8

]! 8u

[

(2.13)

with

the

a (u,v)

=

i( i ! I

so c a l l e d

+ c lul p-2v

c > 0, .

.

.

p > 2 .

u

u

xi

p-2 xi

subspace

Vx. i

+ clulP-2uvldx

of H I 'P(~)

satisfying

c V c H* 'P(~)

,

identity

(Au,v)

(2.15)

a strictly

is c o n t i n u o u s of V'

~u

3X--~

HA'P(~)

defines

as

p-2

~i

(2.14)

the

form,

form

If V is a c l o s e d

then


,

i=I ~ associated

PDO

e.g.,

and

= a(u,v)

T-monotone

f r o m the

is c o e r c i v e

strong

, u6H

I ,P(~) , v E V

operator

f r o m X = HI'P(~)

topology

of HI'P(~)

on V = H~ (~) and,

to V',

which

to the weak t o p o l o g y

if c > 0, on any V s a t i s f y i n g

(2.14). For more

examples

of T - m o n o t o n e

operators

we refer,

e.g.,

to

[32 ]. Given

an o p e r a t o r

(2.16)

I

If A

the

Au

,

not

u

9

- w ) < ~(w)

T-monotone

f r o m V to V', Corollary

exists

- ~(u)

then

convex

l.s.c,

function

- + = ,

of T h e o r e m

operator

3.1

a unique solution u of u =

vwev

which

is c o e r c i v e

by B r o w d e r - H a r t m a n - S t a m p a c c h i a

by (2.18)

for any

variational inequalit E

is a s t r i c t l y

(see a l s o

I), t h e r e

(2.1),

u E V

hemicontinuous rem

as

~ : V ~ ]- ~, + =]

we can c o n s i d e r

(2.17)

A such

a (~)

and P r o p o s i t i o n (2.17).

3.1

We d e n o t e

and theo-

of C h a p t e r

this

solution

124

Note

~(~)

that

By r e l y i n g the

on the

following

occurring

in

section, (2.17).

above

is increasing

order

of the

space

We

We with

in the

the m a p

on the

content

following

c 63~

shall class

define, of

s given

functions

by

~'

of so c a l l e d

.

in

functions (2.18) to the

comparison

section.

denote

by

~, a n d

F0 (V) the

+ ~] ~2

and not

set of all l.s.c, c o n v e x f u n c t i o n s identically

in F0 (V), w e

onV,

+ =.

set

~i + < A u 2 ,

This

(2.17)

+ ~2 (u,V u2)

Au A Av

in V'.

PRoof. By

(3.24)

we have

(3.27)

Au A A v 6 V e

and there exists

i

(3.28)

a unique

z >uAv

solution

,

z of the f o l l o w i n g

z 6V

(Az,

z-w) 0 ,

and

Az > Au A A V

in V'

.

x~0,

we get from

130

We now prove

(3.29)

that

z -- u A v.

z < u

and this

follows

into account.

and

we have

,

by comparing

to show that

z < v

from Corollary

In fact,

u = p (u /~ v, Au)

therefore,

It s u f f i c e s

2 of Theorem

3.1,

by taking

Remark

3.2

trivially

v = p (u A v, A v ) ,

each of these

solutions

with

z = 0 (u /~ v, A u /~ Av)

we obtain

4. D U A L As

(3.29).

ESTIMATES

FOR SOLUTIONS

in t h e p r e v i o u s

X is a r e a l (4.1)

under cone

section

we assume

reflexive

the partial

Banach

ordering

INEQUALITIES

that

space

and a vector

< induced

by a closed

(1.1),

V is a c l o s e d

(4.2)

OF V A R I A T I O N A L

subspace

and a sublattice

of X

of X ,

and

A (4.3)

We

: X ~ V'

is a s t r i c t l y

whose

coercive

and hemicontinuous

then assume

o n V is

g 6 V'

together

I ~ 6 Xv (4.5)

restriction

that a functional

(4.4)

is g i v e n ,

T-monotone

operator

with

an admissible

, i.e.,

for s o m e v 6 V

upper

obstacle

~ q X such that v
_ g /k A~

provided

this

element

g a n d A~ b e l o n g

to the

does

exist.

in V'. can

also

be e s t i m a t e d

fr-Jm

is

,

This

is c e r t a i n l y

order dual V e of V',

see

the

section

case

if b o t h

I.

Theorem 4.1. Under verify

the

assumptions(4.1)(4.2),

(4.3),(4.4)

following

and t h a t

(4.5)

let us s u p p o s e

~ A v q V

if u is the

solution

for e v e r y

of

(4.6),

that A

~ , in a d d i t i o n ,

condition

(4.9)

Then,

and

v 6 V

we have

.

, g and

satisfies

the

~

132

(4.10) where

g > Au h is a n y e l e m e n t

Remark 4.1. C l e a r l y ,

o f V'

> h

in

V'

satisfying

such on h exists

both h ~ g and h ! A~

in V'. 9

iff g - A ~ 6 V ~.

9

Corollary of Theorem 4.1. In a d d i t i o n g and A% belong

to the assumptions to the order

of T h e o r e m

4.1,

d u a l V e of V. T h e n ,

suppose

that both

Au also belongs

to

V ~ and we have

(4.11)

g

> Au

> g

A

A%

in

V'.

the

inequality

Proof of Theorem 4.1. L e t us p r o v e

first

(g-

for a l l v @ V, Now,

Au

v > 0. T h i s

let h 6 V'

, v)

(4.7).

We must

show that

> 0

follows

from

(4.6),

by replacing

w=u-v.

be s u c h t h a t

(4.12)

h 0 ,

~u

(u-h) .~

= 0

on

F.

to the

so c a l l e d

142 To r e p l a c e depending

on the

corresponds remain

the

function

solution

as the

librium,

it is g i v e n

integral

term

the

flux entering In o r d e r

introduce

by its

in o u r

function

initial

the

region

values that

h

problem

pressure ~, b u t

lowered

represents

(1.3),

does

not

, at the e q u i by the

some m e a n

constant value

of

a variational

formulation

to p r o b l e m ( 1 . 1 )

(1.2)we

space

(1.6)

H L (~) = { u 9

and w e

enters

(1.3),

the

the e x t e r n a l

initial

in

with

Q.

to give

the

as

in w h i c h

liquid

appearing

(1.4)

u itself,

to a s i t u a t i o n

constant

h in

recall

that,

H* (~)

by c l a s s i c a l

any u 9 H Li (u) the n o r m a l

:

Lu 9 L 2 (~)}

trace

theorems

(see, e.g. [35] ), for

derivative n

(1.7)

~u 3n

is w e l l

defined

~ i=I

as an e l e m e n t

Ux. + l I

of the

fractional

Sobolev

space

H-I/2(F),

and

(1.8)

lids,

_h-< ~ , ~ )

denotes

the d u a l i t y

pairing

H-V~ (F). We

(I .12)

suppose

that

s

a function

f 6 L 2 (s

a.e.

on

between

s

H~2 (r) and

its

143

is also g i v e n and we c o n s i d e r

quasi-variational


inequa-

lity Ii

I (~)

u6

,

HL

U

6

Q(u)

(I .13) (Lu,

u-w)

_ 0

,

and

Do =

v E H

(~)

The m a i n a s s u m p t i o n s 1) Do is not-empty

to be v e r i f i e d

: In fact,

following Neumann

~-~

2) Do

exists

.

are the f o l l o w i n g ones

the v a r i a t i o n a l

solution

:

u of the

=

0

in

on

as an e l e m e n t

F

of HI (~) and

is convex a n d closed in X0 = H Li (~)

a consequence

on F

problem

i Lu = f

clearly

a.e.

of the l i n e a r i t y

in H Li (~) n o r m e d by

(1.9),

%~-~

(1.8)

S: T h a t

of Do is

; its c l o s e d n e s s

f r o m the c o n t i n u i t y

8~__n : HLI (~) ~ H-I/2 (F), as s t a t e d by 3) Do is stable u n d e r the s e l e c t i o n

to Do ;

: the c o n v e x i t y

of L and

follows

belongs

of

; is,

for e a c h f i x e d

I44

u 9 Do

, the

solution

v = Su

of the Vl

(1.14)

i

v 9 H i (s

,

(Lv,

0 ~n -

on

r .

equation

(1.14)

(1.15),

both

con-

below:

(1 .15)

The

(1.11).

in t h e d i s t r i b u t i o n

sense,

follows

from

by replacing

W = V +

for an a r b i t r a r y

(1.17)

~ E C0 (~)

(Lv,

In o r d e r

to p r o v e

_ h - (~,

v k _~

~u

--~

h - (~,

in

: (~)

it follows

that

is,

s)

have

, we have

since

given

on

F ;

I

vk r

r

u k __~ u i n Xo

from

8u ) Therefore ~ F" (I .20)

w 6 Q(u)

%u >~F ~-~

a.e.

choose

~u > _ _ h - ( ~,

To verify

and

)F

; moreover,

H-,/2

~Uk ~-~ ) F ~ h - (~,

k ~ + ~,

k ) >_ 0 .

~u k >

F ,

on

F ,

,

147

Wk >h-

~u) + (~, ~ u ) ~ F ~

(~"

~~u k ) F

- (~,

=

~u k =

that

h

- (~,

~--n-- ) F

is,

w k 9 Q ( u k) . Moreover 9 since H -I/2 (r) ,

u k __~

~Uk 8u 9 w e h a v e ~-~-- --~ ~

in X0 = H Li (n)

u

in

therefore

w k - w= as k ~ + |

8u k > F ~0 ~~ u > F - (~, ~

(~,

. On

the o t h e r

v k __~ v in Xo = H~.(~)

bounded;

this

implies

lim (LUk,

as k ~ + | , i.e.,

2. A Q U A S I CONTROL

This operator

(2.1)

(2.2)

VARIATIONAL

llVkllH~(~)

that

is b o u n d e d ,

IILVk~H_ ' (n)

UVkUHt (~) is

thus

is b o u n d e d ,

hence

w - wk ) = 0

8) holds.

INEQUALITY

involves


Lu =

~ i,j=1

open

Mu(x)

a function

since

CONNECTED

TO A S T O C H A S T I C

IMPULSE

PROBLEM

problem

a bounded

,

hand,

~ aij ~ i

linear

partial

differential

+bj ~

+ c

u

~ of IRN a n d an o p e r a t o r

= I + inf ~>0 x+~

f defined

order

form

~

region

a second

on

u(x+~)

~, and

,

, M of t h e

form

x E R ,

can be f o r m a l l y

written

as

148

u y01~12

subspace

studied

or

by A.Bensoussan

approach

to some

[ 2 ][ 4 ] a n d

[5 ]9

coefficients

a i j , b j, c 6 L~(~) N [ i,j=1

been

for i n s t a n c e

impulse

con-

of L v e r i f y

>_ c > 0

a.e.

a.e.

xen

,

x 6 ~ ,

u

emN(y0>0)

V of HI (S) s a t i s f y i n g

H~ (~)

(2.6)

the

kind

in a d y n a m i c

problems, We s u p p o s e

For

a boundary

of this

and J . L . L i o n s

(2.5)

- f) = 0

type.

Problems

trol

(u - M u ) ( L u

in a d d i t i o n ,

Neumann

in n

C V

C H I (~)

identity

(2.7)

(Lu,

v > = a(u,v)

u 6 H I (~), v E V

,

,

where

(2.8)

a(u,v)

=

N[

i,j=1 defines

! (aij UxiVxj + b j u x v + c u v ) d x

L as a c o n t i n u o u s

V. M o r e o v e r ,

the

,

j

linear

following

operator

coerciveness

f r o m H* (~) to the d u a l

condition

V'

of


2

(2.9)

(Lu,u

where

> = a(u,u)

U'II d e n o t e s The

operator

the

usual

(1.2)

>_ 7llvll

norm

YvEV

of the

can be m o r e

( y > 0 )

space

precisely

HI (~). defined

for any

u6L~(~)

by setting

(2.10)

Since

(Mu) (x) = I + ess sup ~>0 x+~--~ the

space

L~(~)

u(x+~)

is a c o m p l e t e

,

a.e.

lattice

under

x e ~ .

the

a.e.

ordering,

149

see

e.g.

[

], M u 9 L ~ ( ~ ) ,

(2.11)

M

thus

: L~(~)

(I .10)

is w e l l

defined

as a m a p

~ L~(s

Remark 2.1.

The

operator

blem

(1.3)

cely

on S o b o l e v

a global

In fact, itself, Tartar: with

as

M clearly problem.

by the

set of all x =

, x2 < 0 ; u(x)

The operator

(2.12)

remark

as N 9 2, M does

it can be s h o w n

xl > 0

We a l s o

what

that M does

makes

pro-

not behave

ni-

spaces.

as soon

s the

is a n o n - l o c a l o p e r a t o r ,

(1.11)

u, 0

a.e.

in

[0,u] H = [0,U]L2 (~) is n o t

empty

and

by

(2.11)

m a p of [0,u] H into The Theorem

H,

uniqueness 2.3

following

,

is c o n t a i n e d

below,

of the

For

is left

every

(2.12),

that,

by

solution

M is a w e l l

defined

increasing

(2.13), M0 ! 0. u > 0 of

that

(2.15)

the o p e r a t o r

u ~ 0 and e v e r y

exists

follows (2.10)

from has

the

some

e e [0,1[,

8 6 ]~,I[

such

that

8Mu

< M(au),

as an e x e r c i s e .

Theorem 2 . 3 . ( L a e t s c h In a d d i t i o n

u ~ 0 of

and

o n c e we v e r i f y

there

operator

such

property:

(2.19)

[31 ])

to the h y p o t h e s e s

M satisfies (2.15),

(Q),

(2.17).

. Therefore,

what

5.1

solution

(2.16),

(2.18)

thus

of T h e o r e m

, g ~ f and u the

Uma x

condition

of T h e o r e m

(2.19).

, is the u n i q u e

Then,

2.2,

suppose

the m a x i m u m

solution

u h 0 of

that

the

solution (2.15).

in

151

Proof. L e t us r e m a r k a.e.

in ~, w h e r e

u h 0 of sfying

that

since

u is the

(2.15)

coincides

0 ~ u ! ~

9 Thus,

Now,

let u 6 L=(~),

u ~ Uma x

(as e l e m e n t

any

solution

solution with

of

u of

(2.17),

the m a x i m u m

b y the p r e v i o u s

theorem 9 of

verifies

the m a x i m u m

solution

u > 0 be a s o l u t i o n

of L~(n~.

(2.15)

then

of

(2.15)

u ~ solution

sati-

U m a x exists. (2.15)

By the m a x i m a l i t y

such

of U m a x

that

, we h a v e

0 < u < u -- max Define

a = max{7

Since allowed,

:> _0

7Umax

u 0 , 7 = 0 is a l l o w e d

therefore

8 satisfying

e 6 [0,1[.

~ < 8 < I , such

in ~} .

and

Thus,

since

implies,

s i n c e M is i n c r e a s i n g

(2.20)

and

a.e. a U m a x _
0 ,

,

0

while

(2.23)

Therefore, and

u : a(Mu,

b y the c o m p a r i s o n

(2.22)(2.23)

above,

f)

theorems,

it f o l l o w s

that SUma x ~ u

a.e.

in ~ .

from

(2.20,(2.21)

152

The

inequality

B ~ ~ , hence Remark

above

(2.24)

given

and

u0 = u

property

increasing,

does

9

authors by

k = 1,2,...

show

and

that

, (2.13),

the

0 and weakly

costru-

procedure

a mild

sequence

convergent

conti-

u k is n o n to the

so-

(2.15). whether

the n o n - d e c r e a s i n g

sequence

de

by

0

,

u~=

(from below)

su~_ 1

,

k=

1,2

to the

solution

problem

(2.3)

....

of

(2.15).

SOLUTIONS

again

n o w the b o u n d a r y

(3.1)

condition

u = 0

The weak

is of a m o r e

iterative

(2.11) (2.12)

from below

L e t us c o n s i d e r specifying

these

at the p r e s e n t

u 6=

3. R E G U L A R

u k = SUk_ I , to

iteratively

converge

following

in a d d i t i o n

bounded

known

(2.25)

,

on the

of M,

u > 0 of

It is n o t fined

of ~, t h a t

8 > ~

by B e n s o u s s a n - G o u r s a t - L i o n s

is b a s e d

By using,

lution

with

b y the d e f i n i t i o n

2.1.

type

nuity

implies,

a contradiction

The proof ctive

clearly

formulation

a.e.

of p r o b l e m

u 9 He* (n) N

L'(~)

of t h e p r e v i o u s

to be the D i r i c h l e t

on

section,

by

condition.

F 9

(2.3)(3.1)

is t h e n

,

a.e.

u < Mu

the

following

one:

in

(3.2) a(u,u-

where

a is the

form

If we a s s u m e , f >, 0 a.e. lution

w)

in ~ ,

(2.8). as in the p r e v i o u s

in ~, it is e a s y

u ~ 0 of

u w 9 H* (~), w < M u a.e.

0 is a s o l u t i o n

and Thus,

ty

and

(3.1)

on f, the

(3.2)

it has b e e n

of C h a p t e r

exists

additional

of

section. pro-

2 and

a regular

conditions

the

solution

are

sati-

sfied. By r e g u l a r

solution

of

(3.2)

we mean

i u 6 H~ (~) N H 2'p(D)

,

a solution

u < Mu

a.e.

u of the p r o b l e m

in

(3.4) a(u,

for

u - w)

0 x+~ 6

it is n o t

itself

and

difficult

A necessary can be e x p r e s s e d

u(x+~)

,

(f,w)

is

on ~, C(~).

pointwise

by

the

space

C(~)

into

of a s o l u t i o n

u of

(3.4)

that M carries

for the e x i s t e n c e of the

solution

u of the D i r i c h l e t

(3.7) =

H = 'P(~)

x e

i u e H0* (~)

a (u,w)

space

function

on s e q u e n c e s .

condition in t e r m s

Sobolev

of c o n t i n u o u s

M can be d e f i n e d

to v e r i f y


I,

s u c h p the

space

the o p e r a t o r

{NI

V w e H~ (~)

,

problem

154

marne ly,

(3.8)

u > -I

In fact, (3.4). xo 6 ~

by the comparison

On the other (recall

x 6 r , such

theorems,

t h a t u is c o n t i n u o u s

u(x)

~

in 9 .

~ > u if u is a s o l u t i o n

w e h a v e u > -I ,fo~,if u ( ~ )

t h a t x0 = x + ~ w i t h

u(x)

whereas

hand,

a.e.

0, h e n c e w e w o u l d

have

(Mu) (x) = I + i n f u(x+~) ~>0_ x+~6~

of

at some point

! I + U(Xo)

some

< 0 ,

= 0 for all x 6 F .

A sufficient c o n d i t i o n tion ~ of the Dirichlet u

9

can be given

instead

in t e r m s

of the solu-

problem

H' (~)

(3.9) a(u,w)

involving

the

(3.10)

note

=

function

g(x)

that,

condition

(g,w)

g obtained

r = inf,0, l

if f 6 L = ( R ) , we are

in a s s u m i n g

u w E Hi (~)

talking

f(x+~)~ , )

about,

as T h e o r e m

f by

x a.e.

t o L|

3.2 b e l o w

in ~

. The shows,

;

sufficient consists

that

this

(3.8),

condition

is a c t u a l l y

in

R .

stronger

than the necessary

con-

for we have

(3.12)

always

essinf ~ > 0 x+~-e 0

u > -1

that

dition

from the given

then g also belongs

(3.11)

Note

,

u > u

in c o n s e q u e n c e

in

of the comparison

n

,

results

(g ~ f a.e.

in

~ ).

155

Theorem 3 . 1 . ( J o l y - M o s c o - T r o i a n i e l l o If f 6 L=(s

there exists

[29,30]

a solution

u of p r o b l e m

(3.4),

sati-

sfying

(3.13)

g _ g a.e.

m a p of p r o b l e m

(3.4).

is the dual estimate of Th.4.1

the detail

we refer

much

N H 2,p(~)

the s e l e c t i o n

is used here

which

u and ~ are the s o l u t i o n s

(3.9),

techniques

2 to show that the set

is stable u n d e r

ced here,

in s .

1 to the case at hand and by u s i n g

Do = {u 6 Hlo (s

which

a.e.

of the proof

to [29]

of the content

and,

The basic

section

tool

of C h a p t e r 2.

are too t e c h n i c a l

for an e x p o s i t o r y

of this

in fl}

to be reprodu-

account,

to [40]

on

has been based.

Remark 3.1. Laetsch's have

a direct

the s o l u t i o n interpret

it,

following control

of

(3.4)

(3.2).


Bensoussan-Lions,

problem,

(3.4)

(Theorem 2.3 above)

to the p r o b l e m

It w o u l d be nice

the s o l u t i o n (3.2)

result

u of p r o b l e m

stic impulse is known.

uniqueness application

for w h i c h

makes

as the s o l u t i o n

proof

the fact that

it p o s s i b l e

the u n i q u e n e s s

to have a d i r e c t

and we m i g h t

does not seem to

However,

to

of a stocha-

of the s o l u t i o n

of the u n i q u e n e s s

also ask w h e t h e r

the s o l u t i o n

of

of

is u n i q u e .

9

Remark 3.2. The (2.15) 2.1.

In fact,

Theorem blems

solution

of s e c t i o n

3.1,

(3.7)

of p r o b l e m

(3.4)

can be obtained,

2, by an i t e r a t i v e

it is also p r o v e d

process

in [40],

under

that if u and u are the s o l u t i o n s and

(3.9),

then the s e q u e n c e s

as that of p r o b l e m

such as those of s e c t i o n

{Uk},

the same a s s u m p t i o n s of the D i r i c h l e t {u{},

defined

of

pro-

iterati-

vely by

converge

u0 = u

,

u k = SUk_ I

,

k = 1,2, . . . .

u'0 = --U

,

u~

'

k =

to the

=

(unique)

Su~_1

solution

u of

1,2 . . . . . (3.4), w e a k l y

in H 2 'P(n).

Moreo

156

ver, u k' _< u _< u k for every k; the sequence increasing;

the sequence

{u k} is p o i n t w i s e non-

{u~} is p o i n t w i s e n o n - d e c r e a s i n g ,

and both

sequences converge u n i f o r m l y to u on ~. These results are more com plete than those a v a i l a b l e for p r o b l e m

(2.5), since then, as already

r e m a r k e d in section 2, the c o n v e r g e n c e from below of an iterative process starting w i t h a s u b - s o l u t i o n has not yet been proved

(u = 0 in that case,see(2.25))

(and it may be indeed false, unless,perhaps,

a p r o p e r initial s u b s o l u t i o n is chosen).

9

4. FINAL REMARKS a)

R e g u l a r i t y results

for one p r o b l e m c o n s i d e r e d in sec.2 have been

given, along the lines of T h e o r e m 3.1, by H a n o u z e t - J o l y [20 ] for a n o p ~ rator L of the form

L = -& + bl ~

+ b2 ~

b,, b2, c being constants, b)

+ c ,

and the region ~ of r e c t a n g u l a r type.

Dual estimates of the kind given in section 4 of

been given by C h a r r i e r - T r o i a n i e l l o [14]

Chap. 2 have

[15 ] and by C h a r r i e r - H a n o u z e t - J o l y

in the case of strong and weak solutions,

respectively,

to uni-

lateral p r o b l e m s for p a r a b o l i c linear second order PDO. In [15]

the

estimates for strong solutions to p a r a b o l i c p r o b l e m s have been p r o v e n a l t o g e t h e r w i t h the e x i s t e n c e gular perturbations. proven,

of such solutions,

via a m e t h o d of sin

In [14], the same estimates have been d i r e c t l y

by similar methods than those of section 4 of C h a p t e r 2, for

weak solutions in the sense of M i g n o t - P u e l [39]. c)

A p p l i c a t i o n s of these dual e s t i m a t e s to p a r a b o l i c q u a s i - v a r i a t i ~

nal inequalities,

along the lines of [28]

and [30], can be found in

[15]

and C h a r r i e r - V i v a l d i

[16].

d)

Q u a s i - v a r i a t i o n a l i n e q u a l i t i e s involving decreasing o p e r a t o r M

have been i n v e s t i g a t e d by B e n ~ o u s s a n - L i o n s [ 3 ] and J o l y - M o s c o [28], see also J o l y [25]. e)

For nonlinear QVI's arising in s t o c h a s t i c impulse control theory

see B e n s o u s s a n - L i o n s

[4 ].

f)

Many quasi-variational

now,

involving d i f f e r e n t types of o p e r a t o r s M and in c o n n e c t i o n w i t h

inequalities have been c o n s i d e r e d up to

a v a r i e t y of free boundary problems, Lions, g)

we refer again to B e n s o u s s a n -

loc. cit., and to F r i e d m a n - J e n s e n [17][18]

For problems as those of Ch.1

and B a i o c c h i [ 1 ] .

see also J . P . A u b i n , M a t h e m a t i c a l mo-

dels of game and economic theory, C E R E M A D E Univ.,

Paris IX Dauphine.

INT E G R A L

FUNCTIONALS,

NORMAL

INTEGRANDS

R. Tyrrell

A fundamental mization, operator

notion

probability, theory,

an e x p r e s s i o n

in many

X

tion

(S,A,~)

Classically,

space

only

finite

the a s s u m p t i o n

urable

s

and

efficiently

existence

of m e a s u r a b l e

of "normality".

The p u r p o s e

of the most

of the results

this

as i n d i c a t e d

case,

ample, plete

technically

it is only theory

assumption

therefore,

of the details to search varying

extent

and dualities

for a u x i l l i a r y

case

results

of

values

require

a relatively

further

knows

in some

a distinctand the

that where

these

E = R n.

be ironed

freeing

sequences

For exa com-

is complete,

situations.

one

an

In t r e a t i n g of the multi-

out.

a full and consistent E = R n,

beyond

how to develop space

in

are often more

are the usual p r o b l e m s w h i c h must

f

thorough

restrictions.

the m e a s u r a b l e

through

for

of c o n s t r a i n t s

and are r e f l e c t e d

in the text,

usu-

and meas-

in one way or another

one p r e s e n t l y

to be awkward

x

of m e a s u r a b i l i t y

in a p p l i c a t i o n s ,

to have a v a i l a b l e

func-

from the m o d e r n

kinds

Such i n t e g r a n d s

E, there

The

studied,

in

infinite

are p r o m i n e n t

that

on a meas-

E.

were

However,

questions

and may require

in the basic

S x Rn

important

extensions

to some

spaces

of topologies

sirable,

is meant

defined

space

continuous

is to provide

case

have

assuming

appears

Inflnite-dimenslonal plicity

notes

for R n that

without

which

that

where

common

While many

on was

condition).

selections

of these

functions

to admit p o s s i b l y

be represented.

approach,

a concept

complicated

By this

optiand

Inte~rand.

f(s,x)

it is in this way

ly new t h e o r e t i c a l

treatment

analysis

x E X,

in a linear

integrands

(the C a r a t h g o d o r y

since

can most

values

that

of view it is e s s e n t i a l If,

functional.


is the a s s o c i a t e d

ally under in

including

functional

= ~ f(s,x(s))p(ds),

and h a v i n g

f: S x E § R

point

of m a t h e m a t i c s ,

problems,

of the form

is a linear

ure space

areas

of an i n t e g r a l

If(x) where

SELECTIONS

Rockafellar*

variational

is that

AND M E A S U R A B L E

It is de-

exposition

from the need

of papers

wlth

frameworks.

*This work was s u p p o r t e d in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under A F O S R grant n u m b e r 72-2269.

158

The m a t e r i a l b e l o w is divided in three p r i n c i p l e sections. we present the theory of m e a s u r a b l e quivalent properties, tion of m e a s u r a b i l i t y ,

First

closed-valued multifunctions.

E-

any of which could actually be used as the definiare discussed,

and the basic m e a s u r a b l e

theorem of Kuratowski and R y l l - N a r d z e w s k i

selection

is derived via a stronger

t h e o r e m on the existence of C a s t a l n g representations.

(The proof, w h i c h

is given in full, is simpler for

Rn

usually seen in the literature.)

Much effort is devoted to e s t a b l i s h i n g

than in the more general case

convenient means of v e r i f y i n g that a m u l t i f u n c t i o n is indeed measurable. The second part applies

the results on m e a s u r a b l e m u l t i f u n c t i o n s

the study of normal integrands,

a concept originally

to

i n t r o d u c e d by the

author

Ill in a setting of convexity, but d e v e l o p e d here in more general

terms.

Again the emphasis is on m e a s u r a b i l i t y questions and the manu-

facture of tools w h i c h make easier the v e r i f i c a t i o n of "normality". Normal Integrands are also important multlfunctlons

in the g e n e r a t i o n of m e a s u r a b l e

given by systems of constraints,

subdlfferential mappings

etc. These technical d e v e l o p m e n t s

come to fruition in the theory of

integral functionals p r e s e n t e d in the third section of the notes. is here also that convex analysis

It

comes more to the front of the stage.

This is due to natural c o n s i d e r a t i o n s

of duality, which are always

important in a setting of functional analysis,

as well as deeper reasons

related to Liapunov's t h e o r e m and i n v o l v i n g the w e a k compactness of l e v e sets of integral functionals. For obvious reasons of space, functionals on d e c o m p o s a b l e

the d i s c u s s i o n is limited to integral

function spaces,

such as Lebesgue

spaces.

These are c h a r a c t e r i z e d by the v a l i d i t y of a fundamental result on the interchange of i n t e g r a t i o n and minimization.

The treatment of more

general f u n c t i o n spaces usually relies heavily on this, more basic theory,

as for example the case of B a n a c h spaces of continuous

as d e v e l o p e d in [2], or the spaces of d i f f e r e n t l a b l e ed in v a r i a t i o n a l problems

(cf.

[13],

no attempt to cover the many results

[15],

[26],

functions

functions encounter.

[32]).

in such directions.

We have made

159

i.

Measurable Closed-Valued Multifunctions. In e v e r y t h i n g that follows,

ped with a e - a l g e b r a

A;

thus

S

is an arbitrary nonempty

(S,A)

subject only to the r e s t r i c t i o n that measurable

subsets of

A multifunction

S ~ A.

such that

Unfortunately,

F:S + X,

(s,x)

~ F

where

X

is another set,

for a given

(F(s) = {x}

S • X.

s ~ S

this n o t a t i o n is ambiguous

happens to be a function ly troublesome

E l e m e n t s of

or

thought of as a m a p p i n g a s s i g n i n g to each F

is, llke a

The set of all r(s).

in the special case where F(s) = x?),

s ~ S

F

F

and it is slight-

can really be

a subset

r(s)

of

x.

gives rise to such a m a p p i n g and is uniquely deter-

m i n e d by it, but of course the two are not the same. s § F(s)

are called

is d e n o t e d by

in s u g g e s t i n g more g e n e r a l l y that

It is true that

A

space

S.

function, best defined simply as a subset of x E X

set equip-

is a general m e a s u r a b l e

corresponds,

strictly speaking,

The m a p p i n g

to a subset of

thus the q u e s t i o n of w h e t h e r or not it is measurable,

S • 2 x, and

for example,

p r o p e r l y a n s w e r e d in terms of the usual theory of m e a s u r a b l e and the choice of a m e a s u r a b i l i t y

structure on the space

is

functions

2 X.

This is

not the point of view we want to adopt, and so the d i s t i n c t i o n should be borne in mind. Nevertheless,

it is hard to be a purist on such matters without

h a v i n g a nuisance with basic ways of w r i t i n g things write

F[s]

in place of

element of

F[s]

denoted by

[r]).

technicalities

(e.g. one could

F(s), r e s e r v i n g the latter for the unique

when one exists, In practice,

and the m a p p i n g

s + F[s]

could be

no serious c o n f u s i o n arises even if

are slightly abused in this respect.

We content ourselves with the following n o t a t i o n for m u l t i f u n c t i o n s F:S + X,

which,

if a little redundant,

does serve to e m p h a s i z e the

setting: dom F = {s ~ SIF(s) # ~}, gph F = {(s,x) Ix c F(s)}, F(T)

=

UsE T F(s).

Of course,

gph F

and

is its p r o j e c t i o n on

dom F

is really no different S.

from what we have

We shall denote by

called

the m u l t i f u n c t i o n o b t a i n e d by r e v e r s i n g the pairs c o n s t i t u t i n g

r-l(x)

=

{s

~ Slx

F,

F-I:x § S F; thus

~ r(s)},

r-l(c) = UxecF-l(x)

= {s ~ sir(s)

o c @ Z}.

For the most p@rt, we shall be concerned only with m u l t i f u n c t i o n s F:S § R n

w h i c h are closed-valued,

in the sense that

r(s)

is a closed

160

s~Dset

of

R n for every

measurable the set This

(relative

F-I(c)

alence with lences,

which

that

P

ability

reduces

that

closed

is me a s u r a b l e ,

F

set

is closed

Of course,

of the

of

z

which

w here

I'I

below.

It is

Rn

is re-

closed-valued,

and just

is open to contro-

that the present

definition

(hence

i.e.

P

trivially

closed-

is a function,

measur-

if it is constant:

fact worth

and

T c S

recording

F(s)

is that

is m e a s u r a b l e ,

~ D if

follows,

IF(S)

[

n D if

if

we denote

set

s ~ T

s L T of

P

implies

the m e a s u r -

as dom P = P-l(Rn). by

dist(z,C)

the E u c l i d e a n

C c Rn:

= min{Iz-xIIx

is the E u c l i d e a n

PROPOSITION.

followin~

For

properties

norm.

~ c},

(This

is i n t e r p r e t e d

as

+~

if

a closed-valued

multifunction

F:S

+ Rn ,

P

(b)

F-l(c)

is m e a s u r a b l e

for all open sets

(c)

P-l(c)


for all

compact

(d)

P-l(c)


for all

closed balls

(e)

dist(z,P(s))

is m e a s u r a b l e ;

is a m e a s u r a b l e

f u nc t i o n

C;

sets

of

C; C; s ~ S

for each

z ~ Rn . PROOF.

the

are equivalent:

(a)

C = Uk=lCk ,

F

then the

C=~.) 1A.

may

cases.

inasmuch

from a closed dist(z,C)

adopted its equiv-

such equiva-

down when

is not

the m e a s u r a b i l i t y

set dom F,

In the result distance

first

by

=

ability

C c Rn

concept.

F'(s)

is me a s u r a b l e .

be stated

single-valued

Another

defined

Many

"measurability"

is m e a s u r a b l e D.

D c Rn P~

F

nonempty-valued,

to the usual

It is obvious

multifunction

or if

in such

is actually

and everywhere

will

was

[3] p r o v e d

definitions.

to u n d e r s t a n d

to r e v i s i o n

valued)

for a fixed

space,

the reader

if

in his thesis

they b r e a k

should then be called

We want

well be subject Note

general

that

set

to A).

of m u l t i f u n c t i o n s

who

to know,

however,

is said to be

if for each closed

belongs

of other p o s s i b l e

to realize,

property

versy.

by Castaing,

a number

p l a c e d by a more

A),

(i.e.

of m e a s u r a b i l i t y

context

w h i c h are very useful

important

Such a m u l t i f u n c t i o n

to the ~-fleld


definition

in a general

s r S.

(c) ~

(a).

Let

where

each

Ck

C

be any

is compact,

closed

set in

and hence

R n.

Then

161

= F-I = Uk= 1 (Ck).

F-I(c)

(i.I) We have each

F-I(Ck )

measurable,

(a) ~ (d).

This

is trivial.

(d) ~ (b).

Let

C

hence

so is

F-I(c).

co

a closed ball. conclude

Thus

F-l(c)

(b) ~ (c).

be open.

Then

(1.1) again holds with

F(s)

Ck

Given a compact

is open, cl C k

n Ck ~ Z

and since

F-l(Ck )

where each

Ck

measurable,

is

and we

is measurable. set

C,

C k = {z e Rnidist(z,C) Then

C = Uk= 1 C k,

for all

F(s)

< k -1)

is compact, k

let for

and

C k = C l C k + 1.

if and only if

is closed,

k = 1,2,...

F(s)

n cl

the latter is equivalent

We have

Ck ~ Z

for all

by compactness

k, to

oo

Z ~ nk=iF(s)

n c l C k = F(s)

n C.

Therefore co

r-l(c) and since each F-I(c)

-1

= nk=lr

F-l(Ck )

(C k) ,

is measurable

by assumption,

it follows

that

is measurable.

(d) (e). the ball

We have

z+aB

(1.2)

dist(z,F(s))

{sidist(z,F(s))

Condition

< a

(B = closed unit ball,

if and only if

e > 0).

F(s)

meets

Thus

0.

Theorem

Then

is a c o m p a c t n Te

(xili=l,2,...)

each

relative

every

set w i t h open

Let

For

for m e a s u r a b l e


x;l(c)

is m e a s u r a b l e zero.

k = 1,2 .... ,

m e s ( S k \ T k)

Tk

(S,A)

can be r e d u c e d

set

disjoint

relative

because

is m e a s u r a b l e .

of c o m p a c t

and

of

F

argument

(c) h o l d s

1E,

(F')-I(c)

< k},

. Let T e = Ol=iT e. and

is a u n i o n


is a B o r e l

of the

f o r m of L u s i n ' s

xi

to

there

functions Ti

and


mes(S\Te)

relative

i,

to

Te

~ e.

If

for all

set F-I(c)

is o p e n

If

the

Isl

of the proof, we

usual

union


of m e a s u r e

that

S

a sequence

representation

r

!

continuous

T iE , _ ~ u c h

is closed.

be the

by T h e o r e m

a set

it f o l l o w s

disjoint

S

T

(b)

Then

at m o s t

we d e m o n s t r a t e

relative

is also

Thus

measurable

be

S k < ~.

of the

T

Then

S < =.

~ > 0

let

~ T,

it as the u n i o n

mes

for any

with

x E F(s))

T E.

S k = {s r S l k - 1 and these

{(s,x) Is c Te,

{(s,x)Is

by

and

First mes

Te c S

of sets

F'

r-l(c)


set

T.

C c Rn

from

set

measurable.

to

We have

Let

and d i f f e r s

set

Borel

of

(a).

is c o m p l e t e .

the

is a c l o s e d

Trivial.

e = k -1,

= 0, and the

the r e s t r i c t i o n

I,

be n o n e m p t y - c l o s e d - v a l u e d .

equivalent:

is a c l o s e d - v a l u e d

(Lusin

(d)

F

F:S § R n

is m e a s u r a b l e ;

is a B o r e l (c)

T

of L e b e s g u e

to

T

. C

n T e = Ul=l[x~l(c) Thus

F

Is

lower

n T e] semicontinuous

relative

a

167

to

T . E

It remains only to show (d).

Let

(Cklk=l,2,...)

closed subsets of center and radius. the sets

Ck

Rn

(assuming

Sk

s,

c o n t a i n i n g it.

and

S~

that

c o m p l e m e n t a r y to the open balls

For each

F(s)

Let

(a) implies

are m e a s u r a b l e

Uk

with rational

is then the i n t e r s e c t i o n of all

S k = F-l(Uk )

s~ = s\s k = {slr(s) Then

mes S < ~)

be an e n u m e r a t i o n of all the (countably many)

and

c Ck}.

(by c r i t e r i o n

(c) in P r o p o s i t i o n IA),

and gphr = n~=l[(S k x R n) u (S~ x Ck) ]. Fix

e > 0.

For each

k,

there exist compact sets

Kk c Sk

and

K~ c S k' , such that m e s ( S \ ( K k u K~)) ! E2-k" Let T e = nk=l(K k u K~). Then

Te

is a compact set with

m e s ( S \ T e) ~ e,

and we have

{(s,x) Is c TE,x ~ F(S)} = nk=l[(K k • R n) u (K~ • Ck)]. The latter set is closed,

so (d) is established.

Q.E.D.

The p r e c e d i n g results provide the main direct ability that are convenient

in practice.

criteria for measur-

However, we add for complete-

ness one further condition, which has been used as the d e f i n i t i o n of m e a s u r a b i l i t y by some authors, IG.

PROPOSITION.

Let

such as Debreu

F:S § R n

[9].

b__eenonempty-compact-valued.

Then

F

is a m e a s u r a b l e m u l t i f u n c t i o n if and only if the c o r r e s p o n d i n g m a p p i n g from

S

to the space

M,

c o n s i s t i n g of all compact

under the H a u s d o r f f metric,


functions

space to a metric

from a m e a s u r a b l e

PROOF. Let

in

consisting of all compact

M

tion,

C

be any closed subset of K

Rn

space).

Suppose first that this m a p p i n g from

able.

subsets of

(in the usual sense of

Rn~

such that

to

S

and let

U

M

is measur-

be the open set By assump-

K n C = g.

the set

{s ~ slr(s) is measurable,

and therefore

E u} = s \ r - l ( c )

F-I(c)

is measurable.

Thus

F

is a

measurable multifunctlon. For the converse argument, finite sets in

Rn

let

M0

denote the c o l l e c t i o n of all

c o n s i s t i n g only of "rational" points.

countable and dense in

M,

so that every open set in

M

Then

of a countable family of closed balls whose centers b e l o n g to Therefore,

to show the m e a s u r a b i l i t y of the m a p p i n g from

M 0 is

is the union

S

M 0. to

M

168

associated set

with

F, we need only

{s ( s i r ( s )

and center Then

~ W}

F E M0,

K ~ W

is measurable. and let

if and only

words

It follows finite

that

family

measurable

the

verify

if

B

open.

K c F + eB

The enable

= Z.

set

{s e SIF(s) F-l(x

+ eB)

radius

x ~ F,

(each of w h i c h

~ W}

IA,

since

is measurable,

RnS(F + eB)

of the theory


multifunctions


selections

of the kinds

given

cally,

F

other,

("integrands", properties

in terms

Without

measurable towards

more

relation

results

selections series

w

that

or less

in

of results

by analogous

objects

may be more

about

functions

various

no t h e o r e m

the very

influenced

describes

Typi-

construction

than a p r e l i m i n a r y

that

choice

by such

must

not

operaon

step of the

consideraonly possess

to manipulate.

operations

measurability.

However,

accessible,

under

direction,

as more

is to be

The m e a s u r a b i l i t y

are p r e s e r v e d

in this

is to

for multi-

directly.

as certain

of m u l t i f u n c t i o n s

results

IC).

complicated

w

but also be c o n v e n i e n t

preserve

(cf.

as well

is h e a v i l y

category

and this

easy to apply

It may be r e m a r k e d

of " m e a s u r a b i l i t y "

The next

in

fundamental

can be viewed

the a p p r o p r i a t e

multifunctlons

of a more

be d i s c u s s e d

auxilliary

selections

measurable

is m e a s u r a b l e always

to know how they

applications.

definition

F

in practice,

multifunctions,

w h i c h will

of these

arise

are not

simpler

and it is important tions.

that

above

is g i v e n

involving

that

is

q.E.D.

the e x i s t e n c e

by s h o w i n g

of the is

+ r

by P r o p o s i t i o n

chief goal

F

R n.

or in other

is the i n t e r s e c t i o n

x ~ F

the

E > 0 in

us to verify

the c r i t e r i a

on c l o s e d - v a l u e d

The picture

"integrands"

will

and their

be comintimate

to m u l t i f u n c t i o n s .

IH.

PROPOSITION.

let

F'

F'(s)

has

unit ball

W

and

{s E SIF(s)

accomplished

pleted

e W}

for

such ball

F c K + eB,

K n (Rn\(F + aB))


functions

tions;

and

for each

of sets

Hence

W

the closed

K n (x + eB) ~ Z

by hypothesis)

latter

Suppose

denote

S\F-I(Rnk(F The

t h a t for each

Let

F:S + R n

be the m u l t i f u n c t i o n

= cl coF(s)

(closed

be a c l o s e d - v a l u e d

such that,

convex hull).

for each Then

multifunction,

and

s ~ S,

F'


(and

closed-valued). The closed space

same

cone

is true

containing

~enerated

by

if, in place F(s),

F(s).

of cl coF(s),

or the affine

hull

one takes

the

of

or the sub-

F(s),

smallest

169

PROOF. expressed

We e x p l o i t

as a c o n v e x

(Carath4odory's of

F

(cf.

the fact that

combination

Theorem).

comments

0

and

the f u n c t i o n

xj:

(xjlj

~ J)

is m e a s u r a b l e proposition

by T h e o r e m

IB.

PROPOSITION.

Let

for

j = l,...,m,

and

in

elements

IB).

R n+l,

of

r(s)

representation Let

A

be

such that

many

indices

I (n+l times),

+.-.+

/nXin(S)"

representation

(The p r o o f s

are a n a l o g o u s . )

ii.

of T h e o r e m

can be

by

= k0xi0(s)

Is a C a s t a i n g

cor(s)

be a C a s t a i n g

e J = A • I •

domr § R n

xj(s) Then

(or fewer)

i c I)

the p r o o f

of

For e a c h of the c o u n t a b l y

J = ( k , i 0 , . . . , i n) define

n+l

element

(kO,kl,...,kn)

k =

Z~=0k k = I.

of

(xil

following

the set of all r a t i o n a l kk

Let

e v ery

of

r',

for the o t h e r

and hence cases

r,

in the

Q.E.D.

r.: S § R nj be c l o s e d - v a l u e d and m e a s u r a b l e J n nm for Rn = R l• let F: S ~ R n be d e f i n e d m

by F(s) Then

r


PROOF. for

Let

j = l,...,m.

= Fl(S)•

(closed-valued).

(xili

e lj)


j = ( i l , . . . , i m) let of iJ.

xj = (x i ,...,x i ). I m F,

so

r

Then

Let

rj:

and let

many

E J = ll•

(xjl j e J)


PROPOSITION.

j = l,...,m,

representation

For e a c h of the c o u n t a b l y

of

rj

indices m,

is a C a s t a i n g

representation

Q.E.D.

s § Rn

r: s + R n

b__eec l o s e d - v a l u e d be d e f i n e d

and m e a s u r a b l e

for

by

r(s) = cl(rl(s)+...+rm(S)). Then

r


PROOF. IK.

The a r g u m e n t

COROLLARY.

function,

multifunction

IL. each

Let

and let

measurable

F'

is s i m i l a r

r: S § R n

a: S + R n g i v e n by

Let

(countable

r

for

be a m e a s u r a b l e

be a m e a s u r a b l e F'(s)

= r(s)

iI. closed-valued

function.

+ a(s)

multi-

Then the

(translate)

ri: S § R n

b__eec l o s e d - v a l u e d

index

and let

r(s) Then

to that

is

(closed-valued).

PROPOSITION. i ~ I

(closed-valued).


set),

= clui~iri(s).

(closed-valued).


r: s + R n

be d e f i n e d

for by

170

PROOF.

For each open set

C c R n,

we have

r-l(c) = ni~ir~l(c). Hence by the e q u i v a l e n c e of (a) and

(b) in 1A, F

is measurable.

result also follows immediately via C a s t a i n g r e p r e s e n t a t i o n s . ) THEOREM.

1M.

for each

i E I

Let

ri:S ~ R n

(The Q.E.D.

b_~e c l o s e d - v a l u e d and m e a s u r a b l e

(countable index set), and let

F:S § R n

be defined

b_Z F (s) = niEIFi(s)" Then

r


(closed-valued).

{S ~ SloiEiri(s)

In particular,

the set

~ Z} = d o m F

is measurable. PROOF. set

First we treat the case where

I = {1,2}.

C, and define the c l o s e d - v a l u e d m u l t l f u n c t i o n s

r{(s) Then

F~

and

F~

-- c ~ r l ( s ) ,

r~(s)

are measurable,

c ~ rl(s)

n r2(s)

Fix any closed

F 1'

and

F~

by

-- - r 2 ( s ) .

and one has

~ ~

~

o ~ q(s)

+ r~(s).

Therefore

r-l(c) -- (r{ and we may conclude via P r o p o s i t i o n Thus

F

r))-l(o),

+

IJ that

is measurable.

is measurable.

The validity of the t h e o r e m for its validity I

F-I(c)

for any finite I.

is infinite;

implies by induction

It remains to consider the case where

we can suppose

closed-valued multifunction

I = {1,2}

I = {1,2,...}.

Fk

For each index

k,

the

defined by

k rk(S) = ai=iFi(s) is m e a s u r a b l e by what has already been proved.

For each compact set

C c R n,

Fk(S) # C

we have

F(s)

o C ~ ~

Therefore r-l(c) where

FkI(C )

is measurable,

if and only if oo

P r o p o s i t i o n IA.

Q.E.D.

measurable

--i

and it follows that

the m e a s u r a b i l i t y of

T h e o r e m 1M, a crucial

k.

= ak__lrk (C),

This establishes

proved in the present

for all

F

F-I(c)

is measurable.

by way of criterion

(c) of

fact in several arguments below, was first

f r a m e w o r k in R o c k a f e l l a r

[6]2

Of course,

if the

space is complete, the result is trivial in terms of criterion

(b) of T h e o r e m 1E, and hence it is trivial also in general

contexts

171

where

this

criterion

is a d o p t e d

as the

is new,

least

definition

of the m e a s u r a b i l i t y

of a m u l t i f u n c t i o n . The IN.

next

result

THEOREM.

each

Let

s ~ S

let 9

depending

F:S~R n A

'

on

s

(i.e.

and m e a s u r a b l e ) .

and measurable, with

and

closed

for

graph

the m u l t i f u n c t i o n

Then

G(s)


: gph A s

r':s

i_~s

+ Rm

by


(closed-valued). is b o u n d e d . )

if

F(s)

PROOF.

Let

a sequence

+ Rn x Rn C

C

by

sets

= F(s)

sets

(x,y)

in the

(F')-l(c)

of P r o p o s i t i o n is c o m p a c t ,

operation

R n.

• C k.

Then

k,

Gk

here

C

define

Then

one

~ gphA s with

n Gk(S)

is s u p e r f l u -

is the

union

of



by

II.

latter

F'

sees

~ g} x E r(s),

union

is m e a s u r a b l e a n d we

that

by T h e o r e m

conclude


easily

y c C k}

~ ~}.


1A that

and

in each

= {slc ~ As(r(s))

= Uk=l{SlG(s) of the

set For

closure

we have

= Uk=l{Sl

Therefore

(The

C k.

Gk(X)

is open,

= ClAs(F(s))

be any o p e n

of c l o s e d

(r')-l(c)

Each

generality.

S

ous

Gk:S

stated

be a m u l t l f u n c t i o n

F'(s)

Since

in the

b_~e c l o s e d - v a l u e d

:R n + R m

measurably

closed-valued defined

'

at

(If

A (F(s))

from

F(s)

IM.

condition

is b o u n d e d ,

is closed,

making

(b) it

the

S

closure IP.

operation

COROLLARY.

for e a c h

s ~ S


in

in the

definition

Let

F:S + R n

let

F:S

s

and

b-XY

continuous

r,(s)

Then

F'


PROOF. subset

of

Let R n.

Thus

the h y p o t h e s i s

1Q.

COROLLARY.

for each measurable

s E S in

each

Then G(s)

s

unnnecessary.)

closed-valued

and m e a s u r a b l e ,

be a m a p p i n g in

x.

Let

Q.E.D.

such F':S

that + Rm

and

F(s,x) be d e f i n e d

= clF(s,r(s)).

i

{zili

Let

define E I}

= gphAs, of T h e o r e m

Let

F:S § R n

let

F:S

and

F'(s)

(closed-valued).

A s = F(s,.). For

= (ai,F(s,ai)). multifunction

be

• Rn § Rm

of

(aili zi:S

~ I)

is a C a s t a l n g

which

therefore

continuous

in

u.

dense

zi(s)

representation

for the

(Theorem


a mapping Let

by

1B).

Q.E.D.

be c l o s e d - v a l u e d be

a countable


1N is s a t i s f i e d .

x Rm + R n

be

§ Rn • R m

F':S

such § Rm

and

that

F(s,u)

be

defined

i__~s by

172

r,(s) Then

r'


PROOF.

Clearly

= {u c Rml

F(s,u)

9 r(s)}.

(closed-valued).

r'(s)

is closed for all

s.

Let

A s = F(s,-) -1.

By an argument Similar to the one in the p r e c e d i n g corollary, multifunction T h e o r e m IN IR.

G(s) = gph A s

is applicable.

COROLLARY.

multifunctlon,

Let

has a C a s t a i n g r e p r e s e n t a t i o n , Q.E.D.

F: S § R m • R n • R k

and let

r': S § R n

F'(s) = c l { x l ~ w where

u: S § R k

9 Rm

is measurable.

be a measurable,

F2(s,w,x)

Let

F1

with Then

be the p r o j e c t i o n

= (w,x,u(s)).

(w,x,u(s)) r'

F"


REMARK.

IP.

F(s)

(closed-valued).

is bounded.) and let

Let

is m e a s u r a b l e by IQ, and

is m e a s u r a b l e by

E F(s)},

(w,x) § x,

r"(s) = {(w,x) I F2(s,w,x) Then

closed-valued

be defined by

(The closure o p e r a t i o n here is superfluous i f PROOF.

the and hence

9 F(s)}.

F'(s) = cl FI(F"(s)),

so that

F'

Q.E.D.

Two new articles will be especially useful to those in

need of a more general theory of m e a s u r a b l e m u l t l f u n c t l o n s f u r n i s h e d here.

Wagner

than is

[29] has put together a c o m p r e h e n s i v e survey

of the existing literature.

Delode,

Arlno and Penot

out a new and b r o a d e r framework for the subject,

[30] have worked

from the point of view

of fiber spaces, and have thereby o b t a i n e d extensions of a number of previous results,

for example,

involving a w e a k e n i n g of the "complete-

ness" requirement

in T h e o r e m IE.

173

2.

Normal

Integrands.

For present an i n t e s r a n d

= R u {+_~}. its e p i g r a p h (2.1)

purposes,

on

S • R n.

Corresponding

multifunction

Ef(s)

= epi

We shall

say that

f

is l.s.c.

(lower

closed-valued), Ef

of the

can speak

f

Rn

if

f(s,x)

in

x

certain

f

(2.2)

if

s,

i.e.,

function

set is convex

since

[8],

most

+ x

and d e v e l o p e d

integrands,

change

of t e r m i n o l o g y

classical

satisfying

Various to spaces Valadler

is this,

depends

on the

is possible,

one

and to call

sense

f(s,x) Thus,

by e x t e n d i n g

on f

for every

if

a

s e S. is convex

for a p r o p e r

as

+~

a

set

< +~}. is a convex

the

is proper

integrand.


image

of

Ef(s)

if

Observe f

under

is the pro-

IP). with p o s s i b l y

treated

with

to be noted: normal

infinite

in a series only

advantage

it agrees

precursors

Ef

what

of convexity,

previously integrand.

of normal

integrands

case. was

there was

are

These will

[i],

was [2],

A different employed

in

applied

to

definition

However,

convex

conditions.

of papers

the convex

the present

values

is one

slight

a normal

convex

finite

be shown

Integrands to fit in

case. results, g e n e r a l i z i n g

other [Ii],

Obviously, $: R n + R

f

integrands

the C a r a t h ~ o d o r y

as a special

s

as will be seen below.

is now a p r o p e r

~ +~,

inte~rand

f(s,x)

is Just

taking

f(s,x)

(i.e.,

f(s,.)

in this

by R o c k a f e l l a r

of normality,

if

if, besides

is convex-valued.

if

originally

of this work, but

The

f(s,x)

s § cl dom f(s,.)

of normal

s

choice

on a n o ne m p t y

s,

integrand

normality

function

for each

(Corollary

convex

integrand

Ef

= {x ~ Rnl

for all

[10],that

definition

the and

defined

dom f(s,-)

F:(x,e)

The theory

x

if

by

Intesrand

than one

is

for clarity.

is p r o p e r

is o b t a i n e d


[6],

if more

it,

a ~ f(s,x)}.

for each

is said to be a convex

that

introduced

A;

f(s,-)

This

Jection

x

Of course,

to say that

dom f(s,.)

normal,

in

multlfunction.

f(s,-)

finite

semicontinuous

be called

reals:

determining

defined

~ Rn x R]

is a normal

> - ~ for all

for each

integrand,

f

and completely

= {(x,a)

will

the e x t e n d e d

S § R n+l,

being A-normal,

integrand

Furthermore,

Ef:

f: S • R n + R

denotes f

semlcontlnuous)

It is convenient

proper

R

is a lower

g-algebra

of

function

to

f(s,-)

and that


choice

any Here

than

Rn,

Castaing any

is lower

may be

some of the d e v e l o p m e n t found

in [8] and,

more

of the

semlcontlnuous,

form

f(s,x)

is normal.

notes in

[303.

[24] and D e l o d e - A r i n o - P e n o t

integrand

in these

recently,

e $(x),

The

where

following

results

174

furnish other criteria. 2A. If

THEOREM. f

Let

is normal,

of Borel sets).

f

be a lower s e m i c o n t i n u o u s

then

f

is

A @ B-measurable

inte~rand on (where

B

The converse is true if the m e a s u r a b l e

S x R n.

is the algebra

space

(S,A)

is complete. PROOF.

Necessity.

For

s Then

sB

Ef

measurable.

s

S § Rn

by

For every closed

C c R n,

we have

where =

C8 and since

define

= {x I f(s,x) ~ 8}.

is closed-valued.

r61(C) = Efl(cs) 9

6 ~ R,

E R n+l I x ~ C 9

{(x,~)

is a m e a s u r a b l e m u l t i f u n c t i o n , Thus

F8

= B}, this implies

F61(C)

is

is measurable 9 and it follows from T h e o r e m 1E

that the set gphs is

A @ B - measurable.

= {(s,x)l

f(s,x) ! 6}

This being true for every

8 E R,

f

is

A @ B-measurable. Sufficiency. g(s,x,a)

If

= f(s,x)-a

f

is

on

A | B-measurable,

S x R n+lo

then so is the function

This implies the

A @ B-measurabillty

of the set {(s 9 Assuming Ef

(S,A)

I g(s,x,a) S 0} = gph Ef.

to be complete, we can conclude from T h e o r e m 1E that

is a m e a s u r a b l e m u l t i f u n c t i o n ,

2B.

COROLLARY.

is a m e a s u r a b l e PROOF. (S9

to

measurable, A @ B

If

f

measurable.

is normal.

then the function

The t r a n s f o r m a t i o n A @ B)

4 x B.)

S x Rn

s § f(s,x(s))

6: s + (s,x(s))

(For all sets

T

A x B, T

A @ B,

and

x: S § R n

is measurable. from

~-l(T)

is

in the a - a l g e b r a

We know from T h e o r e m 2A that

function with respect to

Q.E.D.


in

and hence the same must be true for

g e n e r a t e d by

measurable

f

is a normal integrand on

function,

(S x R n

i.e.,

and t h e r e f o r e

f

is a

f~

is

Q.E.D.

As with m e a s u r a b l e m u l t i f u n c t i o n s ,

the

A @ B-measurability property

can be adopted as the d e f i n i t i o n of the n o r m a l i t y of an i n t e g r a n d when the m e a s u r a b l e

space

(S,A)

is complete.

This a p p r o a c h then allows an

easy e x t e n s i o n of much of the theory b e l o w to cases where placed by an i n f i n i t e - d i m e n s i o n a l 2C.

PROPOSITION.

conditions

space;

For an i n t e g r a n d

are equivalent:

f

cf. on

Rn

is re-

[8].

S x R n,


175

in

(a)

both

f

(b)

(Carath@odory

s, and continuous PROOF.

nor

in

Therefore,

-f

condition):

f(s,x)

in

Let

R+, respectively. yj: S § R n+l by

s

D

(YjIJ

P

For each

in

2C;

~ J) Ef

s,

J = (a,B) = f(s,a)

f

f

a Carath@odory

of more general

More generally, Carath@odory

2D. on

family

s

E

f

define

Ef,

so by Theorem

Q.E.D. if it has property of normal

(b)

integrands.

in their own right and in the

integrands. F: S • R n § R m

is measurable

in

s

and continuous

in

in 1P and 1Q.

ties the present

property

a

originally

concept

used to

f

be a lower semlcontinuous, convex inte~rand

is normal if and only if there is a countable of measurable

f(s,xi(s))

(ii)

for

normal).

for convex integrands,

Let

Then

(xili E I)

(i)

and

in [1].

PROPOSITION. S • R n.

Rn

+ B.

are examples

in with the m e a s u r a b i l i t y

define normality

2B.

J = D • P

have already been encountered

The next result, of normality

F(s,x)

f(s,-)

On the other hand,

by

we shall call a function

mapplng,if

Such mappings

normal

the function

dense subsets of

inte~rand

integrands

x.

s

in

In fact, they are among the most important

x.

in

be countable

(i.e.,

thus Carath@odory

construction

neither

is a Castaing r e p r e s e n t a t i o n measurable

We shall call

is finite, measurable

and both are lower semlcontinuous.

for each fixed

and

yj(s)

we have

+=,

is finite and continuous

is measurable

1B

f(s,x)

For each fixed

takes on the value

(b) ~ (a).

Then

are normal and proper;

x.

(a) ~ (b).

-f(s,.)

f(s,x)

and

(xi(s)li

functions

i_~s measurable

i__nn s

~ I} n dom f(s,.)

xi: S ~ R n, such that for each

is dense in

i E I, dom f(s,.)

for each

S.

PROOF.

Necessity.

If

a Castaing r e p r e s e n t a t i o n of the form Then

Yi(S)

f

is normal,

(Yili

~ I)

= (xi(s),ai(s)) ,

(il) holds trivially,

Just the projection

of

Ef(s)

on

by T h e o r e m where

because


R n,

1B, and each

xi: S § R n

dom f(s,-)

Ef

has Yi

is

is measurable.

(defined in (2.2))

while on the other hand

is

(i)

holds by 2B. Sufficiency. subset

D(s)

El(S) E12,w

of

Here we use the fact that, by convexity, dom f(s,-)

= cl{(x,a)

~ Rn+llx

Given a family

any dense

yields E D(s),a ~ f(s,x))

(xili c I)

with the properties

in question,

176

let

Q

be a countable dense subset of

(YjlJ

E J)

for

J = (i,~)

yj(s)

= (xi(s),~).

Then

in yj

R~ and define the family

J = I x Q

as follows:

is measurable,

and for each

s E S

we

have Ef(s) by (il).

= cl[Ef(s)

At the same time, {slyj(s)

is m e a s u r a b l e by by

Ef

and

measurable. 2E.

S x Rn

to

Let

such that f

s

j ~ J

the set

= {slf(s,xi(s))

Thus c o n d i t i o n

e J},

( J}]

~ ~}

(c) of T h e o r e m IB is satisfied

w h i c h allows us to conclude that

Ef

is

Q.E.D.

COROLLARY.

Then

for each

E Ef(s)}

(i).

{YjlJ

n {yj(s)lj

f

be a lower semicontinuous,

dom f(s,.)

has a nonempty interior for every

is normal if and only If for each

PROOF.

f(s,x)

s.

is m e a s u r a b l e with respect

x.

Sufficiency

(xili r I)

convex intesrand o__nn

follows from P r o p o s i t i o n 2D, by taking

to be a family of constant functions with values in a dense

subset of

R n.

N e c e s s i t y is immediate from Corollary

2B.

The e q u i v a l e n c e of (b) and (c) in the next t h e o r e m was p r o v e d by E k e l a n d and T e m a m normality

[13,p.216], who adopted

(with the slight d i f f e r e n c e that they r e q u i r e d

lower s e m i c o n t i n u o u s 2F. A

THEOREM.

Let

in S

x

integrand on

S x R n.

f

There is a Borel m e a s u r a b l e

For every

s ~ S,

c > 0,

function

for all

there is a closed set

such

x c R n.

Te c S

is lower s e m i c o n t i n u o u s

Ef, Let

yields S

(c) ~ (b) ~

with

i__nn (s,x)

(b)

is elementary,

(a) ~ (c).

while T h e o r e m

Q.E.D.

be a Borel subset of some E u c l i d e a n space,

the algebra of Lebesgue

sets, and let

S x R n. Then the following p r o p e r t i e s (a)

f

(b)

(Scorz~-Dragoni property):

closed set

g:S x R n + ~

f(s,x) = g(s,x)

f(s,x)

The i m p l i c a t i o n

COROLLARY. A

be any lower s e m i c o n t l n u o u s

T e x R n.

IF, applied to 2G.

f

is a normal inte~rand.

for almost every

PROOF.

with

Let

(b)

relative to

to be

Then the following conditions are equivalent:

(a)

(c)

f(s,x)

s).

be a Borel subset of some E u c l i d e a n space, with

mes(S\T e) < ~, such that

on

only for almost every

the algebra of Lebesgue sets.

that,

(b) as their d e f i n i t i o n of

f

b__ee~ finit__~e i n t e g r a n d

are equivalent:

is a C a r a t h @ o d o r y intesrand; Te c S

with

for every

a > 0, there is a

m e s ( S \ T e) < e, such that

f

is continuous

177

relative to PROOF.

T

• R n.

This is immediate

Corollary

from T h e o r e m 2F and P r o p o s i t i o n 2C.

2G is the w e l l - k n o w n theorem of Scorz~-Dragoni.

Q.E.D.

Part

(b)

of T h e o r e m 2F complements T h e o r e m 2A in the special case of a complete m e a s u r a b l e space of the form in 2F. Next on the agenda is a further e l u c i d a t i o n of the r e l a t i o n s h i p between integrands and multlfunctlons. 2H. PROPOSITION. F: S + R n, !.~.

Let

(2.4)

~F

b__eethe i.ndicator integrand of a m u l t i f u n c t i o n

O

if

x E F(x),

§

if

x ~ r(x).

~r(s,x)

Then

~F

is a normal integrand if and only if

F


c l o s e d - v a l u e d multifunction. PROOF. E~r(s) 2I.

This is obvious

= F(s) • R+.

PROPOSITION.

from P r o p o s i t i o n

Q.E.D. Let

F: S • R n

be a m u l t i f u n c t l o n of the form

F(s) = (x I f(s,x) where

f

grand),

is a normal i n t e g r a n d on and

IH and the r e p r e s e n t a t i o n

m: S ~ R

i a(s)},

S • Rn

is measurable.

(e.g., a C a r a t h g o d o r y

Then

F

inte-

is c l o s e d - v a l u e d and

measurable. PROOF. Let

Since

A: S ~ R

is lower semicontinuous,

B ~ a(s)}.

Then

A

is measurable,

C o n s i d e r i n g an arbitrary closed set

responding multifunction

F': S § R n+l

is c l o s e d - v a l u e d and m e a s u r a b l e r-l(c)

by

is closed.

n Ef(s)

urable for all closed

C.

is measThen

Thus

F-I(c)

is meas-

Q.E.D.

for normality,

especially

in c o n j u n c t i o n with the

2A~ 2C(b) and 2F(b), an easily

class of m e a s u r a b l e m u l t i f u n c t i o n s

to w h i c h the operations

in the p r e c e d i n g section may be applied. As an illustration,

F'

We have

~ ~),

P r o p o s i t i o n 2I is important in providing, above conditions

a

we define a cor-

F'(s) ~ C • A(s).

(Proposition ii).

= {s[ r'(s)

because

C c R n,

and the latter set is m e a s u r a b l e by T h e o r e m 1M.

recognizable

F(s)

be the c l o s e d - v a l u e d m u l t i f u n c t i o n defined by

A(s) = {B ~ El urable.

f(s,.)

we have the following version of the famous

result in optimal control originally known as Filippov's

lemma.

178

2J.

THEOREM.

(Implicit

multlfunction (2.5)

Measurable

of the general

F(s') = {x ~ C(s)[

Functions).

C: S § R n

Carath~odory integrands

F(s,x)

= a(s)

measurable, Then

i_~s closed-valued

mapping,

(e.g., and F

(filie

Carath~odory ai: S § E

is measurable

urable selection where PROOF.

D

I)

Fi 2I).

and therefore to

(closed-valued),

of normal

a: S * R m

and hence

it is n o n e m p t y - v a l u e d

is

F

has a meas-

(i.e., relative

to

dom F).

F

fi(s,x)

is measurable

dom F

THEOREM.

= a(s)},

are closed-valued

~ el(S)}

for each

and measurable

n D(s)

i c I.

(Corollary

IQ and

ni~ I Fi(s),

by T h e o r e m 1M.

then exists by 1C. to optimization

the following complement

of form

collection

S • R n,

i_~s

We have

For applications

F: S § R n

on

i ~ I},

F: S x R n § R m

is a countable

integrands)

F(s) = C(s)

2K.

for all

Let

and

relative

~ el(S)

is measurable.

Fi(s) = {x e RnI

Proposition

be a

and

and measurable,

D(S) = {x c Rnl F(s,x)

Then

F: S § R n

form

fi(s,x) where

Let

Let

f

to T h e o r e m

be a normal

be a measurable, (2.5), o__rr F(s)

m(s)

selection

Q.E.D. problems,

it is useful to have

2J. intesrand __~

closed-valued

~ Rn).

A measurable

S x R n,

multifunctlon

Then the function = Inf

and let (e.g.,

m: S § R

F(s)

given by

f(s,x)

x~r(s) and the closed-valued

multifunction

M(s) =

M: S § R n

arg min

given by

f(s,x)

xcr(s) are

both

measurable.

PROOF. closed-valued

To d e m o n s t r a t e

the

multifunctlon

measurability

r,:

s § Rn + l

r,(s) = Ef(s) This is measurable

by IM (and II). {sim(s)

which is a measurable able,

defined

n [F(s) For any

m,

we c o n s i d e r

the

by

• R]. 8 E R,

we have

< 8} = (F')-I( Rn • (-~,8)),

set by property

(b) of IA.

Hence

m

is measur-

and since M(s)

the m e a s u r a b i l i t y f(s,-)

of

of

= {x c F(s) I f(s,x) i m(s)}, M

follows by T h e o r e m

is lower semlcontlnuous.)

Q.E.D.

2J.

(M(s)

is closed, because

179

We turn now to the methods for g e n e r a t i n g new normal Integrands from given ones. 2L.

PROPOSITION.

Let

f

be an Integrand o_~n S • R n

of the form

f(s,x) = suPi~ I fi(s,x), or instead, f(s,x)

infie I f i ( s , x ' ) ,

= lim inf Xv~X

where f

(fill ~ I)

is a countable

family of normal integrands.

Then

is normal. PROOF.

In the first case

Ef(s) = ni~ I Eli(S),

is immediate from T h e o r e m 1M. closure of 2M.

uIEiEfi(s),

PROPOSITION.

Let

In the second case,

so the normality Ef(s)

is the

and we can apply P r o p o s i t i o n 1L. f

be an I n t e g r a n d on

S x Rm

Q.E.D.

of the form

f(s,x) = Zi=lm fi(s,x), where each

fl

PROOF.

is a proper,

normal Integrand.

It is sufficient to consider

F: S § R n+l • R n+l

by

F(s) = Ell(S)

A: R n+l x R n+l § R n+l

Then

m = 2.

f

is normal.

Define

• Ef2(s) , and

by I

(Xl'al+m2)

if

x 2 = x1

A ( X l , a l , X 2 , a 2) if so that while

Ef(s) A

inherits

= A(F(s)).

Here

has closed graph.

x 2 # x I,

F

is m e a s u r a b l e by P r o p o s i t i o n II,

We have

lower s e m l c o n t i n u i t y

from

Ef(s) fl(s,.)

closed and

ous from c o n s i d e r i n g the "llm Inf" at any point), m e a s u r a b l e by T h e o r e m IN. Of course,

(since

f(s,.)

f2(s,.),

as is obvi-

and therefore

Ef

is

Q.E.D.

some of the terms in the sum in P r o p o s i t i o n 2M could

be i n d i c a t o r Integrands as in P r o p o s i t l o n

2H (e.g., with

r

as in

T h e o r e m 2J). 2N.

PROPOSITION.

Let

f

(2.6) where

f(s,x) g

S x Rn

= +=). Similarly,

with

Then f

f

of the form

= r

i__ssa proper, normal i n t e ~ r a n d o_nn S • R n

I n t e g r a n d o_~n S • R r

be an i n t e g r a n d on

r

and

n o n d e c r e a s l n g i_~n a

r

is a normal

(convention:

is normal.

is normal if it is of the form (2.6), with

r

inte~rand o_nn S x R m and g: S x R n ~ R m ~ C a r a t h @ o d o r y mappln~.

a normal

180

PROOF. Ef(s)

Obviously

is closed.

f(s,x)

Define As(X,a)

so that

El(S)

because

g

is normal,

because

r

is normal.

To p r o v e F:

S x R n+l

El(S)

COROLLARY.

We have

Hence

r

of

Ef

Let

f

then

PROOF. g(s,x) 2Q.

f

Apply

the

g

urable, Then

f

Let

either

normal

of the

the

from

on

on

Corollary S x Rn

S • R n x R~

assertion

1Q.

Q.E.D.

of the

form

and

u:

S § Rk

i_~s

of P r o p o s i t i o n

2N w i t h

on

S • Rn

of the

form

= ~(s)g(s,x), Integrand

conventions

first

assertion

yields

(measurable)

the

o__nn S • R n,

A: S + R+

0.~ = 0

0.~ = ~

or

Let

f

is a n o r m a l

semlcontlnuous (The

lower

attained: K ~ R n,

in

is m e a s is used.

then

growth

semicontinuous for e v e r y

the

for

k(s)

on

= 0.

f

s ~ S,

.

The

f(s,.)

case

to be

of

identically

Q.E.D. on

S • Rn

S x R n x E k.

of the

form

If

f(s,x)

is l o w e r

is normal.

condition in

0 ....

redefining

2N w i t h

= inf r u~R k

integrand

x,

following

by

be an i n t e g r a n d

f(s,x)

r

of P r o p o s i t i o n

result

simply

set w h e r e

(2.7)

to be

so that

with

9 Er


obtained

PROPOSITION.

where

1N.

= (g(s,x),~),

= r

second

f

this

is then

on the

2R.

Apply

= k(s)e;

0.~ = 0 0

mapping

s,

is normal.

PROOF. $(s,e)

by T h e o r e m

F(s,x,a)

in

Q.E.D.

is a p r o p e r and

let



s is m e a s u r a b l e


f(s,x) where

so

is n o r m a l .

= (x,u(s)).

COROLLARY.

closed-valued

is c l o s e d

follows

integrand

Then

x,

by

Eg

is a C a r a t h 6 o d o r y

is a n o r m a l

measurable.

Ef

assertion,

f(s,x) where

gph A

in

5 r

= {(x,a) I F ( s , x , e )

The m e a s u r a b i l i t y 2P

= {(x,~)18

while

the o t h e r

semicontinuous

A: R n+l § R n+l

= As(Eg(s)).

§ R m+l

is l o w e r

x,

on

and

every

r

is s u f f i c i e n t

for the m i n i m u m

e c R

and

every

in

for

f(s,x)

(2.7)

bounded

set

set {u c Rkl

~x

E C

with

r

! e}

is b o u n d e d . ) More integrand

generally,

if

f

fails

to be

lower

to be

s e m i c o n t i n u o u s , the

181

(2.8)

T(s,x)

is n e v e r t h e l e s s PROOF. E~(s)

= lim inf f(s,x') Xt~X

normal.

For the p r o j e c t i o n

= cl A(Er

c ons e q u e n c e

of T h e o r e m

we of course

have

an e l e m e n t a r y

1N.

If

f = f.

proof.

To conclude lead us into

A:

(x,u,a)

The n o r m a l i t y

of

~

f(s,x)

The

§ (x,e),

is lower

condition

we have

is thereby

seen to be a

semicontinuous

for lower

in

semicontinuity

x, has

Q.E.D.

this

section,

we treat

some aspects

of duality

that

convex analysis.

By the conjugate the i n t e g r a n d

f*

of the i n t e g r a n d

on

(2.9)

S x Rn

f*(s,y)

f

defined

= sup

on

S x R n,

we shall mean

by

{x.y - f(s,x)}.

xER n The b l c o n J u g a t e

integrand

(2.10)

f**(s,x)

According closed

is

to the theory

convex

continuous

(i.e.,

maJorlzed

f**

are proper.

2S.

PROPOSITION.

the conjugate PROOF. and let

by

If

f

integrand Let

-| f.

on

T x Rn

and hence

able

T

y,

relative

f(s,x)

i.e.,

closed both

semi-

o__nn S • R n,

representation

The C a r a t h @ o d o r y

and

then

~ I)


-~

convex f*

integrand

= xi(s).y

to

cones,

relative

= +~

Ef,(s)

= R n+l. to

and hence to

Let

and let

to

so are

f**. of

El,

integrands

- ai(s)

f*,

F*(s)

Thus

On the other hand,

Ef,

is normal.

relative

El,

is measur-

Since

f**

is the

be a m u l t i f u n c t i o n of

= -~

that

be normal.

be the polar

for

f*(s,y)


It follows

it too must

F: S § R n

s e T,

x, and c o n s e q u e n t l y

S\T. f*

for T x R n.

for all

relative

S,

conjugate

COROLLARY.

closed

a lower

take on the value

and proper,

Integrand

= suPiE I gi(s,y)

is normal

and constant

integrand 2T.

f*

we have

for all to

s

is a

give us the r e p r e s e n t a t i o n f*(s,y)

s ~ T

not

f*

and the b i c o n J u g a t e

(measurable). gi(s,y)

[12],

is the greatest

is convex

is a normal f*

does

f**

f

functions

is for each

either

and If

((xi,al)li

T = dom Ef

convex

f*(s,.)

function,which

at all or is i d e n t i c a l l y integrand

= sup {x'y - f*(s,y)}. yeR n

of conjugate

integrand

convex

g i v e n by

F(s).

Q.E.D. whose If

values F

is

are

182

measurable,

then so is

PROOF.

If

F*.

f = @F

(cf.

(2.4)), then

f* = ~F*"

Apply 2S and 2H.

Q.E.D. 2U.

COROLLARY.

Then

F

Let

F: S § R n

be a c l o s e d - c o n v e x - v a l u e d multifunction.

is m e a s u r a b l e if and only if its support function

(2.11)

h(s,y)

Is a normal

(convex)

PROOF.

If

= sup{x.ylx

~ F(s)}

inte~rand.

f = ~F'

then

f* = h

and

f** = f.

Apply

2S and 2H.

Q.E.D. 2V.

COROLLARY.

Let

f

be a proper integrand on

S • R n.

Then

f

is

normal and convex if and onlF i f ther_~e i__ss~ countable c o l l e c t i o n ((ai,~i)li

E I)

a.: S § R,

comprised of m e a s u r a b l e

functions

ai: S § R n

and

such that

I

fi(s,x) Similarly,

= suPi~i{x.ai(s)

a mult!function

F:S § R n

- ~i(s)}.

is c l o s e d - c o n v e x - v a l u e d if

and only if there is such a c o l l e c t i o n y i e l d i n g a r e p r e s e n t a t i o n F(s) = {x ~ Rnl x.ai(s) ~ ai(s) PROOF.

For

f,

the sufficiency

for all

i E I}.

follows from P r o p o s i t i o n 2L (the

functions in the s u p r e m u m being C a r a t h ~ o d o r y

Integrands), while the

n e c e s s i t y is o b t a i n e d by taking the c o l l e c t i o n to be any Castaing r e p r e s e n t a t i o n for the sufficiency

Ef,.

(One has

f*

via any C a s t a i n g r e p r e s e n t a t i o n of grand in C o r o l l a r y

2U.

s r S

f** = f.)

For

F,

Eh,

where

h

is the normal inte-

Q.E.D.

For a convex integrand each

normal and

is J u s t i f i e d by T h e o r e m 2J, and the n e c e s s i t y is seen

f

on

S • R n,

there is a s s o c i a t e d with

the s u b d i f f e r e n t i a l m u l t l f u n c t i o n

~f(s,-):R n § Rn~ defined

by (2.12)

~f(s,x) = {y c Rnl

This is c l o s e d - c o n v e x - v a l u e d , lower semicontinuous.

If

is the cone of normals to

f(s,x') ~ f(s,x)

+ y.(x'-x)

and its graph is closed, if

f = ~F

(cf. P r o p o s i t i o n

F(s) at

2H), the set

The following t h e o r e m was first p r o v e d by A t t o u c h infinlte-dimenslonal

2W.

Let

f

f(s,.)

x'}. is 3f(s,x)

x.

what different THEOREM.

for all

[14] in a some-

setting.

be a lower s e m l c o n t i n u o u s proper convex i n t e ~ r a n d

o__nn S • R n. Then the followin~ are equivalent: (a)

f

is a normal inte~rand;

(b)

(Attouch's condition):

the graph of the c l o s e d - v a l u e d

~aS

multlfunction

Sf(s,')

one measurable

function

measurable

s

in

PROOF.

depends measurably

and

x: S ~ R n Bf(s,x(s))

(a) ~ (b).

o__nn s,

such that # ~

and t h e r e is at least is finite and

f(s,x(s))

for all

s ~ S.

Let

g(s,x,y)

= f(s,x)

+ f*(s,y)

- x.y,

so that gph f(s,-) In view of Propositions representation s

because

2S and 2M,

therefore

(Proposition

2I).

f(s,.)

g

shows that

Furthermore,

1C measurable

such that

y(s)

c ~f(s,x(s))

is finite;

of course,

(b) ~ (a).

Let

the m u l t l f u n c t l o n

gph f(s,.)

functions

for every

is measurable

((xi,Yi)Ii

E I)

f(s,x i (s)) 0

in

s

be a Castaing

y: S § R ~ f(s,x(s))

by Corollary

2B.

representation

this can be chosen so that,

is finite and measurable

is known from [12, Theorem 24.9 and proof of T h e o r e m

s,

Hence there

and

This implies

f(s,x(s))

on

for every

[12,p.217]. x: S + R n

s.

and this

depends measurably

this graph is nonempty

F(s) = gph f(s,.); 10,

~ 0}.

is a normal integrand,

is a proper convex function

exist by Corollary

a certain index

= ((x,y) I g(s,x,y)

in

of for

s.

24.8] that

It

f(s,x)

is the supremum of f(S,Xo(S))

+ (Xil(S)-Xio(S)).Yio(S)+(xi2(s)-xil(S)).Yil(S) +.--+(x-x i (s)).y i (s) m

over all finite families the expressions Carath~odory

of Carath~odory Proposition 2X.

2L.

COROLLARY.

(iklk=l,...,m)

in the supremum,

integrand.

m

Thus

integrands,

f

of indices

in

viewed as a function of

I.

Each of

(s,x),

is the supremum of a countable

and the normality

of

f

follows

is a family

from

Q.E.D. Let

f

be a normal proper convex integrand o_~n S • R n,

and let r(s) where

x: S § R n PROOF.

= Sf(s,x(s)),

is measurable.

In view of

Then

F

is measurable

2W, this is a special

(closed-valued).

case of Theorem 1N. Q.E.D.

184

3.

Integral Functionals

on D e c o m p o s a b l e Spaces.

From now on, we denote by

~

a nonnegative,

G-finite measure on

(S,A). For any normal i n t e g r a n d x: S § R n,

we have

f

f(s,x(s))

on

S • Rn

measurable

and any m e a s u r a b l e

in

s,

function

and therefore the

integral If(x) = f f(s,x(s))~(ds) S has a well d e f i n e d value in

R

under the f o l l o w i n g convention:

neither the p o s i t i v e nor the negative part of the function is summable

(i.e.,

finitely), we set

(3.1)

If(x) We call

f.

If

If(x) = +~.

< +~ ~ f(s,x(s))

the integral

< +~

X

measurable

functions,

functions

X~ if

Among the linear spaces

then,

a.e.

functional a s s o c i a t e d with the integrand


is a convex functional on

s § f(s,x(s))

In particular,

T y p i c a l l ~ we are c o n c e r n e d with the r e s t r i c t i o n of

linear space

if

X

f

x:S ~ R n.

If

to some

Notice that

If

is a normal convex integrand.

of interest,

besides the space of all

are the various L e b e s g u e spaces and Orlicz spaces,

the space of constant functions,

and in the case of t o p o l o g i c a l or

differentiable

spaces of continuous or d i f f e r e n t l a b l e

functions.

structure on

S,

In their role in the theory of integral functionals,

these spaces fall into two very different

categories,

however,

d i s t i n g u i s h e d by

the p r e s e n c e or absence of a certain p r o p e r t y of decomposability. Slightly g e n e r a l i z i n g the original d e f i n i t i o n in [i], we shall say that

X,

able if

a linear space of m e a s u r a b l e S

subsets

measurable

function

S k (k=l,2,...), x': S k ~ R n,

function

~

(3.2)

for

x(s)

Sk,

x:S ~ R n,

is decompos-

can be expressed as the union of an i n c r e a s i n g sequence of

measurable

belongs to

functions

X.

and every

x" ~ X,

Sk

the

x'(s)

for

s E Sk,

x"(s)

for

s r S\Sk,

and b o u n d e d

(measurable)

=

(The original d e f i n i t i o n r e q u i r e d this property, not just

but all m e a s u r a b l e

is G-finite,

such that for every

the sets

Sk

sets

T c S

with

u(T)

finite.)

can always be chosen with

The space of all m e a s u r a b l e

functions,

Orlicz spaces, are all decomposable. functions and spaces of continuous

the Lebesgue

However,

Since

u(S k) finite. spaces and

the space of constant

or d i f f e r e n t i a b l e

functions

furnish

examples of n o n d e c o m p o s a b i l i t y . The concept of d e c o m p o s a b i l i t y

is d e s i g n e d for the following result.

3A.

THEOREM.

Let

f

be a normal Integrand on

be a linear space o f m e a s u r a b l e (3.3)

inf f f(s,x(s))u(ds) xEX S

t_~o hold,

it is sufficient that

inflmum not be X

functions

+~.

such that

PROOF.

If(x)

X

For the r e l a t i o n

X

b__~ed e c o m p o s a b l e and that the first

(These conditions are superfluous

is the space of all m e a s u r a b l e

x

x: S § R n.

and let

= f [Inf f(s,x)]u(ds) S xER n

functions,

satisfies a condition i.mplylng that tion

S • R n,

X

in the case where

or more generally,

i__ff f

contains every m e a s u r a b l e

func-

< +~.)

The e x p r e s s i o n i n t e g r a t e d on the right side of (3.3) is m(s) = inf f(s,x), xcR n

which is m e a s u r a b l e by T h e o r e m 2K; as in the d e f i n i t i o n of integral is c o n s i d e r e d to be tive part of f(s,x(s))

m

+~

is summable.

> m(s)

for all

If,

this

if neither the p o s i t i v e nor the nega-

For each m e a s u r a b l e

s ~ S.

function

Thus the inequality

>

x,

we have

is trivial

in (3.3), and our task is to show, a s s u m i n g f m(s)~(ds) S that there exists

x r X

there is a positive

< 8 < +~,

satisfying

function

If(x)

p: S + R

< 8.

such that

Since

Setting

is G-finite, < ~.

S ~(s) = cp(s) + max{m(s),

for

~

~ p(s)~(ds)

s > 0

such that

-e -I}

s u f f i c i e n t l y small, we have a m e a s u r a b l e ~(s)

> m(s)

for all

s, and

~ a(s)N(ds)

tion

function < 8.

~: S § R

The m u l t i f u n c -

S F(s) = {x E Rnl

f(s,x) ~ a(s)}

is then n o n e m p t y - c l o s e d - v a l u e d and, by P r o p o s i t i o n 2I, measurable. Hence there is a m e a s u r a b l e a(s) ever,

for all x'

s

function

(Corollary IC)

need not belong to

X

x': S § R n

is needed.

(Sklk=l,2,...) each

Sk

x" ~ X

< +~,

If(x")

{s E S I Ix'(s)I ~ k}

to be bounded on

we have for all

k

< 8.

How-

< +~, and let

be as in the d e f i n i t i o n of decomposability.

x'

f(s,x'(s))

so in general a m o d i f i c a t i o n of

be such that

with the m e a s u r a b l e set

we can suppose If(x")

Let

If(x')

(except in the cases covered by the

p a r e n t h e t i c a l remarks in the theorem), x'

such that

and consequently

S k.

Since

If(x')

< 8

s u f f i c i e n t l y large that

/ f(s,x'(s))~(ds) Sk

+ / f(s,x"(s)) S\S k

Intersecting

if necessary,

< 8.

and

Thus

for

x

defined

composibility

as in

assumption,

As an i m p o r t a n t us c o n s i d e r

(3.2), x E X.

(Q)

minimize

where

r

spaces


functional

J:

adopt

the

gated

is w h e t h e r

x: S § R n

~ - ~ = +~

(Q)

all

de-

J(x)

let

x ~ X, u ~ U, X

and

u:

(To c o v e r

all

in

(Q).)

is e q u i v a l e n t

minimize

and by our

3A can be a p p l i e d ,

S • R n • R k,

is a r b i t r a r y .

convention

< B,

form:

over on

functions

(p)

of the

+ Ir

integrand

X § R

of h o w T h e o r e m

problem

J(x)

is a n o r m a l

If(x)

Q.E.D.

illustration

an o p t i m i z a t i o n

we h a v e

The

and

over

are

S § R k,

linear

and the

contingencies,

question

to the r e d u c e d

+ If(x)

U

to be

we investi-

problem

all

x ~ X,

where (3.4)

f(s,x)

Here

f

is n o r m a l

lower

semicontinuous

semicontinuity

by P r o p o s i t i o n in

x (ef.

furnished

in

3B.

COROLLARY.

(Theorem

text

of p r o b l e m s

(Q)

(i) (cf.

the

the

J(x)

< +~

Then

(P)

with

(Q)

this

infimum

always

PROOF.

Fix any Then

follows inf u~U

from Theorem

g

(3.4)

lower

In the a b o v e

is a l w a y s

con-

the

attained

2~R), and

function

one n e c e s s a r i l y

equlvalent,in

= inf uEU

with

sense

yielding haG.

u ~ U.

that

for e v e r y

lr by at J(x)

If the

infimum

u E U,

so let

closed-valued

< +~,

(ii)

one and

u E U. define

(Corollary

g(s,u) 2P),

=

and

that

= f f(s,x(s))~(ds). S

over us

least

integrand

assumption

holds.

The

for

has

by e v e r y

< +~.

is

f(s,x)

that

in P r o p o s i t i o n

is a n o r m a l

2A and

in

x ~ X,

attained

x E X

further

I (u) ~ f [inf g(s,u)]B(ds) g S u~R k

(3.5)

attained If(x)

being

condition


If(x)

= r

Thus

given

are one

(3.5)

assume,

Minimization).

f(s,x)

S + Rk

< +~

as we now

sufficient

suppose

for some

and

J(x)

the

o__nnR e d u c e d (P),

defining

u:

2R if,

2R).

condition

whenever

+ Ir

x E X

and

infimum

sufficient

(ii)

= inf r u~R k

U

is

suppose

multifunction

+~,

it is of c o u r s e

it is not F: S § R k

r(s) = {sl g(s,u) i f(s,x(s))}

+~;

then

defined

by

it

187

is m e a s u r a b l e by P r o p o s i t i o n Hence it has a m e a s u r a b l e lr which entails (3.5) 9

2I and n o n e m p t y - v a l u e d by a s s u m p t i o n

selection

u.

We have

= Ig(U) ~ If(x)

u ~ U

by

(i).

< +~,

(ii), and thus

u

furnishes the m i n i m u m in

Q.E.D.

The wide range of problems where this r e d u c t i o n t h e o r e m can be applied is apparent, are r e p r e s e n t a b l e and

r

if it is r e c a l l e d that very general constraints

in terms of the d e s i g n a t i o n of the elements where

have the value

+=.

The result generalizes,

for example,

J

one

c o n s t i t u t i n g a key step in e s t a b l i s h i n g the existence of optimal traJectories

in control theory;

see R o c k a f e l l a r

in c o m b i n a t i o n with all the m a c h i n e r y

p o w e r f u l tool for the analysis of m u l t i s t a g e problems.

Such problems

[15].

It also furnishes,

for v e r i f y i n g normality,

a

stochastic o p t i m i z a t i o n

can be reduced to "dynamic p r o g r a m m i n g " more

efficiently than has p r e v i o u s l y been shown, e.g. by Wets and the author [16] and E v s t i g n e e v

[17].

In the rest of this section, we denote by spaces of

Rn-valued

(3.6)

X

and

Y

two linear

functions such that

[ Ix(s)'y(s)l~(ds) S

< +|

for all

x ~ X, y E Y.

The b i l i n e a r form <x,y> = / x(s)-y(s)u(ds) S defines a p a i r i n g between

X

and

Y,

in terms of w h i c h the standard o

theory of locally convex spaces can be applied. w e a k topologies

o(X,Y)

and

~(Y,X)

In particular,

are available.

the

(Strictly speak-

ing, these are not, of course, H a u s d o r f f topologies unless we identify elements of

X

p r o d u c i n g the same linear functional on

and similarly for elements of a potential nuisance

Y.

Y

via

This i d e n t i f i c a t i o n is harmless,

for terminology and n o t a t i o n in what follows,

but so

we gloss over it, leaving the details implicit.) An important Lebesgue spaces:

case to be borne in mind is that of the X = LPn

and

Y = L~,

(decomposable)

1 .< p .< ~ . and . 1 < q < =,

where L np= L P ( s , A , u ; R n) The r e l a t i o n

(l/p) + (l/q) = 1

suffices

totally necessary;

for instance,

p = ~

in the case where

and

q = ~

The conjugate on

Y

for (3.6), but it is not

it is o c c a s i o n a l l y useful to employ ~(S)

of a functional

< ~. F: X § R

is, of course,

188

defined by F*(y) = sup{<x,y> - F(x)}, xcX and similarly the conjugate on F**(x) As is well-known, F**

is the

F*

-~,

of a functional

G:Y ~ R;

thus

= sup{<x,y> - F*(y)}. ycY

is convex and

~(X,Y)-l.s.c.

has the value

X

1.s.c.

convex hull of

while otherwise

w i t h respect to

e(Y,X);

F ~ if that functional nowhere

F** ~ -~.

Our aim now is to apply these facts to integral

functionals, m a k i n g

use of T h e o r e m 3A and the n o r m a l i t y of the conjugate integrands and

f**

in P r o p o s i t i o n 2S.

f*

The next t h e o r e m is a slightly improved

version of the m a i n result of R o c k a f e l l a r

[1],as e x t e n d e d in [8].

The

version in [8] was p r e s e n t e d in terms of a separable reflexive B a n a c h space in place of

R n,

but with the m e a s u r a b l e

For a recent generalization, 3C. If

THEOREM. o nn

x ~ X

Let

X.

Suppose

with

If(x)

f

I~f

Y

X

PROOF.

If.(y)

Then

I~ = If.

If.

< +~,

Fix any

[ll].

is d e c o m p o s a b l e , a n d there exists at least one

< +|

o__nn Y

is likewise decomposable,

with

then

o_~n Y,

is

and there exists at least one

I~* = If**

on

X.

= f(s,x) - <x,y(s)>.

The second term in this e x p r e s s i o n constitutes (hence a normal integrand), g,

and hence in particu-

e(Y,X)-!.Z.~.

y ~ Y, and consider the integrand g(s,x)

T h e o r e m 3A to

(S,A) complete.

be a normal i n t e ~ r a n d o_n_n S x R n, and consider

lar the convex functional

y ~ Y

see V a l a d i e r

space

so

g

a C a r a t h @ o d o r y integrand

is normal by P r o p o s i t i o n 2M.

Applying

we obtain

inf / [f(s,x(s)) x~X S

- <x(s),y(s)>]~(ds)

the common value not being

+~.

= /[-f*(s,y(s))]u(ds),

Due to the latter,

it is legitimate

to rewrite the e q u a t i o n as inf{If(x) xEX or in other words, by duality. 3D. If

If.(y)

= If,(y).

The rest of the t h e o r e m follows

Q.E.D.

COROLLARY. Y

If*(y)

- <x,y>} = -If,(y),

Let

f

be a normal proper convex i n t e g r a n d on

S x R n.

is d e c o m p o s a b l e and there exists at least one

y ~ Y

with

< +~, then the convex integral

o_~n X

i_~s o(X,Y)-

lower s e m i c o n t i n u o u s

(and nowhere -~).

functional

If

18g

PROOF.

Apply T h e o r e m 3C to

hypothesis on 3E.

f

is equivalent

COROLLARY.

that

If(x)

Let

f

If,

If, = If**.

to the p r o p e r t y that

x ~ X.

The

f** ~ f.

be a normal convex i n t e g r a n d on

for at least one

< +~

to see that

Q.E.D.

S • Rn

Then for every

such

x c X

the

subdifferential (3.7)

~If(x) = {y e YI If(x') ~ If(x) + <x'-x,y>,

V x' c x}

i_ss given by ~If(x) = {y E YI y(s) PROOF.

~ ~f(s,x(s))

A c c o r d i n g to the d e f i n i t i o n

a.e.}.

(3.7), we have

y E ~If(x)

if and only if N

<x,y> - If(x) = If(y), where

If(y) = If.(y)

by T h e o r e m 3C.

The result now follows from the

fact that <x(s),y(s)> - f(s,x(s))

! f*(s,y(s))

always holds, with equality if and only if 3F.

COROLLARY.

function,

Le.t

r: S + R n

= sup x~r(s)

is d e c o m p o s a b l e and

x.y

C ~ Z,

c l o s e d - v a l u e d multi-

If in a d d i t i o n

Y

PROOF.

Let

f = ~F

(cf.

for

a.e.},

(s,y)

for all

is decomposable, ~(X,Y)-cl

~ r(s)

E S • R n.

then

sup <x,y> = lh(Y) xEC

y E Y.

then

co C = {x E X I x(s) (2.4)); then

follows at once from T h e o r e m 3C.

E cl coF(s)

f* = h,

and the result

In many situations, it is useful to be able to apply

some indirect c r i t e r i o n for the existence of < +~.

a.e.}.

Q.E.D.

without very explicit k n o w l e d g e of the i n t e g r a n d

If,(y)

Q.E.D.

and let

h(s,y)

X

c ~f(s,x(s)).

be a measurable,

C = {x ~ X I x(s)

If

y(s)

3C and 3D

f*, and this requires

y ~ Y

satisfying

One case w h i c h falls out immedlately, ls that where there

is a lower bound f(s,x) with

B

summable;

then

> ~(s)

for all

f*(s,0) ~ -8(s),

x E R n, so

If.(0)

< +~.

Another

criterion is p r o v i d e d by the next result. 3G.

PROPOSITION.

and let

Y = Lp n'

Let

f

be a normal convex i n t e g r a n d o_qn S • R n,

1 < p < ~. --

-

Then for the existence of at least one

190

y ~ Y

such that

for some

If.(y)

~ c Lq

< +~,

(where

the f o l l o w i n g c o n d i t i o n is sufficient:

I/p + I/q = I)

and some

e > 0,

the function

n

s § f(s,~(s) + u) lu] < E,

while

--

PROOF.

belongs t__0_o L~ If(x)

Let

>

m

for each

u ~ Rn

satisfying

.

{al,...,am}

c Rn

be any finite set whose convex hull

contains the unit ball; then (3.8) Let

m I ai'Y ~ maxi= ~ > 0

IYl

be small enough that

y E R n"

for all 16all ~ e

for all

i.

Then each of

the functions ai(s) belongs to T c S T.

L~, as does

with

~(S\T)

For each

n e i g h b o r h o o d of multifunction

a(s)

= 0,

s E T,

= fi(s,~(s)

+ ai),

= f(s,x(s)).

There is a m e a s u r a b l e

f(s,-)

and therefore has

s + ~f(s,~(s))

is finite on a

3f(s,~(s))

~ ~.

Thus the

is almost e v e r y w h e r e n o n e m p t y - v a l u e d ;

since it is also c l o s e d - v a l u e d and m e a s u r a b l e by

2X,

it has a meas-

urable s e l e c t i o n r e l a t i v e to the set where it is n o n e m p t y - v a l u e d Hence there is a m e a s u r a b l e (3.9)

y(s)

set

such that these functions are all finite on

the convex function

~(s)

i = l,...,m

function

~: S § R n

~ 3f(s,~(s))

(1C).

satisfying

a.e.

We then have, almost everywhere, fi(s,~(s)

+ ~a i) ~ fi(s,x(s))

+ ~ai-Y(S) ,

i = 1,...,m,

or in terms of the n o t a t i o n i n t r o d u c e d above,

ai'Y(s)

S ~-l[~i(s)

- ~(s)],

i = 1,...,m.

T a k i n g the m a x i m u m on both sides with respect to we obtain Since

I~(s)I ~ ~(s)

a.e~, where

and r e c a l l i n g

This shows that

(3.8),

y ~ Y.

(3.9) implies

f*(s,y(s)) while

If(~)

> -~,

we have

In T h e o r e m 2C, If.

a c L~.

i

However,

If**

= If,(y)

< +~.

- f(s,~(s)), Q.E.D.

turns out to be the "closed convex hull" of

in an important case c o n n e c t e d w i t h the theory of "relaxed"

v a r i a t i o n a l problems,

If**

other words,

follows from weak lower semicontlnuity.

convexity

is also simply the "closure" of

If;

in

This

case is d e l i n e a t e d next. We shall say that the I n t e g r a n d each a t o m s ~ T.

T c S,

Of course,

the f u n c t i o n

f

f(s,-)

if the m e a s u r e space

this c o n d i t i o n is a u t o m a t i c a l l y

is a t o m i c a l l y convex if, for is convex for almost every (S,A,~)

satisfied.

is without atoms,

191

3H.

THEOREM.

atomically

Let

f

convex.

tain elements

x

be a normal i n t e g r a n d o__qn S • R n

Suppose and

y

X

and

Y

such that

are both d e c o m p o s a b l e and con-

If(x)

Then the proper convex f u n c t i o n a l 1.s.c.

functional on

~(X,Y)-l.s.c. every

X

< +|

Ifm m

m a J o r i z e d b_[y If.

if and only if

which is

f(s,x)

and

Ifw(y)

< +|

is the greatest I_p_nfact,

is convex in

If x

~(X,Y)-

itself is for almost

s. PROOF.

To prove the first assertion,

it is enough,

T h e o r e m 3C, to d e m o n s t r a t e that the weak closure of the epi If = {(x,a) is convex.

in view of (nonempty)

set

c X • R I e ~ If(x)}

R e m e m b e r i n g the nature of the topology

a(X,Y),

one sees

this is equivalent to showing that the closure of the image of

epi If

under any m a p p i n g of the form (x,a) § (<X,Yl> + ~ 8 1 , . . . , < X , Y m > is convex.

Here we have

k(s)

> f(s,x(s)) a.e.,

Let

Z = X • L1

tlons, since

~ = f k(s)~(ds) S

is decomposable),

(a n o n e m p t y set because

M(s)

such that that

is nonempty).

P a s s i n g to

any p a r t i c u l a r element of Let property

C,

(Skl. k = 1,2,...)

T

C o n s i d e r any linear

m • (n+l)

whose components are

z E Z.

C - ~

It suffices to show

if necessary, where

it can be supposed in this that be a family of m e a s u r a b l e

in the d e f i n i t i o n of d e c o m p o s a b i l i t y ,

and m e a s u r a b l e set

a.e.}

z E Z,

is summable for every

is convex.

w h i c h is c o n t a i n e d in

Sk

for all

ly large, let C T denote the set of all m e a s u r a b l e z: T § R n+l satisfying z(s)

E El(S)

and

Iz(s)l ~ r

The d e c o m p o s a b i l i t y p r o p e r t y implies r e s t r i c t i o n s to (since

0 ~ C)

T any

of functions r

z ~ CT

=

/ T

T.

is

r > 0 k

sufficient-

functions

for all

s r T.

is the same as the set of all satisfying

(3.10), and in fact

can be e x t e n d e d to an element of

giving it the zero value outside of

ATZ

r CT

z ~ Z

~

0 E C.

sets with the

and for each

r

(3.10)

R n + l - v a l u e d func-

of the form

is a m a t r i x of d i m e n s i o n

cI(AC)

space of

~ El(S) = epi f(s,.)

epi If

A: Z § R m

M(s)z(s)

such that

and let

Az = f M(s)z(s)~(ds), S where

k r L~

so the q u e s t i o n can be r e p h r a s e d as follows.

C = {z ~ Z I z(s)

transformation

for some

(this b e i n g a d e c o m p o s a b l e

X

+ e8 m)

Thus for the m a p p i n g

M(s)z(s)~(ds)

C

by

192

we have

r AC = ATCT,

where the latter set increases with

For any

z ~ C

~ > 0,

and

IAz - ATZI

ATZ E ATC ~.

< ~

for

and

r.

the set

T = S k n (s I z(s) yields

T

E Ef(s) k

and

and r

Iz(s)I ! r)

sufficiently large, and one has

Therefore cl AC = cl u ATC~,

where

ATC;

increases with

r > 0


T

r

and

such that

T;

the union is respect, to all

T c Sk

for

k

sufficiently large.

The p r o b l e m can therefore be reduced to showing that each of the sets ATC Tr

of the form ponents of over every

M(s)

s c T,

T

(For this purpose, we note that the com-

in the d e f i n i t i o n of

since

z r Z,

tions to

is convex.

M(s)z(s)

AT

must actually be summable

is by a s s u m p t i o n summable over

T

for

and by the d e c o m p o s a b i l i t y p r o p e r t y the set of restricof the functions

in

Z

includes all bounded m e a s u r a b l e

functions.) The convexity of t h e o r e m of Liapunov,

ATC ~

will be shown to follow from the w e l l - k n o w n

which asserts that the range of a nonatomlc

valued measure is convex,

in fact compact.

R n-

(For a short p r o o f of

Liapunov's t h e o r e m using the K r e i n - M i l m a n Theorem,

see L i n d e n s t r a u s s

El8]; the Hahn d e c o m p o s i t i o n t h e o r e m can be used to remove the assumption of L i n d e n s t r a u s s First we p a r t i t i o n relative to

SO

that the component measures are nonnegatlve.)

S

into

SO

and

S1,

where

~

S 1.

Let

and nonatomic r e l a t i v e to

is purely atomic TO = T n SO

and

A c c o r d i n g to our h y p o t h e s i s that f is a t o m i c a l l y convex, T 1 = T n $I. we have El(S) convex for almost every s ~ TO, and hence C r is To convex. Since

AT0;= AT00;~ + AT convexity

of

ATC;

will follow from that of

be any two elements of urable sets

C~l,

ing

AT1

Let

z

and

z'

T, for meas-

E c T1, by

ZE(S) = z'(s) - z(s)

Obviously

ATIC;1.

and define the set function

T(E) = AE(Z' where

l,

w

for

s c E, and

is countably additive

are, as seen above, 9 (E) +

ATlZ

=

- z) = ATlZ E , for

s ~ Tl\E.

(since the matrix components defin-

summable over

ATl( zE

ZE(S) = 0

+ z), with

T1) ,

and

zE + z c

CT1. r

193

Let

D = (range T) + ATIZ.

Liapunov's theorem, r e s p o n d i n g to

ATIC~I;

is convex.

E = ~)

segment j o i n i n g

Then

z

and

and

is a subset of

Moreover,

z'

z'

D

D

ATIC;I

which,

contains both

( c o r r e s p o n d i n g to

z

E = T1).

is t h e r e f o r e c o n t a i n e d in

by

(cor-

The llne

D,

hence in

this shows the latter set is convex.

It remains to d e m o n s t r a t e the final a s s e r t i o n of the theorem. sufficiency of the condition is covered by around a set of measure zero),

If**(x)

= If(x)

for every

(3.11)

Making use of d e c o m p o s a b i l i t y , i n c r e a s i n g sequence of sets (3.12)

f**(s,x(s)) whenever

Fix any F

k

and

Since

= f(s,x(s))

we can express

f** ~ f, a.e. S

this implies

for each

x c X.

as the union of an

Sk~ such that

= f(s,x(s))

x: S k § R n

r > 0,

to the necessity.

our starting a s s u m p t i o n is

x E X.

f**(s,x(s))

The

(with a slight m a n e u v e r

so we direct ourselves

In view of what has already been proved, that

3D

for almost every

s E S k,

is m e a s u r a b l e and bounded.

and consider the

(measurable)

multlfunction

defined by r(s) = Ef**(s)

where

B

n [rB x R],

is the closed unit ball in

a Castaing r e p r e s e n t a t i o n for

F.

R n.

el(S) ~ f**(s,xi(s)) for almost every

Let

Then by

s ~ S k n domF,

((xi,ai) [ i c I)

be

(3.12)

= f(s,xi(s))

so that

(since

I

is countable)

the

relation (3.13)

(xi(s),ai(s))

holds for almost every

~ El(S)

n [rB x R]

s E S k n domF.

F(s) c Ef(s)

for all

Of course,

i ~ I

(3.13)

implies

n [rB x R],

or what is the same thing, f**(s,x)

= f(s,x)

x ~ Rn

for all

with

This e q u a t i o n has been shown to hold for almost every that

F(s) @ ~

Ixl ~ r), f(s,x)

(i.e.

f**(s,x)

< +|

and it holds trivially if

then being

+=).

the conclusion that measure zero.

Q.E.D.

Since

f**(s,.)

k

for at least one F(s) = Z and

= f(s,.),

r

(both

Ix I ~ r. s r Sk x

such

with

f**(s,x)

and

are arbitrary, we reach

except for

s

in a set of

194

There are many situations where it is convenient in direct terms to w o r k w i t h integral functlonals on the space example,

Ln,

because,

for

c o n t i n u i t y with respect to the n o r m is then easier to work w i t h

and to express via local p r o p e r t i e s of the integrand. advantages

However,

such

are often paid for by a troublesome p r o b l e m w h e n it comes ~W

to duality:

the dual Banach space

Ln

cannot

be i d e n t i f i e d with

L I. We shall describe a special result in this d i r e c t i o n which shows n the s i t u a t i o n is not quite as bad as might be imagined, and w h i c h can be used to derive some useful compactness theorems. A (norm) continuous

linear functional

z

sin~ular, lf there is an i n c r e a s i n g sequence urable sets satisfying

S = Uk=iSk,

on L = is said to be n (Skik = 1,2,...) of meas-

such that, w h e n e v e r

function v a n i s h i n g almost everywhere outside of some z(x) = 0.

holds as an isometry functions in

Lln

E Ln1 • Lsing n '

(subject to the usual i d e n t i f i c a t i o n of "equivalent"

and

L~).

For a p r o o f of this result in a much b r o a d e r

(R n r e p l a c e d by an I n f l n l t e - d l m e n s l o n a l

space),

The following t h e o r e m is taken from R o c k a f e l l a r

If

THEOREM. on

one has

to the H e w l t t - Y o s i d a theorem

(3.12) <x,(y,z)> = <x,y> + z(x) for x E L ~ n' (y,z) the relation (3.13) L =N = L1 @ Lsing n n n

3I.

Sk,

is a

The set of these forms a linear space we shall denote by

L slng A fundamental fact equivalent n " [193, is that under the pairing

context

x e Ln

L n" ~

Let

f

see Levin

[20].

[2].

be a normal inte~rand o_n_n S • R n,

and consider

Suppose the set F = {x e LnI

If(x)

< +~} ~N

i_~s nonempty.

Then the conjugate of

If

o__nn L n

is given in terms of

the p a i r l n 5 (3.12) b__yy * If(y,z)

(3.14)

= If,(y)

+ JF(Z)

for all

1 ~sing y ~ Ln, z ~ bn '

where JF(Z) PROOF.

= sup z(x). xEF

Using T h e o r e m 3C and the d e f i n i t i o n of the conjugate func-

tlonal, we obtain If(y,z) = sup{<x,y> + z(x) - If(x)} x~F sup{<x,y> , If(x)) + sup z(X) = Ifi(y) x~F x~F

+ JF(Z).

195

Thus

~

holds

inequality.

in (3.14),

In thls, we can s u p p o s e

for o t h e r w i s e every

If(y,z)

~ +|

Then

that

If(x)

> -~

f(s,x(s))

the o p p o s i t e

for all

is s u m m a b l e

in

x ~ Ln, s

for

s E F. Fix

enough

Y ~ Lln'

z ~ Lsing'n

to show that

can choose

x'

and

8" ~ z(x").

the p r o p e r t y "singular

8' < If,(y)

8' + B" < If(y,z). x"

in

B' < <X',y> and

and the m a i n task is to v e r i f y

Let

F

functional",

z

= / [<x'(s),y(s)> S

that

Z(Xk-X")

other hand, over

= 0,

because

s ~ S,

3C, we

- f(s,x'(s))]~(dS)

be a s e q u e n c e

is d e s c r i b e d

of sets h a v i n g

in the d e f i n i t i o n

of

and define

Xk(S)

Then

of T h e o r e m

It is

such that

- If(X')

to

8" < JF(Z).

By virtue

(Ski k = 1,2,...)

relative

and

[

so that

and

xk ~ F

<Xk,Y>

for

s E Sk,

x"(s)

for

s ~ S\S k.

z(x k) = z(x")

f(s,x'(s))

we have

x'(s)

> 8"

f(s,x"(s))

for all

k.

On the

are both

summable

and

- If(x k) = f [<x'(s),y(s)> Sk

+

/

- f(s,x'(s))]~(ds)

[<x"(s),y(s)>-f(s,x"(s))]u(ds),

S\S k so that lim[<xk,Y> Therefore,

choosing

k

-If(Xk) ] = <x',y> sufficiently

8' + B" < <Xk,Y>

of

we have

- If(x k) ! If(y,z),

Q.E.D.

As a corollary, result

large,

- If(x k) + z(x k)

= <Xk,(Y,g)> as desired.

- If(x').

we state

a slight

generalization

of the m a i n

[i0].

3J.

COROLLARY.

If,

considered

Let

f

be a normal

as a f u n c t i o n a l

on

integrand LI

n'

on

n S • R , such that

is not i d e n t i c a l l y

the set

(3.15)

G = {y E Ln[

If

= {y E L |nl If,(y)

is b o u n d e d

< +~}

below

on

1 L n}

+~,

and

196

i_~s n.onempty.

Let

If

be the convex functional on

terms of the canonical If(x,z)

isomorphism

(3.14)

Ln

defined in

by

I z E Lsing n ' for x ~ L n'

= If**(x) + JG(Z)

where JG(Z) Then

If

is the greatest

maJorized

by

If

on

In fact, if

f

o(L n ~* ,L~ )-l.s.c.

L1

- -

= sup z(y). ~EG

(regarded

n

~* Ln

convex functional o~n

as a s u b s p a c e of

is atomically conve___x,

If

L

).

n

is simply the ~reatest ,

o(L~*,L

) - l . s . c .- - f u n c t i o n a l

on

Ln~ *

m a j o r i z e d by

-on -

If

Ln I.

Then

for each a > inf{If(x)l

x e L~},

one has {(x,z)l PROOF.

If(x,z) ! e} = o ( L n ,Ln)-Cl{xl

We have

If

of the two expressions If,

we get

= If,

for

G

If = If, = If

by T h e o r e m 3C ( j u s t i f y i n g in

(3.15)).

F:X § R

its level sets of the form

every

It is

A p p l y i n g T h e o r e m 31 to

The second result is then immediate

Q.E.D.

A functional

compact.

the equivalence

(with respect to the extended pairing),

and this yields the first result. from T h e o r e m 3H.

If(x) ! m}-

is said to be

o(X,Y)-inf-compact

{x E XI F(x) ! a},

G(X,Y)-coercive

if

a ~ R,

are

if all o(X,Y)-

F- has this property

for

y E Y. Our next objective

is to state a rather complete criterion for

these propertles, in the case of an integral functional and their duality w i t h continuity p r o p e r t i e s that many p r o p e r t i e s , w h i c h might

of

If

If..

on

Lp n

~

It will be seen

in general be expected to be distinct,

collapse into e q u i v a l e n c e when the m e a s u r e space is without atoms. The f o l l o w i n g growth conditions

on an integrand

f

on

S • Rn

will be crucial: (G1): For each every

r ~ 0, there exists

b c L1

such that,

for almost

s ~ S, f(s,x) ~ rix[-b(s)

(Gp)(1 < p < ~): There exist r ~ 0 and

for all 1 b E L1

x e R n. such that,

every s E S, f(s,x) ~ rixlP-b(s)

for all

x ~ R n.

for almost

197

(G):

There exist S

and

< +~ ~

Ixl < r

~ p < ~):

There exist every

f(s,x)

(a~) : 3K.

f(s,x)

< aixl p + b(s) in

s

1 ~ p ~ ~,

X = LPn

If,

Y = Lq'n

and

implications conditions

~~

(a) ~

Let

(c) ~

(d) ~

equivalent

is

(c)

If,(y)

(d)

f

f*

~(L~,L~)-coercive

PROOF.

conditions

o_nn the

valid, with the

space has no atoms:

and proper on

L~;

P Ln;

and proper on

for every

y E L q"

n ~

the ~rowth condition

The convexity

(Gp),

and

If(x)

(Gq),

and

If,(y)

< +~

> -~

of

(b) ~ (a).

f(s,x)

in

x,

at least for almost

for (a) to hold in the case of an atomless

This follows

from T h e o r e m

3H.

Trivial.

In particular,

the function

for any finite

subset

{yl,...,ym }

of

~ ( s ) = maxi= m 1 f*(s,Yi(S))

and we have f*(s,y)

This shows that,

Yi'

~ a(s)

for almost

co{Yl(S),...,Ym(S)}. functions

are always

If

y E L~;

(c) ~ (b).

is summable,

(e)

the growth condition

s ~ S, is necessary space.

Consider

x E L~;

satisfies

REMARK.

b_~e ~ n o r m a l convex integrand

if the measure

If

is finite

f

Then among the following

(b)

(e)

for almost

Rn.

x

(i/p) + (i/q) = I.

__Is q(L~,L~)-inf-compactn

satisfies

that,

such

x ~ R n.

for all

If

f*(s,y)

every

> -b(s).

i b ~ L1

and

(a)

for at least one

q, Ln

(b) ~

all actually

for a t l e a s t one

f(s,x)

for all

(Weak Compactness). and let

measure

for almost

s ~ S,

is summable

THEOREM

and

a ~ 0

o_~n S • R n,

every

such that,

~ S,

f(s,x) (G)(1

b ~ LI

r ~ 0

when every

y r co{Yl(S) ..... Ym(S)}. s,

f*(s,')

Arguing in this way with various

it is easy to see that,

must be finite

Proceeding

is finite on

for all

for almost

choices

every

of the

s ~ S,

y E R n.

after this preliminary,

we show

If

is proper.

Fix

any

~ E L q and let F(s) = Bf*(s,~(s)). Then F is a measurable, n closed-valued m u l t i f u n c t i o n (Corollary 2X) and by the finiteness Just

established,

r(s) # g a.e.

Hence

r

has a measurable

selection by

198

Corollary

IC: there exists

~: S + R n

such that

~(s)

E ~f*(s,y(s)

a.e.

Then ~(s)-u(s) In other words, function.

< f*(s,y(s)+u(s))

- f*(s,y(s))

-

for every

Therefore

u ~ L~,

x ~ L p.

~'u

Since

n

for all

u ~ L q. n

is majorized by a summable

x(s)

~ 3f*(s,y(s))

a.e.,

we also

have f(s,x(s)) and hence

If(x)

= ~(s).~(s)

< +~.

- f*(s,y(s))

Of course,

it is trivially

If(x) ~ <x,~> - If,(~) and hence

If(x)

> -~

for all

(summable),

P x ~ L n,

for all

x E L p. n

true that

Therefore

If

is proper, as

claimed. Since

If

is proper,

a(Lq,LP)-l.s.c, n

and in particular

n

finite

convex functional

necessarily

it follows by Theorem

continuous

l.s.c, [21;

dual Banach space is t h e n w e a k * - c o e r c i v e we have

L~* = L~,

further ado. L nI • Lsihg

and

For

If, = If

is But a

on a Banach space is

7C ], and its conjugate [22],

[21].

For

on the

i ~ q < ~,

(Theorem 3C), so (b) follows without

q = ~, the dual Banach space can be identified with

as in Theorem

n

If,

in the norm topology.

having this property

everywhere

3C that

31, yielding

for the conjugate

functional

the

representation (3.16) where

If,(x,z)

= If(x)

JG (z) = sup{z(Y)l In fact, finite ,

If,

JG(Z)

= +~

throughout

for all

L q. n

are essentially

those of

nothing other than the (e) ~ (c).

Thus

Y ~ Lqn with

If,

throughout

If,

cluded by assumption.

If

the same conclusion)~in choose a finite set

convex,

every

all such

y

f*(s,y)

of

for

we have

y

all

this implies

y ~ Rn

with

observing

{yl,...,yn } IY[ ~ r.

E

L q.n

either

If,

in

m

Rn

T h e ~ since

(summable).

is finite

has been ex-

the following.

satisfy ~ maxi= 1 f*(s,y i)

If, is

If.

i < q < ~, +~

of

q = +~, we again get the finiteness

(and thereby hull contains

,

If, ~ -~; but the second possibility

any

r ~ O,

is being assumed

tells us that the level sets of

I ~(Ln,Ln)-Coerclvity

For the case where

or

If,

< +~}.

and the weak*-coercivity

is a convex functional, L~,

If,(y)

z ~ 0, because (3.16)

IrA(Y) _< allyll q + /bd~ < Since

+ JG(Z),

of

If,

Given

whose convex f*(s,.)

is

199

(d) ~ (G~)

(e).

Condition

is satisfied

is verified

by

(Gp)

f*,

by taking

is satisfied

at least

conjugates

question.

In the case

assertion

that for each

for

by

r > 0

9

if and only if

1 < p ~ ~(1 ~ q < ~);

on both sides

p = l, q =

f

of the inequalities

the exact dual of

there exists

this

b(s)

(G l)

in

is the

(summable)

such

that IYl ~ r This is implied and it implies sumption

by

(G~)9

(3.17)

To complete

< +~

~ <x 9

established 9

of

for every

of

that

(a) ~ (e)

for

If,

is continuous

> -~

in all cases q y 9 Ln9

we obtain

(d) ~ (e).

(e) ~ (d)9 we need only invoke

~

nonatomic,

at

0

the

(a) and in particular

p = I9 q = ~.

has a nonempty

the

functional

on

We have

Therefore 9

topology

If

and

(a) implies

T = T(Ln,L~) 9

and

set

T-interior

and consequently

3C.

in the Mackey

G = {y 9 Lnl

L~

If,(y)

containing

< +~} 0.

Therefore,

which is bounded

corresponds

above

to an element

then, by convexity~

G

of

is all of

every nonzero

on

L I. n L~

9

G

is

T-contlnuous

If there

is no such

and we are done

(in

n 9

view of the additional < +~9

Therefore 9

The as-

yields

for all

(e) implies

to each other by Theorem

in partlcular, the convex

If(x)

y 9 L n.

x 9 LPn

paragraph,

If.

conjugate

functional

< +~

this with the facts just mentioned

If,

linear

~ b(s).

for some

- If(x)

the verification

fact already properness

If,(y)

If(x)

If,(y)

f*(s,y)

as seen at the end of the preceding

in turn that

in (d) that

and combining

~

fact that

and at least suppose

(3.17)

one such

x

holds

for any

is assumed

x 9 L 1 with n in (a) to exist).

1 0 ~ x 9 L n'

(3.18)

~ > B > sup. yeG

Let (ykl k = 1 9149 be a maximizing (3.18). Now define (yk I k = 0,i,...) start, yO ~ 0. Given yk-I let I Yk(S) yk(s)

=

yk-l(s)

Then

yk E G

tive

and nondecreasing

for all

k9 in

if

~(s).Yk(S) if

sequence for the supremum in recursively as follows. To

~ ~(s).yk-l(s),

~(s)-Yk(S)

< ~(s).yk-l(s).

and the expression k, with integral

~(s).yk(s) bounded

is nonnega-

above by

200 according

to (3.18).

Denoting by

e(s)

the limit as

k + ~,

which

exists a.e., we have (3.19)

fedu = llm k+~

In fact,

then

(3.20)

y c G ~(s)-y(s)

for if

y

were a function

a contradiction for

= sup<x,y>. yEG

k

to

~ad~

sufficiently

contradicting

a.e.,

this implicatlon~we

would get

being the supremum in (3.19), by considering,

large,

the function

i y(s) y'(s)

< a(s)

if

y' ~ G

x(s)-y(s)

defined by

> ~(s)'yk(s),

= yk(s)

if

x(s).y(s)

< ~(s)-yk(s).

We thus have (3,21) Since

G c H = (y r G

has a nonempty

H~

the polar set

in

L~ nl

x(s)-y(s)

T-interlor,

Ln1

is

< a(s)

so does

a e.}.

H,

and it follows

~(L},Ln)-Compact. n

that

Applying C o r o l l a r y 3F

with F(s) = {y ~ Rnl ~(s)-y ! m(s)}, one finds that = I k(s)e(S)W(ds) S

sup<x,y> y~H

if

x(s) = k(s)~(S)

with and otherwise

the s u p r e m u m is

functions

of the latter form with

x

+~.

Thus

/ k(s)Ix(s)Iw(ds)

H~

< ~

k(s)

consists of all m e a s u r a b l e

and

f k(s)a(s)w(ds) S

S (where

a(s)

(3.21),

it is in particular

> 0).

Actually,

since

G

is a T-neighborhood

a neighborhood

and there exlsts, therefore, some

e > 0

> 0 a.e.,

of

0

such that

of

~ 1 0

in

in the norm topology, eI~(s) I ! a(s) a.e.

Hence

-1

/ k(s)~(s)~(ds)

! 1 ~ / k(s)l~(s)l~(ds)

S

< e

,

S

and we see that

H ~ = {Ix[ We claim the nonatomic. set

T

Indeed,

with ~

l(s) ~ 0 a.e.,

1 O(Ln,Ln)-Compactness if

~

0 < ~(T)

< ~,

I~(s)l ~ ~-i

and

fl~d~ ~ i}.

of this set is impossible with

is of this nature, we can find a measurable together with number g ! e(s) ~ ~-I

6 > O,

for all

such that

s e T.

201

The mapping

k + kx

is then an isomorphism between the space

(which Is necessarily L1

n'

the unit ball of implies, so

Infinite-dlmenslonal)

with the property

If

and

therefore

for

W

If.

B+

is weakly

implies

(3.22)

nonatomic,

(c) ~ (b)

compact

in

This

LI(T,aw)

If,

1 < p ~ ~, 1 ~ q < |

to each other by T h e o r e m

is continuous for some

is nonatomic, above shows

at

a > 0

IlYll ~ e ~ ~

part of

1 O(Ln,Ln)-Compact.

is relatively

conjugate

In particular,

Because

and a certain subspace of B+, the nonnegative

itself,

is finlte-dlmensional.

(a) ~ (e)

[22].

LI(T,ew],'"

Inadmissibly~that

LI(T,a~)

have

that the image of

Ll(T,~)

0

Again,

3C, and

we

(a)

in the norm topology

and

B ~ R,

[21],

we have

If,(y) ~ B.

the maneuver

at the beginning

(even if the elements

Yi

of the proof of

in that argument

are

U

required to satisfy almost

every

generality (3 23) 9

IlyIN ~ e)

s e S.

that

fi(s,y)

For this reason,

is finite in

we can suppose, wlthout

y

for

loss of

in the rest of the proof, that actually f*(s,y)

is finite

for all

s ~ S

and

y c Rn .

Define (3 9

8(s,n)

= Inf{-f*(s,y) I (lyl/e) q ! n}

(3.25)

8(s,n)

< -f*(s,0)

(3.26)

e(s,~)

= +~

for

(s,n)

and

b ~ L i1

~ S x R,

so that

if

if n
0, 0

It will be enough to show the existence

of

c c L1

such

that (3 9

e(s,n)

> c(s)n - b(s)

since then by the definition f*(s,y)

(3 9

a.e., of

~ Ic(s)In + b(s)

e

we will have

whenever

(lyl/e) q ~ n,

and consequently f*(s,y)

~ alyl q + b(s)

for

a = llcIl~/c q,

We shall obtain this existence of integral

functlonals

to

To see the normality (3.28)

8(s,~)

by applying some of the p r e c e d i n g 1 I e on L 1.

of

e, we look at the r e p r e s e n t a t i o n

= inf r y~R n

where -f*(s,y) r

if

= +~

otherwise 9

(IYl/~) q ~ n,

theory

202

We have

r

itself normal by

al by 2C) and the indicator does not depend on (3.25) and (3.26)

s;

2M,

r

is the sum of -f*

of a closed set of pairs of

hence

that

because

Ie

e L1

on

(3.29)

Ie(n) ! -If,(0)

(3.30)

Ie(n) = +~

is normal by 2R.

(n,Y)

(normthat

It is evident

from

has the properties

< +=

for all

for all

n ~ 0.

for all

n 5 0

n ~ 0,

We claim next that (13.31)

le(~) ~ -8

with

For,

this were violated by a certain

f~d~ ! I. I

suppose

n 9 L~.

The set

r(s) = arg min r y~R n is closed and nonempty is a measurable a measurable

by the continuity

multifunction

function

by

2K

y: S § R n

of

and

f*(s,y) 2P.

such that

in

y,

and

r

Hence by 1C there is

y(s)

~ F(s)

for all

s,

i.e. -f*(s,y(s))

= e(s,q(s))

(lY(S)I/e) q < n(s) The latter implies

y E Lq n

and

for all

for all

s,

s.

JJyJJ m< c,

since

fnd~ < i;

the

former then yields If.(y) contrary

to (3.22).

Now for (3.32) Then

let

) = max{e(s,n),

is another normal

I e, the conditions

(3.29),

considering

,k

(3.30),

measure

U

(by 2L), and Iek

(3.31).

0.

by (3.33), Ie**(n)

We also have

< -8 ~ n

8**(s,n)

~ 0, ~nd~ ~ i.

= +=

for

Ie**(n)

= +=

q < 0 if

by

(3.32),

and hence

n ~ o.

This shows that Ie**(n)

~ -8

q ~ L1

for all

with

Ilqll

_
,.T -

n(E)

-

2,"

,

+

%~ * f lu-*l dr

avec

* I1 s'aglt que

0a

~ present

--~ O

dans

d e L e b e s g u e on v o l t

de passer

Ll(fi), alors

et

~ la limite

a

dx - presque

0

darts ( 2 . 1 8 ) . Par application

I1 est

clair

du Th~or~me

que

y [o., + -A o'j axoa

--~

partout.

.

n'(c>=fo

Igldx,,t

o,(r

c~=s0

c.

E D'autre (~r ,

0a(x)

applicaglon

Finalement

et

part --~

[0 ( x ) ] ~< 1 01(x)

pour tout

x

9

e t COmmp 010~

, y x e 0 c , c'est-~-dire

,-presque

du Th~or~me d e L e b e s g u e n o u s p e r m e t d e v o i r

(2.18)

le r6sultat

-

01

eat

partout.

continue

que

donne ~ la limlte

suit puisqur

q'(c)

+ 2 ~ ( c ) § 3E

~

0

qua~

c

--~

sur

Une n o u v e l l e

0 .

225

2.4. Calcul de

Be .

On consld~re de d6terminer

Be(u)

Soit donc

(~

9

comme une fonctionnelle

sur

L2(G)

et on se propose

quand cet ensemble n'est pas vide.

u ~ BV(~) ~ L2(~) , tel que

Be(u) ~ ~ , et soit

~ ~ ~e(u)

L2(a)) , i)

La d~compositlon de Lebesgue de

par 6 t r i t e

,

s>O . Avec (2.14), on voit que

I

we~(~)

~tant ~crite en (2.13), on commence

e(u+sw) ~ e(u) + s(~,w) ,

(2.19)

V wc.~(~)

Vu

que

. On fair

dx > (~,w)

/l+(~+sVw)2 - I~+~2

s~O

, et par le Th~or~me de Lebesgue on obtient

ce qui entralne que

(2.20)

V

~

~ L2(~) ,

et =-V

(2.21)

8 l+g

Cela entralne d6j~ que

ii)

~e(u)

contient un point a u plus

On ~crit ensuite (2.19) pour s>O

O[,w) = - I

w.V

@

et

w ~ ~(~)

. On note que

dx

l~+g 2 = (par la formule de Stokes g~n~ralis~e

[12] (I))

/1+g2 (1) D'apr~s [12].[ 8] et (2.20),

@'

a

u a seas eomme ~1~ment

de

L=(r) .

226

D'autre part quand

f o ~

o = 0

fow

dr

J

Jr

s

Sgn(u-~) si

,

lu+~-*l - lu-~l dr ~

)r o~

s~O

(o = S g n ( u - ~ ) )

w = u - r = O)

Passant ~ la limite

si

c

u - ~ O ,

u-~=O,

= Sgn w si

.

s~O

,dans

s-l[e(u§

- e(u)]

>i (~,w)

,

on obtient alors

d'ofi

(2.22)

~.v

-

~Sgn

(u-%)

.

~l+g 2

iii)

II nous reste ~ ~crire

(2.19) pour

w ~ BV(~)

quelconque.

Pour garantir

(2.19) il faut et il suffit que

Inf

(2.23)

e(u+sw)

Soit

Vw = h dx + ~'

p' ~ p~ + ~" , p ~ (~

et

p"

- e(u) = lim e(u+sw)

S

S>O

, la d ~ c o m p o s i t i o n

Ll(fl,~)

- e(u) ~ (~,w)

de Lebesgue de

, la dficomposition de Lebesgue

sont ~trang~res

,

Y w e BV(fl) 9

S

8~0

Vw , et solt de

~'

par rapport

et toutes deux sont potties par u n ensemble

dx -

n@gligeable). On a

e(u). f

s

fl

s

+ r

-

.§

s

(2.24) o ~

Sgn(u-~)

0

f

,

,h

on obtient

a V/l+g2 ,

dx + f

Vw~BV(pO

fl

I '""

s

lu+s--~l- lu-*l dr

Jr Quand

f

s

la condition

o.W + f

r

a

pill + ; ,p.,l > _ f a

a

V

g

l+~g2

W dx

.

II est facile de voir que r~ciproquement,

(2.20)-(2.21)-(2.22)-(2.24)

sont

227

suffisantes pour que

Cas partieulier

~e(u) # ~ .

u ~q~l(fl) .

Si

PROPOSITION 2.1.

u e~l(fl) 0 L2(fl) ,

~e(u) ~ ~

8i et seulement si (2.20) et

(2.22) ont lieu. Darts ce vas

~e(u)

~ E ~ 2 ~

~ = O, comme la fonction

: Dans ce cas

elle est dans

aV

Alors

Ll(fl,~ ')

~

(2.24)

Remarque 2 . 1 .

$.v

w dr + Jn

automatiquement varifia

Si

u ~2(~)

, alors

~

~e(u)

On suppose que

born~e sur

fl , di~ f ~ L2(fl) , et soit

Damonstration :

est continue et borage,

;

Par prolongement de

w

en ~vidence une suite de foncti6ns

m

donna par

ont lieu.

(2.21)

w

.

f~(fl)n

est

w ~ BV(~) N L2(fl) . Alors

r f.~w dr+

m

Vw

= r , ~

:

.

est un ouvert de classe ~2 9 que

- Jnv f.w dx = -

W

dx +

quand (2.20)-(2.21)-(2.22)

LEMME 2.2.

(2.25)

~

~ donn~ par (2.21).

et on peut intagrer par pattie (cf. Lemme 2.2. ci-apr~s)

w dx = -

est

se r~duit d u n seul point

sur

Rn

~ ~(fl)

I f. V w d ~ . et ragularisation, on peut mettre

, tels que

,

~

W

L2(~)

--'~

Vw

pour la topologie faible ~toile de l'espace

des mesures born~es. Pour t o u t

m , d'apr~s

[12] ,

fl

m et ~ la limite

m

--~

U_~n exemple o__~ u

est

PROPOSITION 2 . 2 .

Soit

de dimension s~r ~ ,

n -i

m

~ , on obtient (2.25).

q~l

par morceaux.

~ = ~oU[

. Supposons que

en sorte que

~2

~=

[u]~[,

U ~I

"

O~ [ = 3~~

est une vari~t~ de classe ~ 2

u 6 ~ 1(-rio) k)~1(n I) [u]

le saut

Ul-U ~

et admet des discontinuit~8 de

u

sur ~

. Alors

228

ae(u)

si et seulement si ( 2 . 2 0 ) - ( 2 . 2 2 )

~ #

~"

(,~.,~)

Darts ce cas,

~e(u) = ~,

D~monstration

:

et la condition suivante

,s~ [~] d[ p.p.

o~

% e s t donn~ p a r

Ii s'agit d'interpr~ter

On int~gre par parties

~-

utilisant

-I

t rdalis~e~:

~ (1)

(2.21,).

(2.24).

le Lemme 2.2

:

,.,) ~

X§ 2 g

+

i = 0

ou

devient

r

_ Z z _ L w dr

I , E~ = + I , E 1 = - i , ~

orient6e

de

~o

vers

~I " Alors

(2.24)

:

r

/i+g2

Comme

J iX§ 2

I."

-

+

la somme des trois premieres

arbitraire,

I Iu]l "

ce qui est gquivalent suffisante

2.2.

Pour eoZution

f u

(I)

Soit

donn6 dans

Ii est clair que

(2.24),

~

[~] aS p.p. sur

~,")

~ (2.27). pour

2.5. L__eeprobl~me

TI~OREbE

est positive

et que

p

a un signe

il faut done que

c2.27)

condition

int~grales

~'~

compte

(2.29)

(2.27)

est r~eiproquement

une

tenu de (2.20)-(2.21)-(2.22).

d'~volution.

un ouvert borng de claese ~ 2 L2((O,T)

x ~)

et

u

donn$ dane

quand

[u] > 0

o et une 8eule du probldme d r~volution

(2.28)

~: ,

du + a e ( u )

u(O)

= f

et

~

donng dane

,

= uo ,

et (I)

Cela entraEne que

(2)

Ce r~sultat est repris et complgt~ en ~ii].

g.v = + ~ ou

- ~

~(r) j

L2(~) j il existe une

ou

~a IVv[ 2-

- mes(~) m(f) m(v) + 1 lv-122(n)L

Lz(a)

Par l'in~galit~ de Poinear~, la norme de

HI(n)

est ~quivalente

IvulL2(~ ) + lm~u) l 9 Supposons que

llvjII ~

+" ; on est amenfi ~ distinguer trois eas :

a)

m(vj)

reste borna, et done

IVvjl 2(n )

--+ + -.

Alors

J(vj)

est born~ inffirieurement par une quantitfi qui tend vers

+~et

le

r~sultat suit. b)

Si

m(vj)

--+ + - , alors par (3.27), J(vj)

et donc

c)

J(vj) Si

--~

m(vj)

--~

+ =

~

puisque

-| ,

- mes(n)

m(f) m(vj)

re(f) ~ 0 .

alors

mes(~) m(vj)ffi f~ (vj)+ dx- ;~ (vj)_ dx >- I~ (vj)_ dx , Im(vj)l ~ (mes ~1/2 i(vj) IL2(~) si bien que

I(vj)_IL2(~)

-'~ + =

et

I(vj)_IL2(~ ) > I

pour

j

asse, grand. On

235 utillse (3.21) qui entralne (cf. Remarque 3.1) :

(3.28)

pour

lUIL2(Q) ~ c {[Vul2(~)

]u IL2(a ) > I et

lU_lL2(~ )

+

[Vu[2(~)

3/2}

n = 2 ou 3 . I1 en r ~ a u l t e que 1

2

J(v) > ~[Vv]m2(~) a (vj)

+

eat minor6 par une quantir

_

1

IflL2(~) ]VlL2(n) + ~ Iv_[~2(~) qui tend vers +

236

REFERENCES

[1]

E. Bombieri, E. de Giorgi, M. Miranda

Una Maggiora~ione a priori relative alle ipersurfici minimali parametriche. Arch. Rat. Mech. Anal., 32, 1969, p.255-267. ~]

H. Brezis

Op~rateurs maximaux monotones. North-Holland, Amsterdam, 1973. [3]

H. Brezis

Int@grales convexes dans les espaces de Sobolev. Isr. Journ. Math., 13, 1972, p.9-23.

[4]

I. Ekelan~,

R. T e m a m -

Analyse convexe et Probl~mes variationnels. Dunod, Paris 1974, Traduction anglaise, North-Holland-Elsevier, Amsterdam-New York, 1975.

[5]

L.E. Fraenkel, M. Berger -

A global theory of steady vortex rings in an ideal fluid. Acta Mathematica, 132, 1974, p.13-51. [6]

E. Gagliardo -

Caratterizzazioni delle trace sulla frontiera relative alcuni olassi di funzioni in n variabilili. Rend. Semin. Mat. Padova, 27, 1957, p.284-305. [7]

P.R. Garabedian

-

Partial differential equations. John Wiley and Sons 1964.

[8]

C. Jouron -

R~solution num~rique du probl~me des surfaces minima. Arch. Rat. Mech. Anal., ~ paraltre.

[9]

A. Lichnewsky -

Principe du maximam local et solutions g~n~ralis$es de probl~mes de type hypersurfaces minimales. Bull. Soc. Math. France, 102, 1974, p.417-434. [iO] A. Lichnewsky -

5ur le comportement au bord des solutions g~n~ralis~es du probl~me non param~trique des surfaces minimales. Journ. Math. Pures et AppI., 53, 1974, p.397-425. [II] A. Lichnewsky - R. Temam (A paraltre). 12] J.L. Lions - E . M a g e n e s -

Probl@me aux limites non homog~nes et Applications. Dunod, Paris 1968, Springer Verlag, 1970. [13] M. M i r a n d a -

Comportamento delle successioni convergenti di frontiere minimali. Rend. Sere. Univ. di Padova, 1967.

237

[14] M. MirandaUn teorama di Ezitenaa e Unicitd per il problema dell area minima in n variabili. Ann. Scuol. Norm. Sup. Pisa, 19, 1965, p.233-249, [15] J. SerrinThe problem of Dirichlet for quasilinear ellipti= differential equations with many independant variables. Phil. Trans. Royal Soc. London, A.264, 1969, p.413-496. [16] R. TemamSolutions g6n~ralis~es de eertaine8 Squations du type hypersurface8 minima. Arch. Rat. Mech. Anal., 44, 1971, p.121-156. [17] R. TemamOn the theory and numerical analysis of Navier-Stokes equations.

North-Holland-Elsevier Amstardam-NewYork, 1976. [18] R. TemamA nonlinear eigenvalue problem : The shape at equilibrium of a confined

plasma. Arch. Rat. Mech. Anal., ~ paraltre.

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