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
<
by s e t t i n g
x -< y
P . Each cone
P
y - x E P . The r e l a t i o n
iff
is r e f l e x i v e , t r a n s i t i v e , antisymmetric (that i s , an ordering), and compatible with the l i n e a r structure and the topology of
ordered Bo~ach space (OBS), usually denoted by space
E together with an ordering
cone o f
E . We w r i t e
x < y
iff
E . By an
(E,P) , we mean a Banach
_< induced by a cone
P , the positive
y - x E ~ := P ~ { o } , and x < < y
0
y - x ~ P = i n t P . The elemtents of
iff
P are called positive.
The norm of an OBS i s c a l l e d monotone i f and ~mi-monotone i f there exists a constant
o ~ x -< y
implies
~ such t h a t
Uxll < llyll
o -< x _< y
implies
llxll_< = llyll 9 The p o s i t i v e cone is called normal i f the norm is semi-monotone. P is called total i f
= E and generating i f
~
P - P = E , Clearly,
every cone with nonempty i n t e r i o r is generating. L a s t l y , the order interval Ix,y]
[x,y]
is defined by
:= { z E E I x_< z < y }
= (x + P) n ( y -
p) .
Hence every order i n t e r v a l is closed and convex. The f o l l o w i n g proposition contains some important characterizations of normal cones. For proofs we r e f e r to [ 16, 18, 27].
(1.1)
Proposition: Let
equivalent:
(i)
P is .orma~
(E,P)
be an OBS. T ~ n t~e following statements
(ii)
every order interval i8 bounde~"
(iii)
tl~re exists an equivalent monotone norm.
In the following we give a series of examples of 0BSs. For simplicity we r e s t r i c t our considerations to some of the most commonly used function spaces together with t h e i r natural o r ~ n g s . These orderings w i l l be used throughout the remainder of this paper. F i r s t we observe that, t r i v i a l l y , every Banach space E can be i d e n t i f i e d with the 0BS (E,{o})
whose positive cone, the t r i v i a l cone,
is even normal.
(1.2) E m i l e (T~ Real Line): The real l i n e , absolute value as norm and
R+
R , is an 0BS with the
as positive cone. Clearly,
+ = i n t R+ r # and, since the norm is monotone, R+ Observe that
R+
(1.3) E x i l e
is normal.
induces the standard natural ordering in
(Products of OBSs): Let
R
(Ei'Pi) ' i : 1. . . . . N , be
OBSs. Then the product Banach space E := E1 x
...
x EN is an 0BS
with the natural (or canonical) ordering induced by the cone P := P1 x . . . x PN " The cone P is normal or has nonempty i n t e r i o r i f f each of the
Pi's
has the respective property. In particular, the Euclidean
N-space, RN, is an 0BS with the natural ordering whose positive cone, R+N := (R +)N , is normal and has nonempty i n t e r i o r .
(1.4) E=an~le (Spaces of Continuous Bxnctione): Let and l e t
X be a compact Hausdorff space. We denote by
space of a l l continuous maps f : X ~ E with the norm
(E,P)
be an OBS
C(X,E) the Banach
Ilfll := m a x { l l f ( x ) l l E I x E X}
. Then C(X,E)
is an OBS with the
natural ordering whose p o s i t i v e cone is given by C+(X,E) := { f E C(X,E) f f(X) c P} . Clearly,
C+(X,E) is normal i f f
nonempty i n t e r i o r i f f iff
B ~ d 9 In the l a t t e r case,
o
f(X) c P . In particular,
positive cone C+(X) := C+(X,~)
(1.5) let
P is normal, and C+(X,E) has
C(X) := C(X,R)
f E i n t C+(X,E)
is an OBS whose
is normal and has nonempty i n t e r i o r .
Exa~ole (Lebesgue Function ~oaces): Let (E,P) be an OBS and
(X,~,p)
be a o - f i n i t e measure space. For every p with
we denote by Lp(X,~,u;E)
1~ p s |
the Banach space of a l l (equivalence classes
of) strongly measurable maps f : X § E such that
"f"Lp
::
1/p
(
(with the usual modification for the case p = | ). Then Lp(X,~,u;E)
is
an OBS with respect to the natural ordering whose positive cone is given by L~(X,~,p;E) := { f E Lp(X,~,p;E) I f(x) E P for I t is easy to see that I ~ p< |
L~(X,~,u;E)
is normal i f f
then, in general, the i n t e r i o r of
a.a.
x ~ P}
P is normal. I f
L~(X,~,u;E)
is empty. In
P
particular, this is the case i f
X is a Lebesgue measurable subset of
RN of positive measure, and u is the Lebesgue measure. In this case we sinply denote these spaces by Lp(X;E) , and Lp(X) := Lp(X,~) . I t is easy to see that
L;(X)
is generating for every
p ~ [1,|
(1.6) Exa~le
(Sub~oaces and Continuous Imbeddings)= Let E and F
be Banach spaces such that natural injection
F is a vector subspace of
i : F § E , defined by
i ( x ) = x , be continuous.
Then F is said to be continuously imbedded in L e t (E,P)
be an OBS and l e t
Then (F,i-I(P))
E , in symbols, F ~ E .
F be a Banach space such that
is an OBS. Observe that, as a set,
Hence the ordering induced in
E and l e t the
F by
i-1(P)
ordering which F inherits as a subset of
F~ E .
i-1(p) = p n F .
is the same as the E . Hence this ordering is
called the natural (or canonical ordering) and we can use the same symbol
_< in
Clearly,
E and in i-l(P)
F .
is nontrivial i f f
~ n F * ~ , and i-1(P)
has
P is normal, then i-1(P)
is
0
nonempty i n t e r i o r i f
p n F * # . If
not necessarily normal. However, i f E , t h e n i'1(P)
F is a closed vector subspace of
inherits the normality from P .
(1.7) Example (Spaces with Order Uait Norms): Let with normal positive cone and suppose that u{x[-e,e] Ix E IR+}
(E,P) be an OBS
e > 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
x E P \ k e r T . Then Tx E P, which easily implies that
> o . Hence
= p-l
= p-I
> o
9
By means of this lemma i t is easy to obtain more detailed information about the spectral properties of almost strongly positive compact endomorphisms. The following theorem contains some of the most important of these properties. For further results we refer
to
[7| .
O
(1.13) Theorem: Let
(E,P)
be an 0BS such that
P ~ ~ , and let
T E K(E) be a . s . p . . Then
(i)
r(T)
(ii)
T
> o
.
(Hence the Xrein-Rutmc~ theorem applies.
possesses exactly one normalized positive elgenvector
to a nonzero eigenvalue
(iii)
)
X . (Hence
X = r(T) .)
The equation
(1)
),x
-
hes f o r every
Tx
= y
y ~ P exactly one solution
> r(T) , and no solution i n If
X = r(T)
and
P\ker T i f
x ~ P if
~
y ~ (P u ( - P ) ) \ k e r T , then (1) has
l]
no solution at all.
SE K+(E) and S <- T ,
(iv) If
~d/tlon,
then r(S) _< r(T) . I f , in o (T - S)(P) c P \ k e r T , and
S is a.s.p.,
O
pn
ker T = #,
then
o < r(S)
< r(T)
.
O
Proof= ( i ) some
If
x E p% ker T , then
Tx E P . Hence Tx - o x E P f o r
e > o . Consequently, by [19,Thm.6.2]
In the f o l l o w i n g we denote by T ~ to the eigenvalue
r(T)
r
,
r(T) ~ e >
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
=
Hence
x = r(T) .
Tx = ~x
f o r some ~ ~ o
and some x E p . Then
x > o . Consequently,
=
= r(T) , t h a t i s ,
(~ - r(T))
= o .
O
Suppose t h a t ar
o
r ( T ) y = Ty . Since
such t h a t
x E p , there e x i s t s a real number
x + a y e @P . Suppose t h a t
r ( T ) ( x + ay) = T(x + ay) = o , t h a t i s , Hence i f
x
and
y
and, consequently, choice of (iii)
x
x +aye and
y
ker T . Then
are l i n e a r l y dependent.
were l i n e a r l y independent, then
x + ~yE P\ker T
r ( T ) ( x + ay) = T(x + ~y) E ~ , which contradicts the
a If
x > r(T),then
~-k-ITk E L+(E) . This implies
(~ - T) -1 = k=o
the f i r s t
assertion. By ( I ) , (~ - r(T))<@,x> = <@,X - Tx> = <@,y>
,
and the remaining part o f the assertion f o l l o w s . O
(iv)
If
r(S) > o , then there e x i s t s an element y E p
r(S)y = Sy ~ Ty . Hence r ( S ) < r Consequently,
<@,Ty> = < l ~ , y >
such t h a t
= r(T) <~,y>
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
implies
f(x) < f(y) , and f
strongly increasing i f
x
implies
f(x) << f(y) . The map f
decreasing or strongly decreasing i f
-f
is called is called
is increasing or strongly in-
creasing, respectively. Suppose that point
x
E = F , and l e t
Y be a nonempty subset of
X . A fixed
of a map f : X ~ E is called a minimal (or maximal) fixed point
[in Y] i f every fixed point y of
f
{in Y] satisfies x ~ y
(or
y~x). The following simple theorem contains the basic result about the existence of fixed points of compact increasing maps.
(2.1) Theorem: Let interval in ~ c h that point
x
(E,P)
E . Suppose that
y <- f('y)
and
and a m ~ m a l
be an OBS and let
[y,~]
be a nonempty order
f : [y,~] -~E is an increasing compact map
f(~t) < ~ ~ x e d point
.
T~n
f
possesses a minimal ~ x e d
~ . 4oreover, ~ =
lira fk(~) k§
and
18
R = lim
(fk(~))
fk(~) 9 and the sequence (fk(~)) is increasing and
is decreasing.
Proof= Since
follows that
f
f
is increasing with ~ ~ f(~)
maps | y , ~ |
and f(~) ~ ~ , i t
into i t s e l f . Hence the sequence (fk(~))
is well-defined, and i t is easily seen to be increasing and r e l a t i v e l y compact. This implies the convergence of the whole sequence (fk(~)) towards i t s only l i m i t point ~ E [~,~] . Since f = lim fk(~) = f ( l i m fk(~)) = f(-~) . I f of
f , then, by replacing ~
that ~ E { ~ , x ]
. Hence ~
by x
x
is an arbitrary fixed point
in the above argument, i t follows
is the minimal fixed point of
The assertion concerning the maximal fixed point argument
f : P ~ P
minimal
f
in
|y,~] .
R follows by an analogous
9
(2.2) Corollary: Let
let
is continuous,
(E,P)
be an OBS with normal positive cone, and
be a completely continuous increasing map. Then
~ x e d point iff
f
iff tl~re exists an element
f
has a
has a fixed point at all. This is the case ~ ~ P
such that
f(~) -- ~ .
Proof: The assertion follows from the theorems (1.1) and (2.1) and the fact that
f(o) ~ o
9
I t should be observed that we do not assert the existence of a maximal fixed point i n P . Moreover, the existence of a fixed point in the order interval
[o,~]
is an immediate consequence of Schauder's fixed point
theorem. However, for many applications (cf. the remainder of this paper) i t is of great importance that there exists a minimal fixed point. Lastly,
19
i t should be observed that the minimal fixed point can be computed i t e r a t i v e l y since ~ = lim
fk(o) .
k§
If
f(o) = o , then the Corollary (2.1) becomes t r i v i a l . In this case
one is interested in the existence of positive fixed points. For this one has to find positive elements ~ < ~
such that ~_< f ~ )
and f(Y) -< Y 9
To treat this problem, i t is natural to invoke the properties of the derivative at Let
o
(and "far out", that is, the asymptotic derivative).
(E,P) be an OBS. For every
A subset D c E is called a
p> 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
f ( x + ~h) >> 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) << x - Th
if
r(f'(x)) > 1
and
Proof= I f follows from [ 18,Chap.3] that
Theorem (1.13),
~ := r ( f + ( x ) )
f+(x) e K(E) . Hence, by
is positive and an eigenvalue o f
f+(x)
O
possessing an eigenvector h e p . Moreover, + r ( x + Th) , where ~ - I r ( x + Th) -~ 0
as
f ( x + ~h) = f ( x ) + ~f+(x)h +
~ § o+ . Consequently,
f ( x + Th) - (X + ~h) = T [ (~ - 1)h + ~ ' l r ( x + Th) ] O
and since
I~ - 11h e P , there exists
~~ ~
such that
o < 9< c
implies (x-
1)h + T ' I r ( x + Th) >> o
if
~> I ,
( X - 1)h + ~ - I r ( x + Th) <
if
~< 1 .
and
This proves the f i r s t part of the assertion. The second part follows by an analogous argument
9
The following theorem is a simple application o f the above results. For s i m i l a r applications we refer to [ 6 ] .
21
(2.4) Theorem= Let << ~ , and let f~)
= y
Then
f
be an 0BS such that
f : [~,~] -~ E
~
such that
~ r ~ . Suppose that
be an increasing con~oact map such that
f(~) _< ~ . "4oreover, suppose that
and
derivative at
(E,P)
f
has an a.s.p, right
r(f+(y)) > 1 .
has a maximal fixed point
s
~ >> ~ . 'r
and
s is
the l i m i t o f the decreasing sequence (fk(p)) .
O
Proof: By Lemma (2.3) there exists a pair
(~,h) E ~ + x p
such that
O
fCy+ ~h) > > y + Th f o r every
T E (o,e) . Since
a number T e (o,e)
~ < < y := ~ + Th << ~ . The assertion now
such that
~ - ~ E P , there exists
follows by applying Theorem (2.1) to the o r d e r i n t e r v a l
Let
(E,P)
9
be an 0BS whose positive cone is normal and has nonempty
i n t e r i o r , and l e t
D c E be an open subset o f
is completely continuous and increasing, that and that the (Fr~chet-)derivative f ' ( x ) suppose that
{Y,Yl
E . Suppose that x
f : D~ E
is a fixed point of
f ,
exists and is a.s.p.. F i n a l l y ,
r ( f ' ( x ) ) < 1 . Then the inverse function theorem, applied to
the map id - f
at the point
x , implies that
x
is an isolated fixed O
point o f
f . Moreover, by Lemma (2.3) there exists
(T,h) E ~ + x p such
that := x - Th < f ( x - Th) ~ f(x + Th) < x + Th : : ~ , and
T can be chosen so small that
order interval point
x
x
is the only fixed point of
f
in the
[ y , ~ ] . Hence i t follows from Theorem (2.1) that the fixed
can be computed i t e r a t i v e l y by means o f the i t e r a t i o n procedure Xk+1 = f(xk)
In fact, the sequence (fk(~)) decreasingly towards
x .
,
k = o,1,2 . . . . . converges increasingly and
(fk(~))
converges
22 The next proposition gives a partial converse to the above result.
(2.5) Proportion: Let
be on OBS with
(E,P)
be a nonempty order interval in
E
and let
y S f(~)
increasing and compact with
0
P ~ d 9 Let
[~,~]
f : [~,~ ] ~ E be strongly
and
f(~) -< ~ 9 Suppose that
f
has an a.s.p, left (resp. right) derivative at the minimal (resp. maximal)
fixed point ~
(resp. ~ ) o f
r(f+(~)) _< 1 i f
f()) <
9
. Then r ( f ' ( ' ~ ) ) < 1 i f
and
and ~ follows from Theorem (2.1). Since
is strongly increasing, ~ < f(~)
consequently,
~ < f(y)
.
Proof: The existence of ~ f
f
implies that
f(~) << f2(~)
and,
~ <<
Suppose that
r(f~(-x)) > 1 . Then Lemma (2.3) implies the existence of 0
an element h e p such that
~<<~-
h and f ( ~ -
Theorem (2.1), applied to the order interval existence of a fixed point
~ e [~,~ - h]
h) < < ~ -
[ ~ , ~ - h]
h . Hence
implies the
which contradicts the
minimality of ~ . Hence r(f~(~)) g I . The proof of the remaining assertion is similar
9
The above considerations show that, roughly speaking, each fixed point x , which can be computed by monotone iterations, has the property that r ( f ' ( x ) ) < I . Conversely, every fixed point
x with
r(f'(x)) < I
can be
computed by monotone iterations (from above and from below). These facts suggest the following definitions= A fixed point stable i f
r ( f ' ( x ) ) < I , weakly stable i f
r(f'(x)) > 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 << 91 and Y2 <<72 " Y2 ~ x~ ~ 91 "
follows from Theorem (2.1), applied , respectively. Let X := [~1,92 ] .
25 Then X , XI , X2 are nonempty convex closed subsets o f X1 u X2 c X and XI n X , X1 , X2 XI
satisfies
U1 , in in
X2 = #
. Moreover,
f
E such that
maps each o f the sets
into i t s e l f . Since the maximal fixed point Xl << Yl ' i t follows that
X , and f
X . Similarly,
X1
R1 o f
f
in
has nonempty i n t e r i o r ,
has no fixed point on the boundary XI~U 1 o f X2
has nonempty i n t e r i o r ,
no fixed point on the boundary X2\U 2
of
U2 , in
X2
in
X , and f
X1 has
X .
Consequently, by the a d d i t i v i t y property o f the fixed point index, i ( f , X \ ( U I u U2),X ) = i ( f , X , X ) The permanence property implies excision property gives
2 ~ i(f,Uj,X) . j=l
i(f,Uj,X) = i(f,Uj,Xj)
, and the
i ( f , U j , X j ) = i ( f , X j , X j ) . Hence
2 i ( f , X \ (UI u U2),X ) = i ( f , X , X ) - j =sl i ( f , X j , X j ) . Let
xo E X be a r b i t r a r y and define a compact map h : {o,1] x X-~ X by
h(k,x) := (I - k)x o + kf(x) . Then, the homotopy invariance and the normalization property imply
i ( f , X , X ) = i(xo,X,X ) = 1 . S i m i l a r l y ,
i ( f , X j , X j ) = 1 and, consequently,
i(f,X~,(U 1 u U2),X ) = -1 . Hence the
assertion follows from the solution property
(2.7)
f
Corollarzj= Let t ~
hBpot~eses of Theorem
has at least three fixed points
Proof:
9
x I , x2 , x3
This follows from the fact that
a maximal f i x e d point in the order i n t e r v a l
f
(2.6)
be ~a~isfied. Then
such that
x I < x2 ~ x3
possesses a minimal and {y1,~2 !
9
I t should be observed that the f i r s t set of hypotheses of the above
.
26
RI << 91 and 72 << ~2 proviaea f
theorem automatically implies that
is stronglg increasing. Moreover, i t suffices obviously to suppose that
72 ~91
instead of assuming that
Suppose i t is only known that
91 <72
f
maps one order interval into i t s e l f ,
but that the minimal and the maximal fixed point are distinct. Is i t then true that
f
has a third fixed point inbetween? The following theorem
gives an answer to this question. 0
(2.8) Theorem: Let << 9 that
be an OBS such that
f : {7,9] -~ E
and let
y < f(~)
(E,P)
and
P r ~ . S~ppose that
be compact and strongly increasing such
f(Y) < Y 9 Suppose, in addition, that the minimal
and the maximal ~ x e d point
-x and
a.s.p, right and left derivatives at
~ ~
are distinct, and that and
~
. Then
possesses at least three distinct fixed points provided and
r(f'(~))
% I
f
has
f
r(f~))
~ I
.
Proof= I t follows from Proposition (2.5) that ~ and R are weakly stable. Hence r(f~(~)) < I increasing,
and r(f~(~)) < I . Since f
is strongly
~ << ~ , and Lemma (2.3) implies the existence of points
91 , 72 such that ~<<91 <<72 << ~ , f(91) << 91 , and 72 << f(y2 ) . Hence the assertion foTlows from Theorem (2.6)
9
By specializing the above theorem to the one-dimensional case, i t is easily seen that the hypotheses concerning the spectral radii cannot be omitted.
27 The above results suggest the following question: Suppose that f : [yl,Y2 ] ~ E is a compact increasing map such that Y2 ~ f(Y2 ) " Does i t follow that
f
f ( Y l ) ~ Yl
and
has a fixed point? The following
simple example shows that the answer is "no".
Suppose that
P is normal and has nonempty interior. Let
be a.s.p, and suppose that
T has a positive eigenvalue
any y E E which is not in the annihilator of there exists a positive eigenvector
T E K(E)
~ < r(T) . Choose
ker(~-T~). By Theorem (1.13)
xo to the eigenvalue
~o := r(T) .
0
Since xo E P , there exists a positive constant
a such that
-a(~o-~)xo ~ y ~ a(Lo-~)xo . Let Yl := "aXo and Y2 := ~Xo and denote by f : [yl,Y2 | ~ E the restriction of [yl,Y2 ] . Then f
~-IT + y
to the order interval
is compact and increasing, and i t satisfies the above ine-
qualities. However f
has no fixed point since, by the Riesz-Schauder theory
x - Tx = ~y has no solution at a l l .
i(2.9) Notes and Remarks: I t is well-known (e.g. {18J ) that the continuity hypotheses for
f
can be considerably relaxed i f "smaller"
cones are being considered. On the other hand, almost a l l of the above results remain true (with appropriate modifications) i f the compactness hypotheses are replaced by the assumption that
f
be a s t r i c t set
contraction (e.g. [ 3 ] ). For Lemma (2.3) see Krasnosel'skii {1 8] . The remaining theorems of this paragraph are due to the author.
28
3. Applications to Nonlinear Boundary Value Problems
I. Rlliptic BoundarF Value Problems= In the following we use the hypotheses and notations o f the Examples (1.14) and (1.15), and we denote by I a n o n t r i v i a l subinterval of f : ~ x I ~ ~
and
g : e x I § ]R
I~ . We suppose that
are locally Lipschitz continuous
functions. Then we consider nonlinear e l l i p t i c BVPs of the form
(1)
Au = f ( x , u )
in
~
B6u = 6g(x,u)
on
F
,
By a solution we mean a classical solution, that i s , a function u E C2(~) n C6(-~) such that and
B6u(x) = 6g(x,u(x))
(8 = o)
u(~) c I , Au(x) = f ( x , u ( x ) )
for
for
x E ~ ,
x E F . (In the case of the D i r i c h l e t BVP
we consider homogeneousboundary conditions. This is done f o r
simplifying the presentation. I t is easy to generalize the following results to the case that
BoU = g , where g E C2+u(F) is independent
o f the solution ( c f . [ 6 ] ) . A function
u
is called a subeolut~on for the BVP (1) i f
u E C2(~) n C6(~) , u(~) c I , and Au(x) -< f ( x , u ( x ) )
for
x ~ ~
B6u(x) _< 6g(x,u(x)) f o r
x E r
,
A subsolution which is not a solution is called a strict subsolutian.
Supersolutions and s t r i c t supersolutions are defined by reversing the above i n e q u a l i t i e s .
(3.1)
Tl~orem: Suppose that
-v is a subsolution and
solution for the BVP (i) such that solution
-U and a maximal solution
~
is a s~oer-
-v <_ ~ . Then there exists a minimal ~
in the order interval
[~,~] .
29
Z>z~oo.f -
Without loss o f g e n e r a l i t y we can assume that
namely t h a t
I
is compact,
I = [min ~ , max ~I 9 Hence there exists a number ~ _> o
such that (2)
f ( x , ~ ) - f ( x , n ) > -m({ - n)
and (3)
g(Y,~) - g(Y,n) > -~(~ - n)
f o r every
xE ~ , yE r
, and every
Moreover, we can suppose that
~,nE I
ao + m > o
satisfying
and
~> n
Bo + m > o . Then the
BVP ( I ) is obviously equivalent to the BVP (A + ~)u = f ( x , u ) + ~u
(4)
in
~
,
(B6 + 6m)u = 6g(x,u) + a=u on r and ~
is a subsolution and ~
,
is a supersolution f o r (4). Consequently,
without loss of g e n e r a l i t y we can assume that f(x,.)
and
g(y,-)
are s t r i c t l y increasing f o r every
For every compact Hausdorff space and f o r every continuous function ding c a p i t a l l e t t e r ) by
(ao,Bo) > o
X let
and t h a t
xE~
and y E r
IC(X) := { u E C(X) I u(X) c I } ,
h : X x I -- ~R denote by (the correspon-
H the Nemytskii operator
H : IC(X) ~ C(X) defined
H(u)(x) := h ( x , u ( x ) ) . Then the BVP ( i ) takes the form Au = F(u)
in
~
,
Bau = aG(u)
on
r
,
and we can assume t h a t G : IC(F) -~ C(F) for elliptic
(ao,Bo) > o , and that
F : IC(~) - C(~)
and
are s t r i c t l y increasing. By using the r e g u l a r i t y theory
BVPs i t can be shown that the BVP (1) is equivalent to the fixed
point equation
u = @(u)
in
C(~) , where
~(u) := Sa(F(u),6G o ~(u)) By using the results of Paragraph i , i t is e a s i l y seen that
r
is defined
on [ ~ , ~ ] , increasing, and completely continuous. Furthermore, due to the p o s i t i v i t y of
S6 , i t follows that
~ _< ~(~)
and
r
_< ~ . Hence the
30 aSsertion is a consequence of Theorem (2.1). (For a more detailed treatement cf. [ 6 , 7 ] )
9
In practical cases i t remains to find a subsolution
~
and a super-
solution ~ such that ~ ~ ~ . In certain cases this can be done by inspection of the graphs of
f
and g . In fact, the following corollary
gives geometrical conditions which guarantee the existence of comparable sub- and supersolutions.
(3.2) Corollary= S~opose that there are functions b E cl-(r) number
with
(a,b) > o
(aa + ao,ob + 6o) > o for some
such that
~ >- o . Let there exist real numbers
such that
{ f ( . , ~ ) < xa{2 and
a ~ CU(~) and
~ < ~o(a,~b)
{g(.,~)_< xb{2 for every
and ~
to > o
with
l{l -> {o " ~"hen t~e BVP (1) has at least one solution.
Proof: The assumptions imply the existence of a positive constant such that, for every
~~ R ,
ef(.,C) ~ y + cxa{
,
cg(.,{) ~ y + cxb{
,
where E := sign E . I t follows from Theorem (1.17) that the l i n e a r BVP Au = xau + y
in
B6u = ~6bu +6y
on
has exactly one solution
~ ~ o . Hence
, F := -~
is the only solution of
the BVP Au = ~au - y
in
~
B6u = ~6bu -6y
on
r
Since i t is obvious that ~
,
is a subsolution and
is a supersolution
for the BVP (1), the assertion follows from Theorem (3.1)
31
I t should be observed that the above corollary imposes only one-sided growth restrictions for the function if
(ao,Bo) > o and f ( x , . )
and y E r
f(x,-)
and g(y,.)
and g(y,.) . In particular,
are decreasing for every
, respectively, then the BVP (1) has a (unique) solution without
f
~ y growth restr~ctione for
g wl~tsoever.
a~d
(3.3) Example: Suppose that there are real numbers ~ , such that
xE~
f ( . , ~ ) -> ao-~ ,
with T <
g(',-~) >- 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 <~2 < 92 ' Then tl~ BVP (1) ha8 at
least tl~ee d i s t i n c t solutions
Ul, u2, u3
such that
~1 ~ Ul < u2 < u3 ~ 92 "
(3.8) Exa~ole: The BVP
-AU = a s i n
U
in
au = 4 2 cos u + u2 sin lxl @8 has for every
a E ~+
fl
,
on r
at least three d i s t i n c t solutions such that
-2~ ~ uI < u2 < u3 ~ x . This follows from the fact that the constant functions ~I := -2~
' Vl := -~
hypotheses of Theorem (3.7)
' ~2 := o , and V2 := ~
satisfy the
*
We leave i t to the reader to transpose the abstract m u l t i p l i c i t y result given in Theorem (2.8) to the setting of the nonlinear BVP (I) (cf. also [ 2 , 6 ] ).
I I . Parabolic Bo~dar~
~lue Problems: In the following we consider
nonlinear parabolic BVPs of the form
(6)
a u + Au = f ( x , u ) at
in
QT := ~ x (o,T) ,
Bu = o
on
sT :=
u = uo where A , B := B6 , and f
r
x
(o,T)
on ~o := ~ x {o}
,
,
satisfy the hypotheses of Part I , and
By a s o l u t i o n we mean a classical solution, that is, a function
T~+
35
u E C2'I(QT) n C6'~ T u ZT) n C(~T) , satisfying (6) pointwise (where the f i r s t superscript tives and j
Let
k
in
ck'J(...)
denotes the number of x-deriva-
denotes the number of t-derivatives).
p > 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 <_ ~ . Denote by 0
~
the unique solution
be the unique solution of the correspon-
ding parabolic BVP with initial condition converges for
theorem.
~
. Then
~(t) (resp, O(t) )
increasingly (resp. decreasingly) in
C(~)
the ninimal (resp. maximal) solution of the elliptic BVP (8) in
towards
{~,~] .
Proof. By the above considerations i t remains to show that :=
lim u(t) is the minimal solution in [~,~] of the e l l i p t i c BVP t-~(8). This follows immediately from the fact that we can replace ~ by the minimal solution in parabolic BVP (6)
{v,~}
without affecting the solution
-ff of the
9
I t should be observed that the above s t a b i l i t y theorem gives an other justification for the definition of a stable fixed point of Paragraph 2.
(3.12) Notes and Remarks: The above results about e l l i p t i c BVPs are essentially due to the writer although special cases have been proved much earlier by many other authors (cf. the bibliographic remarks in [6] ). For an interesting discussion of the "resonance case" based on the method of suband supersolutions we refer to [ 17] . The results concerning parabolic BVPs have f i r s t been proved by Sattinger {25,26] . The above derivation via the
40 the theory of semigroups is new. For generalizations to quasilinear e l l i p t i c and parabolic equations cf. [22] .
The method of sub- and supersolutions is a very e f f e c t i v e technique in the theory of nonlinear e l l i p t i c
and parabolic equations. Hence i t is im-
portant to know i t s l i m i t a t i o n s . The f o l l o w i n g r e s u l t shows t h a t , f o r a large class of equations, there are solutions which cannot be included in an order i n t e r v a l described by sub- and supersolutions.
Suppose that f(x,-)
f : ~ x ~+ ~ ~+
is s t r i c t l y convex f o r every Au = ~f(x,u)
(1%)
Bu :
where
has the property that
o
f(.,o) > 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
~<
f(x,.)
subsolution
we can suppose that
is a s t r i c t
~ E (o,~)
. Fix
and Lemma (2.3), i t ~
and a s t r i c t
and such that
(I0~) in the order i n t e r v a l
Vl
has no solution for
f o r every
u(~) < ~
i s a second s o l u t i o n of (10x)such that subsolution and
(Io~)
~(~)
superis the
[ ~ , 9 ] . Suppose now that
u E [~1,91] , where T I
u
is a strict
supersolution. Then i t can be shown that
v < ~ 1 " Consequently, by applying Theorem ( 3 . 7 ) , we can
deduce the existence of three solutions of (I0~) which are l i n e a r l y ordered. However, due to a r e s u l t of F u j i t a [14] , the BVP (io~) cannot have 3 comparable s o l u t i o n s . This c o n t r a d i c t i o n shows that no nonminimal s o l u t i o n of (Io~) can be found by the method of sub- and supersolutions. On the other hand, t h e ~ are several s u f f i c i e n t growth conditions f o r
f
which guarantee the existence
41
of at least two solutions for (loL) provided o < ~ < ~ (eg. {5,6] ). (For more details and generalizations we refer to {8] ).
42
4. Nonlinear Eigenvalue Problems and Bifurcation
Let
(E,P) be an OBS with nontrivial positive cone. In this paragraph
we study equations of the form (I)
x : f(~,x)
where, for simplicity,
f : ~ + x P-~ P is a completely continuous map.
The set z := {(~,x) E ~+ . p I x : f ( ~ , x ) } is called the eo/ut/on set of the equation (1), and i t s projection into the f i r s t coordinate space is denoted by
A , that is,
A := {~ E ~+ I f ( ~ , ' ) has a fixed point in P} I t is easily seen that
z n A is compact for every closed bounded subset
Aof]R+xP. Recall that a nonempty closed connected subset of some topological space X is called a 8ubcontinuum of
(4.1)
Theorem: Suppose that
su~ontinuum emanating from
Proof: For every S+ u
u> o
is the boundary of
P
X .
f(o,. ) = o . Then
~
contains an unbounded
(o,o) .
let
S+ := {x E p I
llxll = ~}
in (the relative) topology of
Qu := [o,u] x l~u . Then the boundary ~Qu of
Denote by C the component of
z
Q~ in
containing
(o,o)
and observe that P . Let
IR+ x p equals
and suppose that
C is bounded. Then C n @Q~ = ~ for some u > o . Since D := @Qun and C are d i s j o i n t closed subsets of the compact metric space X := s n Q
43
a result from point set topology (e.g. [3o]) implies the existence of disj o i n t compact sets
KI ~ C and K2 D D with
X = KI u K2 . Since Q~
is a metric, hence a regular topological space, there exists an open subset
U of
Q~ with
K1 c U and ~ n (K2 u @Q~) = ~ . Consequently,
is a bounded open subset of (~,x) E 3u
[o,u] x p such that
x r f(X,x)
U
for a l l
.
Hence by the homotopy invariance and the normalization property of the fixed point index (cf. Paragraph 2), I = i(o,Uo,P ) = i ( f ( o , . ) , U o , P ) = i ( f ( u , . ) , U ,P) , where U~ denotes the slice of choice of theorem
U . Hence
U at
~ E [o,~] . But
U = ~ by the very
i ( f ( u , . ) , U u , P ) = o . This contradiction proves the
9
I t should be observed that the condition
f ( o , . ) = o can be replaced
by much more general conditions guaranteeing that i(f(o,.),Uo,P) r o
(o,o) E z
and
(cf. [ 6 ] ).
The following corollaries are easy consequences of the fact that the coordinate projections map connected sets onto connected sets.
(4.2) Corollary: Suppose that there exists a positive number f(~ .)
that
has no ~ x e d point. Then
unbounded subcontinuum emanating from
(4.3) that
~. n ([O,~] x p)
(0,0)
~
i4ore precisely~ emanating from
x
for every
~ > o
and every
9
x ~ S+ P
Then
p
.
such
A = ~ +
r n (R + x ~ ) contains an unbounded subcontinuum (o,o)
such
contains an
Corollary: Suppose that there exists a positive number
f(~,x)
~
44 In order to verify the hypotheses of Corollary (4.2) i t is often possible to use additional information about the map f . For example. O
suppose that such that
P $ ~
and that there exists an a.s.p, linear operator
f(~,x) > ~Tx _
E n ([ o,I/r(T)I x p)
for every
(~,x) E ~R+ x p . Then
contains an unbounded subsolution emanating from
(o,o) . To see this, we l e t
fE(~,x) := f(~,x) + c~e , where E E ~+
e E P \ k e r T . Then Theorem (1.13) implies that if
T
f (~,')
and
has no fixed point
~ > 1/r(T) =: Let
Uc p
be an arbitrary bounded open neighborhood of zero. Then, by
Corollary (4.2), there exists a pair x E = fc(~ ,xE) . By l e t t i n g continuity of
c
(~E,x) E [o,~]
x @U such that
tend to zero and using the complete
f , i t follows easily that
E n ([o,~] x BU) ~ ~i . Hence
the assertion follows by involving the separation theorem used in the proof of Theorem (4.1)
9
I t is easy to see how the ideas of the above proof can be used to prove much more general results in this direction (cf. [6,1o]). Theorem (4.1) becomes t r i v i a l i f
f(.,o) = o
contains the " l i n e of t r i v i a l solutions"
since, in this case,
IR+ • {o}
E
. In this situation
the existence of nontrivial solutions follows provided i t can be shown that bifurcation occurs. Let
f ( - , o ) = o . Then ~o E IR+ is called a b i f u r c a t i o n p o i n t (for
the equation
x = f(~,x)) i f , for every neighborhood U of
~ + x p , there exists a point
(~,x) E Uns
proposition gives a necessary condition for point.
with
(~o,O) in
x > o . The following
~o E ]~+ to be a bifurcation
45 (4.4) Proposition: Suppose that
f(',O)
: 0
and s~ppose that
A0
i s a bifurcation point such that (the right partial derivative) D~f(~o,O ) exists. If the mop
f ( ' , X ) l l x l l ' 1 : ~ + - P is continuous at
~o J uniformly on null sequences in
D~f(~o,O) Proof:
in
P j then
1 is an eigenvalue of
w~th a positive eigenvector.
The assumptions imply the existence of a sequence ( ( ~ j , x j ) )
(R+ x P) n E which converges to
(~o,O) . Hence, l e t t i n g
yj := xj llxjll - I E S+ , yj - D~f(~o,O)yj = [ f ( ~ j , x j ) - f(~o,Xj)] llxjll - I §
+ { f(~o,Xj) - D2f(~o,O)XjI Ilxjl1-1 Since the right side of this equation tends to zero as j ~ | , i t follows + that o belongs to the closure of the set [ i d E - D2f(~o,O)] (S+) . By a result of M.A. Krasnosel'skii [18] ,
D~f(~o,O)IP is completely
+
continuous. Consequently, l i d E - D2f(~o,O)] (S+)
is closed in
E . Hence
the assertion follows 9
Suppose that such that result,
o
P r ~ and l e t there exist
D~f(~,o) = ~T for every
an a.s.p, linear operator
~ ~ ~+ . Then, again by Krasnosel'skii's
T E K(E) . Hence, by Theorem (1.13), the spectral radius
the only positive eigenvalue of sequently, in this case, point for the equation
~o
T
r(T)
is
T possessinga positive eigenvector. Con-
:: i / r ( T )
is the only possible bifurcation
X ~ f(~,X) .
I t is the purpose of the f,,llowing considerations to prove, in some sense, the converse of Pr,,p,,~ition (4.4).
46
(4.5) LenTna: Let g+(o)
such that g~(o)
P
such that
is not an eigenvalue of
i(g,P ,P) = 0
for every
g+(o)x =
lim 3§
3-1g(Tx)
f o r every
x E P , i t follows
g~(o) E L+(I~-F) . Moreover, by K r a s n o s e l ' s k i i ' s r e s u l t ,
P . Therefore,
contain
( i d - g+(o))(S+)
Choose
oo E
for every
(o,p]
is a closed set which does not
< ca/2 , and every
I I g ( x ) - g~(o)xll_< a l l x l l / 2
o e (O,Oo] X E [o,1]
possesses no f i x e d p o i n t on
a such t h a t
x E P .
such t h a t
x e l~Oo Then, f o r e v e r y
g+(o)IP
is closed on bounded subsets
o E E . Hence there e x i s t s a p o s i t i v e constant
IIx - g + ( o ) x l l _>a Ilxll
Ilyll
, provided
O+
i s completely continuous. Hence id - g+(o) of
~ E (0,~
possesses a positive eigenvector to an eigenvalue greater than one.
Proof= Since
that
1
exists. S~ppose that
and
to a positive eigenvector. Then there exists a constant
c 0 E (O,pI g~(o)
g(o) = o
g : TS § P be a compact map w i t h
S+
9
every
y E p
,. the map
satisfying
(1-x)(g~(o)
Indeed, for every
for all
+ y ) + xg
x e S+
s
llx - (1-X)(g+(o)x + y) - Xg(x)ll -> llx - g+(o) xll -IIg(x) - g+(o)xll
Ilyll
Hence, by t h e ho~otopy i n v a r i a n c e
-> ~ ( a - a / 2 - Ilyll / o ) > o .
p r o p e r t y o f the f i x e d p o i n t i n d e x ,
i(g,Pa,P ) = i(g+(o) * y,P ,P) . Denote by
h E S+
an eigenvector of
Then we claim t h a t , f o r every
g+(o)
to an eigenvalue
B > o , the equation
x - g+(o)x = Bh has
no p o s i t i v e solution. Indeed, suppose t h a t there exists a solution f o r some B > o . Then there exists a nonnegative number 3o x >_ 3oh
and
x ;~ 3h
for
x = g~,(o)x +
3 > 3o . Hence ' Bh-> g+(o)3 oh + Bh-> (3 0 + B)h ,
which contradicts the maximality o f
3o 9
~> 1 .
x > o
such that
47 Now, by setting
y := Bh with
of the fixed point index implies
In the following we l e t
o < B < aal2 , the solution property i(g,P ,P) = i(g$(o) + Bh,P ,P) = o
9
E+ := cl(E n ( ~ + x p)) , and we call this
set ( s l i g h t l y incorrectly) "the set of positive solutions" of the equation x = f(A,x) . Hence z+ consists of the union of
X n ( ~ + x p)
and
{(A,o) 9 R+ x p I A is a bifurcation point} .
(4.6) Theorem: Let
f(o,') = o . Let there exist an operator
r(T)
radius
such that
that the m a p
f(',x)
s e q u e n c e s in
P .
Then
1/r(T)
f(',o)
P be t o t a l and suppose t h a t
T e K(E)
D2+f(A,o) = AT f o r a l l
IlXlI - I
: ~+
~
P
ponent emanating from
Proof: Observe that
(1/r(T),o)
by C the component of
T
Z+
. F i n a l l y suppose
u n i f o r m l y on n u l l
c o n t a i n s an u n b o u n d e d com-
Ao := 1/r(T)
and l e t us call the
"characteristic values" of
z+ containing
bounded (in particular, that
A9 R
.
T e K+(E) . Let
reciprocals of the eigenvalues of
with positive spectral
is continuous,
is a b i f u r c a t i o n p o i n t a n d
C1 := Cu ([O,Ao] x { o } )
T . Denote
(Ao,O) and suppose that
C is
C = • ). Then, by using the notations of the
proof of Theorem (4.1), there exists a number ~ > Ao such that Let
and
= 0
C n BQ~ = ~ .
and C2 := z + n @Q~ . Then C1 and C2 are
d i s j o i n t compact subsets of the compact metric space X := Qu n (z+ u ([O,Ao] x {o})). Hence there exist d i s j o i n t compact sets with
Cj c Kj
and Kl u
K2 = X .
Kj , j = 1,2 ,
48 Let
~I
be the largest characteristic value in
smallest characteristic value in and l e t
~2 = p otherwise. Let
K1 , l e t
~2 be the
(~1,~] , i f there exists such a value, c
be a positive number such that
2E < min {~2 - ~I ' dist (KI,K2) , dist (KI,BQ~)} and denote by
UE the c-neighborhood of
K1 in
Qp . Then
(~I - E, ~1 + E) x {o} c UE and, due to Proposition (4.4),
2~ :=
dist(Kl,[~ 1 + E,~] x {o}) > o . Finally l e t U:=U c\([~1+E,~]x~a). Then U is an open subset of Let
P := ~1 + E/2
[o,~] xp
with
C1C U and
BUn (s+ o @Q~) = # .
and l e t 2o := d i s t (KI , [p,~]x {o}) . Then, again by
Proposition (4.4), e>o and, by making o smaller, i f necessary,we can assume that
{p} x~
c U . Hence, by the a d d i t i v i t y property of the fixed point index,
i ( f ( p , . ) , ~ , P ) = i ( f ( p , . ) , P o , P ) + i(f(p,'),Up~l~o,P) 9 Let
V := u n ( [ p , , ] x P \ F )
Vp = U p \ l ~ , V = d , and
. Then V is open in s n @V= d
[p,u] xP ,
. Hence, by the homotopy invariance
property, i(f(p,.),Up~.~,P) = i(f(,,.),V Since
§
D2f(P,o ) = pT , the Krein-Rutman theorem implies that
has a positive eigenvector to the eigenvalue since
,P) = o
1 is not an eigenvalue of
D f(p,o)
pr(T) = px~l > I . Hence,
+
D2f(P,o ) , Lemma (4.5) and the excision
property of the fixed point index imply that
i(f(p,o),Po,P ) = o . Hence
i(f(p,.),Up,P) = o . On the other hand, by the normalization property and the homotopy invariance property (as applied to the open set
U n ([ o,p] x p)
{o,p] x P ), i t follows that I = i(o,Uo,P ) = i ( f ( o , . ) , U o , P ) = i ( f ( p , . ) , U p , P ) = o . This contradiction proves the theorem
~
of
49
(4.7) Corollary: Suppose, in addition to the hypotheses of the preceding theorem, that
~ $ ~ ,and that
T is a.s.p.. Then
point and the only one. Yoreover, Z+ emanating from
(I/r(T),o)
is a bifurcation
I/r(T)
contains an unbounded subcontinuum
.
Proof= The assertion is an easy consequence of Theorem (4.6), Proposition (4.4), and Theorem (1.13)
9
I t should be remarked t h a t , in the above theorem, the hypotheses that P be t o t a l has only been made in order to apply the Krein-Rutman theorem. By using the cone spectral radius
rp(T)
and a r e s u l t of Bonsall
instead of the Krein-Rutman theorem, the t o t a l i t y
f 9]
hypothesis can be
dropped (cf. also [ I o ] ). The above theorems are extremly general siince i t has only been assumed that
f ( ~ + x P) c p and that
i t is not presupposed that
f
f
be completely continuous. In particular,
be increasing.i Consequently, the above
theorems can be applied to quasilinear e l l i p t i c BVP (or even systems) of the form N
a~u(x,u,grad u)DiDkU = a(x,u,grad u,~) i,k=l"'"
in
~
,
u=o
on
r
,
provided the c o e f f i c i e n t matrix
aik
is uniformly p o s i t i v e d e f i n i t e and
a ~ o . For d e t a i l s we r e f e r to
[28,29| .
S i m i l a r l y as in the beginning of t h i s paragraph i t is also possible to prove the existence o f an unbounded subcontinuum of the form
[o,pI
x p , provided
minorant (cf. [ 6 ] ).
f
s+
in a " s t r i p " of
is known to possess an appropriate
50 In the remainder of this section we consider the case where f(~,o) > o for
~ > o . Let
~ : = sup A . The following proposition, which is an
immediate consequence of Theorem (1.13), gives an easy sufficient condition for
~# to be f i n i t e .
0
(4.8) Proposition= Suppose that compact endomo~ohism
f(~,x) _> ~Tx + g(~)
T
of
for
E
P ~ ~ . Let there exist an a.s9149
and a map g : ~+-* P ~ k e r
(~,x) E ~ + x P
By imposing the condition that
f
. Then
~E
T
such that
1Jr(T) .
be increasing i t is possible to
obtain much more information about the solution set. The following theorem is the easiest result in this direction 9 For much deeper theorems, which are based on Theorem (4.9) and which are concerned with lower estimates for the number of solutions, we refer to [ 5 , 6 ] .
(4.9) Theorem= Let f : ~+
x P ~ P
$~opose that Then f(~,.)
A
(E,P)
be an OBS w~th normal positive cone and let
be a completely continuous map such that
fl ([ o,~~) x P)
is increasing.
is an interval containing
o
possesses a minimal ~ x e d point
increasing and left continuous. If {~(~) [ 0 ~ ~ < ~ }
f(o,o) = o .
~
andj for every ~(~) . The map
< |
j then
is bounded. Ifj for every
~ E A j the map ~(.) : A § P
~ E A
is
iff tl~ set
x E P j the m a p
f ( . , x ) : [O,X ) ~ P ~s strictly (st.rongly) increasing~ then ~he m ~ ~(') : A § P
is also strictly (strongly) increasing.
Proof= Clearly,
o E A and zero is the minimal fixed point of
I t follows from Corollary (2.2) that
f(~,-)
f(o,.) 9
has a minimal fixed point
51 ~(k)
for every
f(~,~(k))
x ~ k . If
o ~ p < x , then
and, again by Corollary (2.2),
~(x) = f(x,~(k))
f(p,-)
has a minimal fixed
point ~ ( , ) , and ~(~) ~ x(k) . Clearly, ~(~) < ~(k) ~(~) <<~(k)) Since ~ ( . )
if
f(.,~(x))
is increasing,
is s t r i c t l y (or strongly) increasing. {~(p)Io~p
~ A . Hence, by the normality of by the complete continuity of and is a fixed point of follows that
c
[ o,~(k)]
for every
P , this set is bounded. Consequently,
f , i t follows that
x x :=
f ( k , . ) . Since, by continuity,
lim ~(u) uf k xk ~ ( k )
exists
, it
x x : ~(k) . Hence 7(-) : A § P is l e f t continuous. The
last argument applies also to assertion
(or
k~ to give the remaining part of the
9
(4.1o) Notes and Remarks: The global existence theorems (Theorem (4.1) and Theorem (4.6)) have f i r s t been derived by e n t i r e l y different methods by Dancer [lo] and Turner [29] . The above proofs, based on the fixed point index, are due to the author (cf. also [ 11] ). They are in s p i r i t very close to the global bifurcation results by P.H. Rabinowitz [24] . The fundamental Lemma (4.5) is due to the author [ 4 ] . For further results, generalizations, applications, and bibliographical remarks we refer to {6] .
52 Bibliography
[I)
H. AMANN:
On the existence of positive solutions of nonlinear e l l i p t i c boundary value problems, Indiana Univ. Math. J., 21 (1971), 125-146.
[2]
H. AMANN:
Existence of multiple solutions for nonlinear e l l i p t i c boundary value problems, Indiana Univ. Math. J., 21 (1972), 925-935.
[31
H. AMANN:
On the number of solutions of nonlinear equations in ordered Banach spaces, J. Functional Anal., 11 (1972), 346-384.
[4]
H. AMANN:
Fixed points of asymptotically linear maps in ordered Banach spaces, J. Functional Anal., 14 (1973), 162-171.
[5]
H. AMANN:
Multiple positive fixed points of asymptotically linear maps, J. Functional Anal., 17 (1974), 174-213.
[6]
H. AMANN:
Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, to appear.
[71
H. AMANN:
Nonlinear e l l i p t i c equations with nonlinear boundary conditions, Proc. 2nd Scheveningen Conf. Diff. Equ., August 1975, ed. W. Eckhaus, North-Holland, Amsterdam, 1976.
[8]
H. N~NN:
Supersolutions, monotone iterations, and stability. J. Diff. Equ., to appear.
53
[9]
F.E. BONSALL:
Linear operators in complete positive cones, Proc. London Math. Soc. (3), 8, 53-75 (1958).
[10]
E.N. DANCER:
Global solution branches for positive maps, Arch. Rat. Mech. Anal., 52 (1973), 181-192.
[11 1
E.N. DANCER:
Solution branches for mappings in cones, and applications. Bull. Austr. Math. Soc., 1_~I(1974), 133-145.
[12 ]
A. FRIEDMAN:
"Partial Differential Equations", Holt, Rinehart and Winston, New York, 1969.
[13]
A. FRIEDMAN:
Partial Differential Equations of Parabolic Type, Prentice Hall, Englewood C l i f f s , N.J. 1964.
[14 ]
H. FUJITA:
On the nonlinear equations Au + eu = o and av/@t = Av + ev . Bull. Amer. Math. Soc., 75 (1969), 132-135.
[15 ]
E. H i l l e and R.S. PHILLIPS: Functional Analysis and Semi Groups, Amer. Math. Soc. Coll. Publ., Vol. 31, Providence, R . I . , 1957.
[16 ]
G. JAMESON:
"Ordered Linear Spaces", Lecture Notes, Vol. 141, Springer Verlag, New York, 197o.
[17 ]
J.L. KAZDANand F.W. WARNER: Remarks on some quasilinear e l l i p t i c equations, Comm. Pure and Appl. Math., to appear.
[18 ]
M.A. KP.ASNOSEL'SKII: "Positive Solutions of Operator Equations", Noordhoff, Groningen, 1964.
[19]
M.G. KREIN and M.A. RUTMAN: Linear operators leaving invariant a cone in a Banach spaces, Amer. Math. Soc. Transl., Ser. I , 1o (1962), 199-325.
54 [20]
O.A. LADYZENSKAjA, V.A. SOLONNIKOV, and N.N. URAL'CEVA: Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. Transl. of Math. Monographs, Vol. 23, Providence, R.I., 1968.
[ 21 ]
A. PAZY:
Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Univ. of Maryland Lecture Notes~lo, 1974.
[22]
J.-P. PUEL:
Sur des probl~mes quasilin~aires elliptiques et parabolic d'ordre 2, C.R. Acad. Sc. Paris, 275, Sbrie A, (1972), 179-182.
[23]
M.H. PROTTERand H.F. WEINBERGER: "MaximumPrinciples in Differential Equations", Prentice Hall, Englewood Cliffs, N.J., 1967.
[24]
P.H. RABINOWITZ:
Some global results for nonlinear eigenvalue problems, J. Func. Anal., ~ (1971), 487-513.
[25]
D.H. SATTINGER:
Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J., 2_~I(1972), 979-1000.
[26]
D.H. SAI'TINGER:
"Topics in Stability and Bifurcation Theory", Lecture Notes, Vol. 309, Springer Verlag, Berlin, 1973.
[ 27 ]
H.H. SCHAEFER:
"Topological Vector Spaces", Springer Verlag, New York, 1971.
55
[28]
R.E.L. TURNER:
Positive solutions of nonlinear eigenvalue problems, C.I.M.E. Conference "Eigenvalues of Nonlinear Problem", Varenna, Italy, 1974.
[29]
R.E.L. TURNER:
Transversality and cone maps, Arch. Rat. Mech. Anal., 58 (1975), 151.179.
[30]
G.T. WHYBURN:
"Topological Analysis", Princeton Univ. Press, Princeton, N.J., 1958.
Mathematisches I n s t i t u t Ruhr-Universit~t D-4630 BOCHUH Universit~tsstraBe 150 W.-Germany
QUELQUES
PROPRIETES DES
DES
OPERATEURS
SEMI -GROUPES
MONOTONES
ET
NON" LINEAIRES
H. BREZIS
Ce cours est divis~ en deux parties. propri~t~s
des op~rateurs monotones
Le chapitre I concerne l'~tude de certaines
et de louts images
quelques r~sultats relatifs aux semi-groupes la r~solution d'~quations
; au
non lin~aires.
aux d~riv~es partielles
Chapitre
II on prasente
De nombreuses
sont incorpor~es
applications
~ l'expos~.
PLAN I.
Op~rateurs monotones.
1.1
Quelques r~sultats plus ou moins classiques
1.2
Etude de
R(A+B)
pour
A = ~
1.3
Etude de
R(A+B)
pour
Aet
B
1.4
Etude de
R(A+B)
pour
Aet
B monotones avec
1.5
Applications
1.6
Etude de
pour
A
R(A+B)
Etude de
R(A+B).
et B
= ~
monotones
. avec
B = ~
et
D(B) - H .
(Au,Bu) > 0
lin~aire non monotone et
B
monotone
nonlin~aire. II.
Semi groupes non lln~aires
If.| Rappel de quelques r~sultats usuels II.2 Formulation de certaines minimisation
;effet
r~gularisant
~quations d'~volution
convexe.
II.3 Comportement
au voisinage de
t = 0
II.4 Comportement
au voisinage de
t =
comme probl~mes de
57
I
-
I.I
OPERATEURS MONOTONES.
Quelques Soit
monotone
r~sultats
H
ETUDE D E
plus ou moins classiques
u n espace de Hilbert.
si
Vu,v(Au-Av,u-v)
n'admet aucun prolongement
> 0
Equivalences
A
On pose alors H
dans
H)
et
I.
A
A 1 = ~(l-J%)
de
2.
: H
+
P(H)
est monotone
A
est monotone,
~=~ V I > 0 , R(I+IA)
r~solvante
r~gularis~e
D(A) = H
A
et
A
de
A
est
et
alors on
= H .
(qui est une contraction
Yosida de
monotone
aux segments de
Soit
q
: H
D(q) = {u ~ H ; q(u) ~(u)
H
A ;
h~micontinu
est continue
u
~
Lemme
I. u
~(u)
§
- q(u) >
A1
est m o n o t o n e
de
et
de
H
dans
~ valeurs dans
H
H (i.e. la faible).
une fonction convexe
r ~ +~. On pose
s.c.i.
u e D(q)
(f,v-u)
V v e H} 9
est maximal monotone.
(cf [ 7 ] p.27) , fn-.=~ f
Notation.
(-~,+~]
< +~}, et pour
= {f 9 H ; ~(v)
Alors
--~
A
A
est maximal monotone.
Exemple
Un
de Minty on sait que si
maximal monotone
Jl = (I+IA) -I
Soit
restriction Alors
si
1 (de constante ~).
Lipschitzienne
Exemple
On dit qu'une application
est maximale m o n o t o n e
strict monotone.
D'apr~s une caract~risation ales
R(A+B)
et
. Soit
A
maximal monotone.
Soit. [Un,f n] e G(A)
lim sup (fn - f'Un-U) < 0 . Alors
Etant donn~s
S 1 , S 2 C H , on note
S 1 = S2
f e Au
si
avec
.
~I = ~ 2
et
Int S I = Int S 2 .
Lemme 2.
On a
D~monstration I*) (1)
D(~)
= D(q)
Ii suffit de v~rifier
Soit u
u e D(~) E
et
soit
u
+ e~(u)~u
On a alors (U) --~ (Uc) >
que
D(~) C D--~-) et que
e D(~)
(c > O) B--Us (-q-- , U--U )
et donc
lue-ul2 < ~ ( u ) (noter que par Hahn-Banach
- ~(a,u)
on a
~(u~ >
- E b
(a,uL)+b >
la solution de
Int D(~) C D(~P).
58
D'o~ il r~sulte 2~
que
Soit
s
§
u
et donc
u e Int D(~)
On reprend quand
uE
us
.
solution
de
(I). On va montrer
0 . Ii en r~sultera
-~
u e D(~)
l ~ (u-us ) ~
que
n
lemme
l) et donc
h e H
avec
u e D(~a).
lhl < r
Comme
-~(ue)
D(@),
que
aussi fortement Fonction
=
3.
a,
~
Soit et ~
:
1
~*
et par suite
~(u-u
)
est faiblement
born~
= (~)-|
monotone
Vu e D(M)
et soit
F C H . On suppose
que
Vfe
H
tels que
(Mu-f,u-a)
et
Int[conv
(cf. [~i]).
avec
i
Pour tout
- ~(u)}
si l'on remplace
f = Z t.f.
B(u,r) C D(~).
on pose
maximal
conv F C R(M)
v~rifi~e
On a
M c
Demonstration.
on a
borne.
V u e D(M)
Alors
le
!
Sup {(f,u) ue H
(2)
(par
I> (-~-e, h)
(--~-e,h) ~ C(h)
II est bien connu que
Lemme
f r ~(u)
U--U
I> (--~--e,u+h-ue)
con~u~u~e
~*(f)
avec
rests born~
on a
D--U
On en d~duit
f
le [uE-ul
n
u e Int
U--U
~(u+h)
que
par
et
1
F] C R(M)
.
Notone d'abord F
t. > 0
> C .
Zt.
que l'hypoth~se
F = cony F. En effet =
I
soit
(2) est encore f s F
i.e.
9
1
p
(Mu - fi ' u-ai) > Ci
'
soit (3)
(Mu,u)
Multipliant
- (Mu,a i) - (fi,u) > C!
i
(3) par
t.
on obtient
apras addition
(Mu,u)
- (Mu,a)
- (f,u) > C'
1
o~
affi
E
t.a. 1
, 1
On est donc ramen6 ~ ~tablir I~
Soit
f e F
et
que soit
F C R(M) u
et que
la solution
Int F C R(M) de
E
(4)
eu
+ Mu E
~
f
(e > 0).
E
Ii suffit de prouver que
r u
§ 0
quand
e § 0 .
.
; done
59
Or on a
i.e.
(Mue-f,ue-a)
> C
et donc
~[ul 2 < c + ~[u[[al 2 ~)
Soit
lhl < r
V u e D(M) Reprenant
u
~lu~I
et donc
f e Int F
il existe
(f-eue-f,ue-a)
et soit
ah e H
et
<
>
0
c'
B(f,r) C F . Donc pour tout
Ch
h e H
avec
tels que
(Mu-f-h,u-a h) > C h . solution de (4) il vient
s
eluel 2 + ( h , u ) On en d~duit que
< ~lue[
uE
lal + Ifl
est faiblement
lahl + C h 9
borne, donc fortement
borne.
Soit
ue
---% u ; n
grEce au lemme Remarques. ~-M)
et
]il
I)
2)
inf
> 0
et que
(6)
Soit
Appliquant
(7) Soit
Aet
f e H
de
f .
o
B
A+B
A et B
sont m a x i m a u x
est maximal m o n o t o n e o
sont m a x i m a u x monotones
e D(A)
et soit
et que
u%
la
% + 0
pour en d~duire ainsi que
A+B
(5) et (6) on obtient
; on a
solution
de
(6)
[ ~ ] (cf aussi [7 ]), il suffit de m o n t r e r
O n prouvera
+ l xuxl 2 v
n'est pas maximale
f
reste born~ quand ~
lorsque
tel que
V u e D(A), V I > 0 .
D'apr~s un r~sultat
u + Au + Bu
a e H
On suppose que
D(A) A Int D(B) # @ . Alors
u~ + Au~ + B~u~ ~
IBlu%[
~
est maximal monotone.
D~monstration.
:
:
On suppose que
A+B
V f ~ F
et aussi [7 ] p.36).
(Au,B~u) > 0
Alors
bien connus
I
on a l e s
(cf. [ ~ ]
Lemme 5.
(2) par
la somme de 2 op~rateurs m a x i m a u x monotones
N~anmoins
monotones
f ~ R(M).
Nous ne savons pas si la conclusion du Lemme 3 est valable
( ~ a )
Lemme 4.
et
sont convexes.
l'hypoth~se
En g~ngral monotone.
Mu ~ f
On retrouve ~ partir du len~ne 3 deux r6sultats
Int R(M)
l'on remplace
(5)
vient
Ill
IBxuxl.
que
u% § u
et que
est maximal monotone.
que
60
(Au
- Av ~ , u I - v o)
>
0
ou encore (f-BAuA - uA - Av ~ , uA-v o) ~ 0 . D'o~
(s)
lux[2
+
(Bxu~ , u x) < C~
(dans route la suite, Soit
v I e D(B)
+ c 2 lu~1 + c 3
les constantes
C. 1
IBxuxl
sont ind~pendantes
x).
de
; on a
_(B~uI - B~v I , u I - v I)_ ~ 0 et donc
(9)
(BAu%,uA) >
(noter que Combinant
- c4 - %[u~I
I~~
IB%vlI<
- c 6 lhu~l
.
(8) st (9) il vient
I%12 < % + c a luxl + c 9 Iexuxl
;
grace ~ (7) st (9) on a
c4
IBxuxl 2 <
+
On en d~duit enfin que
1.2
ETUDE~FR(A+B)
c5
luxl
[Blu~[
POUR
cl0
+
IBxuxl.
demeure born~ quand
A = ~0
ET
=
U
(Au+Bu)
Aet
est b i e n plus petit que
(prendre par exemple dans
o
B = 3~
Etant dorm, s deux op~rateurs monotones R(A+B)
I § 0
B
o n notera qu'en g~n~ral
R(A) + R(B) =
H = R 2 , A = rot(+ ~~ ) et B - -
U (Av+Bw), v~w
rot(+ ~) ) . N~anmoins
on se propose d'~tablir que dans u n grand hombre de situations R(A+B)
= R(A) + R(B)
Au + Bu ~ (v st w
f
. Dens ces cas on peut affirmer bri~vement
a une solution d~s qua
ind~pendants).
sent extraits de
Les r~sultats
H.Brezis
Soit
Th~or~me
6.
monotone
(ce qui ~quivaut
Nous indiquons fonctions
conjugu~es
on a
f
que l'~quatlon
admet une d~compositlon
pr~sent~s
aux
1.2
, 1.3
f e A v + Bw , 1.4 et 1.5
et A. H a r a u x [ ~ ] .
A = ~
et ~
B = ~$ ; on suppose que
~+~)
= ~
deux d~monstrations
+ 3~)
. Alors
A + B R(A+B)
est maximal = R(A) + R(B).
: la l~re est tr~s simple et u t i l i s e
; la 2~me s'~tend ~ des op~rateurs
plus g~n~raux
les
61
D~monstration (10)
I.
D~
En effet,
On a toujours
) + D($*) C DC(r
si
f e D@*)
et si
Vu e H
(f,u) - ~(u) <
~*(f)
rue
(g,u) - ~(u) ~
~*(g)
H
*
Donc
g e D($*)
(r
on a
*
(f+g) < ~*(f)
+ ~ (g) .
On d~duit de (10) que (ll)
R(A) + R(B) = D ( ~
) + D(35")
C D((r
.
A partir de (10) on obtient R(A) + R(B) C D((~+$)*)
= D(~(r
(Lemme 2)
Par consequent R(A) De m~me
+ R(B) C R(~(~+~))
(Kl) implique
+ R(B)] C Int [R(A+B)]
Remarque.
La d~nonstration maximal
monotone
La conclusion et que
C Int [R(A)
I
montre
+ R(B)]
En effet on a pour
alors
C Int R(A+B).
a e D(A) N D(B)
V~rifions
et
des ~galit~s
M = A + B
en particulier
et
que
?
F = R(A) + R(B).
R(A) + R(B) C R(A+B)
que l'hypoth~se
(2) est satisfaite.
fix~
(12)
(Au-Av,u-a)
> (Av-Aa,a-v)
Vu,v e D(A)
(13)
(Bu-Bw,u-a)
> (Bw-Ba,a-w)
Vu,w e D(B)
Prouvons
par exemple
(12)
; on a
~(u)
- ~(v) > (Av,u-v)
~(a)
-~(u)
>
R(A+B) C R(9(r
sont-elles
le lemme 3 avec
du Lemme 3 impliquera
Int JR(A)
toujours
Ces inclusions
On applique
+ R(B)]
que, m~me si l'on ne fair pas l'hypoth~se
" , on obtient
Int R(A+B) C Int R(~(r
D~monstration.ll
.
aussi
Int [R(A)
" A + B
= R(A+B) C R(A) + R(B)
(Au,a-u)
(v) - ~ (a) > (Aa,v-a) et par addition ~vidence
on obtient
par Rockafellar
Soit
(12)
[~]
f e F = R(A) + R(B)
(c'est une propri~t~
, cf
aussi
; on a donc
de monotonie
cyclique
mise en
[ 7 ] P" 38). f = Av + Bw . Par addition
de (12) et
(13) il vient Wu e D(M) = D(A) ~ D(B) Remar~ue. utilis~
(Mu-f,u-a)
> C
Notons que dans la d~monstration le fair que
A+B
est maximal
monotone
II du Th~or~me et la propri~t~
6 nous avons
seulement
62
vf 9 R(A) (*)
et
Va
D(A)~
e
(Au~f,u-~a) > C(f,a)
La propri~t~
(*)
plus faible.
Ainsi
Amann)
tout op~rateur C
our
satisfaite
lin~aire
A
et pour
lorsque
angle born~
B/).
A = ~
mais elle est bien
(notion
introduite
par
tel que
l(Au,v) - (Av,u) I < C [ (Au,u)
v~rifie
tel que
vu 9 D ( ~
es= en particulier
i.e. il existe
(14)
C(f,a)
+ (Av,v)]
(*).
En effet on a, par monotonie
de
A
(Au,v) + (Av,u) < (Au,u) + (Av,v) et par addition
avec
(]4) il vient
(Au,v) < C'[ (Au,u) D'o~ l'on d~duit (15)
(Au,v) < (Au,u)
Soient
+ (Av,v)].
(en remplagant
f 9 R(A)
et
(Au-f,u-a)
R(B)
(*)
a e D(A)
ETUDE DE
Th~or~me
7.
On suppose
Soient que
f e F
(16)
(17)
A
part,
Corollaire VA > 0
pour tout
B
maximal B
monotone
maximal
alors
monotone
A = BI
tel que
A
ET
B
MONOTONES
AVEC
maximaux monotones
et que
B
v~rifie
le le,lne 3 avec
avee
(*).Alors
M = A+B
B = ~
ET
A+B
maximal
R(A+B)
et de
D(B)
= H
monotone.
= R(A) + R(B).
F = R(A) + R(B). A
il vient
comme
B
v~rifie
(*) et que
v
e
D(A) C D(B)
on a
> C(v,w).
de (16) et (]7) on obtient
8.
R(A+B)
>
f e Av + Bw . Par monotonie
> C
On d~duit directement
Alors
que
> 0 .
(Au+Bu-f,u-v)
Corollaire
que
B
On applique
(Bu-Bw,u-v)
Par addition
l ) % = ~C'
o~
il vient
f = Ab
> 0 . Enfin tout
et
de sorte que
D'autre
I ~ v
par
= C(f,a).
POUR
D(A) C D(B)
(Au-Av,u-v)
v
(*).
R(A*B)
D~monstration. Soit
; posant
pour tout
soit born~ v~rifie
1.3
%u , et
= (A(u-b),(u-b)-(a-b))
facile de v~rifier
a la propri~t~
par
+ (C')2(Av,v).
(C')2(A(a-b),(a-b)) Ii est aussi
u
Soient
du Th~or~me
A et B
7 les
maximaux
monotones
avec
maximaux
monotones
alors
= R(A) + R(B).
9. R(A+BI)
Soient
A et B
= R(A) + R(B%)
= R(A) + R(B).
D(B) = H
et
B = 8~ .
63
10.
Corollaire
Soient
A et
B
maximaux monotones avec
POUR
A B
R(B)
borne. Alors
R(A+B) = R(A) + R(B).
1.4
ETUDE DE
Th6or~me
R(A+B)
Soient
I!.
A et B
R(A+B) = R(A) + R(B).
On salt d~j~ (Lemme 5) que
D~monstration.
(AU,BxU) > 0
maximaux monotones avec
Vu e D(A). Alors
(Au,BIu) > 0
MONOTONES AVEC
A+B
est maximal monotone. On utilise
le Lemme 12.
Soit
eue
(18)
f r H
la solution de
IBue[ < c ( f )
et
D~nonstration du L e m e (19)
us
+ Au r + Bu e ~ f .
IAu I
Alors
et soit
12.
Soit
ue, X
la solution de
Eur X + Aur X + Bxue, X ~ f
On salt (cf. d~monstration du Lemme 5) que pour i
ue'~ (~ O)
u
e
r > 0
fix~, alors
solution de (18). Afin de simplifier les notations on pose
u = uE9 A
Elevant (19) au carr~ on a (20) Soit
s21u] 2 + IAul 2 + IBxul 2 + 2s(u,Au+Bxu) < If] 2 v
o
e D(A) N D(B)
fix~ ; on obtient
(AU+BxU-AVo-Bx Vo, U-Vo) > 0 et par suite (21)
(Au+Bxu,u) ~ C(]u] + [Au I + IBxU I + I) 9
-
Combinant
(20) et (21) il vient 2
lu]2 + ]Aul2 + [B~ul2 < If [2 + 2 c(lul + IA.[ + l xul + *>.
D'o~ l'on d6duit
~21u12 § IAul 2 + IBxul2 < c (f) D~monstration du Th~or~me Int [R(A) + R(B)] C I~
Soit
ll(fin~
.
Montrons que
R(A) + R(B) C R - - ' ~
et que
R(A+B).
f e R(A) + R(B) ; on a
On obtient (Aur
, ue-v)
>
0
(Bur
, ur
>
0
f e Av + Bw . Soit
u
la solution de (18).
64
et par addition (f-eue-f
, ue) >
D'o~ l'on d~duit que 2~
Soit
Pour tout
(Aue-Av,v)
eluel 2
reste born~ quand
f e Int [R(A)
h e H
avec
+ (Bue-Bw,w) > C
lhl < r
+ R(B)] on a
s § 0 .
de sorte que
B(f,r) c R(A) + R(B)
.
f + h e A v h + Bw h .
On obtient encore (AuE-Av h , ue-v h) ~ 0 (Bue-Bw h , uc-W h) > 0 et par addition (f-eue-f-h,ue)
>
o~
C(h)
u
est faiblement borne.
e
est ind~pendant
1.5
APPLICATIONS.
Exemple
|.
que
ou bien
K
(22)
Solent F
(Auc-AVh,Vh) de
+ (Bue-BWh,W h) > C(h)
~ . Par consequent
(h,us
On o o n c l u t g l'aide du Leone
K : H + H
v~rifie
et
F : H + H
(*). Alors
Vf e H
~ C(h) ] .
monotones ~u
et l'on d~duit que
e H
h~micontinus
. On suppose
solution de l'~quation
u + KFu ~ f .
En effet si Appliquant
K
v~rifie
le Th~or~me
(*)
on pose
7 avec
v = Fu
A = F -| et
et (22) s'~crit
B = K
F-Iv + Kv
9
f .
on a
R(F-]+ K) = R(F-])+ R(K) = D(F) + R(K) = H 9 Lorsque
F
v~rifie
Appliquant
R(A+B)
encore
(*) le
on
~crit
Th~or~me
la
Au = -
forme K-I(f.u)
-
K-l(f-u)
+ ~u
, Bu = F u
on
~
0
.
a
= R(A) + R(B) - - D(K) + R(F) = H . r~sultats
Exemple
S
2.
Soit
8 : R
+
R
sur les ~quations
- Au + B(u) = f ~u = 0 ~n
de B a m m e r s t e l n
un ouvert born~ r~gulier de fronti~re
une fonction monotone
lement u n graphe maximal monotone).
(23)
sous
7 avec
On trouvera d'autres
Soit
(22)
croissante
On consid~re
et contlnue
dans
~
l'~quation
~
Notons que si (23) admet une solution
] .
.
(ou plus g~n~ra-
sur sur
[~
u , alors on a n~cessairement
85
Inversement
l[ IN
(241 alors
montrons
I
f 9 Int R(8)
(23) poss~de
En effet dans Au = - Au
que sous l'hypoth~se
une solution.
~ = L2(~)
aver
posons
: ~u ; ~n = 0
D(A) = {u 9 H2(~)
sur
~}
; Alors
est maximal
A
i Soit
Bu = 8(u)
monotone Convexe
avec
D(B) = {u e L2(~)
et l'on a m~me telle que
B = ~
avee
-~ R(A) + R(B). Etant
g= (g-lgl D'o~ l'on
on a
Des probl~mes
a
analogues
Soit
8
v~rifie
avaient
alors
f
(25) poss~de
En effet dans Au monotone
e
prac~demment
dans
avee
~3,L~$],-- [a~]
2 . On consid~re
l'~quation
sur
[g-flL 2
.
eo~mle ~ l'exemple
diff~rentes.
(0,T)
on a n~cessairement
~
f,
0
R(8)
9
que sous l'hypoth~se
Int R(8)
une solution.
~{ = L2(0,T)
A # ~).
On peut appliquer
g 9 tt
enti~rement
posons
du Au = ~ , D(A) = {u e H|(O,T) (mais
pour tout
par des m~thodes
que si (25) admet une solution
-T
| | car
f 9 I n t [ R ( A ) + R(B)]
~t~ trait~s
Notons
(26)
est une fonetion
"semi-coercifs"
(25)
montrons
le Th~or~me
j
est maximal
g-f)"
(24) a l o r s
f du i ~ + 8(u) = f , u(0) = u(T)
Inversement,
o~
B
on ~crit
l'~I
g e R(A) + R(B) i . e .
sous le nom de probl~mes
Exemple 3.
g 9 H
I~I f
j(u)
; alors
~i )
donn~
a
d ~ d u i t que s i
assez petit
~(u) = f J
8 = ~j . On peut appliquer
a " R(A+B)
; 8(u) e L2(~)~
Soit
le Th~or~me
Bu = 8(u) |l car :
; u(0) = u(T)} avec
; alors"
A
e s t maximal
D(B) = {u e L 2 ; 8(u) e L 2} .
06
du "S~(u) ffi f T ~-{3~(u) d .. ~-[ = 0 , o~ 0
(Au,BIu) = f T 0
0j~ = 8 x . Cette technique
s'~tend a la r~solution de syst~mes. On trouve de m~me que le syst~me d~+L
dt l~i
= ~
,
~(0) = ~(T)
189f0 Tf I <
poss~de une solotion d~s que
l (cf
aussi [~.~] ).
Exemple 4, limites
Soit
u = 0
sur
correspondante.
eomme
la l~re valeur propre de
~
et soit
~
vl(v ! > 0
sur
-A
relative ~ la condition aux
~)
une fonction propre
On consid~re l'~quation
- AU-AlU + 8(u) = f u = 0
i (27) 6
%1
I'Ex.
2
avec
8(0)
sur sur =
0
~ .
f~fv I
Notons que si (27) admet une solution on a n~cessairement
e
R(B)
9
/ ~ Vl Inversement montrone que sous l'hypoth~se f~fv I -' e Int R(8) fflv I
(28)
alors (27) possade une solution. En effet darts
H - L2(~)
posons
Au = - Au - %1 u ,
D(A) = H2(~) A H~(~)
Bu = 8(u) , D(B) = {u e L 2 ; B(u) e L 2} , A
est maximal monotone et l'on a m~me l f (u) = ~ n
B
~ i
(3u ~ . >9 2 z
est maximal monotone avec Enfin
A+B
A + B + ~i I
~lu 2 , D @ )
= HIo
B = 3~ , ~(u) = f~ j(u) (3j -- 6) 9
est maximal monotone car
R(A + B + (A]+l)l) = H
(rioter que
est maximal monotoneppar exemple d'apr~s le Leone 5). On d~duit alors
du Th~or~me 6 que entraine
-
A = ~4
R(A+B) = R(A) + R(B). Enfin, on note que l'hypoth~se
f 9 Int [R(A) + R(B)]
car
g = (g-k) + k
(28)
o~
ffl gv 1 k -
appartient ~
R(B)
si
Ig-flL 2
est assez petit.
ffl v 1 Si l'on remplace
~I
par une autre valeur
propre
Xk
l'op~rateur
-A-~ k
n'est plus monotone et les techniques d~crites ne s~appliquent plus. L'~tude de ce probl~me fair l'objet du w 1.6 .
67
1.6
ETUDE DE
R(A+B)
POUR
A
LINEAIRE
NON
LINEAIRE
Dans toute cette partie n6cessairement (29)
I
monotone.
A : D(A) C H
§
[ N(A) = N(A*) II r~sulte
- dim N(A)
et
{u e D(A)
h6micontinu
B
MONOTONE
un op~rateur
lin~aire
(non borne)
Don
l'hypothgse , G(A)
; lu] < |
ferm6 et
IAu] < I}
est compact
de (29) que
< =
N R(A)
Les r6sultats
13.
A
ET
dense
est ferm6 et donc
- AID(A)
Th~or~me
d~signe
H , D(A)
classiquement
- R(A)
A
On falt sur
NON MONOTONE
est bijectif
qui suivent
lim
sur
R(A)
et d'inverse
sont dGs ~ H. Brezis
On suppose que avec
R(A) = N(A)
A
v~rifie
compact.
et L. Nirenberg
B : H
(29) et que
+
(cf. ~ i ] ) .
H
est monotone
IBvl
I v l ~ ~ Iv-W~ = 0 Alors
R(A+B)
= R(A) + c o n v
D~monstration. suivant Pour tout sur
Pour
R(A)
H
et
soit
e ~ 0 , l'op6rateur
et d'inverse
f = fl + f2 = Pl f + P2 f
sa d~composition
compact. f e H
u e D(A)
~-~ cu 2 + Au
Appliquant
et tout
le Th~or~me
e > 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 §
<
o2)
I'%1
I-] 1 < %(Ill * IBu l).
On obtient enfin
clu2J 2
<
C4(l+[Suel 2)
et par suite
~lu 12 < On en d~duit que lim
tvl-
I Bv l m
C4(]+IBu
12) + Elu]r 2 < Cs(]*ISu
EIuEI -~ 0
12)
(raisonner par l'absurde et utiliser 1'hypoth~se
= 01
2~
f 9 Int[ R(A) + cony R(B)]
Supposons que
born~ quand
et montrons que
En effet pour tout f+h " Av(h) + ~
h
avec
ti(h) Bwi(h)
[hi ~ r
on a une d~composition
; reprenant la d~monstration pr~c~dente on obtient
(33)
(f-eu2 -Aur
Combinant
(31) , (32) et (331 on est conduit
- (f+h - Av(hl,ur
> - C|(h)
.
(h,u) ~ C2(h)(] . [Su 12) . Choisissant
h = • re. 1
lu2~] <
o~ les
{e.} 1
forment une base de
N(A) on a
c3(1 + [Su 121
E n f l n , g r a c e ~ (32) on c o n c l u t que
(34)
t u G } demeure
s -> 0 .
lull <
c 4 (! . IB%121
D'o~ il r~sulte que
IuEI
reste born~ quand
Le passage ~ la limite est iEan~diac : u
-~
n
~§
u
0
avec
.
Au + Bu = f .
69
Th~or~me
14.
h~micontinu
On suppose que avec
IBvl
lim
lvI~ Alors
R(A+B)
A
(29) et que
B ffi ~
est monotone
= 0
~l
= R(A) + cony R(B)
On reprend
D~nonstration.
v~rifie
.
u
solution
de (30) mais on utilise
maintenant
E
aulieu de la monotonie
Lenrne 15.
Soit
de
B
B = ~
la conclusion
avee
lim
V~ > 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
<
il vient
~]v] * ~/~ (~u-~v,u-v) ]/2 § M~
Donc IBuJ 2 < Rempla~ant
12 ~
D~monstration 1~ quand
3(621vi 2 + 4~(Bu-Bv,u-v) par
6
du Th~or~me
Supposons
que
+ M~)
on conclut.
14 (fin). f e R(A) + Cony R(B) et montrons
que
e + 0 .
En effet
f - Av + Z t. Bw. ; appliquant z l (Bu e - BW i , u E - W i)
>
~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
<
,, + C6
C IBucI 2
et
e . Choisissant
restent
born~s
6
quand
f e Int [ R(A) + c o n v
D'autre part
pas intervenue
de
- c~
(341 reste valable
g
assez petit on conclut
e~0
R(B)].
.
IBul
On salt daja que
(l'hypoth~se
dans la d~monstration).
que
lim
On conclut
que
[Bv]
]u [
= 0
reste borne.
~ la limits est imm~diat.
R emarque.
Ii est souvent
commode
d'utiliser
les Th~or~mes
13 et 14 sous la
forms suivante. Notons t
d'abord,
~-+ (B(tu),u)
grace
~ la monOtonie
est c r o i s s a n t e ;
de
on pose
B , qua la fonction
JB(U)
=
lim
B(tu),u)
.
Alors (35)
JB(U) > (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
et
du d--{+ Au 9 0
u(0) = u
p.p. t > 0 , d+u d - ~ + A~ - 0
Enfin on a
Kato, Crandall-Pazy
] p.54).
uest
d~rivabLe p . p . o .
~t>O
Notons que si no, Qo e D(A), alors les solutions correspondantes =(t) Q(t)
v~rifient
In(t) - G(t) I < lu~ - QOI. E1 ~
et
r~s~Ite que pour cheque t > O,
l'application u
~'+ u(t) est u~ contraction de D ~ ) ~ans lui-m~me q~e Iron prolono ge par continuit~ ~ D - ~ ) e n u n e application notre S(t).S(t) est Is semi groupe engendr~ par -A.
Ii est ais~ de montrer que l'on a S(t I + t 2 ) = S(tl) o S(t 2)
V tl,
t 2 ~0
IS(t)u 0 - S(t)~ol < lU 0 - G o l
Vt o
lim Iu~ - S(t)Uol = 0 t+o Inversement, fiant
~tant donn~
{S(t)}t> 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 <~I
luo-~(T)12
La conclusion du T h @ o ~ m e D(A)
eb done
. 17 exprim~ q~e si
A - ~r
alors
Vt > 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<~2
x e D-~
tout
~o
, o~ a
I <
~2lx-yl I IX
+ tlA~ -
j~xl
o~ o b t i e n t
77
D~monstration.
Supposons d'abord que
x e D(A) e t posons
u(t) = S(t)x. On a,
par monotonie d e A du , (Av + ~-f(t)
v - u(t))
Vv e
>~ O
D(A)
i.e, 1
(Av, v-x) § (Av , x - u(t)) > Int~grant sur
[O,T]
on
d
~" ~E l u ( t ) - vl ~
a
T
(50)
T(Av,v-x) + (Av, Tx -
Prenant
v = JT x
on
i I~-JT~I =
<
o~(t)d~) ~ ~'1~<~>~1 ~ - ~ 1 ~ 1
~
conduit
est
IA~
2
et donc
IX-JTXl
I
2
~ y I~ -
l I T x - "IT o
IT s(~)x
u(t)atl 2
a~l <
o
IT I x - s ( t ) ~ l a t 0
Enfin (49) s'en d~duit ais~men~ par densit~ pour tou~ Re marque.
Pour un op~rateur A maximal monotone g~n~ral une estimation du
typ~ I~-Jtxl
< c]x-s(t)~l
exemple dane
H = R ~ , A = rot(+ ~~ ) et noter qu'il existe
que
w e b-G) ,
x-S(t)x = O). Toutefois lorsque
Th~or~me 22.
On suppose que
Ix-Jtxl
v~ > o
A = ~
A = ~
n'est p~
w~ab~e
< p ~ e ~ r ~ p.~
x ~ 0 ett
> 0 tels
o~ a
; alors pour tout
~
D-"~)
ona
~ 21x-s(t)xl
D~monstration.
Soit
Ix-Jtxl - l(x-J=x) -
y e D(A) ; on ~cri~
(y-Jty)
+
(on utilise le falt que A t e s t
(y-Jty)l ~
lx~i + tlA~
lipechi~zien de constante
~). Prenant
y - S(t)~
on conclut ~ l'aide du Th~or~me 17. Lorsque A est op~rateur diff~ren~iel, les r~sultats precedents montrent qu'il y a
un lien tr~s ~troit entre lee 2 probl~mes sulvan~s :
a) u ~ ~tant donned on consid~re le probl~me (~e perturbation singuli~re) u
+ ~ Au
= u~
(r > 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<~)
Come
89
(52)
[v(t+h) + v(t) - 2u(t+h) l2 > 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
<
(o(t) - T I
(56)
fl
v(s)ds , ~-~) < ix-PxI. Ts
l~-~<s)lds
o
On d~duit de (56) que (s Prenant
-
~
, y - s
y = ~'
~
Vy e
0
on voit que
F
% = 9~',
et par consequent
(y(t)-~ s
quand
t -~ +~.
Remarques. I) Une version "discrete" du Th~or~me 23, due ~ Baillon [4 une contraction avec point fix§
slots V~
I affirme qua si T §
CF - ~(~ § Tx § ... Tn-Ix) converge
faiblement vers un point fixe de T qu~nd n -~ 2) Si dans le Th~or~me 23 on fait l'hypoth~se supp~mentaire Vt > O, slots
O(t) converge for~tement quand ~ ~
Le r~sultat suivant dQ ~ Br~ck [ ~ Th~orame 24.
Soit A = ~0
avec
+co
Sit)(-u) = -S(t)u
I con~erne le cas 05
0 e R(A). Alor$
%h~,
A = ~@.
~5( e C , S(t)x converge fai-
blement quand t § o= vers l'un des points d'~qt*ilibre de S. D~monstration. 1~
Indiquons 2 m~thodes
Bolt
S(tn)X---%s
i.e.
s
; comme
:
A~
§ O, d ' a p ~ s
le Th~or~me 17 oI~ a O e As
~ F.
D'autre part (Cf d~monstration dt~ Th~or~me 23) o~% sai~ qUe PS(t)x -~ s
Enfln
on a (5(t)x - P S ( t ) x
, y-PS(t)x) ~ 0
Yy r F.
D'o~ il r~sulte que (~"
- s
, y - ~) <
et par consequent 2~ (57)
s
0
W
(
- i. On concl~t que
S(~)x ~
~.
Montrons q u e S('c)x d,'~I § 0
[S(t)x - T
quand
t * ~.
o
oo
On pourra d~duire le r~sultst du Theorems 23. Er~ effe~, o~ sai~ q~e (Cf.(40))
-o
~T/2 Comme la fonction quand T § §
.
d u . t .)I ~-{~
du est d~croissante on d~duit que ~3 T z I-~(T) I -> 0
Enfin notona que
80
u(t) - g1
It
o u(z)d~
= g~
It
o
~
du(~" dE
dr"
et (57) en r6sulte ais~ment.
Remar~ues. l~
Si l'on fair l'hypoth~se suppl~mantaire
converge fortement quand t ~
~(-u) = @(u)
Vu e H, alors S(t)x
+= (ce r~sultat est d~ ~ ~ruck [ ~
]; on peut aussi
l'obtenir ~ partir de la Remarque pr~c~dente). 2 ~ ) Si l'on fair l'hypoth~se suppl~mentalre et
~(u) < M}
~M ~ R, l'ensem~le {u e H ; lu] < M
est compact, alors il est facile de v~rifier qua S(t)x co~verge
fortement quand t § +co . 3~
II semblerait, d'apr~s Komura, q~'en g~n~ral S(~)x ne =onverge pas fortement.
Un example.
Soit B : R § R
~ne fomction c ~ o i s s a ~ e st continua avec 0 e R(6)
On consid~re l'~quation ~ - ~ - - Au ~ o
-
~n
~(u)
u(x,O) = ~ (x) On peut mettre (58) sous la forme (u) = ~I Ifl ~+ ( ~"i ) ' et -
r
= §
suz
fax [0, + ~ [
sur
r x [0, §
s~
~
.
du + ~ ( u ) ~-~
;s J ( ~ ) ( o ~ j ' - 6 )
= 0
dans
po~r ~ e
}I = L 2~Q)
~16~) avea j(u) e Li(s
ailleurs. On montre (Cf [ ~ 1) q~e Au = -d~
ana--u= S(u) sun r }. comte
{u ~ ~ ; [=[L= < M
~ver
et ~(u) < M
~vec
D~%) = {u
e
l~2(fi);
} est compact po=r
tout M, les consid~ratiorts pr~c~dentes morttrent clue ti(.,t) -> 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
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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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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 ,
For
each
fixed
u 6 Ci,
the
u
9
s e t S(u)
is n o w t h e
set of all vectors
v
satisfying
I
(1.13)
Therefore, term,
be seen,
veQ(u)
(1.8)
yw6Q(u)
in t h i s Q by the
(2.1)
f(u,-,-),
we
problem
Problem
sense
(1.4)
of the
can thus
of f i x e d - p o i n t s
of the
u E C.
9
AND VARIATONAL
INEQUALITY
FO~4S
shall
two given
in t h e u s u a l
(1.11).
: a) T H E F ~ ' S
FOR BILINEAR
section
involving
mapping
as a s e l e c t i o n
functions
PROBLEMS
INEQUALITIES In t h i s
case,
.
in n o w a s e l e c t i o n ,
given multivalued
2. V A R I A T I O N A L
own,
,
f ( u , v , w ) <_0 ,
the map
of the
mapping
v 6 C
study problem
as a p r o b l e m
(1.7)
on its
functions
: C ~ ]-~,+~]
,
not
- +
and (2.2)
g
: C x C ~ ]-~,+~[ , w i t h
for e v e r y
We shall
(2.3)
assume
v 6 C
g ( v , v ) <-0
.
that
,
C is a c l o s e d
I
~ is c o n v e x
convex
subset
of a Hausdorff
t.v.s.
E
and that
(2.4)
A s t o the (2.5)
l.s.c,
g, w e
o n E.
shall
first make
the
1
(i) F o r e a c h v 6 C,
g(v,.)
is c o n c a v e
I
(j) F o r e a c h w 6 C,
g(-,w)
is l . s . c ,
We also tisfied
function
and
require
that
the following
following assumption
o n E.
coercivene88
condition is sa-
92
There such
exists that
a compact
subset
B of E a n d a v e c t o r
w.6 B AC
,
~(w0) < + =
and (2.6)
(v) + g(v,w0)
> ~(w0)
for all v 6 C \ B
Then,
THEOREM
# and g be g i v e n
respectively.
satisfied.
the
following
2.1
L e t C, (2,5),
w e can p r o v e
Then,
Let
the
that
the
set of a l l
~
satisfy
(2.3),(2.1),(2.4)
coerciveness
condition
solutions
(2.6)
and
(2.2)
be a l s o
v of t h e p r o b l e m
vEC
(2.7)
is a n o n - e m p t y
(v) + g(v,w)
compact
subset
u
< ~(w)
of B ~ C,
B being
any
set that
verifies
(2.6).
9
Let
us r e m a r k
that
(2.7)
as a non-empty intersection
can be s t a t e d
problem for the f a m i l y of sets (2.8)
if
G(w)
,
w E C
,
we define
(2.9)
G(w)
In fact, rized
the
as the
= {v E C : ~(v) + g ( v , w ) <_~(w)} set of all v e c t o r s
v that
;
satisfy
(2.7)
can be c h a r a c t e -
set
(2.10)
N
G(w)
.
wqC
Proof of Theorem 2.1. U n d e r the a s s u m p t i o n s family
of sets
G(w) lower
the
of the
is oompaot,
(2.6)
set B ~ C
has
following
is closed for e v e r y
semicontinuity G(w,)
fact,
(2.8)
;
implies
where
that
w 6 C
functions Wo
G(w0)
: this
theorem,
is a c o n s e q u e n c e
~ and g(.,w)
is the v e c t o r is a
of the
the
properties:
(closed)
appearing subset
of the
; in
of the
(2.6)
: In
(compact)
93
If W , , . . . , w n is an a r b i t r a r y
n [ i=I
w =
Xiw i
finite
,
with
subset
X
> 0 i --
of C a n d
u i ,
n [ i=I
X
= I, i
then we have
w 9
In fact,
if the
i = I,...,
n
U i=1
contrary
G(wi)
is true,
, t h e n we h a v e
%(W) for all
i = I,...,
this
that
implies,
%(w) < + =
%(w i) < + ~
+ g ( w , w i)
is if w s G ( w i)
for all
and
> %(W i)
n. T h e r e f o r e ,
~(w) and
that
+
n [ i=1
by the
l i g ( w , w i) >
concavity
n [ i=I
l i ~ ( w i)
of g(w,-)
and
the c o n v e x i t y
of
~,
and
(w) + g(w,w)
> %(w)
hence,
g(w,w)
that
contradicts The
the h y p o t h e s i s
conclusion
lemma
below.
Lemma
2.1
(i) T h e
(ii) Then,
are
subset
satisfied
convex in the
F(w0)
is t h e n
function
g.
a consequence
if Ky F a n ' s
F(w)
h u l l of e v e r y
is c o m p a c t
of E be g i v e n ,
set
in a t . v . s . E .
such
that
the
To e a c h
following
:
corresponding
finite
union
for at l e a s t
subset
{ w l , . . . , w n} of T is c o n -
F(w,) U ... U F ( W n )
;
one w0 E T
we h a v e
(2.11)
Remark
2.1
on the
[3~ ) L e t T be an a r b i t r a r y
w E T let a c l o s e d
tained
(2.2)
9
(Ky F a n
conditions
of T h e o r e m
> 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
Proof. A p p l y T h e o r e m (2.12)
theorem
For
compact
subset
on C • C
variable
u
and
u that
of a
Hausdorff
, which
l.s.c,
t.v.s.
is <_ 0 o n t h e
in t h e
first va-
satisfies
.
2.1 w i t h
is o f t e n
2.1.
9
r - 0 .
referred
9
t o in the
as Ky Fan's
literature
(minimax) inequality, COROLLARY
9
2 of THEOREM
2.1
(Stampacchia
Let V be a real Hilbert of V,
a(.,.)
every
continuous
vector
a coercive
u that
(2.13)
linear
g(v,w)
In p a r t i c u l a r , since that
(e) W e
bilinear o n V,
< (v',
v'
2.1
u-w)
with
= a(v,v-w)
It is e a s y t o c h e c k
fact
(~)continuous functional
convex
f o r m o n V.
there
exists
Then, a
[36 ] ). subset for
(unique)
YweC
9
E = V endowed
with
,
.
that
all
the
recall
assumptions
f o r e a c h w 6 E,
i t is c o n t i n u o u s f o r m a(-,-)
that
v, w E C
the
its w e a k
o n V and,
XII v II.U d e n o t e s
112 < a(v,v)
topology,
the norm
of t h e
on a Hilbert
y > 0 such
v 6 V
s p a c e V.
are
is c o n v e x ,
and non-negative,
f o r m a(-,-)
for a l l
2.1
v ~ a(v,v-w)
moreover,
is b i l i n e a r
a bilinear
of T h e o r e m
function
coercive if t h e r e e x i s t s a c o n s t a n t
where
closed
and
(2.14)
E,
[42 ], L i o n s - S t a m p a c c h i a
C a non-empty
e C
a(u,u-w)
Proof. A p p l y T h e o r e m r 5 -v'
space,
satisfies
-u _
its
138 ~ )
a vector
,
of t h e o r e m
defined
second
exists
-u E C g(u,w)
Remark 2.3.
convex
function
in the
there
fixed-point
loc.cit.
,
that
satisfied.
is l.s.c, due i.e.
on
to the a(v,v) ~ 0
s p a c e V is s a i d
95
for all v 6 V. As to the c o e r c i v e n e s s tisfied
by taking w0
to be an a r b i t r a r y
~R = {v E V
with R > 0 sufficiently coercive,
hypothesis vector
: ,.v.,<_R}
large.
(2.6),
it is now sa-
of C and B = ZR' w h e r e
,
In fact,
since
the form a(.,.)
is
we have
1-a(v,v- w0) -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
<_ r
introduce
(3.9)
when
~ ~(w)
if it is a s o l u t i o n
(3.8)
If we
+ g(v,w)
H(w)
= {v6C
the e q u i v a l e n c e
cidence
of the
set
of (2.10)
: %(v)
(3.7) with
with the
< %(w)
+ g(W,V) },
(3.8)
is n o t h i n g
set
else
than
the
coin
98 (3.11)
,"1 weC
LEM~4A 3.1.
Let C satisfy
(2.3),
(2.2) (3.3). L e t the sets G(w), by
(2.9),(3.5)
Then,
and
(3.10)
H(w)
r satisfy F(w)
(2.1)(2.4)
and H(w)
and g satisfy
be defined,
for e a c h w 6 C ,
respectively.
we have
(3.12)
N F(w) w6C
and this
common
=
O H(w) w6C
intersection
=
O G(w) w6C
is a c o n v e x
closed
subset of C.
Proof. Since G(w) C F(w) for e v e r y w q C, in order to p r o v e suffices
to prove
(3.13)
n weC
Let us prove H(w)
is concave
closure which
(3.12)
it
that
F(w)
C
N w6C
the first
is c o n v e x and closed,
g(w,.)
9
H(w)
C
inclusion
n w6C
G(w)
above.
For every w 6 C, the set
for r is c o n v e x and l.s.c,
and u.s.c,
of G (w), it suffices
by
(3.3)(i).
to prove
Therefore,
that G(w)
by
(2.4)
and
since F(w)
is the
C H (w) for e v e r y w 6 C,
is to say that
(v) + g(v,w)
< d~(w)
implies
r
for any v 6 C. This that holds
that
is not true that
(3.8)
of the m o n o t o n i c i t y
(3.7)
esists
inclusion
a vector
for some v e c t o r w E C w i t h
~(w) < + |
Consider
in
(3.13),
for any v 6 C. Suppose
thus
(3.14)
(v) + g(v,w)
v t = t~ +
(2.4)
and
(1-t)v
we have
> r
(3.3)(j),
,
that
t e[0,1
the f u n c t i o n
].
is,
let us
in fact that this
v 6 C satisfying
the v e c t o r
By the h y p o t h e s e s
relation(3.2)
(3.3) (j).
the s e c o n d
implies
: There
+ g(w,v)
is a c o n s e q u e n c e
by a s s u m p t i o n
Let us now prove prove
< r
(3.8),
such
%(v t) + g(v t, W) of the real v a r i a b l e t 6 [0,1 ] is l.s.c,
as t ~ 0 +. Therefore,
(3.14)
implies that for some { > 0 we have
(3.15)
~(v t) + g(vt, w)
>
on the other hand, we have by
(3.16)
%(w)
u
(3.8), w h e r e w = v t ;
~(v t) + g(vt, v) >_ ~(v)
Inequalities
(3.15)
and
and the c o n c a v i t y of g(vt,
EL .
u
].
(3.16) together,
by the c o n v e x i t y of
-), imply that for every t 6 ] 0 , ~ [
(vt) + g(v t, v t) > %(v t) hence ~(vt) < + =
we have
,
and g(vt, vt) > 0, w h a t c o n t r a d i c t s a s s u m p t i o n
(2.2) . .
We are now in a p o s i t i o n to give the
Proof of Theorem 3.1. We must prove that the set convex closed
(hence, c o m p a c t ) s u b s e t of the
(2.10)
(compact)
B is any set that v e r i f i e s the c o e r c i v e n e s s h y p o t h e s i s By Lemma 3.1, the set
(2.10)
is c o n t a i n e d in B A C since, by
Gr
= {v~C
is a non-empty
set B N C, w h e r e (2.6).
is convex and closed. Moreover,
it
(2.6), the set
,(v) + gCv,w0) <_,(w01}
is c o n t a i n e d in B N C. It remains to prove that the set
(2.10)
is non-empty.
A g a i n by
Lem~a 3.1, this is the same thing than p r o v i n g that the set
(3.6)
is
non-empty. We prove that by a p p l y i n g Ky Fan's lemma to the family(3.4). In fact,
each F(w),
w 6 C, is closed and F(w0),
being the closure
of G(w0) w h i c h is c o n t a i n e d in the compact B N C, is itself compact. Moreover,
if w belongs to the convex hull of a finite subset
{w, ,..., w n} of C, then we can prove e x a c t l y as in the proof of T h e o r e m 2.1
(where only the c o n v e x i t y of % and the c o n c a v i t y of g(W,o) were
used)
that n w 6
U
i=I hence,
"a fortiori",
G (W i)
100
n U F (w i) i=I
W 6
Therefore,
the
family
of sets
(3.4)
satisfies
the a s s u m p t i o n
of L e m m a
3.1. T h e o r e m 3.1 y i e l d s ties
involving
Theorem
following
result
that
for v a r i a t i o n a l
generalizes
inequali-
Corollary
2 of
2.1.
We r e c a l l Banach
the
monotone o p e r a t o r s ,
space
that
an o p e r a t o r
X to its d u a l
X'
A from
a convex
subset
C of a r e a l
coercive if the f o l l o w i n g
is said
condi-
tion holds (3.17)
There
exists
I- ( Av, as COROLLARY
UvU ~ =
of T H E O R E M
L e t X be a r e a l of X, A a c o e r c i v e dual
a vector
w0 6 C ,
such
that
v-w0 > / U v113 ~ + , v e C 3. I
.
(F.E.Browder
reflexive
monotone
[ 13] , H a r t m a n - S t a m p a c c h i a
B a n a c h space, C a c o n v e x
and h e m i c o n t i n u o u s
mapping
closed
[ 22] ) 9 subset
f r o m C to the
X' of X. Then,
for e v e r y
functional
v' 6 X ' ,
the
set of all v e c t o r s
v sa-
tisfying (3.18)
-v 9 C (Av,v-w)
is a n o t - e m p t y
<_ ( v ' , v - w )
bounded
Proof. A p p l y T h e o r e m and g given
by
coerciveness of
(3.17)
with
As
hypothesis
(2.6)
bY taking
(2.7).
this
We use
whenever
to D e f i n i t i o n
convex
subset
E = X with
can
of C.
the w e a k
topology,
of the p r e v i o u s n o w be s a t i s f i e d ,
to b e t h e v e c t o r
~ = -v'
theorem,
the
in c o n s e q u e n c e
appearing
in
(3.17)
and
uniqueness r e s u l t
for
large. with
9
the
following
the
3.1
g is strictly monotone and,
moreover,
the
if it is m o n o t o n e
sign
< holds
in
(3.2)
3.1.
solution
and/or
g
v of p r o b l e m is s t r i c t l y
Proof. L e t the v e c t o r s and
convex
v ~ w.
PROPOSITION The
Wo
section
Definition 3.3. A f u n c t i o n according
9
in the p r o o f
R > 0 sufficiently
We c o n c l u d e problem
3.1
(3.1).
above,
B = ZR w i t h
closed
YweC
vl,
v2
(2.7)
is u n i q u e ,
provided
r is s t r i c t l y
monotone. both
be
solutions
of
(2.7),
i.e.
v i6C
I01
r
i) + g(vi, w) < % ( w )
i = 1,2. By taking w = that C is convex)
(v, + v ~ ) / 2
u
,
in the two i n e q u a l i t i e s above(recall
and summing up we get, by the c o n c a v i t y of g(vi,.),
I +~l~(vl ,vl) +g(vl ,v2) + g ( v 2 ,v,) + g ( v 2 ,v,)3 <_24((v1+v2)/2) .
(v,) + r
Since g(vl ,v,) -- g(v2 ,v2) = 0 (recall the remark f o l l o w i n g D e f i n i t i o n 3.1) , we find
r
I + ~ l - g ( v , ,v2) +g(v2,v,)- I <_ 2r
+r
+v2)/2)
,
and this implies, by the c o n v e x i t y of ~ and the m o n o t o n i c i t y of g,that both the i n e q u a l i t i e s b e l o w hold
:
(3.19)
%(v])
< 2%((v, + v 2 ) / 2 )
(3.20)
g(v,,v2)
Thus,
+ r
+ g(v2,v,)
the c o n c l u s i o n vl = v 2
convex and from
,
< 0
follows f r o m
(3.19) if ~ is s t r i c t l y
(3.20 if g is strictly monotone.
9
4. E X I S T E N C E RESULTS F O R THE G E N E R A L IMPLICIT P R O B L E M OF SECTION I Let us
go
back to our implicit s y s t e m of i n e q u a l i t i e s
(1.4). As
we already r e m a r k e d in section 1, the first step for solving p r o b l e m (1.4) consists blem
(1.7)
in solving,
for each fixed u 6 CI, the v a r i a t i o n a l pro-
for the couple of functions
(1.5)
and
(1.6). This can now
be done by simply a p p l y i n g T h e o r e m 2.1 or T h e o r e m 3.1. In order to carry out the second step, n a m e l y p r o v i n g that the s e l e c t i o n map
(1.8) has a f i x e d - p o i n t in the given set C0, we c l e a r l y
need some i n f o r m a t i o n about the m a p p i n g s
(4.1)
u
~
~(u,.)
,
u
~
f(u,-,.)
,
In this chapter we shall only c o n s i d e r the m a p p i n g s lection map
u
6
Co
topological
p r o p e r t i e s of
(4.1), which will be first d e s c r i b e d in terms of the se(1.8)
of the functions
and then d i r e c t e l y in terms of t o p o l o g i c a l p r o p e r t i e s ~ and f.
We shall confine o u r s e l v e s to functions g(',.)
= f(u,.,')
,
u 6 C,,
102
are monotone,
which
according
cases we are i n t e r e s t e d notone
in,
to D e f i n i t i o n
3.1,
for this
as q u a s i - v a r i a t i o n a l
covers
inequalities
the
for mo-
operators.
It turns out to be useful set Co
, appearing
in p r o b l e m
that i n d u c e d by the Co
(4.2)
in these a p p l i c a t i o n s (1.4),
a possibly
space E. Therefore,
is a n o n - e m p t y
locally c o n v e x
closed
t.v.s.
we
convex
E0 w h i c h
to a l l o w
for the
stronger t o p o l o g y than
suppose
that
subset of a real H a u s d o r f f has a c o n t i n u o u s
injection
E 0 ~ § E into E, w i t h C0C-~ CI
THEOREM
4.1
Let C, Ci, (1.3)
~ and f be given,
respectively.
satisfies verify
(2.4),
Suppose
the f u n c t i o n
the c o e r c i v e n e s s
Suppose, that both
g = f(u,.,.)
that
problem
(1.1),(1.2) the f u n c t i o n
satisfies
(3.3)
to Co.
that
there
exists
and stable under
(1.4)
(u,v)
admits
the set S(u)
In the p r e s e n t
if v is a s o l u t i o n (4.3)
and both
a set Co
as in
(4.2),such
the s e l e c t i o n
S
6 Co • Co, w i t h v 6 S(u),
a solution
is c o n t a i n e d
situation,
is closed
u.
9
this m ea n s
in Co w h e n e v e r that,
S, we u belongs
for any g i v e n u 6 C 0 ,
of the p r o b l e m
- v 9 C
~(u,v) + f ( u , v , w ) ! ~ ( u , w )
_
then v also belongs Eo = E
%=~(u,.)
(2.6).
Remark 4.1. By saying that Co is 8table under the s e l e c t i o n clearly mean
and
b e l o w hold:
is compact
(b) The set of all pairs Then,
(2.3),
for each u E Ci,
condition
furthermore,
conditions
(a) Co
satisfying
that,
to Co.
,
Ywec
If the set C is c o m p a c t
C0 = C, then the a s s u m p t i o n
in E and we choose
(a) of the t h e o r e m
above
is t r i v i a l l y
satisfied.
9
Proof of Theorem 4.1 By a p p l y i n g = ~(u,.) S(u)
Theorem
3.1
and g = f(u,-,.)
of all solutions
to the set C and the couple
, for each
of p r o b l e m
of functions
fixed u E CI, we find that
(1.7)
is a n o n - e m p t y
convex
the set
compact
subset of C. Assumptions
(a) and
(b) of the p r e s e n t
restriction of the s e l e c t i o n maps an u.s.c,
mapping
Therefore,
of Co
into
by K a k u t a n i ' s
2 C~
(1.8)
t h e o r e m both
to the c o n v e x
with non-empty
fixed-point
compact
convex
theorem,
imply that the
closed
set Co is images.
the f i x e d - p o i n t
pr~
103
blem
(1.9),
hence
In m a n y by u s i n g LE~
the
(1.4),
applications
has
a solution.
condition
9
(b) of T h e o r e m
4.1
can be q h e c k e d
following
4.1 Let @, Cz,
respectively. satisfies
(4.4)
(4.5)
and
that
following
(4.3)
~ and
the
function
For
each wCC,
hold
the
For every
generalized with
wqC,
exist
v
6C
,w ) + f ( u
(b) of T h e o r e m
Proof. S u p p o s e t h a t
functions satisfies
is a l s o
and
(1.3)
% = ~(u,-) (3.3).
given,
Suppose,
such
that
:
on Co • Co
,
f(-,w,.)
is u.s.c.
sequence
(u ,v ) ~
), and
for e v e r y
6S(u w
(4.2)
function
,
(2.3) (1.1), (1,2)
the
g = f(u,-,.)
~ is l.s.c,
in Co x Co,
u 6 Ci
satisfying
on Co x Co
then
satisfying
for e a c h
function
properties
The
Then,
such
(u,v)
that
,v ,w) - f ( u a , v a , w
)~ ~ ( u , w )
.
4.1 holds.
(u ,v ) ~
(u,v)
in Co x Co,
with
v
6S(u
). We m u s t
that veS(u). We have,
(4.6)
~(u
Let
(4.7)
and,
that
a set C0
limainf19(u
prove
f be given,
Suppose
(2.4)
furthermore, the
problem
for e a c h
,v ) + f(u
us fix w 6 C
~(U
since
,-
be as
f(u
(4.8)
,w)
in
,v ,w)
and
(4.7)
,
hence,
by
(4.3)
~(u e , v and
we
also
have
~ -f(u
,w,v
)
into
) - l i m sup
(4.4)
,
By
.
(4.6),
we h a v e
for e a c h
,V ) ,
account,
~(u,w) _> l i m a i n f l ~ ( u e ,w ) - f(u e ,v ,w ) - f(u _> l i m a i n f
YwEC
(4.5).
,V ,w ) ~ ~(u
.) is m o n o t o n e ,
(4.8)
By t a k i n g
,v ,w) ! ~ ( u
and w
,w ) - f(u
f(u
e ,
f(u
we o b t a i n
,w,v
,w,v e)
,
)3
from
(4.5),
104
(u,w) >_~(u,v) - f(u,w,v) which
is to say,
since w E C v 6
where
H(w)
is given by
N H (w) wEC
(3.10),
(I .5)(I .6). Since these Lemm a
is arbitrary, ,
% and g in
functions
b e i n g the functions
% and g satisfy
the h y p o t h e s e s
of
3.1, we find that
v
the set G(w) This
being
means
6
N w6
given by
G(w)
,
(2.9), w i t h
that v v e r i f i e s
is,
THEOREM
u
9
4.2
respectively. satisfies
~ and f be g i v e n
Suppose
(2.4),
that,
Suppose, that both
(a) Co
Then,
functions
problem
and both ve-
(2.6). u set C0 as in
(4.2) such
:
~ and f v e r i f y admits
(1.3),
% = 9(u,.)
(3.3)
and stable u n d e r the s e l e c t i o n
(1.4)
and
the f u n c t i o n satisfies
that there exists
b e l o w hold
is compact
(b')The
(2.3),(1.1),(1.2)
g = f(u,.,-)
condition
furthermore,
conditions
as in
for each u 6 C i ,
the f u n c t i o n
rify the c o e r c i v e n e s s
S,
(4.3), (4.4), (4.5).
a solution
u .
9
4.1
If ~(u,.) of the p r e v i o u s satisfied
does n o t d e p e n d on u 6 C 0 , hypothesis
by taking w
In particular, of Ky F a n ' s Remark
,
v6S(u)
Let C, CI,
Remark
~ and g as above.
the i n e q u a l i t i e s
~(u,v) + f ( u , v , w ) <_~ (u,w)
that
(3.10)
~w
when
inequality,
then
(2.4). Moreover,
(4.3)
(4.5)
is a c o n s e q u e n c e
can be t r i v i a l l y
. ~ ~ 0, T h e o r e m
see
4.2 yields
the f o l l o w i n g
Theorems
an
"implicit"version
6.1 and 6.2.
9
4.2
If f ~ 0, then ction ~ d e f i n e d
(4.5)
to the c o n d i t i o n
for e v e r y v E C by setting
(4.9)
is l.s.c,
is e q u i v a l e n t
8(V)
on the set C0.
= inf w6C
~(v,w)
that the fun-
105
This
is c e r t a i n l y
8(v) ~ 0. L e t us a l s o dition
(4.3)
Therefore,
states
true whenever
remark that
that
~ is o f t h e
for s u c h
the m u l t i v a l u e d
in t h i s c a s e ,
Theorem
form
(1.10),
for then
a 9, if C0 = C I = C , mapping
(1.11)
4.2 r e p r o d u c e s
con-
is u . s . c .
Kokutani's
fixed-point
theorem.
9
In the Theorem
4.1
following a n d 4.2.
sections For
Co = C, = C in t h e a b o v e trivially
consider
sake of simplicity,
theorems,
properties
for instance,
of t h e
the c r u c i a l
then we might
functions
~(u,.)
in o u r a p p l i c a t i o n s
quasi-variational point
in f i n d i n g
5. N A S H
in m o s t
This
what makes
many
allows
special
cases
cases we
shall
of
us to t a k e E0 = E,
the c r u c i a l
assumption
(a)
satisfied.
If C is n o t c o m p a c t ,
sists
shall
t h a t C~ = C a n d t h a t C is compact.
assume
ness
we
EQUILIBRIA
suitable
and
f(u,.,-),
of T h e o r e m
of Sections
in v e r i f y i n g
the assumption
UNDER
6 and
is l e f t
as
it is t h e case,
4.2 t o v a r i a t i o n a l
inequalities
o u t a set Co w h i c h
uniform c o e r c i v e -
use
8.
In t h e s e
and
problems,
(a) of T h e o r e m
4.2
con-
stable b y t h e s e l e c t i o n
S.
CONSTRAINTS
Let
C = C1x...xC
be a subset
in a p r o d u c t
space
E = E,x...xE
J1,...,J n l e t Qi(u)
n real-valued be a
tisfies
the
functions
(non-empty)
A Nash equilibrium following
n
subset
is t h e n
n defined
of C i ,
any vector
o n C and,
for each
(ul .... ,Un) E C
that
sa-
conditions
u i E Qi (u) Ji(ul , . . . , u i .... ,u n) <_J(ul , . . . , v i , . - . , u n)
In o t h e r w o r d s , c t i o n Ji w i t h
the v e c t o r
respect
constraints An existence
ry 2 a b o v e ,
,
i = 1,...,n. u=
~
(5.10)
cit)
u 6 C
to the
u is s u c h t h a t
i-th variable
u vi6Qi(u)
it m i n i m i z e s
only,
subject
each
to t h e
.
fun(impl~
u i6Qi(u). result
by taking
for this
~ to b e t h e
(U,W)
problem
c a n be o b t a i n e d
function
= 6 (Q(u),w)
+ J(u,w)
,
u,weC
from Corolla
106
with
Q(u)
= Q(u)x...XQn(U) I
and
J(u,w) We have, THEOREM
indeed
n [ J i ( u i , .... u i _ 1 , w i , u i + 1 , .... Un). i=1
=
, the
following
5. I
Let locally
C be a n o n - e m p t y convex
Let
space
convex
subset
of a real
Hausdorff
C.
~ be a f u n c t i o n
on the p r o d u c t
values
in ] - = ,
+ = ], w h i c h
(5.6)
For
uqC,
(5.7)
~ is l.s.c,
(5.8)
The
each
compact
has
9(u,.)
or C x C
real-valued
the
following
is convex,
extended
real
properties:
not-+=,
,
function
8(v)
set C x C w i t h
=
8 defined
inf wqC
on C by
~(v,w)
vEC
,
,
is l.s.c. Then,
the
set of all v e c t o r s
(5.9)
~(u,u) < ~(u,w)
is a n o n - e m p t y
compact
subset
u6C
,
such
that
u
of C.
Proof. Apply taking
Theorem
Remark
6. I M P L I C I T
4.2
4.2 w i t h into
KY F A N ' S
INEQUALITY
HARTMAN-STAMPACCHIA L e t us s u p p o s e quality consider (6.1)
(2.12) the
does
E. = E
THEOREM
that
the
depend
following
f , 0, ~ as above,
, Co = C I = C ,
by
account.
FOR MONOTONE FOR
"MONOTONE
function
on the
g that
solution
FUNCTIONS. PLUS
appears
u itself,
problem I uec f(u,u,w)
<_ 0
COMPACT"
u
C
OPERATORS
in Ky F a n ' s that
is,
ine-
let us
107
F r o m T h e o r e m 4.2 we o b t a i n the following. T H E O R E M 6.1 Let C be a n o n - e m p t y c o n v e x c o m p a c t s u b s e t of a real H a u s d o r f f l o c a l l y c o n v e x space E. Let f be a r e a l - v a l u e d f u n c t i o n on the p r o d u c t
set C x C x C, w i t h
the f o l l o w i n g p r o p e r t i e s :
(6.2)
For e v e r y p a i r u , v 6 C ,
f(u,v,-)
(6.3)
For each u 6 C ,
is m o n o t o n e and h e m i c o n t i n u o u s ,
f(u,-,-)
d i n g to D e f i n i t i o n s
(6.4)
For each v 6 C ,
Then,
3.1 and 3.2
f(.,v,.)
is c o n c a v e
,
accor-
,
is u.s.c.
the set of all v e c s
u that s a t i s f y
(6.1)
is n o n - e m p t y
and compact.
9
Proof. A p p l y T h e o r e m 4.2 w i t h Ee = E, take R e m a r k 4.1
Co = C l = C
~ ~ 0 , f as above and
into account.
9
It is i n t e r e s t i n g to c o m p a r e e x p l i c i t e l y C o r o l l a r y
I of T h e o r e m
2.1 w i t h the f o l l o w i n g C o r o l l a r y of the t h e o r e m above, w h i c h n e d w h e n the f u n c t i o n COROLLARY
f(u,.,.)
does not d e p e n d on u E C
is o b t a i -
:
I of T H E O R E M 6.1
Let C be as in T h e o r e m
6.1
and g a r e a l - v a l u e d
f u n c t i o n on C x C
,
such that
(6.5)
For e a c h v 6 C ,
(6.6)
g is m o n o t o n e and h e m i c o n t i n u o u s .
Then,
problem
Notice
that,
g(v,-)
(2.12)
is c o n c a v e and u.s.c.
a d m i t s a s o l u t i o n u.
in C o r o l l a r y
9
due to the m o n o t o n i c i t y of g, it has b e e n p o s s i b l e
to w e a k e n c o n s i d e r a b l y the h y p o t h e s i s pears
,
I of T h e o r e m 2.1
t h a t g(-,w)
be l.s.c,
t h a t ap-
(see also the r e m a r k s p r e c e d i n g
T h e o r e m 3.1). The existence
i m p l i c i t Ky F a n ' s
inequality
r e s u l t for v a r i a t i o n a l
compact operator,
(6.1)
can be u s e d to o b t a i n an
inequalities
i n v o l v i n g a monotone
+
that g e n e r a l i z e s the C o r o l l a r y of T h e o r e m 3.1.
B e f o r e s t a t i n g this result, the i m p l i c i t Ky Fan's
inequality,
let us state the f o l l o w i n g v e r s i o n of in w h i c h the c o m p a c t e n e s s
assumption
108
for t h e
s e t C is r e p l a c e d
by a coerciveness
condition
on the
function
f. THEOREM
6.2
L e t C be a c o n v e x s p a c e E,
hypotheses exists
closed
f a real-valued (6.2)(6.3)
a non-empty
subset
function
and
(6.4).
convex
compact
of a real Hansdorff
defined Suppose,
locally convex
on C x C x C t h a t furthermore,
s u b s e t Co
satisfies the
that
there
of C and a vector
w0 E C
,
such that
(6.7)
f(u,v,w0)
Then,
problem
(6.1)
> 0
has
for every
uqC
and all vqC-C0
a solution.
9
Proof. Apply Note,
Theorem
4.2,
in p a r t i c u l a r ,
by any
function
Moreover,
the
associates
the
E0 = E, CI = C ,
g = f(u,-,-)
selection
map
s e t S(u)
(6.8)
with
with
, uqC,
any uqC. uEC,
COROLLARY
Therefore
any vector of THEOREM
L e t X be a r e a l
for the p r o b l e m
the
< 0
following
(6.9)
(6.10)
Then,
(6.11)
is t h e m a p
that
satisfying
is s t a b l e (6.8)
under belogs,
Banach
space,
S
: in fact, by
(6.7),
C a convex
closed
for
to C o . 9
[22])
from C to the dual
X'
of X that
subset satisfy
conditions:
A is c o e r c i v e
monotone
B is c o m p a c t ,
i.e.,
in C t o t h e
the
B(C)
image
the v a r i a t i o n a l
~
hemicontinuous
Continuous
sequences
strong
from
topology
o f C is b o u n d e d
,
weakly
convergent
of X',and,
moreover,
in X'.
inequality
uEC
(Au+Bu,
admits
is v e r i f i e d
B = C0
at hand
(Hartman-Stampacchia
reflexive
as a b o v e .
(2.6)
VwEC
set Co
of X and A and B two mappings the
vqC
v that verifies
6.2
hypothesis
by taking
of all vectors
f(u,v,w)
every
~ ~ 0, f a n d Co
that the coerciveness
a solution
u-w>
u.
< 0
ywC
C
9
109
Proof. Apply
Theorem
(6.12)
6.2 w i t h
f(u,v,w)
a n d Co of t h e
E = X weak,
=
v-w)
C as above,
,
u,v,w6
f given
by
C,
form
Co = C N z R with
ZR = {vqX:llvll ! R }
lar,
that
ded
in X',
and R > 0 s u f f i c i e n t l y
s i n c e A is c o e r c i v e ,
UvD ~ =, v 6 C ,
vided
7.
We (7.1)
suppose
in t h i s
in p a r t i c u B(C)
is b o u n -
space
has
t h a t we are the
subset
: C xC ~
and
we want ,
g(u,w)
is a s p e c i a l
6.2
B.
FUNCTIONS
of a r e a l
Hausdorff
that
associates
each
given
uqC
a real
a non-empty
convex
.
valued
function
g on
3.2
and hemicontinuous a n d g(v,.)
according
is c o n c a v e
to
a n d u.s.c,
for
.
uec
E
3.1
of T h e o r e m
of the m a p
properties
is m o n o t o n e
v E C
The problem
also
following
Definitions every
Q(u)
pro-
E
of C w i t h
subset
(6.4)
(6.12),
that
mapping
considered
and
function
compacteness
BY M O N O T O N E
section
by the
the h y p o t h e s i s
closed
(7.4)
large.Note, holds,
~ + ~
of the
Q is a m u l t i v a l u e d
g
This
Moreover,
compact
convex
We suppose
(7.3)
v - w0>
in c o n s e q u e n c e
C is a c o n v e x
C • C which
(3.17)
is s a t i s f i e d
OF F I X E D - P O I N T S
locally (7.2)
(6.7)
enough.
satisfied
SELECTION
+ Bu,
thus
R is l a r g e
is n o w
is
then
(Av
as
that
is the
following
one
:
ueQ(u) ! 0
case
in R e m a r k
to c o n s i d e r
,
of the
1.1.
For
YweQ(u)
fixed-point sake
selection
of s i m p l i c i t y ,
problem
w e are
(1.12)
assuming
here
110
that
f(u,.,.) We will
= g(-,-)
use
for every u 9
.
the f o l l o w i n g
definition
the mapping
: C ~ 2 C is continuous
Definition 7.1 We nuous
say that
below
are
satisfied
Q i8 u . s . e . , (7.5)
to
(u,v)
we have
in t h e
Q i8 l.s.o, (7.6)
THEOREM
if b o t h
in t h e
conti-
converging
v o 9 Q(u
(u ,v ) c o n ) ,
;
following
9
sequence
satisfying
limit v 9
sequence
sense
to u in C,
) such that w
: If u
then
is a g e n e r a l i z e d
for every w 9 C there
converges
t o w in C.
7.1
(7.3)
t h a t C, Q a n d g v e r i f y
the assumptions
(7.1), (7.2)
respectively.
L e t us s u p p o s e ,
(7.7)
for every generalized in C x C a n d
exist w
Let us suppose and
i.e.
verging
Q
:
in a d d i t i o n ,
Q is c o n t i n u o u s
that
according
to D e f i n i t i o n
7.1
,
and
(7.8)
g is l . s . c ,
Then,
on C x C
problem
(7.4)
Theorem
4.1
.
admits
a solution
u.
9
Proof. Apply
and f(u,-,.) problem (7.9)
E = E0
associates
the
f i x e d u 9 C. Now,
let
,
u9
C = C, = C 0 Note
that
,
6
Q(u
by
(1.10) S of t h e
v satisfying
Vw 9
(u ,v ) in C x C a n d
v
, ~ given
the selection
s e t of a l l v e c t o r s
is,
(7.10)
and
at hand
with
for every
r v9 , v9 L g(v,w) ! 0
to each that
= g(.,.)
)
satisfying
v
9
S(u a)
111
(7.11)
By
g(ve,w)
(7.5), we have
< 0
in the
limit
(7.12)
By we
YwEQ(u
from
).
(7.10),
v 9 Q(u)
(7.6),
for a r b i t r a r y
converges
fixed w 9 C we can find w
to w in C. Since
(7.11)
e) such that
9
implies
g ( v e , w e) < 0 for every
e, and
the h y p o t h e s i s
(ve, w ) c o n v e r g e s
(7.8)
(7.13)
Since w 9 C was
to
of the theorem,
g(v, w)
< 0
(7.12)
and
arbitrary,
(v,w)
that
in C x C
, we o b t a i n
from
in the limit
(7.3)
together
yield
v 9 S(u) Thus,
the h y p o t h e s i s
(b) of T h e o r e m
If g ~ 0, the c o n c l u s i o n exists
a fixed-point
r equi r e s (7.1)
and
only
of the m a p p i n g
that Q be u.s.c.
(7.2),
the content
was used for the proof give a v e r s i o n u.s.c,
and,
4.1).
l.s.c,
that this r e s u l t
under
fixed-point
Therefore,
above w h e r e
9
is simply that there
Q, and we know
of K a k u t a n i ' s
of T h e o r e m
in addition,
is satisfied. 7.1
(this is indeed,
of the t h e o r e m
ding to the d e f i n i t i o n
4.1
of T h e o r e m
the a s s u m p t i o n theorem,
that
it is w o r t h w i l e
the map Q is s u p p o s e d
with respect
to the f u n c t i o n
to
to be
g, accor-
below.
Definition 7.2. We say that the m a p p i n g where
property
l.e.c, with respect to g,
Q
: C ~ 2C
real v a l u e d
function
on C x C
For every g e n e r a l i z e d
sequence
(ue,v ~) c o n v e r g i n g
g is a g i v e n
is
, if the f o l l o w i n g
holds:
in C ~ C
to
(u,v)
and s a t i s f y i n g
i
(7.10)
v e E Q (ue) g(ve,w) <_0
and for e v e r y w e Q ( u ) lim a inf
g(ve,w)
-
YweQ(u
e) ,
, there exist w e 6 Q ( u g
(v a
,w
<
0
.
) , such that
112 Note w
that
to be,
every
if
g ~ 0, t h e n
for each
(7.10)
e, a n y v e c t o r
is t r i v i a l l y
of Q ( u
)
(recall
satisfied
by taking
t h a t Q(u)
# @ for
u E )
THEOREM
7.2.
Let us suppose
t h a t C, Q a n d g v e r i f y
and
(7.1),(7.2)
(7.3) ,
respectively. L e t us s u p p o s e ,
in a d d i t i o n ,
(7.11)
Q is u . s . c .
,
(7.12)
Q is l.s.c,
with
that
respect
to g,
according
to D e f i n i t i o n
7.2.
Then,
problem
(7.4)
admits
Theorem
4.2 w i t h
a solution
u.
U
Proof. Apply f(u,.,.)
= g(-,.)
hypothesis
(b')
for every
of t h a t
quence
of the present
(4.5),
for
8. Q U A S I
E = E0,
u 6 C. N o t e ,
theorem
L e t us s u p p o s e
(8.1)
X,
(8.2)
closed
and a subset
A multivalued u E CI,
(8.3)
is t h e p r e s e n t
a coercive
of C
mapping
a non-empty
A mapping
A that
A functional
We want
inequality
Q that
(7.4)
and
(7.12).
9
OPERATORS
following
data:
associates,
closed
associates,
monotone
is a c o n s e -
from
Banach
,
convex
with
hemicontinuous
f r o m C to the d u a l X'
(8.4)
the
follows
(1.10),
that the
(4.3)
s e t C in a r e a l r e f l e x i v e Cl
by
hypothesis
FOR MONOTONE
that we are given
A convex
since
(7.11),(4.4)
INEQUALITIES
~ given
in p a r t i c u l a r ,
is s a t i s f i e d
hypothesis
~ and f as above,
VARIATIONAL
C = CI = Co
of X
subset
with
each
Q(u)
of C
,
e a c h u E C,, operator
A ( u , .)
,
v' E X'
to f i n d a s o l u t i o n
u of the
following
quasi-variational
113
(8.5) I
uEC1
,
uEQ(u)
(A(u,u), u - w ) <
(v', u - w )
u
.
The selection map of this p r o b l e m is the map S that associates, with u E C~, the set S(u) (8.6)
I
v6c
,
of all vectors v satisfying
vqQ(u)
(A(u,w), v - w >
<_ (v', v - w )
Yw6Q(u)
We shall use the following
Definition 8.1. If we are given A real reflexive Banach space X0 that has a continuous (8.7)
injection C~
into X and a non-empty
subset Do of X0
, such that c0C-~ Cz,
then we say that the mapping Q is we~kly if property
(8.7) b e l o w holds
For every sequence
v e Q(u)
(ii)
Yw6Q(u)
(8.8)
(Uk, Vk)
u
on Do
converging w e a k l y to
(u,v)
S(Uk), we have in the limit
, ~ W k 6 Q (Uk)
limksup(A(Uk,Vk), (iii)
(A,v')-continuou8
:
in Do • Do and satisfying v k 6 (i)
convex closed
such that
w - w k) >_ 0
,
C ,
limkinf(A(Uk,W)
, Vk-W)
_>(A(u,w), v - w >
THEOREM 8.1. Under the assumptions and Do satisfying (8.7)
suppose that there exists Xc
(8.7), such that both continuous
below hold
Do is stable under S and the image S(D0) is bounded Q is weakly
(8.9)
(8.1)-(8.4),
in X0
,
(A,v')-continuous
to Definition
8.1.
on Do, according
:
114
Then,
there
exists
a solution
u of
(8.6)
a n d u 6 D.
,
Proof. Apply ction
function
=
E = X weak,
~(u,w) = 6 ( Q ( u ) , w ) - ( v ' , w )
(8.11)
Eo
4.2, w i t h
C a n d C,
as a b o v e ,
the fun-
~ given by
(8.10)
the
Theorem
f given
f(u,v,w)
Xo
weak
,
uqC,
, w6C
,
,
u6Ci
, v,wEC
by
= (A(u,v),v-w)
,
, and
Co = 6 o s (Do) Since
S (Do)
weakly Co
is b o u n d e d
compact
is s t a b l e
in Xo,
in Xo, w h i c h
under
S,
by Krein-Smulian
is to say,
since
S(D0)CD0
Co
t h e o r e m , Co
is c o m p a c t
implies
is
in Eo. M o r e o v e r ,
Co = ~ S < D 0 ) C D o ,
hence
S (Co) C ~ o S(Do ) = Co.
Remark 8.1. In many
applications
S (Do)
is bounded
perty
of the
sets Q(u),
u E Do,
In fact,
There
in X0,
family
by exploiting
of operators
the norm
it s u f f i c e s
exists
we will succeed in proving that
involved that
a vector
(8.12)
some uniform
A(u,-)
the
w0 6
with
being
( A (u,v) ,v " w 0 >
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) =
,
, w)
,
v,w E V 9
We c o n c l u d e this section by stating e x p o l i c i t l y
an important
special case of T h e o r e m 8.1 that is the basic result used in many applications. We suppose that
a(.,-)
(8.16)
(8.17)
is a coercive c o n t i n u o u s b i l i n e a r
form on a H i l b e r t space V
v' is a c o n t i n u o u s
,
linear functional on V
117
and that
Q is a m a p
(8.18)
that associate
with
of a convex
closed
subset
convex
subset
~(u)
closed
The QVI we are
i u 9 C
interested
,
each vector
u
C of V a non-empty
of V.
in c a n n o w b e
stated
as
u 9 Q(u)
(8.19) a(u,u-w)
therefore, any
the
<_ ( v ' , u - w )
corresponding
fixed vector
u 9 C the
u149
selection
(unique)
,
map
S
: C ~ V associates
with
solution
v = Su
of t h e
following
(8.20)
I
VI
v 9 V,
v 9 Q(u)
a(v,v-w)
<_ < v ' , v - w )
By r e c a l l i n g
Remark
8.1,
case of the above
result
:
COROLLARY
2 of T H E O R E M
The Corollary the hypothesis condition
we
Yw 9
can also
state
the
following
holds
provided
special
8.1.
q of T h e o r e m
that
,
S(D0)
8.1
still
is b o u n d e d
with
the
following
we
replace
coerciveness
:
There
exists
a vector
Wo 9
N V 9
Q(v)
,
Do
such that
a (v,v)
-~ + ~
Uvlivo as
[Ivllv0 ~
=
,
v 9 S(D0)
9
9
118
CHAPTER
VARIATIONAL
AND
FOR MONOTONE
QUASI
2
VARIATIONAL
OPERATORS
INEQUALITIES
IN ORDERED
BANACH
SPACES
I. ORDERED BANACH SPACES Let X
X be
a real
by
a closed
induced
(1.1)
two
noted
assume
by
X
is
u and
u V v,
Every
{v 6 X
that
vectors
and
vector
space,
positive
P =
We any
Banach
and
<
a partial
order
relation
in
cone
: v >_ 0}
a
lattice
(w~ctor
v of
X ]have a
a common
greatest
v 6 X can
then
be
for
common lower
this
ordering.
least bound,
decomposed
upper
denoted
That
bound, by
is, de-
u A v.
as
+ (I .2)
V = V
- V-
where +
V
positive
are
the
the
positive
= v V
and
negative p a r t
and
cone
0
(1.1)
decomposition
(I .3)
is a c o n s e q u e n c e of
X,
of
the
v,
P = X
and,
we
In o t h e r
words,
have
.
generally,
,
translation
0
respectively.
i.e.
more
+ v A w
= -v A
v,weX
invariance
the
identity
the
order
,
of
relation
via t h e i d e n t i t y
(1.4)
v V w
In that
(I .2)
v + w = v V w
of
generating~
is
P-
The
v
fact,
(1.3)
(-w) V (-v) From
+ z =
(I .4)
= we
(v +
z) V ( w
follows
-(w
/~ v)
also
from =
obtain
-(v
+ z)
(I .4)
,
when
/k w)) o
the
identities
v,w,zeX.
we
take
z =
-v
- w
(note
11g
(1.5)
v V w = v+
(w-v)+=w+
(1.6)
v /% w = v -
(v-w)+=w
that will
be o f t e n
We r e c a l l Z with
used
that
an u p p e r
in the
bound
possesses
V is a s u b s p a c e
X if for any formed
the
any
a least
(closed)
complete if e v e r y s u b s e t
upper
bound,
denoted
by
X
,
we
say
t h a t V is a sub lattice of
the e l e m e n t s
v A w and v V w
,
to V. shall
subspace
denote
by V e the
of the d u a l
space
order dual of V, t h a t
V' w h i c h
is g e n e r a t e d
by
cone
P' = {v' 6 V '
the p a i r i n g
to be
.
v and w of V,
belong
such V, we
the p o s i t i v e
(1.8)
of
two v e c t o r s
in X, a l s o
For
appearing
: (v',v)
~ 0
yvEP}
in
being
the d u a l i t y
(1.8)
(1.9)
pairing
V and V ' . T h u s ,
V e = P' - P'
The order space
Under
V'
dual V ~ will
positive
cone
In p a r t i c u l a r , v ' V w',
not,
(see the E x a m p l e
dual ordering,
the
(closed)
and
following. X is said
V z
If
dual
- (w-v) +
a lattice
(I .7)
is,
(v-w) +
both
that (1.8),
in g e n e r a l ,
with
the w h o l e
below). is, V*
for a r b i t r a r y elements
coincide
the p a r t i a l is a v e c t o r v'
and w'
ordering
in V ~, we c a n
of V ~, and w e h a v e
(1.10)
v ' V w' = v ' + (w' - v ' ) + = w
' + (v' - w ' ) +
(1 .11)
V' A w' = v ' - ( v '
' - (w' - v ' ) +
-w')+=w
induced
by the
lattice.
as a b o v e
f o r m v ' A w'
the
identities
Example 1.1. In our lities space open any
for
applications
X will subset
closed
to v a r i a t i o n a l
linear s e c o n d o r d e r be
the S o b o l e v
of A N, subspace
space
and V w i l l of Hk~)
(elliptic)
PDO
HI (~), w h e r e
be e i t h e r
such
and q u a s i
that
the
variational
in d i v e r g e n c e
inequaform,
~ is a s m o o t l y
Sobolev
space
the
bounded
H~ (~),
or
120 (1.12)
HA (~) C V C H I (s
Under
the ordering,
(1.13) the
u < v
s p a c e X = Ht(s
iff
see e.g.
lattice
o f L2 (~), w h i c h
subspace
|37].
Note
conditions
to be a sublattice
a.e.
lattice
and the positive
cone
(1.1)
with
a sub-
(1.12),
lattice
which
to Dirichlet
under
is u s u a l l y
for the problem
of X = HI (~). T h i s
V = HA (~), c o r r e s p o n d i n g
x E ~ ,
t h a t H* (~) c a n be d e n t i f i e d
is a complete
V satisfying
the b o u n d a r y
< v(x)
is a v e c t o r
closed,
The
u(x)
at hand,
is c l e a r l y boundary
hence dual
ordering
a positive
(Radon)
o f HA (~) is t h u s
H-I (s
that
if a n d o n l y measure
will
subspace
in terms
of
then be assumed
is a p o s i t i v e
element
distribution,
to H-I (~). T h e
of a l l d i s t r i b u t i o n s
as t h e d i f f e r e n c e
[41].
the case when
is a p o s i t i v e
in ~, b e l o n g i n g
the closed
can be written
if v'
defined
see
condition.
If V = HA (~), h e n c e V' = H -I (~), t h e n v' 6 V , for the dual
(1.13);
is
of two positive
order in
measures
in ~. For
a detailed
in c o n n e c t i o n
with
discussion
trace
In the applications the
spaces
H~,P(~),
H~ (~), HI (~)
H* 'P(~)
2. T - M O N O T O N E
to non-linear , L2 (s
, LP(~)
properties
we refer
, will
of Sobolev
spaces,
t o [21].
second
order
be assumed
PDO,
by the
the role
of
spaces
p ~ 2.
OPERATORS
We consider lattice
of o r d e r
theorems,
a real
for the partial
reflexive ordering
Banach induced
s p a c e X, w h i c h by a closed
is a v e c t o r
positive
cone
(I . 1 ) . We
suppose
a sublattice We
then
that
a closed
consider
A
space
which
is
: X ~ V'
X to t h e d u a l V'
to B r e z i s - S t a m p a c c h i a
(2.2)
[11],
we
( A u - AV,
of V. A c c o r d i n g
u,v6X,
such that
to a d e f i n i t i o n
s a y t h a t A is T-monotone
(u-V) + ) +
~or e v e r y
V of X is a l s o g i v e n ,
a map
(2.1)
from the
subspace
o f X.
(u-v)
6V.
> 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
+-
inequality,
- Av,(u-v)+>
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
therefore
,
striotly T-monotone o p e r a t o r
f r o m the
(2.11)
v)
by
(2.11)
- Av, ( u - v ) +) = a ( u - v ,
and
( u - v ) +)
(2.8)
a (u - v, (u - v) +) = a ( (u - v) +, (u - v) +) > 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
in d i v e r g e n c e
,
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 <_~2
either
(3.2)
~,=
~2 + c o s t a n t ,
or
~i (v A w) + ~ 2 (v V w) <_~, (v) + ~ 2 (w) for all v , w q V
(cfr.identity
this
not
,
(1.4)).
It can be s h o w n For
that
order
is the
(3.1)
if:
of V, we
relation
then prove
in the
for a r b i t r a r y
structure
an o r d e r
V. T h i s
in ]- ~,
any
+ c)
THEOREMS
shall
values For
lattice
from that
theorem8 we w i l l p r o v e 3. COMPARISON
= ~(9
that
trivial
result
(3.1) we
is a refer
(partial)
order
relation
in F0 (V).
to [24].
If
(3.3)
~i =
where
Q,
holds
if a n d o n l y
a n d Q2
are non-empty
~(Qi' 9 ) , i =
closed
convex
1,2,
subsets
of V,
if:
either
Q!
-- Q 2
or
(3.4)
we t h e n
(3.5)
vqQ,
, wqQ2
imply
v A wqQ,
also write
Ql <_ Q2-
, v V wEQ2
.
then
(3.1)
125
Note
that Q ! V
in the
iff Q A v 6 Q ,
function
v which while
are the
lattices
Ze88
whereas
mentioned than
or e q u a Z
set Q of all v ~
V !Q
iff Q V V E Q
above,
a given
the
set Q of all
% satisfies
satisfies
: for e x a m p l e ,
the
the o p p o s i t e
functions
relation
relation
Q~V,
V~Q.
If
(3.6)
~i = - gi
where
gl ,g~ E V , ,
then
(3.1)
(3.7) (note
that,
because
order).
to
for t h e d u a l
of the m i n u s
We h a v e
i = 1,2
is e q u i v a l e n t
g] ~ g 2
reverse
'
in fact,
sign
in
by the
ordering (3.6),
! and ~ c o r r e s p o n d
linearity
~, (v) - ~I (v A w) = - ( g , , 0
V
in V'
of g,
and
g~
(v-w) = - ( g , , ( v - w )
in
:
+)
and
92 (v V w)-~2 (w) = - ( g 2 ,
More
generally,
~i =
(3.1)
0) = - ( g 2 , (v-w) +) 9
if
(3.8)
then
(v-w)V
holds
if a n d o n l y
6(Q,
.) -gi
'
i = 1,2,
if:
either (gl,v)
=
(g2,v) + c o n s t a n t
for all v E Q
,
or
Q is s t a b l e all v E
Therefore, whenever
under
(Q-Q)V
for f u n c t i o n s
V and A
, with
(gl ,v) <_ (g2 ,v)
(3.8),
the
relation
(3.1)
follows
Q is a l a t t i c e .
If w e n o w h a v e
(3.9)
then either
~i = 6(Qi' (3.1)
holds
for
0.
if a n d o n l y
if:
") - gi
'
i = 1,2,
from
(3.7)
126
QI = Q 2
with
gl = g2 + c o n s t a n t
on QI-Q2
,
or
we have
Therefore,
(3.4)
and
in p a r t i c u l a r ,
(g, ,v) <_(g2 ,v)
for all vq
(QI-Q2)V
0.
we have
Lemma 3.1.
Let then
~i
b e as
(3.9).
If
(3.4)
is s a t i s f i e d
a n d g,<_g2
in V',
~l < ~ 2 -
9
We now consider section
and
the m a p
we prove
T h e o r e m 3.1. ( H a u g a z e a u The map u i -- a(~i)
a given
for
~ defined
at t h e
e n d of t h e p r e v i o u s
the f o l l o w i n g . [23 ] )
by
i = 1,2
(2.18) and
is i n c r e a s i n g
~1 <_~2,
from
F0 (V) to V,
i.e.,
t h e n ul _< u2.
if 9
proof.
Replace a n d w = u,
w = u, A u2
V u2
in t h a t
~, (ul)
+
in t h e
satisfied
~2(u2)
+ (Aul , ul - u I A ! ~' (uIA u2)
By
(3.2),
by
b y u = u2
implies,
(ul-u2
)+
sum
up
b y u = ul
:
u2 - u ] V u2 )
we get
u2)+ (Au2,u2-u,V
u2 > <_ 0 ,
(I .5) (I .6) ,
s i n c e A is a s s u m e d
< 0.
to b e s t r i c t l y
= 0 , hence Ul
When
and
satisfied
+
u2 > + < A u 2 ,
This
(2.17)
+ ~2 (u,V u2)
therefore,
inequality
< U2
T-monotone,
that
127 (3.10)
~ = 6(Q,
Q being a non-empty solution
u of
(3.11)
convex closed
(2.17),
.) - g ,
s u b s e t of V and g 6 V', we d e n o t e
the
i.e.,
I-u
l
6 Q (Au, u - w ) < (g,
u-w )
yw6Q
by
(3.12)
u = o(Q,
Corollary
I of Theorem 3.1.
Let QI, Q2 be n o n - e m p t y elements If
g)
closed
convex
subsets
of V and gt, g2
of V'.
(3.4) a n d
(3.7)
We r e c a l l
h o l d and u i = ~(Qi
that w e are a s s u m i n g
' gi )' i = 1,2,
then u,<_u2.
that V is a s u b l a t t i c e
of a space
X. W e n o w d e n o t e by Xv the s u b s e t of a l l v e c t o r s of % b e l o n g i n g (3.13)
to V
% 6 X such that there
: ~ v6V,
We ca l l a n y s u c h % an a d m i s s i b l e % 6 X V,
(3.14)
The
(3.15)
wille
(3.16)
= {v E V
closed
solution
i
u!+ (Au,
v <_ ~}
upper obstacle. (with r e s p e c t to V).
then
Q(%)
is a non-empty
some m i n o r a n t
:
xV = {~6X
If
exists
: v <_ r
convex
u = a(Q(%),
,
s u b s e t of V. g) of the v a r i a t i o n a l
ueV
u - w ) ! (g, u - w )
be a l s o d e n o t e d
Yw<
by
u = o(~, g)
~, w 6 V
9
inequality
128
Clearly,
the
set Q(~)
depends
(3.17)
Q(r
for e v e r y
r E XV
Corollary
2 of Theorem 3.1.
The map every
. Therefore,
(3.16)
~(r
u = o (V, g)
r 6 X v. M o r e o v e r
the
f r o m X v • V'
to V. M o r e o v e r ,
for
we h a v e
(3.18)
where
on
<_ v
we h a v e
is i n c r e a s i n g
r 6 X v and g 6 V '
increasingly
is the
g)
solution
< u
of the p r o b l e m
iuv
(3.19)
(Au,
It is e a s y
w) = (g,
to v e r i f y
form that
the v a r i a t i o n a l
coincides
with
that
w)
VwEV
the v a r i a t i o n a l
inequality
the w h o l e
space
.
(3.11)
equation
takes
when
(3.19) the
is the
convex
Q
V.
Remark 3.1. then
If A is the o p e r a t o r
of E x a m p l e
~ above
solution
is the weak,
2.1,
and V = HI (~) [ V = H~ (~)] ,
of the N e u m a n n
[Dirichlet]
problem
in ~.
Remark 3.2. A similar involving
(3.20)
Here
any
r
uEV <_ ( g , u - w )
of the
and g6V',
that
2 above
: ~vEV
Yw>_
for
inequalities
r , weV
set
, V >_ r
we denote
the
u = p (r
p (r
holds
r
u >__ r ,
Xv = { r
(3.22)
Note
than Corollary
(Au,u-w)
r is any v e c t o r
(3.21)
For
result
a lower obstacle
= o(Q(~) ,g) w h e r e
solution
.
u of
(3.20)
by
129
(3.23)
Q(~)
As above,
= {vqV
: v ~ %}
the m a p p i s l i n c r e a s i n g
We c o n c l u d e Tumonotone
this
operators
from X V • V' to V'.
section w i t h an i m p o r t a n t that
follows
property
from the c o m p a r i s o n
of strictly
theorem
above.
whose
restri-
Theorem 3.2. Let A
: X ~ V' be a s t r i c t l y T - m o n o t o n e
ction on V is c o e r c i v e
and h e m i c o n t i n u o u s .
operator,
Let u , v be v e c t o r s
of V
such that
(3.24)
V e being
Au 6 V e
,
Av q V e
,
the order dual of V. T h e n , we have
(3.25)
A ( u A v) 6 V e
and
(3.26)
A ( u A v)
for the dual o r d e r
> 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) <_ ( A u A Av,
u
u A v, w E V
By r e p l a c i n g
z-w)
w = z + x for a r b i t r a r y
x6V,
(3.28)
(Az - A u
therefore,
by
(3.27),
Az6V e
VI
A Av,x)
> 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 <
lattice positive
131
We n o w c o n s i d e r
Eu<_r (4.6)
the
following
variational
inequality
, uEv
(Au,
Under
the
u - w > <_ (g,
assumption
u-w)
above,
Yw<_
%, w q V
we know
that
there
exists
a unique
solution
u = c(~,
of
(4.6).
As we
shall
see below,
g)
it is n o t d i f f i c u l t
to s h o w t h a t
the
element
A u E V'
from above, in t h e d u a l o r d e r i n g
can be e s t i m a t e d g itself,
that
by the g i v e n
is,
(4.7)
g >_ A u
this m e a n s
~f V',
in
V'
,
that
g - AuEP'
where
P'
is the d u a l
What we shall
positive
prove
below in V' b y the e l e m e n t (4.8)
cone
below
(I .8)
is t h a t A u
g A A%
, that
A u >_ 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 <_g
and
(4.13)
in V'.
h < A%
We must prove
that
(4.14)
Au
In o r d e r
to p r o v e
tion z satisfies
(4.14)
we consider
Az
The
afterwords
auxiliary
~
z
(4.16)
under
the
The proof
that
problem
> u
(Az,
that,
9
an a u x i l i a r y
problem
whose
solu-
the inequality
(4.15)
and we prove
> h
,
z = u,
is t h e
z - w } <_ (h,
(4.15)
following
VI
z6V
assumptions of
> h
z-w)
of t h e
is q u i t e
Yw>_
u
theorem, similar
, weV
has
a unique
to t h e a r g u m e n t
solution
z,
just given
for the proof
of
(4.7):
p l a c e w = z + v in
given
(4.16)
(4.15)
- h, v)
> 0 ,
that
in o r d e r
to p r o v e
that
z = u it s u f f i c e s
that
(4.17)
z <
In fact,
once
The vector the
, v ~ 0, w e c a n re-
holds.
L e t us n o w r e m a r k to prove
v 6 V
and we obtain
(Az
hence
an arbitrary
(4.17)
has been
proved,we
w = z c a n be r e p l a c e d
in
can proceed
(4.6),
by
as f o l l o w s .
(4.17).
This
gives
inequality
(Au,
and,
since
u-z)
u - z <_ 0 a n d h <_ g,
(4.18)
(Au,
On the other (4.16),
therefore
(4.19)
u-z)
hand, we
(4.18)
and
implies
this
< (h,
implies
u-z)
w = u can obviously
be replaced
in
also have
z-u)
(4.19)
(Au-
and this
u-z)
the vector
(Az,
From
< (g,
< (h,
z-u)
it f o l l o w s
Az,
u-z)
u = z, b y t h e
that
< 0
strict monotonicity
of A
(recall
Lemma
2.1). Notice have
that we have
not used yet
comes
in the
strict
T-monotonicity
(4.20)
following
T-monotonicity
In o r d e r
used above
the additional of
assumption
(4.17),
h < A%.
together
h ~ g, h o w e v e r , This
with
assumption
(4.9)
and
the
o f A.
to p r o v e o f A,
proof
the
condition
(4.17)
it s u f f i c e s
that
(z - ~ ) + 6 V
to p r o v e ,
by the
strict
we
134
and
(4.21)
(Az
- Ar
(z -
~)+)
< 0
+ for t h i s
implies
Now,
(z - ~)
z < ~.
= z - , A z
(z - r
(4.22)
and
= 0, h e n c e
since
z 6 V,
(4.20)
is a c o n s e q u e n c e
of our
assumption
(4.9)
on
%, b y
which we have
(4.23)
% A z 6 V
As
a further
consequence
w = % A z can be replaced and
in
z a r e ~ u, a n d w 6 V b y
taking
(4.22)
also,
since h < A~
is t h e
R~mark
(4.16) (4.23).
(z -
inequality
r
in V'
(Az,(z
which
(4.23),
of the
if t h e a d d i t i o n a l
satisfied,
then
mate
in t h e d u a l
of Au
(4.24)
-
we
w h u
find
the vector , since both
from
(4.16),
by
%)+)
,
%)+)
(4.21)
< (A%,(z
we had
-
%)+)
to prove.
s p a c e X is s u c h
norms
dual
of the
of t h e
estimate
corollary
(4.11)
implies
above the
are
also
following
esti-
n o r m of V'
<_ llg V 0Uv,
estimates
il.ii , p r o v i d e d
Estimates
properties
that
f o r a l l v 6 V,
assumptions
the order
llAuliV,
Similar
norms.
: in fact, Thus,
< ( h , (z -
[Iv+li < ilvii
larger
find that
4.12.
If t h e n o r m
and
we
into account,
(Az,
hence
of
.
+ ilg A A ~ A 01Iv,
can be derived iig V 0U
and
of this
kind are very
solution
u, d e p e n d i n g
from
(4.11)
with
Ug A A ~ A 0[I a r e useful on the
in studying regularity
respect
finite
to
in t h a t
regularity
of the data
135
g and
~. In fact,
tion about results
if s u c h e s t i m a t e s
u can then be obtained
for t h e
solution
We shall
Chapter
nection
a detailed with
in t h e
following tool
arising
Chapter
in a p p l y i n g
in t h e
3, t h a t
the estimate
the general
result
applications.
BIRKHOFF
and
of 9
section
(4.3)
Moreover,
X,
inequality
problems
as f o r r e f e r e n c e s
on the
(4.11)
involving subject
we
the
in c o n Sobolev
refer,e.g.~to
THEOREM
we assume
that we
of t h e p r e v i o u s we now
assume
as a v e c t o r
are given
X, V a n d A as
in
(4.1),
section.
that
lattice,
vactor
a complete
Example
value
[20].
- TARTAR
(5.1)
of a l i n e a r
boundary
HI (~), as w e l l
In t h i s
is a s u b l a t t i c e
lattice
of
H
5.1.
X = HI (~) fied with Given
we denote
, H = L2 ( ~ ) , s e e
a subspace
o f L2 (~)
Example1.1,~h~space in t h e
usual
H I (~)beingnow
distribution
two vectors
(5.2)
u0
< v0
,
u0 , v0 6 H
by
(5.3)
the
regularity
Au = f
discussion
elliptic
Hanouzet-Joly
(4.2)
from the corresponding
informa-
u6V
important
I to m a n y Q V I ' s
For
5.
see,
a very
regularity
4.3.
Remark
space
also
is a l s o
available,
o f the p r o b l e m
i (4.11)
are
interval
[u0,v0] H
of all elements
v 6 H
,
such that
u0 < v < v0
,
sense.
ident! 9
136
We
suppose
u E [Uo,
that
v0] H
such
an
a function
interval
~(u,.)
(u,-) the
set
Fo (V) b e i n g
Moreover,
we
defined
suppose
UO
<
that
Ul
<
is g i v e n
is a l s o
and
given,
for
such
every
that
e ro(V)
in section
3.
if
U2
<
t
Vo
Ul
9 U2
E
H
9
then
(u,9
in Fo (V),
in t h e
In o t h e r
(5.4)
of
we
~ : [u0,
Example
section
suppose
Vo] H
~
")
3.
that
we
are
given
F0 (V)
is
increasing.
a map
~ such
V = H~ (~)
increasing
(5.5)
map
Mu
Then,
,
X = H' (~)
of a given
>_ 0 o r
if g 6 V'
Q(u)
~
,
[u0,
for
all
H = L 2 (~)
v0] H
into
u 6 [Uo,
X = H I (~)
Vo] H
, such
.
,
= {v 6 v
: v <_ M u
a.e.
in
~}
and
~(u,
then
~(u,.)
Q(u)
~ ~ More
6
) and
.) =
F0 (V)
.) - ( g ' ,
(in p a r t i c u l a r ,
~ verifies
generally 9
~(Q(u),
~(u,.)
(5.4).
if
Ho*(a) C V C H'(n)
we
that
5.2.
Let
M an
sense
words 9
") < ~(u2,
replace
(5.5)
with
the
condition
9}
is n o t
~ + ~,
since
that
137
(5.6)
M u 9 XV
where
XV
section
is t h e
u E [u0,
set of all admissible
v0] H
,
upper
obstacle
defined
in
3.
With tional
f o r all
9 the above
data,
we want
to s o l v e
the
following
quasi-varia
inequality
i
u E [u0,
(5.7)
~(u,u)
v0] H
+ (Au,
A
V
u-w)
<_ ~p(u,w)
YwEV
Example 5.3. In t h e
special
case
described
in E x a m p l e
5.2
above,
problem
(5.7)
becomes
u a [u0 , V 0 ] L 2 (~)
,
u <_ M u
a.e.
YvEV
,
in
(5.8) (Au,
The
selection
u-w)
map
< (g,
u-w)
S of p r o b l e m
(5.7)
v <_ M u
in
c a n n o w be d e f i n e d
as t h e
map
(5.9)
Su = o(~(u,-))
where
o is t h e m a p
of H i n t o
the
identified
as a s u b s p a c e
with
a subspace
the map
(5.9)
(5.10)
the p r o b l e m
t h e fixed-point
ticular
that
The
o f X, b y o u r
: [u0,
v0] H
theorem
notions
Given
the
[u0,
sub-solution of S
like if
(5.1)
v0] H
V too can be
H. T h e r e f o r e ,
we
can
solutions (5.10).
can be dealt with
u of
a vector
(5.7)
In fact,
to t h a t of
note
in p a r -
then u E V
.
b y a p p e a l i n g to c l a s s i c a l
maps d e f i n e d below.
supersolution
(5.10),
,
u = Su = s(~(u,.)),
f o r increasing
of s u b - a n d
a map
lattice
H
in H o f t h e m a p
if u 6 H s a t i s f i e s
latter problem
~
lattices, as the o n e w e s h a l l m e n t i o n lowing
,
interval
assumption
of the vector
of f i n d i n g
finding
fixed-points
S is a m a p of t h e
v0] H
as a m a p
S
and reduce
Thus,
u 9 [Uo,
s p a c e V.
However,
consider
(2.18).
,
have
in complete
In t h i s
respect,
an i m p o r t a n t
u 6 [u0,
vector the
fol-
role.
ve] H is s a i d
to b e a
138
u < Su
Similarly,
u E [u0,
vo] H
a super solution of S if
is s a i d
Su
Note u < u
that
for an i n c r e a s i n g
of b e i n g
equivalent itself
a subsolution
to the p r o p e r t y
supersolution]
Theorem 5.1. Suppose
that such
not
that
: [~'~]H
~ H
, the p r o p e r t i e s
a supersolution,
S carries
the
of
respectively,
interval
[2,
U]H
is into
.
If S is t h e m a p
(5.4),
S
and
of p r o b l e m (Tartar that
exist
that
u < ~
Then,
the
empty
we
say a l s o
(5.7)
itself.
t h a t u is a s u b s o l u t i o n
[u a
[43])
X, V, A,
u0 ! v0 b e i n g
there
(5.9),
H and
two g i v e n
a subsolution
~ verify
vectors u and
(4.1) (4.2) (4.3) (5.1)
of H. S u p p o s e ,
a supersolution
and
in a d d i t i o n , u of p r o b l e m ( 5 . 7 ) ,
. set of all
and p o s s e s s e s
solutions
a minimum
u of
and
(5.7)
satisfying
a maximum
u < u ! u is
element.
9
Proof. H is a c o m p l e t e problem
(5.7),
property Theorem
considered
follows
as a m a p
and
the
selection
(5.10),
from our assumption
conclusion
of T h e o r e m
4.1
Knaster-Kantorovich-Birkhoff
Theorem 4 . 2 . ( G . B i r k h o f f Let itself. then
lattice
(5.4)
map
is i n c r e a s i n g . and
(5.9) The
the c o m p a r i s o n
of
latter theorem,
3.1.
The lowing
vector
the
exist
a minimum
map
vectors
set of f i x e d - p o i n t s
and possesses
a consequence
the
fol9
of a c o m p l e t e
vector
u < u in H, s u c h t h a t
u of S s a t i s f y i n g
and a m a x i m u m
lattice
u < u ! ~ is n o t
element.
Iu @ H
: u <_ u _< u, u _< Su
1
and
IuE. u u<
empty 9
sets
l-=
H in
u < Su a n d S u ! u ,
Proof. Consider
of the
theorem.
[7 ]).
S be an i n c r e a s i n g If t h e r e
is t h e n
su< uI
139
These there
are not empty,
since u 6 E
, u 6
Z+.
Since
H is c o m p l e t e ,
exists
(5.11)
z = vZ
We now show that
z E
S is i n c r e a s i n g ,
S u < Sz. T h e r e f o r e ,
We have
thus
shown
Z . In fact,
for every
u E Z , u < z, h e n c e , s i n c e
it f o l l o w s
t h a t u ~ Sz f o r e v e r y
from u < Su that u<Sz.
u 6 Z , hence,
z < Sz
On the other Sz E Z
hand,
since
, therefore,
z E Z
by
a n d S is i n c r e a s i n g ,
we also
have
(5.11)
SZ < Z. Thus,
z = Sz.
Moreover,
u < u ! u belongs maximum
of all
The
to
since
Z-,
any
such fixed-points
existence
fixed-point
it f o l l o w s
from
u of S s a t i s f y i n g
(5.11)
that
z is i n d e e d
the
o f S.
of t h e m i n i m u m
fixed-point
is s h o w n
along
the
same
lines.
9
Remark 5.4. The
of a maximal
existence
u < u ~ u is p r o v e d ordered
structure
o f X. T h i s
tion of Birkhoff's see [43 ] o r [44]
In t h e
by Tartar
theorem
for more
following
cha
a n d a minimal
in [43] result
solution
under weaker
relies
to an a r b i t r a r y
of
(5.7)
assumptions
on a parallel inductive
on the
generaliza-
ordered
set H
,
details.
ter w e
9
shall
use
the
following.
Corollary of Theorem 5.1. Let
the
spaces
H, X, V a n d t h e o p e r a t o r
A b e as
in T h e o r e m
5.1,
with A0 = 0 . L e t g . E V'
, g > 0, a n d u b e t h e
solution
of
i u_E V (5.12)
(that e x i s t s
and
= (g,w)
is > 0).
Let M be an increasing H i n t o H, Then,
VwEV
such that M0
map of the
(non-empty)
> 0.
the quasi-variational
inequality
interval
[ 0 , U ] H of
140
i
(5.13)
has
a maximum
u 6 V ,
(Au,
u~Mu
u - w ) !
and a minimum
solution
u - w )
u,
Y w 6 V ,
satisfying
w
< Mu
0 ~ u ! ~.
9
Proof. Apply
Theorem
5.1 w i t h
u0 = u = 0 a n d v0 = u Note,
in p a r t i c u l a r ,
3 and the present notation
of that
(see a l s o E x a m p l e of
from the comparison
5.3),
(5.12). theorems
of section
section:
>_ o(V,0)
= 0
{0,~] H is n o t e m p t y ;
u is a s u p e r s o l u t i o n
SO
hence
5.2
solution
A 0 = 0, g ~ 0 a n d M 0 ~ 0, it f o l l o w s , w i t h
S u = o(Mu,g)
hence
in E x a m p l e
u is t h e
that
hypotheses
u = c(V,g)
hence
~ as
, where
=
o(M0,g)
0 is a s u b s o l u t i o n .
< o(V,g)
= u
,
>
=
,
;
o(0,0)
0
9
141
CHAPTER
3
SOME APPLICATIONS
I. A Q U A S I - V A R I A T I O N A L
INEQUALITY
WITH
IMPLICIT
OBSTACLE
ON THE B O U N D A
RY We
suppose
boundary on
that
F, a n d w e
consider
(1.1)
with
~ is a b o u n d e d
F, t h a t a f u n c t i o n
open
f is g i v e n
Signori:ni
(1.2)
implicit
U > MU
where
for some
(1.3)
,
given
Mu(x)
of ~ N ,
with
a smooth
h is g i v e n
the p r o b l e m
-Au + u = f
the
region
in 9 a n d a f u n c t i o n
in
boundary
~
condition
~-~ >_ 0 , ( u - M u )
function
~ > 0 on
= 0
on
F,
F we have
- I ~ ~~u dc
= h(x)
a.e.
,
x 6 F ,
F n being
the
exterior n o r m a l
Roughly stationary
speaking,
F that allows
vents
the
liquid
liquid
inside
liquid
to f l o w out.
~ a n d h(x)
F, t h e n
, where
gion
R, t h e c o n d i t i o n s
In fact,
the l i q u i d
u=
to g o out,
hand,
G
describe
applied
boundary
the
by a mem-
~, w h e r e a s
it p r e -
is t h e p r e s s u r e
pressure
of the
on the boun-
to e n t e r
the re-
h
on that portion
however
(1.1)
surrounded
region
if u(x)
liquid crossesthe
on the other
tends
the
is t h e e x t e r n a l
the
conditions
in a r e g i o n
to e n t e r
~u -- > 0 ~n are satisfied;
s h the
of a l i q u i d
the
dary
F .
w h e n M(u)
equilibrium
brane
on
the membrane
of
F where
u(x)>h(x)
F p r e v e n t s its o u t c o m ~
hence ~u --
~n
The
above
Signorini
(1 .4)
conditions boundary
u-h
_> 0
=
together,
where
0
are
u(x)
clearly
> h(x).
equivalent
condition
'
__
~n~U _> 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,
<
8null-I/2 (s
--
c Us,
H~ (n)
where ,/2
(1
We
suppose
(1.10)
(1.11)
where
that
the
h 6 H'/2 (r)
are given,
dual
ullH,
--
and
Q(u)
<- , .)
F
for e v e r y
§ IluN
functions
,
~ 9 H'/2 (s
u E H L* (s
we define
= {v 6 H l (~): v >_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
the f o l l o w i n g
inequa-
lity Ii
I (~)
u6
,
HL
U
6
Q(u)
(I .13) (Lu,
u-w)
<_ (f,
We do n o t s h o w in d e t a i l
u-w)
that
u
Q(u)
(1.13)
is i n d e e d
the w e a k f o r m u l a -
tion of problem(1.1)(1.2).This can be d o n e as in the case M(u) 5 h and we refer,
for i n s t a n c e ,
to [32]
(see a l s o the p r o o f of the t h e o r e m below).
Theorem 3.1.([28] ) If ~ ~ 0 a.e.
on F, then p r o b l e m
(1.13)
has a s o l u t i o n
u .
9
Proof. We s h a l l a p p l y the C o r o l l a r y
X = H* (~) = C
of T h e o r e m
8.1
,
C , = H~ (~)
as a s u b s e t of H* (~)
X o = H i(~)
n o r m e d by
(1.9)
: Lv = f
in ~ , ~-~ >_ 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,
<_ (f,
v-w)
also belongs We must
to Do.
prove
ditions
v 9 Q(u)
Here
that the
(1.15)
and
v-w
u
)
Q(u)
is t h e
solution
(1.16)
eQ(u)
convex
cone
(1.14)
satisfies
v of
Lv = f
in
~ ,
(1.16)
~__vv > 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
<_ 0 o n ~
by using
(f,~)
(I .16),
, and replace
(Lv,
hence,
~)=
~)-
Green's
; this
we
gives
u
take
(Q)
an a r b i t r a r y
w = v - ~ in
(f, ~)
(I .14).
~ 6 C=(~), This
gives
< 0
formula
and
taking
(I .17)
into account,
we get ~v
< ~, that implies
4)
S(D0)
(1.16),
~lr by the
is bounded in X0
)
r-
< 0
arbitrariness
:By C o r o l l a r y
of
~,
~ < 0
2 of T h e o r e m 8.1it s u f f i c e s ~o
145 verify
that
(1.18)
and
there
exists
(1.19)
below:
(1.18)
a vector
u
<~v,
E
tq Q(v) V6Do
v-u>/llvll HL
(1 . 1 9 ) as
NvU
~ co
u satisfying
,
both
conditions
,
~ + =
(~)
v ~ S(D0)
H~ (n)
We now verify
that we can take u to be the variational
of t h e D i r i c h l e t
u = 0
i
_
In fact,
in
h
on
u 6 H* (~)
u > h-
As
to
(1.19),
]Ivl] H ~
Thus since
~ =
v-~,>
--
, and this
F ,
condition
~v
~
~ 0 satisfied
is v e r i f i e d .
L is c o e r c i v e
,,llvll
H i
o n HI (~) w e h a v e
(~)
implies
--,- + =
(1.19)
because
o n S(D0)
the
(~)
Iivll
for every
to t h e H * - n o r m
,(~)
2
(~)
: in fact,
by
(1.15),
1 ,/2
v ~ S(D0) we must
5) T h e m u l t i v a l u e d on Do
on
~ ~ 0 and the (1.18)
H~.-norm is e q u i v a l e n t
Finally,
a.e.
(g, ~av )r
<~.~, as
F
and
d u e to t h e h y p o t h e s i s b y a n y v 6 Do.
solution
problem
verify
mapping
for the weak
that
Q defined
topology
of t h e
by
(1.11)
space
is
(L,f)-continuous
' (~) , a c c o r d i n g X0 = H L
146
to the This
definition
means
that
given
in C ~ p t e r
if Uk,
v k 9 Do
I.
satisfy
for
each
k = 1,2,...
V k 9 Q (u k)
(LVk,V k-w)
and
Uk
- u,
vk
__x v
!
(f,v k - w )
weakly
in Xo,
Yw 9
then,
k)
in the
limit,
we
have
a)
v 6 Q(u)
and 8)
u w 9 Q(u)
such
,
~ w k 9 Q ( u k)
that
l i m s u p ( L v k, w - w
Now,
v k E Q(Uk)
(I .20)
since
means
v
i n Xo = H
i n HI/2 (F) C L 2 (F) ~Uk
v k E H I (~) ~u k ~
v k >_ 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 > _ <~, ~ F
w k 6 HI (~) a n d ,
a.e.
on
in t h e
that
is s a t i s f i e d .
8),
w k = w + (~,
We
a.e.
, we
have
limit
as
(F) , t h u s
v >_ 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
in d i v e r g e n c e
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 <_Mu
i
(2.3)
Lu - f < 0
with,
(2.4)
see
condition
have
that
c a n be of D i r i c h l e t
that
the
every
aij(x)~i~j
closed
recently
programming ref.
,
c(x)
>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
is s a t i s f i e d
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, <_ u2
=
following
(x~, x 2 ) 9 Ix[*P,
the
simple
~2
hence
increasing,
is
a.e.
not c a r r y
space
example
satisfying
u9
H* (~) into due
to
Ixl=(x~+x~)V2<1,
(~) , w h e r e a s
i.e.
in s i m p l i e s
Mul~
u 9 0
(a.e.
Mu2
a.e.
in ~,
moreover,
(2.13)
Mu
> 0
we n o w c o n s i d e r
if
problem
(1.3)
with
the
in s
Signorini
boundary
condi-
tion
(2.14)
u ! Mu
a__uu < av L -
,
0
(u - Mu).
au = 0 av L
'
a.e.
x 9 s
where N
au ~
= aVL
is the c o n o r m a l The weak nal
i,j=1
derivative
formulation
inequality
is the
au .~ ~-~ c o s ~ v , x 4 ) aij ~i J
of u on
s associated
of p r o b l e m
following
with
(2.3)(2.14)
the o p e r a t o r
as a q u a s i
one:
i u 9 H l (~) A L'(~)
,
u
<
Mu
a.e.
in
(2.15) a(u,u-w)
We also
(2.16)
assume
that
<_ (f,u-w)
u
H* (~), w
< MU
f satisfies
f 6 L~(n)
,
f > 0
a.e.
in ~
,
a.e.
L.
variatio-
D,
150
and
we denote
by
u the
solution
of the N e u m a n n
problem
i_
u @ H' (~)
(2.17) a(u,w)
=
u
(f,w)
Theorem 2 . 1 . ( B e n s o u s s a n - G o u r s a t - L i o n s L e t M be the o p e r a t o r exists
a unique
u < u
a.e. The
solution
[ 6 ], T a r t a r
and
u ~ 0 a.e.
f as in
[43])
(2.16).
in s of p r o b l e m
Then, (2.15).
there Moreover,
in s .
9
existence
of T h e o r e m
(2.10)
H I (e) .
5.1
of a s o l u t i o n
of C h a p t e r
of
(2.15)
follows
f r o m the C o r o l l a r y
2, via the
Theorem 2.2. L e t M be any o p e r a t o r be as in
(2.16).
solution,
Then,
satisfying
satisfying
problem
(2.11),(2.12)
(2.15)
0 < u < u
admits
and
a maximum
(2.13)
and
f
and a minimum
.
9
Proof. We apply
the C o r o l l a r y
X = V = HI (s Note
, A = L
that,
by
6 H' (~) 6~ L~(~)
L~(s
the
interval
of C h a p t e r of
2, w i t h
H=L2
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<_
a.e.
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 _<
8MUma x < M u
O n the o t h e r
hand 9
since
, 7 = I is n o t
(2.19),
there
exists
that
8 M U m a x <_ M ( U U m a x) that
u ~ Uma x
by a s s u m p t i o n
a.e.
f > 0 and
(2.21)
in ~,
U
9
in ~.
0 < 8 < 1 ,
8f < f
L e t us n o w r e m a r k
that
from
U m a x = o ( M U m a x,
(2.22)
f)
it f o l l o w s 9
8Uma x = ~ ( S M U m a x,
8f)
since
8 > 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.
<_ (f, u - w )
(3.2),
to c h e c k
given
l u 9 He* (n)
,
section, that
there
that
f 9 L~(~)
exists
and
a unique
so-
by the Vl
u <_ 1
a.e.
in
(3.3) I_ a(u,
u - w)
<_ (f, u - w)
u w 9 H~ (n) , w <_ I a.e.
in n
153
In fact, hand,
if u > 0 a n d
b y our
assumption
H~ (~) N L~(~) Viceversa, above
(3.2)
say t h a t
holds,
(3.2)
is u > 0 ; t h e r e f o r e ,
reduces
to
assuming
(cfr.
Remark
n o r w e can 3.2 b e l o w ) .
if w e try to s h o w the
by f o l l o w i n g
the p u r e
However,
topological of
(3.2),
order
by u s i n g
methods
provided
belongs of
to
(3.2).
t h e n M u = I as r e m a r k e d
both
This
I, t h a t some
exhibit
any
sub-solu
is a s e r i o u s d i f f i c u l t y of a s o l u t i o n
constant
methods
there
inequali-
we c a n n o t
in the p r e v i o u s
of L are
and
however,
easily
used
the o r d e r
of C h a p t e r f 6 L~(~)
case,
existence
argument
if the c o e f f i c i e n t s
in [ 2 9 ] [ 3 0 ] ,
(3.3)
the q u a s i - v a r i a t i o n a l
f > 0. In this
t h a t we M e e t
ved
(3.2),
to s t u d y
0 is a s u b - s o l u t i o n ,
tion whatsoever
of
u of
u is a s o l u t i o n
(3.3).
it is i n t e r e s t i n g
without
t h e n M u ~ I in e; o n t h e o t h e r
solution
if u > 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)
<_ (f, u - w)
HA (~) , w <_ M u a.e.
in
some
(3.5)
p > max
L e t us r e c a l l
that
continuously
~mbedded
On the
C(~),
space
(3.6)
and
Yw@
for any
in the
(Mu) (x) = I + inf ~>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
is c o n t i n u o u s
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.
<-I
o n ~) , t h e n w e c o u l d
fin~
~ > 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 <_ Lu <_ f
In particular, richlet
~ ! u ! u, w h e r e
problems
This
(3.7)
and
result ~ a n b e
8.1 of C h a p t e r of C h a p t e r
obtained
of the Di-
respectively.
by a p p l y i n g
However,
9
the C o r o l l a r y
of T h e o r e m
the order
: f >_ Lu >_ 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).
is c o n t i n u o u s
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.
of m e a s u r a b l e
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)
is m e a s u r a b l e
for all
compact
(d)
P-l(c)
is m e a s u r a b l e
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
is m e a s u r a b l e
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
<_ a} = F-I(z+~B).
(e) means that all the sets of the form on the left in (1.2)
are measurable,
while
(d) means all those on the right are measurable.
Q.E.D. lB.
THEOREM.
In 5 conditions
For a closed-valued
(a)
F
(b)
(Castaing representation):
F:S § R n, the follow-
is measurable;
countable
(or finite)
xi:dom F + R n (1.3) (c)
multifunction
are equivalent: dom F
family
(xi[i ~ I)
of measurable
functions
such that F(s) = cl{xi(s)li
There is a countable xi:S § Rn~
is measurable, and there is a
family
~ I}
for all
(xili e I)
s s domF;
of measurable
functions
such that
(1.4)
{s r SIxi(s)
(1.5)
F(s)
(F(s))
n {xi(s)ll
~ I)
is measurable is dense in
for all F(s)
i E I,
for all s E S.
162
PROOF:
(b) ~ (c).
(c) ~ (b).
We can suppose without loss of g e n e r a l i t y that
I = {1,2,...). Then
Trivial.
Let the m e a s u r a b l e sets in (1.4) be denoted by
Ui=lSi = domF
by (1.5), so
the function w h i c h agrees with x3
on
S3\(S 1 u $2), etc.
for all
s e domF.
agrees with xi
xi
xI
Si
on
Then i
S l,
let
with
xi
on
and with
are also m e a s u r a b l e
is measurable 9 x2
is measurable,
Now for each
on
domF
and satisfy
on
and
~
be
S2\S1, with ~(s)
~ r(s)
be the function which
(domF)\S i 9
xi(s)
Let
S i-
E F(s)
The functions
for all
s.
Moreover
(1.5) implies
In other words, (b) ~ (a).
F(s) = cl{xi(s)Ii = 1,2,...}
for all
(b) holds for the c o l l e c t i o n
(x i i = 1,2,...).
For any open set
C,
we have by (1.3) that
F-I(c) = uIEI{S r domFlxi(s) and hence
F-l(c)
is m e a s u r a b l e F
s ~ dom S.
e C),
is a countable union of m e a s u r a b l e
for all open sets
C,
sets.
Thus
F-l(c)
and from P r o p o s i t i o n 1A we see that
is measurable 9 (a) =~ (b).
For every n o n e m p t y closed set
C c Rn
and every
z c R n,
let Pz C = {x ~ CIdist(z,x) (a n o n e m p t y compact set) Rn
Observe that if the points
9
are a f f i n e l y i n d e p e n d e n t
the set
P
P z0
~-- P zI
C
= dlst(z,C)}. z0,zl,...,Zn
(i.e. not c o n t a i n e d in a hyperplane),
consists of a single element.
of
then
This follows
zn
from the fact that the set in q u e s t i o n is contained in the i n t e r s e c t i o n of a family of n-spheres w i t h centers i n d u c t i o n argument, is, for some of
z0,zl,...,Zn;
any n o n e m p t y i n t e r s e c t i o n of
by an e l e m e n t a r y
k+l
n - s p h e r e s in
Rn
m < n-k, an m - s p h e r e in an m - d i m e n s i o n a l affine subset
R n. Let
I
be the c o u n t a b l e index set c o n s i s t i n g of all
i = (z0,zl,...,z n)
such that
and are affinely independent,
z0,zl,...,Zn and for such
xi(s)
be the unique element of
xi(s)
is one of the points of
xi(s)
(1.3) holds.
is m e a s u r a b l e
F(s)
showing that if
F'
Hence to obtain in
s
and each
Pz0Pzl 9 " "P ZnF(S ).
ranges over all "rational" points of see that
have r a t i o n a l coordinates i
for each
nearest to Rn
as
i
s ~ domF
let
In particular, z n,
and since
ranges over
zn
I, we
(b), it will suffice to show that i E I.
But this will follow from
is any m u l t i f u n c t l o n of the form
F'(s) = PzF(S),
18S
where
F
is m e a s u r a b l e and
z
is a fixed point in
R n, then
F'
is
measurable. To prove the latter, we introduce function
Fk:S + R n
such that
for
Fk(S)
k = 1,2,...,
consists of all
the multix ~ Rn
satisfying (1.6)
dlst(x,F(s))
Observe that s E domF).
Fk(S) Let
< k -I
and
is open for all
C
(1.7)
to
On the other hand,
Rn
by the o p e n - v a l u e d n e s s
(1.6)
for all
D
C.
of
Fk
(a) ~
F'
and hence we have
C
that = Ux~Dr~l(x).
Fkl(x)
for fixed
Thus
k,
any countable dense subset of
x (e)
consists of the elements
and
z
is measurable.
S
s
and hence is m e a s u r a b l e by virtue
in P r o p o s i t i o n IA for
(1.7) and (1.8) c o n f i r m the m e a s u r a b i l i t y of closed
C n PzF(S) @ Z
itself has a countable dense subset), we have
But every set of the form satisfying
(nonempty if and only if
= nk=iF k (C).
FkI(C) = F~I(D)
of the i m p l i c a t i o n
+ k -I.
--1
d e n o t i n g by
since
< dist(z,F(s))
The c o n d i t i o n
C n Fk(S) @ Z
(F')-I(c)
(1.8)
s
be any closed set.
is obviously equivalent
(which exists
dist(z,x)
F.
In this way,
(F')-l(c)
for arbitrary
Q.E.D.
The important e q u i v a l e n c e of (a) and (b) in T h e o r e m IB was first e s t a b l i s h e d by C a s t a i n g a family tion of
(xili ~ I) F.
[3].
It is for this reason that we shall call
with the p r o p e r t i e s
in (b) a Castaln5 representa-
The e x i s t e n c e of such r e p r e s e n t a t i o n s
in many situations.
provides a handy tool
The following fact, w h i c h is the focus of all the
theory of m e a s u r a b l e ~ u l t i f u n c t l o n s
p r e s e n t e d here,
is an immediate
consequence. I~.
COROLLARY
measurable
closed-valued multifunction,
able selection, for all
(Theorem on M e a s u r a b l e Selections).
i.e., a function
If
F:S § R n
is a
there exists at least one measur-
x:domF + R n
such that
x(s)
g F(s)
s c domF.
This result may be credited to K u r a t o w s k i and R y l l - N a r d z e w s k i
[4];
a l t h o u g h C a s t a l n g a r r i v e d at it i n d e p e n d e n t l y at about the same time, he did not publish any details until m u c h later by Rokhlin
[5] is now known to be invalid
[3].
An earlier proof
[29]. A c t u a l l ~ these citations
all refer to a more general s e l e c t i o n t h e o r e m than IC, namely where is r e p l a c e d by an a r b i t r a r y
separable complete m e t r i z a b l e
space
Rn
(Polish
164
space,
in the Bourbaki
this case
terminology).
T h e o r e m 1B also remains true in
[3], but the proof is more complicated because a finite
number of nearest point
"projections"
P
zj does
not suffice,
and the
compactness arguments must be r e p l a c e d by something involving Cauchy sequences. Another consequence of T h e o r e m 1B is a simpler condition for measurability in certain cases,
g e n e r a l i z i n g a criterion of R o c k a f e l l a r
[6] for c o n v e x - v a l u e d multlfunctions. 1D.
COROLLARY.
s E S,
n-dlmensional if
Let
F:S § R n
F(s) = cl(intF(s))
F-l(x)
be a m u l t i f u n c t i o n such that,
(as is true,
closed convex set).
Then
is m e a s u r a b l e for every
PROOF. {aili E i}
F
is m e a s u r a b l e
T h e o r e m 1B fulfilled,
and hence
F
(S,A), A
form
A ,
is an
if and only
R n, and let
functions:
xi(s)
is measurable. space
let (xill g I)
~ a i.
we have c o n d i t i o n
i ~ I,
The completion of the m e a s u r a b l e space
is m e a s u r a b l e
dense subset of
for every
for all
r(s)
For the sufficiency,
be the c o r r e s p o n d i n g family of constant F-l(ai )
if
x ~ R n.
The n e c e s s i t y is trivial. be any countable,
for instance,
If
(b)
of
Q.E.D.
(S,A)
is the m e a s u r a b l e
being the i n t e r s e c t i o n of all the a-algebras of the
where
~
is a nonnegative,
q-finite measure on
A
and
A
consists of all u - m e a s u r a b l e sets
(or equivalently,
form
is a subset of a set of u-measure
T A To ,
zero in
A,
where
and
T ~ A,
A
denotes
To
all sets of the
symmetric difference).
One says that ^
(S,A)
is complete if
complete.
Moreover,
Thus and
A
(S,A)
for example,
if
It is elementary that is complete if
S
If instead
A
(S,A), A
A = A
(S,A)
is always
for some
~.
is a Borel subset of some E u c l i d e a n space
is the algebra of Lebesgue sets in
S,
we have
is the algebra of Borel sets in
plete but could,
sets in
A = A.
for many purposes,
S,
(S,A)
(S,A)
complete.
is not com-
be r e p l a c e d by its c o m p l e t i o n
being in this case the algebra of all u n i v e r s a l l y m e a s u r a b l e S.
The most important p r o p e r t y of complete m e a s u r a b l e spaces for present needs, is that for each m e a s u r a b l e set
T
i__n_n S x R n
p r o j e c t i o n o_~f T
o__nn S
means,
that
T
belongs to the a - a l g e b r a in
by products of sets in
A
and Borel m e a s u r a b l e subsets of
of course,
is measurable.
(S,A),
The m e a s u r a b i l i t y
of
S • Rn
the
T
here generated
R n.
recent p r o o f of this p r o j e c t i o n t h e o r e m in the general case where is r e p l a c e d by any Suslin space, IE.
THEOREM.
properties
Let
F:S § R n
the implications
see S a i n t e - B e u v e
be closed-valued. (c) ~ (a) ~ (b)
For a Rn
[7]. Then among the following
are always valid, with
165
full
equivalence
among
the three
if the m e a s u r a b l e
(S,A)
space
is
complete;
(a (b
F
is a m e a s u r a b l e
gph I' B
(c
(a) ~ and
i
of Borel
is m e a s u r a b l e
(c) ~ (b).
(a).
Let
Cik
of
sets
C c R n.
from the definition.
be
sequence
the
in
R n,
We have
x E F(s)
if and only
if for all
such that
x E Cik
and
n Cik ~ ~.
But this
Each of the tells
(b) ~ set in
R n.
= nk=lUi=l[F
F-l(Cik)
gphF
(c),
projection fact
sets
us that
Then
set,
cited just p r i o r
adopted
space
X,
the
of the
facts
some of the central integral it must
be r e a l i z e d
essential venient,
see
that
able
spaces
Rn
here
This
and p e r f o r m
be any Borel
A • B,
replaced
remain
of
and hence
the
by the
of
F
given
does
and
However,
space
always
deal with
manipulations
which
of
integrands there.
may not
one must
be
as a m u l t i f u n c t i o n ,
For a summary
of the m e a s u r a b l e
frequent
(such as taking products,
can usefully
as regards
completeness where
by an i n f i n i t e - d i m r
true.
approach
completeness
in cases
C
Q.E.D.
of
of this
the
and exists
formula
is m e a s u r a b l e
[8] and the references
in such a context. as for instance
ily of Borel m e a s u r e s
to
of the m e a s u r a b i l i t y
aspects
functionals,
so this
Let
r-l(c),
A | B- measurability
developed
ai
there
says that
A • B,
belongs
is just
context
as the d e f i n i t i o n
and many
to
is complete.
to the theorem.
general
k
A ~ B.
n (S • C)
which
center
• Cik].
belongs
to
(S,A)
(gphr)
of this
(Cik)
• Cik
belongs
assuming
For the more ensional
~(s)
and for each
closed ball with
k -I.
gphF
(where
S x Rn
sets);
be a dense
let
subset
for all Borel
Trivial
{a i}
k = 1,2,...,
radius
A @ B-measurable
is the algebra
F-I(c)
PROOF.
i
is an
multifunctlon;
is
be con-
a whole
fam-
on the measur-
not p r e s e r v e
completeness). The next extra S
theorem,
structure
is a t o p o l o g i c a l
upper
topology.
compact
set
is open.
if
space,
semicontlnuous
product
S
set in
a multifunction graph)
is equivalent
F-l(c)
and lower
[3], R m.
r:s ~ R n if
gphr
exploits Recall
is closed that
for every
open
if
in the for every
On the other hand,
semicontinuity
the
that
is said to be
to the c o n d i t i o n
is closed.)
semicontinuous,if
If both upper
to C a s t a i n g
is a Borel
(or of closed (This
C c R n,
said to be lower
due e s s e n t i a l l y
present
set
C c R n,
are present,
r
is
F-l(c) r
is
said to be continuous. THEOREM
1F.
Let
S
be a Borel
subset~ of some E u c l i d e a n
space,with
A
166
the a l g e b r a Then
the
following
(a)
F
(b)
There
gphF'
Te c S S
sets.
Let
conditions
are
with
set
in
multifunction
S x R n,
property).
For
m e s ( S \ T e) < e,
and
F(s)
every
such
= F'(s)
c > 0
that
F,:S
F
§ Hm
such
for a l m o s t
there
every
is a c l o s e d
is c o n t i n u o u s
that s.
set
relative
to
. For
m e s ( S \ T e)
such
(b).
closed
that
(d).
For
corresponding
mes(S\T)
e > 0, there
(c) ~
(d) ~ the
every
< e,
PROOF.
sets,
(b) ~
sequence
hence
F-l(c)
F
(a) ~
(c).
the
case w h e r e
m,
we
can e x p r e s s
k = 1,2,...,
can
have
find
is c o n t i n u o u s
finitely
many
Therefore, closed
F
x ~ F(s)}
closed.
subset
of
In the rest be a C a s t a l n g exists
by the
compact
set
Since
mes(SkT
)
relative C c Rn
~ to
Tc,
is open,
so t h a t
to
with
the
that
relative
Tk
of
with of
with
F'
sets
mes(S\T)
xi
Te
we have
in
Hence
Rm
Borel
relative
touch
to
for some
sets
to
to e v e r y
sets
S k,
T k c S k,
< E2 -k.
we
such
that
No m o r e
than
any b o u n d e d
region.
T = u~=IT k,
which
is a
< |
assume F.
mes
Let
S < ~.
e > 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
is c o n t i n u o u s
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 s a t i s f i e d
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
is c o n t i n u o u s
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
is m e a s u r a b l e
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)
is m e a s u r a b l e ,
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,
is m e a s u r a b l e
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
of m e a s u r a b l e
multifunctions
of m e a s u r a b l e
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))
is m e a s u r a b l e
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'
is m e a s u r a b l e
(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
is m e a s u r a b l e
PROOF. for
Let
j = l,...,m.
= Fl(S)•
(closed-valued).
(xili
e lj)
be a C a s t a i n g
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)
is m e a s u r a b l e .
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
is m e a s u r a b l e
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).
is m e a s u r a b l e
set),
= clui~iri(s).
(closed-valued).
and m e a s u r a b l e
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
is m e a s u r a b l e
(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)
the m u l t i f u n c t i o n
: gph A s
r':s
i_~s
+ Rm
by
is m e a s u r a b l e
(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
the m u l t i f u n c t i o n
is m e a s u r a b l e
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
is m e a s u r a b l e .
easily
y c C k}
~ ~}.
is m e a s u r a b l e ,
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
is m e a s u r a b l e
in
in the
definition
Let
F:S + R n
let
F:S
s
and
b-XY
continuous
r,(s)
Then
F'
is m e a s u r a b l e
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
and m e a s u r a b l e ,
a mapping Let
by
1B).
Q.E.D.
be c l o s e d - v a l u e d be
a countable
is m e a s u r a b l e
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'
is m e a s u r a b l e
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"
is m e a s u r a b l e
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.
is m e a s u r a b l e
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
the m u l t i f u n c t i o n
[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
is a m e a s u r a b l e
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.
is m e a s u r a b l e
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,
the f o l l o w i n g
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
the m u l t i f u n c t i o n
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
is a m e a s u r a b l e
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
be an i n t e g r a n d
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
and m e a s u r a b l e ,
and m e a s u r a b l e
s is m e a s u r a b l e
be an i n t e g r a n d
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)
be a C a s t a i n g
-~
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)
is m e a s u r a b l e
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
of m e a s u r a b l e
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
of m e a s u r a b l e
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
is a m e a s u r a b l e
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**
= <~(s),~(s)> 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
and m e a s u r a b l e
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>
= {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-<.,y> 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<~,y>. 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<~,yk> 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 <
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).
<_-e(s,O)
as a constant
Ze~(-k) The last part of Corollary
e(s,O)-kn}.
integrand
ek(S,-k) so that,
< B,
(3.31) holds as claimed.
k = 1,2,... ek(S,
ek
Thus
=-Ie(n)
3J
being nonatomic,
<_-Ie(O)
In addition,
for all
satisfies, like we have
s E S,
function
in
L;,
we have
< S.
can therefore be applied to and this yields
e k,
for every
a > inf{IA**(~) j ~ ~ L~} vk the relation {q E L I [
Ie**(n)
< a} = ~(L
,Ll)-Cl{q
~ L I
(n) < a}.
the
203
In particular, {n c LII as follows
if also Ie{,(n)
i ~} r {n E L
from (3.30),
the right of (3.35). (3.33) But
~ < -B, the latter implies
(3.31) and the weak closedness
and
(3.25)
< -8 ~ n
5 8 k (s,n)+e**(s,n)
where the first term is summable convergence Ie**(n) Therefore
h 0, fqd~ h 1.
imply
-f*(s,0)
Lebesgue
of the set on
Thus
le~,(q)
(3.32)
n A 0, fnd~ ~ I},
in
s,
for
so that,
n ~ 0, as ensured by the
theorem,
= llm I ,(n) k§ eL
for all
q > 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
_<
1
9
Since also by (3.29), we have Ie**(q)
< -If,(o)
and we are able to conclude lim inf
< +~
for all
~ ! 0,
that
Ie**(n)
is finite.
II~ II+o This implies LII, = L I=
for the convex functional
is proper.
1O, ,
But by Theorem 3C
18, , = le*** at least one
b(s) = 8*(s,c(s))
(summable),
we obtain
condition
for compact
Theorem
3K by (e) with
and generalized and
(S,A)
Valadler
[25].
and necessary, versions
c ~ L1
Rn
For versions
18,(c)
(3.27)
finite.
as desired.
level sets in
L n1
Taking
Q.E.D. given in
Proved in Rockafellar
[2],
replaced by a Banach space
For related results,
see Berliocchi
generalize
with
q = =, was originally
in [8] to cases with
complete.
on
is
= 18,.
Hence there exists
The sufficient
that its conjugate
this conjugate
see also Castaing
[23] and
of the condition which are both sufficient and Lasry
the classical
[26] and Clauzure
[27].
theorem of LaVall~e-Poussin.
These
204
Theorem
3K and its proof yield, with small effort, the following
theorem on continuity. measures
reflects
Here the equivalence
and Clauzure
[27].
3L.
(Continuity).
THEOREM
S • Rn~ and consider conditions
Let
If
__~
the implications
the conditions atoms and
of (b) and (c) for nonatomlc
facts noted in more general
f
1 _< p <_ ~. (b) ~
equivalent
(c) ~
(d)
always hold, with
if the measure
is finite on a neighborhood
(b)
If
is finite and continuous
at an element
~ ~ LP'n
(c)
If
is finite and continuous
everywhere
P L n.
(d)
f
satisfies
If(x) PROOF.
> -~
3K that
of an element
for at least one (a) ~
(b) ~
refinement
(c) ~ (b).
(c).
The proof of (a) ~ (b) can be of the argument
shows at the same time that
to the seemingly weaker assertlon,(c'), that
everywhere
on
in T h e o r e m
(b) ~ (d) g
3K that
for
~
Lpn,
I
is finite
and this implies
of the
(e) ~ (c).
nonatomic,
p < ~.
Let
g(s,x)
(Proposition
is finite and continuous
g
If
in
(c) is
(d) ~ (c'), as shown by the beginning
is a normal convex integrand
functional in
But
n.
and
equivalent
L p. n
~ ~ Lp
x ~ LP'n
(localization)
This
o nn
(G*p),
the srowth condition
Trivially
effected by a slight
Then
space is without
p < ~. If
argument
on
Then among the following
(a)
Theorem
[28]
be a normal convex integrand
L pn, (a) ~
all actually
contexts by Bismut
2N).
= f(s,x(s)+x). The convex
on a n e i g h b o r h o o d
that the conjugate
Ig = Ig.
of the origin
by 3C.
Applying
,
Theorem
3K to g*, we see that
argument If
Just given,
Ig
is finite and continuous
be retraced with
~
g
satisfies
(Gp),
is finite and continuous everywhere,
replaced by
0,
and hence by the everywhere.
and the preceding
showing that
f
Hence
argument
itself satisfies
N
(Gp). 3M.
Q.E.D. PROPOSITION.
and consider neighborhood measurable
If
f
be a normal
convex
o__nn L~,
Let
I ~ p ~ ~.
Suppose
of an element
function
therefore
every such
furnishes
If
on
S • Rn ,
is finite on a
x ~ L p. Then there exists n satisfying
at least one
y: S § R n y(s)
and m o r e o v e r
integrand
y
~ ~f(s,x(s))
belongs
an element
of
to
Lq n
Sir(X).
can
a.e., (i/p + 1/q = l)
and
2O5
PROOF. that
This is obtained by reincarnating
the proof in Theorem
3K
(c) -~ (b). For generalizations
and Clauzure
[27, Prop.
Further properties
of Proposition
BM, see Bismut
E28, Theorem 2~
5]. of integral
with the theory of liftings,
functionals
on
may be found in Levin
L~ ~31].
spaces,
connected
206
References i. 2. 3. 4. 5. 6. 7. 8. 9. I0.
ii. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
R.T.Rockafellar, "Integrals which are convex functionals", Pacific J.Math. 24 (1968), 525-539. R.T.Rockafellar, "Integrals which are convex functionals, II," Pacific J. Math. 39 (1971), 439-469. C.Castaing, "Sur les multi-applicatlons mesurables", th~se, Cain, 1967. This has partly been published in Rev. Franc. Info. Rech. Operationelle I (1967), 3-34. K. Kuratowski and C. Ryll-Nardzewski, "A general theorem on selectors," Bull. Polish Acad. Sci. 13 (1965), 397-411. V.A.Rokhlin, "Selected topics from the metric theory of dynamical systems", Uspekhi Mat. Nauk 4 (1949), 57-128. See also Amer.Math. Soc.Translations 49 (1966), 171-240. R.T.Rockafellar, "Measurable dependence of oonvex sets and functions on parameters", J.Math. Anal.Appl. 28 (1969), 4-25. M.F.Salnte-Beuve, "Sur la g@nCralization d'un th@or@me de section mesurable de yon Neumann-Aumann", J.Math.Anal.Appl. 17 (1974), 112-129. R.T.Rockafellar, "Convex integral functionals and duality", Contributlons to Nonlinear Functional Analysis (E. Zarantonello, editor), Academic Press, 1971, 215-236. G.Debreu, "Integration of correspondences", Proc. Fifth Berkeley Symp. on Statistics and Prob., Vol. II, Part i, (Univ. California Press, Berkeley, 1966), 351-372. R.T.Rockafellar, "Weak compactness of level sets of integral functionals", Trolsi@me Colloque d'Analyse Fonctionelle (CBRM, Li@ge, 1970), H.G.Garnir (editor), 1971, 85-98. TravauxM'Valadier'du S6minaire"C~ ~ n ~ S c ~ ~ v ~ i ~o ~ e~ n ~ 3~ P ~ no. 2. R.T.Rockafellar, Convex Analysis, Princeton University Press, 1970. I.Ekeland and R.Temam, Analyse Convexe et Probl~mes Variationelles, Dunod, 1974. H.Attouch, "Mesurabilitg et monotonle", th@se, Paris-Orsay, 1975. R.T.Rockafellar, "Existence theorems for general control problems of Bolza and Lagrange", Advances in Math. 15 (1975),1312-333. R.T.Rockafellar and R.Wets, "Nonanticipativlty and L -martingales in stochastic optimization problems", Math. Programming Studies, 5 (1975). I.V.Evstigneev, "Measurable selection and dynamic programming", Math. of O.R. 1 (1976). J.Lindenstrauss, "A short proof of Liapunov's convexity theorem", J.Math.Mech. 15 (1966), 971-972. K.Yosida and E. Hewitt, "Finitely additive measures", Trans.Amer. Math. Soc. 72 (1952), 46-66. V.L.Levin, "Lebesgue decomposition for functionals on the vectorfunction space LX, Funkt. Anal. + Appl. 8 (4)(1974) 48-53. R.T.Rockafellar, "Level sets and continuity of conjugate convex functions", Trans. Amer. Math. Soc. 123 (1966), 46-63. J.J.Moreau, "Sur la polaire d'une fonctionelle s6micontinue sup4rieurement", C.R.Acad.Sci. Paris 258 (1964), 1128-1130. C.Castaing, "Quelques resultats de compacit~ li~s a l'iht4gration", C.R.Acad. Sci. Paris 270 (1970), 1732-1735. C.Castaing, "Int~grandes convexes duales", Travaux du S@minaire d'Analyse Convexe, Montpellier, 1973, Expos~ no. 6. M.Valadier, "Contribution a l'analyse convexe", Th&se, Paris, 1970. H.Berliocchi and J.M.Lasry, "Int~grandes normales et mesures parametr@es en calcul des variations", Bull.Soc.Math. France, 1974.
207
27. 28. 29. 30. 31. 32.
P.Clauzure, "Quelques proprlet@s des espaces de K6the g~n ralis@s", Travaux du S~minaire d'Analyse Convexe, Montpelller, 1973, Expos~ no. 4. J.M.Bismut, "Int@grales convexes et probabillt~", J.Math.Anal. Appl. 43 (1973), 639-673. Daniel H. Wagner, "Survey of measurable selection theorems", manuscript of 80 pp. (authorw address: Station Square l, Paoli, Pennsylvania 19301, USA) C. Delode, O. Arino and J.P.Penot, "Champs mesurables et multisections," manuscript of 37 PP. (Penot's address: D@partement de Math~matlques, Boite Postale 523, 64010 Pau-Universit@, France) V.L.Levin, "Convex integral functlonals and the theory of liftings", Uspekhl Mat. Nauk 31(2) (1975), 115-178. (For English translation, see Russian Math. Surveys (1975).) R.T.Rockafellar, "Existence and duality theorems for convex problems of Bolza", Trans. A.M.S. 159 (1971), 1-40. Department of Mathematics GN-50 University of Washington Seattle, WA 98195, USA
APPLICATIONS DE L'ANALYSE CONVEXE AU CALCUL DES VARIATIONS
R. TEMAM
Nous d~veloppons ici des applications de l'Analyse Convexe ~ l'~tude de probl~mes de calcul des variations et de certains probl~mes aux limites non lin~aires. Le paragraphe I e s t consacr~ ~ l'~tude de probl~mes aux limites rattach~s l'~quation des surfaces minima avec des conditions aux limites mixtes du type Dirichlet-Neumann. II s'agit i~ d'une extension de l'~tude faite en
[16] (cf. aussi
[4]>. Le paragraphe 2 est une ~tude pr~liminaire du probl~me d'~volution du ler ordre en
t
rattach~ ~ l'op~rateur des surfaces minima. Une ~tude plus compl~te
fera l'objet de ~l~. Ce paragraphe contient aussi le calcul de certaines fonctionnelles convexes d~finies sur
BV(G)
et qui sont ~troitement li~es aux probl~mes
abord~s dans le paragraphe precedent et celui-ci. Le parasraphe 3 est consacr~ ~ l'~tude de certains probl~mes ~ fronti~re fibre d'un type nouveau que l'on rencontre en physique des plasmas (~quilibre du Tokomak).
Le plan est le suivant : I. Probl~mes mixtes rattach~s ~ l'~quation des surfaces minima. I.i. Position du probl~me. 1.2. Le probl~me primal - Suites minimisantes. 1.3. Le probl~me dual. 1.4. Construction de suites minimisantes particuli~res de ~
.
1.5. Comportement au bord. 1.6. Le r~sultat final.
2. Probl~mes d'~volution rattach~s ~ l'~quation des surfaces minima. 2.1. Description du probl~me. 2.2. La fonctionnelle 2.3. Calcul de
e
2.4. Calcul de
~e
sur
e BV(R) .
2.5. Le probl~me d'~volution.
209
3. Un nouveau type de probl~mes ~ fronti~re libre. 3.1. Une remarque. 3.2. Application. 3.3. Une in~galit~ fonctionnelle.
w
PROBLEMES MIXTES RATTACHES A L'EQUATION DES SURFACES MINIMA.
i.i. Position du probl~me. Soit zienne. Soient ~(r)
et ~(~)
F~
R
un ouvert born~ de
une partie ouverte de
RE
dont la fronti~re
r , g
et
~
Y
est Lipschit-
respectivement dorm,s dans
. On s'int~resse au probl~me d'optimisation
(i.i)
Inf
{I
u e ~+w~'l(n)
/l+(Vu)2 d x -
I
n
gu dF}
rI
o o~
WI'I(~)
est l'espace de Sobolev des fonctions u ~ L I ( R ) , dont les d~riv~es _I,i LI(G) , W ,I(G) (resp. W F (G)) le sous espace des fonctions o dont la trace 7oU est nulle sur F (resp. F o) ; YoU est d~fini co=,ne un
premieres sont dans u
~l~ment de
LI(F)
d'apr~s
[6]
; enfin
Si elle existe, une solution
r I = F~F ~ u~2(-~)
du probl~me (I.I) est solution
classique du probl~ma aux limites mixte
(1.2)
Au
=
-
V
V____~u
=
0
"
/1+vJ (1.3)
u = ~ ~u ~A
(1.4)
sur
=~(I+(Vu)2) -I/2 ~u ~A
r
o = g
sur
rl '
vecteur unltaire ext~rieur normal ~
Cela englobe les cas limites ci-apr~s -
F
o -
FI
r .
:
= r , on retrouve le probl~me non para~trique
des surfaces minima.
- r , on obtieut un probl~me de Neumann pour l'~quation des surfaces minima, et en particulier si D'apr~s
g = cos ~ , ~ ~ ~ , un probl~me du type capillarit~.
(1.4) une condition n~cessaire pour
g
est
210
(1.5)
Ig[L.(r) ~ I
Dans le cas o~ sur
g
.
r I = r , d'apr~s (1.2) et (1.4), une autre condition n~cessaire
est f ~ g dr = 0 Jr
(1.6)
i
D'autre part on assurera dans ce cas l'unicit~ de
u
en imposant que
f |
(i.7)
u dx = 0
P
J
l'unicit~ ~tant ~vidente quand
1.2. L e p r o b l ~ m e LEMME i.i.
Si
rI # r .
primal - Suites minimisantes.
IgIL~(rl) est assez petit,
(1.8)
< 0o(r
]gl | L (r I)
l'infimum dans (1.i) est fini et les suites minimisantes de (1.1) sont born~e8 da~s ~I
(~) . r I # r , nous avons une variante de l'in6galit~ de Poincar~ :
(1.9)
] rI
ue
wl'l(~)
]u[dr~< cl(~) {I ]Vu[dx + I ]uldx} , n ro
. Si (1.9) n'~tait pas vraie, il existerait une suite
u
telle que n
i= f l%]dr > n~f, lvunldx § f rI Alors
Vu
n
--~ 0
dans
LI(~) ~
et si
if
m m e s-
en
[UnNr) ro
s
Un
l'infigalit~ de Poincar~ classique montre que
dx
j
u
n
-e
n
~
0
dens
WI'I(R)
. Mais
on a aussi :
'
I
le=Nr r
o
-
f
]Un-%ldr r
et co~aue la seconde int~grale tend vers Alors
un
--~ 0
dans
1
< ~,
o
wl'l(s
0
d'apr~s [ 6 ] , on volt que
, et donc utilisant encore [ 6 ] ,
e
n
--~
0
.
211
IFI [Unlar
0
en contradiction avec le choix de la suite
u
n (i.I) par
On minore alors la fonctionnelle
II
(1.10)
,
uld
-Igl.
c1( ) "
i*ld
L (rI) Si
rI
=
F
,
mes r o = O
alors
9
o
et par un raisonnement
analogue au precedent,
on trouve que
pour
lF luldr.
~u~wl'l(fl)
qui v~rifie
(1.71, et, pour ces
u , on minore la fonctionnelle
(1.1) par
(1.12)
igIL.(r)C2(o)) . Inlvu[dx '
(l -
et le r~sultat suit.
1.3. L_~eprobl~me dual. On utilise le cadre habituel de dualit~ comme en [4 ] 9 Soient convexes
V - WI'I(~)
, Y = LI(~) s , A = V , et soient
s.c.i, propres Bur
F(v) = - I
V
et
Y
gv dr , si
F
et
G les fonctions
d~finies par
r + w.l,l(fl) r
v~
rI
|
+
o0
sinon,
Y v~
V
o G(p) = "J J fl
l~+p~-dx
,
~p ~Y
.
Le probl~me
(e)
[nf {F(v) + G(Av)) uaV
est alors identique ~ (I.i). Soit
V*
le dual topologique de
gnent les fonctionnelles
conjugu~es de
V
et F
et
Y~ = L|
F~
et
G~
G , alors le probl~me dual de
s'~crit
(~)
~ . Si
Sup {- F~(h"p *) - Gm(-p*')} p~_. Y *
d~si(~)
212
Le calcul de
G ~ est fair en [16] :
G~(p') - I I/--Ip~*12dx Pour
Ip'(~)l .< 1 p.p.,
si
+
~
sinon~
F *:
si
ep*
=
0
et
p ~ . ~ + g -- 0
r1 sur
rI
et
+|
autrement.
D_~_m_o_n_s_t_Ea_t_i_o_n F"(A'p*) =
Sup ve
{[J p'.Vv dx + [J
~+wlr'i(~)
gv dr} r1
0
= I p'.V, dx + I
fl p ~ . V w d x + Sup g, d r + rI w~wlr'l(~)
I
gwdr)
0
Le dernier supremum est sup~rieur Sup [ p*.Vw dx , w e ~ (~) #~ qui vaut
+|
si
Vp* # 0 . Supposons alors
Vp ~ = 0 ; par la formule de Stokes
g~n~alis~e de [ ] ,
Ainsi Sup
I
w6w~'l(a) rl
rI
(p~.~+g)w dF
0
et le r~sultat suit.
9
On peut alors expliciter ( ~ )
Sup {- I p'.V, dx + I~ 1-,/~.[p~J 2 dx}
(1.13) pour
p~
:
L~(~) s
Ip~(x)l ~< i
p.p op
Vp ~ = 0
O
p~
+ g = 0
, sur
rI .
A present on est en mesure d'appliquer les Th~or~mes 4.1 - 4.2 de [4 ] pourvu que (1,8) soit v&rifi&e en sorte que (~)
inf ~ > - 6 ;
il existe une solution
, et cette solution est unique par la stricte concavit~ de
p~
- G ~ . On a
de
213
inf ~ -
et si le probl~me
a une solution
sup 9
E
~ R
,
alors on a la relation d'extr~malit~
G(A~) + G*(-~ ~) = -
que l'on interpr~te comme en ~ 6 ]
(1.14)
~(x)
= -
(x)
~
P.P.
/1-Ip'(x) 12 On va ensuite donner des propri~t~s de
~
et en d~duire des propri6t~s de
quand ce probl~me ne poss~de pas de solution.
1.4. Construction de suites minimisantes particuli~res d__ee ~ On construit des suites minimisantes de
~
particuli~res, et on en d~duit certaines informations sur Cort~e que de
~(~) ~
~(~)
et que
est dense dans
~I(~) N W i,I(~) r o
.
poss~dant des propri~t~s ~ ~ .
est dense dans
_I,i w F (~) o
on voit
~ + W~'I(~)- , et donc il existe des suites minimisantes o
qui sont form~es de fonctions continues dans
~ . Soit
{Vm}meN
une telle
suite. Soit maintenant on peut d~finir
w
@
une boule ouverte, telle que
w
et dans
= v
I Aw = 0 w = v
L'existence de
w~9~(~)
Consid~rant la suite w m . On notera que m
= ~
sur
~ . Si
v~(~)
~ WI'I(~)
dans
~ \
0
est solution du probl~me de Dirichlet
~w
(1.15)
w
~c
de la mani~re suivante :
F
o
N WI'I(L~
dans sur
~~ 9
, qui est analytique d a n s @
, r~sulte de El4].
v
pr~c~dente onlui associera de cette mani~re une suite m W m ~ ( ~ ( ~ ) ~ WI'I(~) est aussi une suite minimisante de ~ car
et
Notons aussi que d'apr~s [9 J, w m ~
L~oc(0)
, et plus pr~cis~ment pour toute
214
01
boule
@ ,
relativement compacte dans
{WmlL|
(1.16) et d o n c p a r
le
~ c3(O,~I,lW,nILI(O))
Lemme I . I ,
lWlL~(~l) ~ =4 o~a
C4
" c4(~,~ I)
m
ind~pendant de
est
'
o
On arrive ~ present ~ un le-~e important
LEMME 1.3. Pour tout eomoact
K c ~ ,
~p
[p"(x) { <. n(x) < I .
K
~9~!~E~E!~ Soit w
~
: Ii suffit dtfitablir ce r~sultat quand
une autre boule avec
dbc 0 c ~c
K = @5
o~
@i
est une boule.
~. On utilise alors la suite minimisante
ci-dessus. On pose n
I
m
s
dx = inf
+ r
,
m
e
m
> 0 ,
e
m
0 9
D'apr~s la Proposition V.l.2 d e [4]
O*(,-p-)--~-
Vw ~ ~c m
m
et d'apr~s le thfior~me sur les sous-diff6rentiels 2), ii existe
Pm ~ LI(~)~
' Pm ~ L~
{P"
-
~
~ pros (ef. [ 4 ] ,
Th~or~me 1.6.
~ , avec
.<~
P~IL|
z
'
et Pm ~
~O*(-
p:)
,
c'est-~-dire
p~(x)
(1.17>
P'(~)
/~--Ip,~<~) 12 P'P"
D'apr~s une estimation a priori de Bombieri, de Giorgi, Miranda FI ], (el. aussi
215
Sup IVWm(X) [ ~< c 5 = c5(C~,~ic 4)
(1.18)
01
I~1
(xl9)
~< c 6 = c6(~)'~I,C 4) "
IH2(~)
w
Par extraction d'une sous-suite, on peut supposer que limite
u , pour
et dans
HI(@)
m fort
--~ ~ , d a n s
LI(~)
w
fort (car
(par 1.19)). Alors
Pm
converge vers une
m
est born~e dans
wl'l(~))
m
--~ Vu
dans
LI(~)
et p.p., et
(1.17) donne a la limite
c5
,Z-Ip'<~)l ~ D'ofi
p.p.
dans
~
.
2
c5
sup l~(x) I2 ~ x e ~
<
f
i
.
l+c 5
A l'aide de la d~monstration du Lemme 1.3 on volt qu'il existe ~(~)
(1.20)
Vu(x) = -
,
p (x)
~•
r e t conrne
~ c
~
existe
analogue dans tout
u , u n i q u e ~ une c o n s t a n t e
I Vm vn
on o b t i e n t
additive
xe~
par recollement
pros,
telle
du Lerrme, on v o i t
--~ u
dans
--~ Vu
quelle que soit la suite minimisante
p.p.
~ . A i n s i m~me s i
. P a r un r a i s o n n e m e n t a n a l o g u e ~ c e l u i
(1.21)
i~m
e s t une b o u l e q u e l c o n q u e ,
(1.20) une r e l a t i o n il
u~wl'l(~)
tel que,
dans
~
n'a
que ( 1 . 2 0 ) aussi
des relations
p a s de s o l u t i o n , ait
lieu
dans
que
LI(~) /R L~oe(~)~
,
v m
1.5.
Comportement
au bord.
On se propose de pr~ciser le comportement au bord de la fonction introduite ci-dessus
(sur
F
en particulier). 0
On suppose pour simplifier que precautions
construire des suites minimisantes uniform~ment
g = 0 , le cas
techniques suppl~mentaires
en
u
m , plus pr~cis~ment
m
g # 0
n~cessitant quelques
sera ~tudi~ dans ~I]. Dans ce cas on peut form~es de fonctions continues et born~es
216
Ii suffit pour le v~rifier de consid~rer la suite et
v m
introduite au paragraphe 1.4,
de poser
U(X)
0ha
lr
i M Vm(X) -M
m
l~+IVVm12
si Vm(X) > M si [Vm(X) l ~ si Vm(X) < -M .
um = ~
p.p. et
Bur
ro
Soit maintenant r' une partie ouverte de F dont la courbure moyenne o o ext~rieure est strictement positive ou bien F' sera une partie ouverte de F~ o de classe ~ 3 et de courbure moyenne ~ 0 . Pour z ~_.F' et pour c>O fix~, on o peut d~finlr une fonction barri~re sup~rieure (of. [15][10]) : il existe ~ d~finie dens l'intersection de
~
avec une boule
i
(1.22)
B(z,p) ,
p>O , telle que
A~ > 0
dans
~ N B(z,p)
~ > M
darts ~ ~ '[B(z,p)XB(z,~/2)}
~(z) - E ~ ~(z) ~ r ~(x)
Alors minizdsante
~+e um
~ ~(x) - z
+ c
sur
r ~ B(z,z) .
est une sur-solution du probl~me et si on consid~re la suite ci-dessus on volt que
tement d~finie dans
~
~m = inf{Um'~+~}
est une fonction parfa i-
et appartenant ~ ~(~) N wl'l(~)
grace ~ la deuxi~me
condition (1.22). En outre (of. [iO] pour les d~tails),
f f ~ si
bien
que
~
m
est
aussi
dx ~
suite
2 dx
.
minimisante.
Alors avec (1.21) et en rempla~ant ~ventuellement on retrouve
%
(1.23)
--~ u
dans
LI(~)
u ~ ~ + r
dans
~ ~ B(z,p)
Utilisant une barri~re inf~rieure, sous-solution ~ trouverait que
(1.24)
u ~ ~ -
u
par
u + c p c
convenable,
et donc
r
dans
G ~B(z,p)
.
E
pros du probl~me, on
217 d'o~
(1.25) Connne
z
~(=)
est arbitraire,
- E .< u ( z )
la relation
.< ~(z)
(1.25) v a u t
+
~ .
d r-p.p,
sur
r 'O ' et
comme
~>0
est arbitralrement petit, on a en fair
(1.26)
u - r
r' 0
sur
Pour c o m p l e t e r l e s informations sur l e comportement de
u
sur
r 0
, signalons
que l'on peut comme en ~6] ~tablir que
(1.27)
lira sup ]Vu(x) I = +| X-~Z
quand
z ~r
et
u(z) ~ r
. Plus pr~cis6ment,
~U
I Tj(z) = +|
,i
u(z) < r
zi
uCz)
(1.28)
~-~(z) = - =
> ~(z)
.
1.6. Le r~sultat final. Regroupant tout ce qui precede on arrive ~ ceci : THEOREME I.i. et
g ~ ~(r)
Soit
~
u n ouvert Lipschitzlen de
Rs
e# 8oient
r ~(~)
n W ~, 7(n)
donn~s.
Lea probl~mes (1.1) et (1.6) 8ont en dualit~ et (I. 28)
Inf $~ = Sup
Le probl~me
~
5~.
poss~de une solution unique
Il ex~ste une fonction
u
p*
qui est analytique dans
d~finie ~ u n e oonstante pros
analytique dans
te l le que u ~- L|
(1.30)
0 wl"l(fl)
(1.31)
/1-1p qui v~rifie ex~ste.
Au = 0
da~s
I*
, e t q u i e s t s o l u t i o n de ( 1 , 1 ) s i o e t t e s o l u t i o n
218
Dana tous lee ca8, minimisante
um
u
est solution g6n6ralis6e au 8ens suivant : toute suite
de (1.1) converge vers
(1.32)
v
(1.33)
Vm
Si
g = 0
et si
u
pour lee topologies ci-apr#s :
--,- u
m
dane
--~ Vu
LI(~)/R
dans
F~' est une partie ouverte de
ext6rieure est strictement positive, ou si
F' o
L 1loc (R) ~ Fo
est ~ 3
dont la courbure moyenne est de courbure moyenne
positive ou nulle, on a (1.34)
Alora
u Si
u
Y' o
sup
est unique et (1.32) est valable dane u/
~
sur
r~
LI(~) .
alors
~u I ~-~ = +~
(1.~5)
= @
,
~u ~-i
-~
lorsque
u < @ ,
lorsque
u > @ .
On eu d~duit le COROLLAIRE I.i.
Si
fl est un ouvert lipschitzien de
une partie ouverte de
F
@ ~ ~(~) si r est a " o de courbure moyenne extSrieure strictement positive, le
probl~me mizte de Diriehlet-Neumann
La ~m~ conclusion vaut si
P
o
R~
(1.2)-(1.4) admet une solution
est
~3
u
unique dane
et de courbure m~yennep o s i t i v e ou
nulle. ~u
Pr~cisons que la condition (1.3) est entendue au sens du th~or~me de trace de Gagliardo
[6], done dana
de trace de Liona-Magenes (I) Si
p ~L2(R) s
H-I/2(F)
et
LI(F) ~2]
. La condition (1.4) est prise au sens d'un th~or~me
(el. aussi ~7])
divp~L2(R)
. Si en outre
p~L~(R)
Ip.~]
, alors s , alors
(i)
p.9IF
p.~ ~ L|
~ Ipl L~(F)
a un sens comme ~l~ment de
L~(n) %
et
219
w
PROBLEMES D'EVOLUTION RATTACHES A L'EQUATION DES SURFACES MINIMA.
2.1. Description d u p r p b l ~ m e . On se propose de consid~rer ici le problame d'~volution suivant :
(2.1)
du d--{+ Au ffiO
(2.2)
u = ~
(2.3)
dans
sur
~
~ x (O,T)
(O,T)
x
u(O) = u o
oO
~
et
~
sont donn~s comme au w
u
est donn~ dans
o
L2(~) .
Le principe de notre ~tude sera bas~ sur la remarque suivante : sl l'op~rateur A
d~fini en (1.2) ~tait remplaca par l'op~rateur
l'~quation de la chaleur dans
~ x (O,T) . Une des mani~res de r~soudre (2.1)-(2.3)
consiste alors ~ in~roduire la fonctionnelle
e (u) =
~'In(Vu)2 dx
i 1
+ ~
II est bien connu que
-& , alors (2.1) se r&duiralt
e
d~flnie sur
L2(R)
par
ol
si u ~
+ H (n)
sinon .
e
est convexe s.c.i, sur
L2(~)
et alors (2.1)-(2.3)
est ~quivalent au probl~me d'~volution abstrait
i d~ du. ~e
(2.4)
u(0) ~ ue qui admet une solution unique (dans un sens pr~cis~ ci-apr~s) d'apr~s la Th~orle des semi groupes non 1in,aires [ 2 ] ,
de
~tant maximal monotone dans
L2(~) 9
Si on proc~de de mani~re analogue pour (2.1)-(2.3), on est amen~ ~ introduire la fonctionnelle
e~
d~finie sur
L2(R)
I
(2.5)
~
par :
2
eo(U) = + ~
et (2.1)-(2.3) est alors ~quivalent
S inon
dx
sl
u e $ + W 'I(R)
220
d~
t 3-s
(2.6)
~%(=)
u(o)
= o
= u~
Malheureusement on verra que la fonctionnelle s.c.i, sur
L2(~)
ni sur
Ll(a)
~e
et donc
e
O
qui est convexe n'est pas
n'est pas maximal monotone. O
Notre objet ci-apr~s est de remplacer
(2.7)
e
=
e
e~
0
par une r~gularis~e s.c.i,
e :
,
de remplacer (2.6) par ~u
(2.8)
u(o) ~ u0 et de pr~ciser en quel sens (2.8) est une forme ~ n ~ r a l i s ~ e de (2.1)-(2.3) (la situation sera similaire a celle du ~.I).
2.2. La fonctionnelle
e 9
Nous allons consid~rer la fonctionnelle
e
cl-apr~s, introduite par
De Giorgi, Guisti, Miranda, pour l'~tude des solutions g~n~ralis~es de (I.i), et montrer ensuite que
e = O
Nous posons n
88 i
(2.9) i=l
i=l n
le
sup
~tant pris pour
80 . . . . ,
On~(~)
avec
.~
8"21& i
(OU = i). On v~rifie
lmO
ais~ment que l'on a aussi
(2.10)
e(u) " SuP (I s 1 i=l
o~
81 , .... 8 n e ~ ) ( ~ )
avec
n ~ e
i=l
2 i .< i .
i=l Si
u ~ WI'I(~)
, ~alors par integration par parties utilisant
[ 6 ] , on obtient
221
e (u) =
Sup
0 i ui(~-u)
~(~)
o. ~ 1
dr)
i=l
n
I o2-'1
soit
(2.111
e(u) = [ / I + Jfl
Notons que 8. ~ ( ~ )
e(u)
dans
1
< + |
f lu-%I dr . r
Vu 2 d x +
si et seulement
si
u e BV(~)
. En effet,
prenant
(2.10) on trouve
e(u)
t
~-
~es ~ + /
I~[.
J
R~ciproquement
(2.12)
si
u ~ BV(~)
~i=l .
, alors d'apr~s Miranda
~x i u d x = -
fl 0i ~~u
=
dx +
[13] ,
r
8i ~iu dr
e(u)~< f IVul + mes fl+ I lu-~Idr r et donc
e(u)
D'autre
< + | part
LEMME 2.1.
e = e- , darts
D~monstration Alors um~
:
Co,me
e
on dolt seulement dp + wl'l(fl)o ' La fonction
(cf. [6 ])
~,2(fl)
O
. Pour
v~rifier
Um ~
est convexe
--~
u
et s.c.i,
que si
dans
u~
wl'l(fl)
, on a
e ~< eo
, il existe une suite
u
m '
.
dans 9
, soit
WI'I(~)
WI'I(~)
peut ~tre prolong~e
~>O
sur
.
~
en une fonction
.d(x)
~E(x) = mlnt--~---, 11 , o~
de
d = distance
WI'I(Q) ~
r
, et
soit
u e = 6eu + (l-~e)~ il est clair que la suite
u
convzent," s
2.3. Calcul de
e
On se propose
~
0
.
m
m
sur
~
;
BV(~)
d'expliciter
. e(u)
pour
u ~ BV(~)
(e(u) = + | si
222
.
u~LI(fl)~BV(~)) Pour
3u
i = l,...,n ,
~x-'-~ est une mesure born~e sur 1
~u ~
d~composition de Lebesgue de
~ , et on pent done ~erire la
: 1
3u "= gi ~x'-"~
(2.13)
o~
gi~
LI(~)
THEOREME 2.1.
et
~i
dx +
~i
'
ast singuli~re. Le r~sultat est le suivant
u ~ BV(~),etsM~ (2.1$) ~ d d c o m p o s i t i o n
Soit
d e L e b e s g u e de
~U
~
.
~-i
(2.24)
~ s= + f. t~,1 + f r /.-~lar,
,c.) = f. 1r
g = (gl .....
gn ) " ~ = (~1 . . . .
~9~!~i9~
:
On
"~n ) "
v~rifie tout d'abord que
droite de (2.14). Utilisant
e(u)
(2.10) et (2.12), on a
est inf~rieur au membre de
(e = (o 1 ..... On ) ~ ( ~ ' ) n ,
le(~)l 2~ 1) : f~2
[-Ve.u + 1-/~ozj dx- I rO.V~ dr.=
R~eiproquement
en s'inspirant d'une d~monstration de f3 ] on va montrer que le
membre de droite de (2.14) est sup~rieur ~ Soit
e>O
existe un ouvert 81~(fl)n
fix~. Comme ~
tel que
E
S
et de mesure ~ e
lel(X)[ ~ I , et
If2l'l O1 "~ f[2 IB I
(ef. [ ])
Sup 0 ~ Ll(fl) n
lek
6>0
est port~e par un ensemble
qui eontient
(2.15) Con=lie
~
e(u) - 6,
- E.
arbitrairement petit. S
n~gligeable,
il
. D'autre part il existe
223
il existe
02 ~_ ~(Q)n , tel que
La r~union des supports de ouvert
Q'
01
et
relativement compact dans
82
est compacte, et il existe donc un
fl , qui contient ces supports et tel que
mes(~-fl') ~ r . Ii es~ facile enfin de trouver O ~ ) ( - ~ ) n supp 8 3 c
~\fl'
J e3.v(u-r dr >
lu-r
r
Soit
maintenant
8 ffi 01
de f o n c t i o n s
pa(x) = 0
st
dans
R\@
~=
82
R , e ~ ~(R)n
dans
~
( p o e ~ (R n )
et soit 0 '
support compact dans
et
T ffi 0
sur
i R '
, 0 ~ pa ~ 1 ,
= P a * ~ ; comme
e
est a
a
lorsque
est assez
petit et m~me
8 c ~' On note ~ present que
Donc
,
r~gularisantes
Ix I ) a)
Rn
le(=)l
~ .
r
e,
Pa ' u n e s u i t e
Pa dx = i ,
supp
183(x) I ~ i ,
et
(2.17) soit
, tel que
- I%(*)1 e(u)
8 = ee + 8 3 e ~
+ le3(=)l
et
est sup~rieur ~
II
I%(=)1~
(~)n
i
et v~rifie
(1~(=)1 ~ l)
le(x) l 4 1 , puisque
.
I ffi 11 + 12 + I3 '
;
=
n [(e3+ea)g + /i-(83+8a)2] dx
I2 = I~ (ea+B3)~
(ea+e3) ~(r
I3 ffi I
dr .
r
On se propose de minorer convenablement ces diff~rents termes. On a ais~nent
I 1 >.- i [8a.g + i~-82j dx + n(E) o~
n(r
--~
0
quand
e
--~
n(E) -- I D 'apr~s (2.15)
[vl(h~)
0 :
,[gldx + mes(~\~') >
Iv[(~)
-
~ , et
donc
.
224
ce qui entratne
If 12 > f Enfin
0a - 0
sur
fi
o3.~l~c,
Oa~ - c .
F , et par la d6finitlon
x3 '" I lu-,I dr
de
03 ,
c
-
r
Finalement
(2.$8)
e(u)
;~ Z > ,.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
<
O.
229
/-{du -~-{,
(Z. aO)
Si en outre
ae(uo) ~ ~ , alors
(2.31)
Si
~ L 2((O,T) x n)
~ae(u)
d_uu aeCu) ~ L2CCO~T) x ~) dt " u ~LI(O,T;~'I{~))
(2.3Z)
, alors la fonction
au_
v ...Vu
at
= f
dane
u
v~rifie
n • (O.,T) .,
I/~+Vu2
et ~U au av a--~a - ~
(2. Sa) e 'est-~-dire
,=-
u = ~ 8ur une pattie de
s~ru-~,)
r
,,~
et
r ,, { o , e ~
= I
et
,
I~-E[ = + ~
8ur l'autre
pattie. Si en outre
uc~1(~•
{Ojr))
alors
u
est la solution classique de (2.2)-
(2. a).
~2~E~2~
:
convexe s.c.i,
On a p p l i q u e propre
Pour presque Quand
tout
sur
s i m p l e m e n t l e Th~or~me 3 . 6 de [2 ] ,
L2(~)
t>O ,
u c LI(o,T;WI'I(~))
(2.21)-(2.22).
Quand au
I~I<=, Remarque 2.2.
uc~l(~•
, et done
ae(u(t)) cela
= f(t) entralne
(O,T)) au
De
est
que
e
est
- u'(t). (2.32)
, on d o i t
I ~-~AI
notant
maximal monotone.
et
(2.33)
en raison
de ( 2 . 2 0 ) -
avoir
r• (O,T) etdo=
u=,
Le probl~me (2.1)-(2.3) poss~de une solution au plus. S'il poss~de
une solution alors c'est n~cessairement la fonction
u
donn~e par le Th~or~me 2.2.
Sinon le Th~or~me 2.2 donne une sorte de solution g~n~ralis~e du probl~me, analogue au cas stationnaire (Th~or~me I.i). Noter en particulier qu'on retrouve pour les conditions aux limites une situation similaire ~ (1.34).
230
w
UN NOUVEAU TYPE DE PROBLEMES A FRONTIERE LIBRE.
3.1.
Une remarque. Soit
V
V
un espace de Banach r~flexif.
Soit
J
u n e fonction de
--~ R , convexe, s.c.i, et coercive au sens
O.1)
(l].lJ = norme de
JCu)
-~
Inf v~V
et qu'il exlste au moins un
On v a c o n s i d ~ r e r
u~
)JU~ --+ + |
J(v) > - |
V
pour lequel l'infimum est attelnt.
le cas particulier
(3.3)
W
si
V-). IZ ,st bien connu que dans ces c o n d i t i o n s
(3.2)
oQ
+ |
suivant
de c e t t e
situation
classlque
:
v c w ,
est un autre espace de Banach r~flexif, l'injection ~tant compacte. D'autre
part
(3.4)
J(v) = Jl(V) + J2(B(v))
(3.5)
J1
: V
(3.6)
J2
: W --~
,
o7
~
R
est
convexe, s.c.i.
R
est
continue
,
.
D)autre part
(3.7)
8 : W
--~ W
es~ un op~rateur continu,
et enfin,
(3.8)
J1(v) + J2(B(v>)
-~
+ -,
llvl~ - ~
Dans c e s c o n d i t i o n s ,
(3.9)
Inf
[~1(v) + J2(SCv))] > - - ,
v~V et l'infimum est a~teint en un point
u~V
au moins.
+
-.
231
3.2. Application. Ce qui suit est un probl~me que l'on rencontre en physiqu e des plasmas. On se reportera ~ ~8]
pour un probl~me d'un type un peu different en
physique des plasmas ; of. aussi [5]
pour des probl~mes du m~me type en m~canique
des fluides. Soit
n
un ouvert born~ de
Sobolev d'ordre sur
~ n , n = 2 ou 3 , et soit
I . On considare
a(u,v)
HI(~)
l'espace de
une forme bilin~aire sym~trique continue
, et semi coercive au s e n s
Hi(n)
(3.10)
-'~a>O ,
On se donne encore
a(v,v)
f~L2(n)
> aIVv]22 V v e Hi(n) L (n) n '
.
avec
I I n f(x) dx < 0 m(f) ffimes'----~,
(3.11) (3.12)
V ffiHi(n) ,
Soit d'autre part
(3.13)
Jl(V) = ~I a(v,v) - (f,~)L2(n)
W = LP(n) , o~
i ~< p < + =
si
2n n = 2 , 2 ,< p < n - ~
si
n>2 .
On p o s e
1 ivl 2
(3.14)
J2(v) = ~
L2(n)
et
(3.15)
B(v) = v
II est clair que les conditions (3.5)-(3.7) sont satisfaltes. La propri~t~ (3.8) sera d~montr~e dans la Section 3.3. Ce point ~tant admis pour l'instant, on obtient l'existence de
(3.16)
u~
Hl(n)
tel que
Jl(u) + J2(B(u)) ~ al(V) + J2(~(v))
On remplace
a(u,v)
ce qui signifie
par
u • %v ,
- (f,v)
%>0 , v ~
- (u_,v)
= 0 ,
, V v ~ Hl(n)
HI(~) , puis on fair
V v ~ HI(~)
.
% ~ 0 ; on trouve
232 (3.17)
-~u
(3.18)
8u 8--j= 0
-- u
= f
sur
~
dans
~ ,
,
a
o~ ~ est l'op~rateur diff~rentiel rattach~ ~ a ~ et ~ la d~riv~e conormale correspondante. (I) a Formellement , u est solution du probl~me a fronti~re fibre
I
(3.19)
- ~u
+ u = f
dans
~u
= f
dane
,
n+ = (x, u(x) > O} ,
(3.20) n_ - {x, uCx) < O} .
3.3. Une i n ~
THEOREME 3.1.
Soit
pn q = n-p
Soit quelconque
n
fonctionnelle.
un ouvert born~ de
si
p < n j
J ~ q < ~
Rn
de fronti~re r~guli~re
q~eloonque si
p >n
(2) et
~ et soit
p~l . r
p ~ r < q .
Alor8 il existe une constante
c~.22)
c
ne d~pendant que de
I,.,,JLqc~J ~ o.Clv~l .pc~.~ + ==[I._1 rcj
~j p, q, r
telle que
i. i~/q~ L2Cn)
D~monstration :
La d ~ m o n s t r a t i o n q u i s u i t
e s t due ~ H. B r e z i s
; of.
[18]pour u n e
d~monstration diff~rente. On ne restreint pas la E~n~ralit~ en supposant i)
Si
~
connexe.
p
Sobolev
(3.22)
[u - m(u) lLq(~) ~ c 1 [VulLp(~) ,
(I) Cf. [18] pour un r~sultat plus precis, quand on suppose plus de r~gularit~ sur les donn~es. (2) Frontiers assez r~suli~re pour que le Th~or~me d'inclusion de Sobolev pour WI'P(a)
soit vrai.
233
o~
fn
1 m(u) = mes ~
et
u(x) dx ,
I/q - I/p - 1/n . On distingue
alors deux (as :
a)
m(u)
<
0
;
comme
1 m(u) > - mes--'--~I
(3.23)
u_(x) dx ,
on obtient
(3.24)
i/r + i/r' = I , et (3.21) b)
m(u)
lU-lLr(~ )
en r6sulte.
> 0 ; on d6duit de (3.22)
que
I
Fu.~o] [u_ + re(u)Iq dx ~ c2 lvulq. ~ (~>
(3.2s)
On sait qu'il exlste Alors
~l/r
[m(u)[ ~ (mes
k = k(q,r)
> 0
tel que
(a+b) q >~ k a r b q-r
pour
a>O
,
b>~)
.
on obtient
~
Lr(fl)
[vu[~P(fi) q
~ c3
Ivul~p(a) (3.26)
[ulr/(q -r) Lr(fi)
D'ofi (3.21)
en combinant
ii)
Si
p>n
(3.22)
et (3.26).
, on a (3.22)
avec
q
arbitraire,
i ~ q < | , et on continue
comme ci-dessus.
Remarque 3 . 1 .
Pour
p - r = 2 , (3o21) devient
ivu n/2
lul 2=
Ln-~f(~)
sl
n>2
~
c~
]v"l~2(~)
+=~lu_l
,i~i~_2/2
L2(fl)
L2(fi)
jl
234
iVu l+c ~2(~) ] } lulLq(~ )
.< e{lvu[2(a)
+ max [lU_lLq(~),
lu_l~ L2(~)
q , l.0
sin
ffi2
(c
d~pend de
e ). Faire pour ce dernier cas
--q--= i + ~ . q-r Preuve de (3.8). On termine avec la d~monstration de (3.8) dans le cas de l'exemple (3.10)-(3.15). On ~crit : 1 i 2 J(v) = ~ a(v,v) - (f,v-m(v)) - mes(~) m(f) m(v) + ~]V_IL2(~ )
(3.271
(par (3.1o) et l'in~galit~ de Poincar~) > ~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.
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[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.
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