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,
(2.1