Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
922 Bernard Dacorogna
Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals
Springer-Verlag Berlin Heidelberg New York 1982
Author
Bernard Dacorogna Departement de Math6matiques Ecole Polytechnique F#derale de Lausanne 61, Avenue de Cour, 1007 Lausanne, Switzerland
AMS Subject Classifications (1980): 46-XX ISBN 3-54041488-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-11488-2 Springer-Verlag New York Heidelberg 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 c954 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. 9 by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
These notes are the result of a graduate course given at Brown during the first quarter of 1981. They should be considered as an introduction subject.
They are not intended to be a complete presentation
to the
of all the re-
sults in this area. The results presented here are not all new and obviously a large part of the first and second chapter owes much to various works of F. Murat and L. Tartar on compensated
compactness.
I would like to thank, particularly, ragement
and his many,
always helpful,
notes would never have been written. fully reading and correcting MacDougall
C.M. Dafermos for his constant encousuggestions.
Without his help these
I want also to thank W. Hrusa for care-
the manuscript.
Finally my thanks go to Kate
for the very nice typing of these notes.
B. Dacorogna Providence,
July,
R.I.
1981
Note. This research has been supported in part by the National Foundation under Contract#NSF-Eng. CME80-23824.
Science
WEAK CONTINUITY AND WEAK LOWER SEMICONTINUITY OF NON-LINEAR FUNCTIONALS
by B. Dacorogna
ABSTRACT
These notes deal with the behavior of nonlinear functionals with respect to weak convergence.
In the first chapter we investigate
several necessary and sufficient conditions in order that a nonlinear function is weakly continuous or weakly lower semicontinuous.
In
Chapter II we give some applications of the results of Chapter 1 to partial differential equations and to nonlinear elasticity,
in the
last chapter we deal with dual and relaxed variational problems.
TABLE OF CONTENTS Page Introduction ....................................................... Chapter I.
i
Compensated Compactness
Preliminary Result (Case without Assumptions on the Derivatives) ...............................................
7
w
Case with Assumptions on the Derivatives ...................
ii
w
Legendre-Hadamard Condition and other Necessary Conditions.
19
w
The Quadratic Case and Some Generalizations ................
31
w
An Important Example:
The Variational Case ................
39
w
Parametrized Measures ......................................
52
w
Chapter II.
Applications
w
Nonlinear Conservation Laws ................................
59
w
Existence Theorems in Nonlinear Elasticity .................
68
Chapter III.
Dual and Relaxed Problems
w
Dual Problems ..............................................
74
w
Relaxed Variational Problems and Applications ..............
80
Appendix ...........................................................
i00
References .........................................................
113
Index .............................................................
117
INTRODUCTION
These notes deal essentially with the behavior of nonlinear functions with respect to weak convergence.
Before describing the problem pre-
cisely, let us give some hints on where this type of problem
may arise.
One standard method for proving existence of solutions to a given nonlinear partial differential
equation
(or a system thereof)
of the general
form f(x,u,Vu,...,Vku)
consists in approximating
fE
be obtained by standard means. {u E}
original equation on the sequence given function and
fe
of solutions (0.1).
{u e} u.
(0.2)
uC
of (0.2) may
The problem is then to discuss whether to (0.2) converges to a solution of the
Usually the only information
that one can get
is that it converges weakly in a Banach space to a
Therefore
the question arises for what nonlinear
f
do we have
f~(u E) ~ when
= 0
has been chosen in such a way that solution
the sequence
(0.i)
(0.i) by a new equation
fe(x,ue,~ue,...,vkue)
where
= 0
u
u
(by
f(~)
(0.3)
9 , we denote weak convergence)?
Similar types of problems may occur in the context of the calculus of variations.
In the direct methods of the calculus of variations one
attempts to minimize a given functional by constructing a minimizing sequence.
As before, in general, this minimizing sequence is only weakly
convergent (in a certain Banach space) so it is important to prove that the functional is lower semicontinuous with respect to weak convergence, i.e. , lim inf I f(x'ug(x)'?ue(x) ..... vkue(x))dx e-+O I la f(x'7(x),V-~(x) ..... Vk~u(x))dx
whenever
u
E
(0.4)
9 -u.
More generally, even if the nonlinear functional
f
is neither weakly
continuous (as in (0.3)) nor weakly lower semicontinuous (as in (0.4)), a precise knowledge of the weak limit of the sequence
f(u E)
is important
and is often used in order to define "generalized solutions" of problems which do not have solutions in the usual sense.
This approach has been
fruitful in different types of problems in the calculus of variations, optimal control theory, etc. Finally there are also some physical reasons why one is interested in the behavior of nonlinear functions with respect to weak convergence, since weak convergence measures some kind of averages and often in physical models only averages of microscopic physical quantities are actually measured. For example in nonlinear evolution equations one is interested to know that given an initial data which is only an average of some quantities how the solution behaves as time evolves; hence the necessity of knowing which nonlinear functions are weakly continuous.
Let us now describe more precisely the problem under consideration.
cA n
be a bounded open s e t and ue
%
-u
us
~-----> A m
be s u c h t h a t
Lp (~), p > i, a s
in
Let
E § 0
(0.5)
m
where
%
denotes weak convergence in
f
~
dx
for every
>
~ 6 LP (a) m
in
A m.
In the case
p ffi~
L p, 1 < p < ~ ,
I <~(x);~(x)>dx, as
i.e.,
e § 0,
~ + = i and <.;.> denotes scalar product P we will denote the weak * convergence by
which means that
u
9 u
in
~ ) m
if
f dx > for every
f
as
E-+ O,
~ s L_I(~). m
Suppose now that a continuous function
f: ~m._.=> ]R
is given and
that f(uC) ' ~ s
I n t h e s e n o t e s we w i l l
in the sense of distributions.
study the relationship
between
s
and
(0.6)
f(~)
and,
in particular, we will investigate
(i)
when i s
f
sequentially
weakly continuous
(ii)
when i s
f
sequentially
weakly lower semicontinuous
(iii)
what is in general the relationship between
(i.e.,
s
s = f(~)),
and
(i.e.,
~ ~ f(~)),
f(~)?
In the remaining part of the notes we will omit the word sequentially in order to simplify the notations.
Before proceeding further let us see on a simple example that the problem is not trivial. s x U iX) = sin--. g
Choose
m = n = i, p = ~, ~ = (0,2~)
Then it is well known that
u
Define now
and
f: ]R
>
~
0
in
L (0,2~).
by
f(u)
2
=
-u
and observe that we have neither s = f(u) nor s ~ f(u), since
x2 f(ue(x)) = -(sin ~)
*
' s = -
89 (
L (0,2~).
Therefore in ,,rder to obtain weak continuity or weak lower semicontinuity one has to impose some restrictions on the sequence nonlinear function
{u e}
and on the
f.
We now give a brief outline of the content of the notes together with some historical comments.
The notes are divided in three chapters.
The
first one studies various necessary and sufficient conditions so that the function
f
is weakly continuous or weakly lower semicontinuous;
these
conditions will depend strongly on the type of bounds on the derivatives of the sequence
{u e}
one assumes.
In the context of the calculus of
variations the problem of specifying conditions of this type has received considerable attention and some of the results presented here go as far back as Hadamard [Tal],
([Hal],
[Ha2]).
Recently Murat and Tartar ([Mul]-[}~3],
[Ta2]) have set the problem in a more general context; they assume
some boundedness on some combinations of the derivatives of mere general than those of the calculus of variations. will adopt their setting.
u e, which are
In Chapter I we
In Section i we will give some well known results when no restrictions on the sequence
{u c}
(i.e., no bounds on the derivatives) are assumed.
In Section 2 we will give a general necessary and sufficient condition that we will call quaslconvexity which corresponds to the condition isolated by Morrey [Mol] in the context of the calculus of variations.
The
inconvenience of this condition is that it is not in a pointwise form. In Section 3 we will turn our attention to find polntwlse necessary conditions; one such condition was isolated by Murat and Tartar and in the setting of the calculus of variations is known as the Legendre-Hadamard (or elllpticity) condition. In Section 4 we will show, following Murat and Tartar, that if the nonlinear functional
f
is quadratic then the Legendre-Hadamard condition
of Section 3 is also sufficient.
This result is of importance to isolate
a large class of weakly continuous functlonals. In Section 5 we will see how some results of the calculus of variations may be deduced from the theory developed in the above mentioned sections. These results which appear as partlcular cases of those of Sections 2, 3 and 4 are historically prior to the above ones and the proofs of the varlatlonal results have been used as guides to the proofs of the more general results obtained in Sections 2, 3 and 4. Finally in Section 6 we will introduce the notion of parametrlzed measures which underlies the analysis of Sections 2-5.
The result pre-
sented in this section was proved by Tartar and follows the ideas developed by Young and MacShane. In Chapter II we will give some posalble applications of the theory developed in Chapter I.
In Section i we will show, following Tartar [Ta2], how the results of Chapter I may be used in order to get existence of solutions for a nonlinear conservation law. In Section 2 we will present the analysis of Ball [Be2] which uses some of the results proved in Chapter I in order to get existence theorems in nonlinear elasticity. In Chapter III we will consider minimization problems in the calculus of variations and we will introduce the notion of dual and relaxed problems. In Section i we will give some well known results of convex analysis relating dual and original problems. In Section 2 we will see how some of the ideas developed in Chapter I may be used in order to define "generalized solutions" of some variatiomal problems which do not have solutions in the usual sense.
The type of re-
sults described in Section 2 are known as relaxation theorems.
Relaxation
of non-convex problems finds its origins in the work of Young and MacShane ([Yol]-[Yo4],
[Mal], [M.a2]) and was also developed in the framework of con-
trol theory (see Warga [Wall
and the references quoted there).
The more
specific results described in this section were established in [De3] (see also [Dal]) and they generalize those of Ekeland and T~mam ([Ekl], [ETI], Lhap. X). We have tried to present these notes only as an introduction to a subject which is still developing and where several important problems are still open.
Moreover we have not tried to present all the results already avail-
able in this area.
We have also left out several closely related results,
in particular those involving higher derivatives (see [Mel], [BCOI]), the problems of
F
and
G-convergence, of homogeneization (see [DGI], [DGS1],
[CSI]o [BLPI], [BLP2]).
CHAPTER I COMPENSATED COMPACTNESS
w
Preliminary Result (Case without Assumptions on the Derivatives) In this section we will state and prove a well-known result.
us first restate the problem. {ug}Ll
and
~
(uV,~: ~ c ~ n
u
Finally let
u
f: ~m
Theorem i.i.
Let
~ c~n
> ~m)
L=(~)- -
in
be a bounded open set, let
be such that
as
m
But let
> ~R be continuous.
~ ~ =.
Then we will prove the following
Let
F(u;~)
=
[~
f(u(x))dx;
(i.i)
then, under the above hypotheses and notations, (i)
F
is continuous, for every
if and only if (li)
F
f
is afflne.
is lower semicontinuous, for every
convergence if and only if Remarks.
~, wlth respect to weak * converRence
(i)
f
~, with respect to weak *
is convex.
The above theorem is well-known in the calculus of varia-
tions (see for example, Tonelli [Tol]). (il)
Since in Theorem i.i
~
is arbitrary, the above theorem implies
that if f(uv)
*% s
in
L~
as
~ ~ ~
for every sequence if and only if
f
{u ~}
such that
u~
is affine, w h i l e
u
*
i > f(u)
in
L , then
i = f(u)
if and only if
f
is convex.
In the next sections we will see that by imposing some further restrictions on the sequences
{uV~
then there will be, in general, more w e a k l y con-
tinuous and lower semieontinuous (iii)
Finally
functions than those of T h e o r e m i.i.
it is important
if we replace weak * convergence p > i
(see for example, Morrey
to note that Theorem i.i is still valid in
L~
by w e a k convergence
Lemma
1.2.
Let
D
Lp
with
[Mo2]).
In the proof of necessity and throughout use the following standard
in
these notes we will very often
lemma.
be a hypercube of
be extended by periodicity
~n
(in each variable)
and let
f s LP(D),
from
to
D
~n
p ~ i,
then
[ f(~x)
If
~
i | f(x)dx meas D JD
in
LP(D),
as
9 ~ =.
"~ - j f(x)dx meas D D
in
L (D), as
~ + ~.
p = ~, then
f(ux)
Proof: p = ~.
We sketch the proof only in the case
Then
and since
f
f s L (0,1)
and so is
f
n = i, D - (0,i)
(defined as
f (x) - f(~x))
is periodic of period I, we deduce that
IIfgIIL~ - IlfIIL- .
it is then equivalent
(1.2)
to show that
fv
,
~
~ "
11
f(x)dx
0
and t h a t
and
( a p p r o x i m a t i n g by simple f u n c t i o n s ; see [DS1])
(1.3)
~0 f~(x)dx for every
0 < e < i.
But (1.4) is easy to verify since
if~
fv(x)dx =
f(vx)dx - ~
and hence, using the periodicity of fv (x)dx = [
]
0 (where
[~]
(i.4)
>~
f(y)dy,
0
(1.5)
f, we deduce that
f
1 f(y)dy + ~1 o
f(y)dy,
(I.6)
[~a]
denotes the largest integer less than
A).
Passing to the
limit in (1.6) we deduce (1.4) and the lemma,
o
We now proceed with the proof of the theorem. Proof:
Part (1) of the theorem is a direct consequence of (ll) (apply-
ing (il) to (il)
f
and
Necessity:
-f). Assume that for every
lira ~nf
u~'~
I
~
{u~}~= 1
we have
f(uV(x))dx > I~ f(;(x))dx"
We want to show that for every
v,w s
(1.7)
~ C [O,l],
(1.s)
f(Av + (l-A)w) < Af(v) + (l-l)f(w). Let of
D
D
be the unit hypercube of ~ n
and let
D1
be an open subset
so that meas
Now define
X1 " XD 1
DI
ffi
~.
(1.9)
to be the characteristic function of
XI(X) =
I 1
if
x E D1
0
if
x s D - D I.
DI, i.e.,
(i.i0)
10 Extend
X1
by periodicity (of period i) in each variable from
the whole of ~n
D
to
and then apply Lemma 1.2 to get
Xl(X) = Xl(VX) _ ~ V
fDXl(X)d x = meas D I = I, in L~(D).
(I.ii)
Finally define (1.12)
uV(x) = Xl(VX)V + (l-Xl(VX))W, and observe that by (i.ii) and by the definition of v
X
we have
*
i U
9 Iv + (l-I)w,
in
L~(D)
fCu v) = x~f(v) + (l-x~)fCw)
*
(1.13) If(v) + (l-I)f(w), in
L~.
Therefore, using (1.7), we get lim ~nf
uv
"~
/ f(uV(x))dx = [If(v) + (l-l)f(w)] meas D D
I f(~(x))dx = f(tv+(1-1)w)meas D. D Sufficiency.
Let L - lim inf
uV-~ We want to show that if
f
L > f
f Jn
I f (uV(x))dx.
<1.14)
is convex then
f(u(x))dx.
We may choose a subsequence of
v
~
{u } ~ I
(without loss of generality we
will assume that this subsequence is the sequence itself) such that L=
v lim ,% _ I u u
f(uU(x))dx"
(1.15)
11 We now apply Mazur's lemma (see p. 6 in [ETI]) to get that a convex N
combination of
N
~ ~k uk k=v u, i.e.,
strongly to
u i),
(with
~k_> 0
and
~ ~k = i), converges k=v
N
~k uk
> u
a.e.
(1.16)
k=9 We may therefore find, for every
e > 0, a sufficiently large
k
so
that
I
f(](x))dx <
fl
Using the fact that
f
I
and since
e
f
f( I ekuk(x))dx + g. k=V
(1.17)
is convex and the definition of
L
we get that
f(~(x))dx ! L + e,
(1.18)
is arbitrary, we have indeed established the result.
The above theorem is used to prove the existence of weak solutions of elliptic equations
(e.g.,see [Mo2]).
In the next sections we will try to
find the equivalent theorem when one assumes that some combinations of derivatives of
w
u
are known to be bounded in an
Lp
space.
Case with Assumptions on the Derivatives We now assume that we have some information on the derivatives, namely
we will assume that
I u~
*%
u
in
L~(Q) m
(H)
m n ~u v. Au v = ~ ~ aij k ~x3 bounded in j=l k=l k
f(u ~)
*'~ s
in
L~(~),
L2(~) q
(i = I ..... q)
12
~lere
aij k 6 ~
are constants.
Our goal is to find the equivalent which
f
we have
Remark.
% = f(u)
or
of T h e o r e m i.i (i.e.,
% ~ f(u)) when hypothesis
(H) holds.
All the following results are still valid if we replace
w e a k * convergence in that case,
in
L=
by weak convergence
in order to ensure that
f(u)
impose some growth condition at infinity on with
to find for
b > 0) wbile in
L~
in
Lp
(p < ~).
is a distribution f "(e.g.
However, one has to
If(y)I ~ a + blyl p
the only requirement will be that
f
is con-
tinuous. Examples.
(i)
Variational
A
is
Au ~ ~x 2 (in this case (ii)
- 0 ~x I -
q = i).
A n o t h e r example to w h i c h we will refer is the following:
u
~)
= (v~),wv) 6 L~176
• L=(~)
m
with
and
t ~ ~, vV ~ ~v9 ~Vgh = ~Ul,U 2) = grad = , 9 \3x I ~x2/
u
Then the natural
case, m = n = 2
m
m = n, and where
Au ~) = (div v , curl w~)).
In the context of the calculus of variations corresponding sufficient order
that,
A) Morrey
condition, for every
([Mol],[Mo2])
= Vv V
with the
has isolated a n e c e s s a r y
called .~uasiconvexit[, ~ c l R n,
(i.e., u
on the function
f
and in
13
lira inf ] f(ug(x))dx _> I
(i.e., s > f(u)).
f(u(x))dx
(2.1)
It is our aim to extend his results to the more general
setting of this section (i.e., under hypothesis (H)).
We will isolate
below (Theorem 2.1) a necessary condition that we will call A-quasiconvexity, by analogy with the variational case; this condition will turn out (Theorem 2.3) to be sufficient at least in some particular cases, including the variational case. Definition.
f
f: ]Rm
> ~
is said to be A-quasiconvex if
f(~ + ~(x))dx > f f(~)dx - f(~) meas D JD
D for every
A function
~ s IRm, for every hypercube
D c lqn
L(D) = {~ s L~(D); f ~(x)dx = 0 JD (by
~ E Ker A Remarks.
(i)
we mean that
~ aijk ~Vk ^ j,k
We will see in w
and for every
and
s L(D)
where
~ E Ker A},
0).
that the above definition corresponds,
up to a minor change,to that of Morrey ([Mol]) when (il)
(2.2)
u
= Vv ~.
Although, as seen in the following theorem, A-quasiconvexity appears
quite naturally, this condition is unsatisfactory since it is not a pointwise condition, as is convexity or the other conditions we will examine in the next section (w
Furthermore the definition of A-quasiconvexity
given here is probably not yet the best possible, in fact one would like to further restrict the set L(D),for example by adding a condition on the support of
~
as in the definition of Morrey.
But by adding this condition
it does not seem to be obvious how one would prove then that this condition
14
is also sufficient;
although in the particular case of the calculus of
variations this can be done (see w (iii)
Finally it may be useful in order to compare convexity and A-quasi-
convexity,
to write (2.2) in the following way (if
f since
f f(~ + ~(x))dx ~ f(~ + | ~(x)dx) = f(~) D JD
(2.3)
~ s L(D). Suppose that (2.1) holds
Theorem 2.1 (Necessary condition).
s > f(u))
(i.e., Then
meas D = i)
f
u
for every sequence
satisfying hypothesis
([i) (p.ll).
is A-quasiconvex.
Proof of Theorem 2.1: be a unit hypercube and
We adapt here Morrey's proof ([Mol]). ~ 6 L(D).
in each variable and define for
V
Extend
~
Let
D
by periodicity of period i
an integer
~V(x) = ~(Vx).
(2.4)
0
(2.5)
We therefore get that
~V * "
in
Lm(D )
~v 6 Ker A ~'~ [
aljk : ~
= O,
i = l,...,q.
(2.6)
j,k Observe also that
f D
f(~ + ~V(x))dx = i f f(p + ~(y))dy V n UD = f
f(~ + ~(y))dy,
(2.7)
D since
~
is periodic of period i.
Finally take the limit inferior as
v + ,~ of the left hand side of (2.7) and use (2.1) to get
15
f
f f(~ + ~(y))dy - lim inf ~ f ( ~ + ~V(y))dy Z f(~)meas D. D ~D By a change of variable the cube
(2.8)
above inequality is true for every hyper-
D.
m
Combining Theorems i.i and 2.1 we have the following diagram convexity ~ ~eak lower semicontinuity ~ A-quasiconvexity.
As a matter of exercise we will prove in a slightly different way that convexity implies A-quasiconvexity. Proposition 2.2. Proof:
Let
convexity ~
f: R m
> R
A-quasiconvexity.
be convex.
By a well known property of convex
functions (see Theorem 23.4 in Rockafellar F s
A(F) = (AI(F) ..... Am(F)) E ~ m
[Ro2]) there exist, for every
so that m
f(F + ~) > f(F) + for every
w E R m.
So choose
f f(F+w(x))dx > I f(F)dx + D
D
w E L(D)
(2.9)
7 Ai(F)n i i-i and integrate (2.9) to g e t
m~ AI(F ) iDWi(x)d x - f(F) meas D, i=l
the last equality following from the fact that
(2.10)
w C L(D).
We now establish the sufficiency of quasiconvexlty for lower semlcontinuity in a particular case (which includes the variational case), by adapting Morrey's proof ([Mol]). Theorem 2.3 (Sufficiency condition).
Suppose that
sis (H) as well as
__(H O)
u
- u { Ker A.
uV,u
satisfy hypothe-
18 If f is A-quasiconvex,
then (2.1) holds for every bounded open set, ~ c ~n ,
i.e., lira inf [ f(u\~(x))dx
>- ( f(u(~))dx .
I Proof: Let ~ be approximated by a union of hypercubes D k of edge length [, i.e., I H
= k
I <JDk. i=l l
meas Dki
(2.7.1)
as k § ~
meas([2-H k) § o
=!
1 ~ i ~ I .
kn
For x E H k , let
Uk(X)
i [ u($)d~ , for x 6 Dki meas Dki JDki
I g i g I .
(2.12)
Observe that we trivially have f(u ~) - f(u) = f(u+ (u~-u))
- f(uk + (u~-u))
(2.13) + f(~k + (u~-u))
- f(~k ) + f(~k ) - f(u).
Note that from (2.12) we may find, for every ~ > o, k sufficiently large so that
I
If(u+ (u~-u)) - f(u k + (u~-u))Idx ~ ~2
(2.14)
Hk Hklf(u) - f(~k) Idx g ~ "
(2.15)
Combining (2.13), (2.14) and (2.15) we obtain ]
f(u v(x))dx - [ Hk
JH k
f(u(x))dx + c >. I [f(uk+ (u~-u)) - f(uk)]dx . Hk
(2.16)
17
But from Hypotheses (H) and (Ho) we have that and ~
-=u
,,
-u
,,
oinL
E Ker A. We then define
for
qV(x) = 6V(x) - I
x 9 Rk ,
(2.17)
Hk a n d we c o n c l u d e
that Q
q
v
"'
"
0
C Ker A
InV(x)
in Lm(Hk) ,
~ § ~ ,
(2.18)
IHk
and
- ~V(x)1
as
q~(x)dx = o ,
+ o
as
(2.19)
~ + ~
(2.20)
Therefore returning to (2.16) we get
f(u(x))dx + c ~ Hk
Hk
[f(uk+~(x)) Hk
(2.21)
f(uk+(x))Idx+ I f(uk+(x))dx [ f(u )dx .R k
k
The two last terms on the left hand side of (2.21) are non negative on each I u Dk.. Therefore taking i=l l the limit as ~ § co in (2.21) and using (2.20) we obtain
of Dki , since f is A-quasiconvex, and hence on H k =
limv+ooinf IHkf(u~(x))dx >. IHkf(u(x))dx - E .
(2.22)
Using (2.11) and the arbitratiness of E, we have indeed established the theorem.
D
We now turn our attention to the problem of weak * continuity and we immediately obtain as a consequence of Theorems 2.1 and 2.3.
Corollary 2.4. Under tile hypothesis (i)
(H)
If for every ~Q c IRn and for every ~u } satisfying (H)
18
lira inf / f(uU(x))dx = I f(u(x))dx then
f
is
A-quasiaffine,
(ii)
If
f
is
that is,
f
and
-f
A-quasiaffiue and hypothesis
(2.24)
are A-quasicOnvex. (Ho) holds (~.e.,
v u -u E Ker A) then (2.24) holds. Remarks.
(i)
In the variational case Ball ([Ba2]) calls
A-quasiaffine
functions null Lagrangians since in this case the Euler Lagrange equations associated to the variational problem reduce to an identity (see w (ii)
Trivially if
f
is affine then
f
is
A-quasiaffiae.
We end this section by identifying a very simple class of functions which are
A-quasieonvex
(in the variational ease, Bali [Ba2] calls such
functions polyconvex). Corollary 2.5.
Let
g: ]Rs
> ]R
be convex an4 suppose that
f(u) = g(~l(U) ..... ~ (u))
(2.25)
s
where
#l,...,~s
Proof: every
are
A-quasiaffine.
Then
f
is A-quasiconvex.
As seen in Proposition 2.2 since
g
is convex we have for
a s ~s s
g(tl,''',t s) ~ g(a I ..... a s ) + i-i Therefore for
F s
and
~ 6 L(D)
Ai(a I ..... a s ) (ti-al) 9
(2.26)
we have
f(F+~) = g(~l(F+~) ..... ~s(F+~)) s
> g(#l(F) ..... *s(F)) +
~ AI(FI(*I(F+~) i-I
- $i(F)).
(2.27)
19 After integration over
D
(a hypercube) we get
ID f(F+~(x))dx> ID f(F)dx + i=ls~. Ai(F) [iD#i(F+~(x) )d x Using the fact that the
#i are A-quasiaffine, i.e., ~i
/D~i(F)dx] .
and
-$i
(2.28)
satisfy
(2.2), we deduce the result.
w
Legendre-Hadamard Condition and Othe r Necessar 7 Conditions We now turn our attention to pointwise necessary conditions (as con-
trasted to those of w tinuity.
for weak * lower semicontinulty and weak * con-
Before stating the main result of this section we need
duce some notation. First recall that I (H)
basic problem is
~ ~ ~n ~ u in Lm(~) (u ,u: ~ c m n 3u~ Am ~ = [ [ a J bounded in J=l k=l i J k ~ x k u
~
our
*
f(u V)
*~ s
> m m) L2
t
i
= 1,
in L~(~).
Define now an operator B(~):~ m
> ~q
(for
~ E~n)
by m
B(~)% =
n
~ ~ aijk%j~ k. j=l k=l
Finally let ~2/= {(%,~) s and
m •
B(~)% = 0} = ~ m
xiRn
~ 1 7 6
,q
to
intro-
20
A = {k E m m :
Since the set
A
_ {0}
with
(l,~) E ~
c ~ m.
will be playing an important role in our analysis we
give now some examples w
B~ 6 ~ n
to
which we will refer later in this section and in
and w Examples.
(i)
Case without assumptions on the derivatives:
(w
Then
A = ~m. ~ u~
Compact case:
(ii)
Suppose that
~--~ bounded in
L2
for every
= {0}
(since
%JSk = 0
(iii)
for all
J,k).
Single variational case:
Suppose that
u = (Ul,...,u n) = grad v
m = n
and
= (By . 8v ) ~Xl, ..,~xn
then au i _ ~uj) Au = (~-j ~x i = O,
i,J = i ..... n,
hence B(~)l = Ai~ j - lJ~i = 0
Using the definitions of ~/ and o~= {(k,~) E ~ m x ~ n = ~ 2 n :
~
for every
we get XlI~ }
therefore we find that in this case the set Example
(i).
i,J - 1 ..... n.
A
is the same as that in
J,k, then
21
(iv)
General Variational
Case:
Suppose
3v I
that
~v I
m = np ~v
9
p > I)
Bv P
u = (grad v I .... ,grad Vp) = ( ~Xl,
(with
, ,
..,Bx n ..... Bx I,
.,~x~)
then
Au = ( c u r l
grad v 1 ....
,curl
g r a d Vp) = 0
so B ( ~ ) I = xk~j - l ~ f
We then conclude
~/=
{(t,~)
= 0
for
all
k " i . . . . . p, and
i,J
= i . . . . . n.
that
E]R np x ~ n :
tkll~
for
k = 1 .....
p} = ~ m
XXRn
and thus
A = {I E~{nP:
(v)
~i II~j
for all
u = ( v , w ) E ~ . n x]R n
(so
i,J " 1 ..... p} c ~ np "JR m 9
m = 2n) a n d
Au = (div v, curl w) then n
I
!l
~j~i -
therefore
~-"= {((,~,]a),~)
-
0
ui~.i
= o,
C ~ 2n
•
i,J = l,...,n
>, J- ~,
and A = {(t,U)
[~2n:
X I ~} c ~ 2 n .
22
(vi)
u = (V(Xl,X2),W(Xl,X2)) 6 ~ 2
and
Au = (~v ~w ~x I' ~x 2) therefore
~/= {((II,12),(CI,C2)) 6 m 2 x m 2 : II~ 1
12~ 2 - O}
and A = {(Ii,% 2) 6 ~ 2 :
ii = 0
or
12 = O} c ~ 2 .
We now establish the main result of this section which gives a pointwise necessary condition in order that
Z > f(u).
In the context of the
calcul~s of variations such a condition is known as the Legendre-Hadamard (or ellipticity) condition and was isolated by Hadamard ([Hal],[Ha2]).
Under
our more general setting, the following result was established by Murat and Tartar ([Mul],[Mu2],[Ta2]).
Theorem 3.1.
(i)
hypothesis (H), then f(H + tA) (ii) f
If f
is convex in If
Z = f(u)
i > f(u)
{u ~}
is convex in the directions of t
for every
H q~ m
for every sequence
is affine in the directions of
Proof:
for every sequence
and
{uD}
satisfying
A, i.e.,
A 6 h.
satisfying
(H), then
h.
The proof of Murat and Tartar ([Mu2],[Ta2]) is a direct one,
while ours will use the notion of A-quaaiconvexlty defined in the previous section. As usual (ii) is a direct consequence of (1). for
Let
I E (0,I)
and let
A E A
(3.1) G=H-
A.
23 We then have that I IF
(I-I)G
+
ffi
H (3.2)
F-GEA. Since
A E A, there exists m
be a unit hypercube of ~ n
> ~
be an
L~
for all
[ aijkAj~k = 0 kffil
j=l
D
such that
n
[ Let
0 # ~ E~n
D1 c D
(3.3)
i = l,...,q.
with
meas D
i
= I
and
function such that I
~(x.~)
i
if
x E D1
if
x E D - D I.
(3.4)
=
- l-I
Finally let ~(x)
and observe that ~ s
~ E L(D)
L (.D);
f
D~(X)dX
(3.5)
"= A ( h ( x ' ~ )
(see Theorem 2.1) i.e., ffi 0
and
[ a iJk ~~xJk J,k
Therefore we can apply Theorem 2.1 (which says that if is A-quaslconvex) I
(3.6)
ffi 0 .
s
>
f(u)
then
f
to get
f(H + ~(x))dx ~ f(H)meas D = f(H)
(3.7)
f(F)dx + f f(G)dx t f(IF + (l-l)G). ~D-D I
(3.8)
D i.e., I DI
We then deduce that If(F) + (l-I)f(G) > f(IF + (I-I)G).
(3.9) D
24
If
Corollary 3.2.
A -JR m
then
(i)
A function
f
is convex if and only if it is A-quasiconvex;
(ii)
A function
f
is affine if and only if it is A-quasiaffine.
Proof:
A direct consequence of Prop. 2.2 and Theorem 3.1.
m
Let us now return to the examples preceding Theorem 3.1. Examples.
(i) and (iii) case without assumptions on the derivatives and
single variational case:
then
A =~m
and therefore the only weak *
continuous functions are the affine functions
(and similarly the only weak *
lower semlcontinuous functions are the convex ones). (ii)
Compact case:
then
A = {0}
vergence is strong we deduce that any (iv)
General variational case:
A = {% 6 m n P :
%illlJ
and since in this case the conf
is weak * continuous.
m I np
for all
(p > i)
then we saw that
i,J = i .... ,p} ciRnP = ~ m
therefore (a)
A necessary condition for weak * continuity is that
be afflne in u = Vv
t
for all
f(a + t%)
% 6 A; in particular, all subdeterminants of
satisfy this condition (we will see in the next section that in
this case the condition is also sufficient). (8)
A necessary condition for weak * lower semicontinuity is that
f(a + tA) (v)
be convex in
t
u = (v,w) E ~ n x ~ n
for all with
% s A. (div v, curl w)
we saw that A ~ {(~,~) 6 ~ 2 n :
~ • ~}.
bounded in
L2
then
25
(~)
The necessary
condition of Theorem 3.1 for weak * continuity
satisfied by the scalar product in
~n,
is
; we will also see in the next
section that the condition of Theorem 3.1 is in this case sufficient. (8)
Similarly in order that
is a c o n v e x (vi)
function of
%
s > f(u), we must have that
for all
u = (V(Xl,X2),W(Xl,X2))
f(a + tA)
% 6 A.
6~R 2
and
~v ~w ~x I ' ~x 2
bounded in
L2;
then we have A
(a)
-
{ (~,~) s ]R 2
Theorem 3.1 implies that if
separately affine; such an (8)
f
for example,
or
~ = 0
s = f(u)
f(v,w) = vw
~ = 0}.
then
f(',')
has to be
(here also we will see that
is actually weak * continuous).
Similarly
Remarks.
s ~ f(u)
(i)
implies that
f
is
be separately
convex.
In the variational
in a
C 2, namely
f"(a)(A,%) ~ 0
(or ellipticity)
f(',')
The condition of Theorem 3.1 can be reformulated
better known form if
(ii)
:
for all
a E~
TM, %
E A c ~ m.
case (3.10) is nothing else than the Le~endre-Hadamard
condition.
Anticipating
the results of the next section, we see from the
above examples that when we have some information on the derivatives the sequence tinuous)
u v) then we have more weak * continuous
functions
than in the case of Section
the case as seen in Example (iii)
(3.10)
Unfortunately
in general sufficient,
the
(of
(or lower semicon-
i; but this is not always
(iii). necessary conditions of Theorem 3.1 are not
as seen in the example below; we will see, however,
26
in the next section that in some cases they are. Counterexample
([Mu2],[Ta2]):
Q c~2,
m = 3
and
I u(x,y) = (~(x,y),B(x,y),y(x,y)) [ (~x' " ~ ~y'28~ x + ~ v )
are bounded in
L2
and let f(xyz) = xyz. We want to show that is affine on
f
satisfies the conclusion of Theorem 3.1 (i.e., f
A) but is neither weak * continuous nor weak * lower semi-
continuous.
~2/= (((XI,%2,X3),(CI,~2))
6~3
x~2:
ll~l ~ 12~2 = ~3~i + ~3~2 = 0}
and therefore
A = {(%1,%2,A3) 6 ~ 3 :
Hence, trivially,
f
at least two of the
%i
is affine in the directions of
are zero}.
A.
Now let us choose
a particular sequence e
Z = sin s
~E =
y
s
Observe that, t r i v i a l l y ,
But
cos
= sin
~~,
x s
--
x,y s
*%
0 = a
* ~
0
*
=
% 0 = %/.
~y , ~x
~y
are all bounded in
L 2.
27 r ~ e a~Beye Z 2 x f(<, ,~ ,Y ) = = sin g cos Z sin 7x cos ~x - sin 2 y~ cos i sin 2x sin 2y =
--
-
-
1
....
~-# 0
but was
3.1;
satisfying
3.3.
If
(H), then
I for all
< 0).
to find some other necessary
that for the variational
j
/'roof:
f E C~ f
satisfies
case
(see,
conditions Tartar
than
([Ta2]),
for example,
i)
for every sequence
the following
conditions
for
(u v} r > 2:
with
for all
y C ~m.
We have seen in Theorem
3.1 that
,,,I,%2 6 k
We will
Consider
~ = f(u)
rank(~l,...,~ r) ~ r - i "''~r = 0
f(2)
for every
and
(%l,gl) .... ,(%r,r ) E ~
f(r)(y)~l'
2)
--
[Ba2]).
fheorem
(NCr)
2 x C O S
the result below was proved by Murat,
known before
[Moll,
~
(and even
We now turn our attention that of Theorem
sin 2 v --
with
then prove
(y),~1%2 = 0 rank(~l,g 2) ~ i.
(NCr)
for
r = 3, the cases
r > 3
are
similar.
a sequence
uV(x) = u + t { ~ a ~ a ( v r
+ aB~B(v~B-x) + ~r~Y(v
where I (A~'ga)'(}'8,r162165 { rank{~Sl,r162 u
< 2,
E~7 ~
(3.11)
28
and
~a, ~ ~ ~y
are
Lm
functions, periodic, with average zero.
By (3.11)
we deduce therefore that u,~
For
t
*
small we expand
~
f
u
near
in
u
(3.12)
e~ .
to get
f(u v) = f(u) + tf'(u)(lar & + I~r B + IY@ Y) o
t
.(2)
+ ~-., t
(u)(~o$a + ~8~8 + !YSY)2
t3
+ 7.' f ( 3 ) ( u ) ( t ~ ~ + t ~ B + >Y~Y)3 + o ( t 3 ) .
As
~ ~ ~, by choice of
(3.]3)
~,~fl,~Y, we deduce that
f'(u)(l~ ~ + ~B~
+ IY~ Y)
~
0;
(3.14)
therefore we have to estimate the third and fourth terms in the right hand side of (3.13). Step i.
If we expand
the third term in (3.13) we get
f(2) (u)(~ar ~ + ~8~8 + ~v r f (2) (u) (li, lJ) ~i('o~i.x) ~J ( ~ J.x).
(3.].5)
i,j Case i.
If
rank{~i,~ j} = 2, then it is easy to see that as
~i(~i'x)@J(v~J'x)
Case 2.
If
* %
0.
~ § oo
(3.16)
rank{~i,~ j} < i, then by Theorem 3.i we get
f(2)(u)(li,lJ) ffi O.
(3.17)
29
Thus by (3.16) and (3.17) we deduce that f(Z)(u)(la@a + %8r
+ %y~Y)
0.
(3.18)
Step 2.
It therefore remains to estimate the last term in (3.13).
Case I.
If
rank{~a,~8,~ 7) ! 1 (in particular
3.1
by Theorem
rank{~i,% j} < I), then
we know that f(2)(u)(li,lJ)
by making the particular
choice
= 0;
u + t% k
(3.19)
in (3.19) and differentiating
we get f(3)(u)(%J,%J,%k)
If
Case 2. erality that
apply Case i).
rank{~,~,~
(3.20)
~ 0.
Y} = 2, we can suppose without loss of gen-
rank{~i,~ j} = 2
for every
i,j = a,~,7
It is then possible to choose ~(~.x)r
(3.21)
preceding Theorem 3.3) since rank{~i,r j} = 2.
Therefore returning to (3.13) using (3.14), that
to satisfy
c # 0
(as in the counterexample
the hypcthesis
~,@8,@Y
(otherwise we
~ = f(u)
(3.18),
(3.20),
(3.21) and
we deduce that
cf(B)(u) (X~,~ ~,~Y) = 0
and since
c # 0, we obtain the result.
In fact we have a more explicit form of Theorem 3.3 ([Mu3]).
(3.22)
30
Corollary 3.4. by
A (c~m)
E = (u 9 6 ~m nomial in
Let
be the dimension of the subspace
E
generated
and suppose that coordinates have been chosen so that Ud+ 1 . . . . .
yl,...,y d Yd+l,..~
Proof:
u E~
Let
generate
u
= 0} .
m
If
of degree at most
functions of
el,...,e d
d
and let
TM
E; then
~ = f(u), then inf{n,d}
f
is a poly-
whose coefficients are
el,...,e m be the basis of ~ m m u = _ ~lujej. But since f(a+tb)
such that is afflne
jfo* every
b E A
(Theorem 3.1) we have
m ~ u.e.) J=2 J j m m = ulf(e I + ~ u.e.) + ( l - u ) f ( ~ u.ej) ~=2 j j 1 j=2 J
f(u) = f(ule I +
= ui{u2f(e I + e 2 +
+ (l-Ul){u2f(e 2
and so on so that
f
+
m m ~ u.e.) + (l-u2)f(e I + ~ u~ J-3 j 3 J*3 j 3 m m j!Bujej) + (l-u2)f(j!Bujej)}
is a polynomial in
whose coefficients are functions of
Ul,...,u d
yd+l,...,y m,
of degree at most
n.
s Rn,
d
but also less
But this follows from Theorem 3.3, condition (NCn+]). then
rank(~l,...,~n+ I) ~ n
Vll,...,In+ 1 6 A
and therefore
Vu 6 ~ m
f(n+l)(u)ll .... 'ln+l = O.
d
It now remains to show
that the degree of the polynomial is not only less than than
(3.23)
Since
31
w
The Quadratic Case and Some Generalizations We now turn our attention to sufficient conditions and we obtain here
an important result, namely
that the Legendre-Hadamard condition ob-
tained in the preceding section is also sufficient when
f
is quadratic.
This result was established by Van Hove [VHI] in the variational case and later generalized by Tartar [Ta2] (see also [Mull, [Mu2]). Theorem 4.1.
Let
M: A m
> A TM
be a symmetric matrix and let
f(a) = <Ma;a>
where
a 6 A TM
J (H)
u
f(u g) "
% ~
=
"
in
denotes scalar product in A TM.
in the sense of distributions is compact in
denotes the dual of
{I 6 A m :
3 6 6JR n - {0}
W-I, 2 loc (~)
s.t.
I s A
then i > f(u). If
f(1) = 0
for all
then =
f (u).
i = i .... ,q,
[ aijklj~ k = 0}. J,k
If
f(1) > 0 for all
(il)
for
W~'2(~); see [Adl]) and let
Then
(i)
Assume that
L2(~) Tn
~ aij k ~X k j,k
W-I'2(~)
A
<'; .>
u C __a
Au s
(where
and
(4.1)
i 6 A
32 Remark.
The hypothesis
the hypothesis sections.
Au s
Au e
in a compact set of
in a bounded set of
L2
T-I'2 Elo c (~)
is weaker than
assumed in the preceding
We make this weaker hypothesis in view of the applications of
Chapter II. Proof: (i)
As usual (ii) is a consequence of (i).
Step i.
We start by making a translation and then a localization
of the problem.
Let v E = us - u
and then let for
(4.2)
6 c0(~) w
E
=
~v
g
.
(4.3)
It therefore remains to show that if
I wE
~ 0
in L2(~) g m aw. aijk ax ---->0 in
W-I'2(~), i = i
q
(4.4)
j,k we
have support in a fixed compact set
K
of ]Rn,
then lim inf rn<MWE;wE>dx ~ 0.
Step 2.
We now apply Fourier transform to get
^E
w
Using
(4.5)
~m
g~0
(O
=
f~
n
we(x)e-2~i~.Xdx.
the hypotheses (4.4) we get (since
e-2~i$'x C L2(K))
(4.6)
33
I $c(~)
> 0
a.e.
(4.7) where
~ > 0
is a constant.
w
) 0
Therefore
(strongly) in
(4.8)
L ~ (]Rn).
Furthermore, if we use the hypotheses (4.4) on the derivatives of
W
E
j we
obtain
^c(~)~ k I ~ aijkW] I+--~T j, k Step 3.
Extend
f(w) = <Mw;w>
)
0
from ~m
in
to
(4.9)
L 2 ( m n) q ~m
by
(4.10)
f(w) = <Mw;w>. Observe that Re f(~) = Re<M~;~> > 0
since if
if
~ 6 A + iA
(4.11)
% = h I + i% 2 C A + iA, then (4.12)
Re<M%;~> = <M%I;%I > + <M%2;%2 > which is positive since we assumed that We,
<M%;%> > 0
for all
~CA.
now, use Plancherel's formula to get
I~n f(wE(x))dx = ~ n
f(wg(~))d~ = ~ n
Re f(w~(~))d~.
(4.13)
Therefore it remains to prove that lim inf f f(we($))d~ > 0 ~-~0 ~Rn
(4.14)
34 in order to deduce (4.5) and thus the theorem. S__tep 4. c
> 0
We also have that for all
a > 0, there exists a constant
such that q Re
7(%) >-~I~I 2
c( ~ r
-
(4.15)
aij k%j Nk I2)
i=l j ,k for all
% E cm
and for all
n c~n
with
In[ = I.
To prove (4.15) we proceed by contradiction. exist
~0 > 0, c
In~[ = i
= ~, %~ s cm
with
I%~I = 1
Suppose that there and
D~ 6 ~ n
with
so that
Re f(% v) < -~01%~] 2 - ~ El ~ aijkAjDk[ i j,k We then extract convergent subsequences
(still denoted b y %
(4.16)
~
and
n ~) so
that %~
>~,
D~
..> n~.
(4.17)
We now use (4.16) to get that
~2
I[ ~ aijk%j~kl i j,k
)
0
as
~ § ~;
(4.18)
hence
J,k and therefore %~ E A + iA.
Using the hypothesis on
f
(4.20)
and (4.20) we deduce that
Re f(%=) > 0.
(4.21)
35 But returning to (4.16) we get (4.22)
Re f(l~) ! -So< 0, a contradiction, therefore (4.15) holds. Step 5.
We now conclude the proof.
I~l<_lRe
Re f(we(~))d~ = ^c
Using (4.8) (i.e., w
>
0
Returning to (4.14) we have
f(w )d~ +
strongly in
Re f(w )d~
I~l>l Re
f(we)d~.
(4.23)
L~oc (~n)) we obtain e § O.
(4.24)
^e ~,~k t2 . aijkWj(%)~,
(4.25)
>
0
as
I~1~1 Using Step 4 (equation (4.15)) we get Re f(we(~)) > -elwe(~)l 2 - c [[ ~ -~iJ,k After integration we get Re f(w (~))d~ > -~ lw (~)i2d$ I l~l>~ ~ ^g - I I~I>1 ^g f
-% J
[l I aijkO~(~)~l 2d~. I~I>i i j,k
(4.26)
Using (4.9) we deduce that
fI~I>~
Re f(wE(~))d~ > -~[
but since
-
is arbitrary and
that lim inf f
[w~(~)]2d~;
(4.27)
jl~l>l |f lwe(~)I2d$ J J~i>1
is bounded, we obtain
Re f(we(~))d~ =~ O.
(4.28)
36 Combining (4.24) and (4.28) we obtain the claimed result.
O
From the above theorem we can draw the following conclusions
(the first
one should be related to Corollary 3.4). Corollary 4.2. by
A
that
(so E
ffi
Let
d
be the dimension of the subspace
E
generated
d < m) and suppose that coordinates have been chosen such {u E Am: Ud+ I
ffi
9 ..
=
=
u
O}
.
If
ug
*'--u
in
L~(~)
m
f: ~m-d
>
Proof:
JR
and
m
is continuous then
Using Theorem 4.1 we get that (u~)2 ~
--2 uj
but this (and the fact that
uj
>
in
us
strongly in
L2
VJ = d+l ..... m,
*~ u) Just means that
L2
VJ ffi d+l ..... m.
We now return to the examples of the previous section except the variational ones which will be dealt with in the next section. Corollary 4.3. in
L2(~)
Let
g c g E ue(x I .... ,Xn) ffi (v I .... ,Vn,Wl,...,Wn)
and suppose that
I div v C
=
8vl i!l ~--~i is bounded in
curl w~ " (~xj ~w~ - ~w! ~x ) then
L2(~)
is bounded in
L2(~)
(v,w)
37
----%
Proof:
in the sense of distributions.
We have seen in this case that A
{(~,~) 6 ~ 2 n
=
:
~•
We have by Theorem 3.1 that a necessary condition for continuous is that
f(a+t%, b+t~)
(%,~) s A, which is the case for
is affine in
t
f(v,w) = .
f
to be weakly
for every Since
f
a,b 6 ~ n , satisfies
the hypotheses of Theorem 4.1, we deduce the corollary. Corollary 4.4. and suppose that
ue(xl,x2 ) = (ve(xl,x2),we(xl,x2)) \ (v,w) ~v e ~w e ~ and ~ are bounded in L2(~), then
g g v w
Proof:
%
vw
In this case
f(v,w) = vw
Remark. u e___a have
L~(~)
in the sense of distributions.
A = {(l,~) 6 ~ 2 :
and
in
u
% = 0
or
~ = 0},
satisfies the hypotheses of Theorem 4.1.
m
Before proceeding further, it is important to note that if in
f(u E)
L 2 (Q) ~
f(u)
and if
f
is quadratic, then, in general, we only
in the sense of distributions and not in a better
sense (see for an example Murat [Mu2]). Finally in this section we mention without proof a result of Murat [Mu3] which is an extension of Theorem 4.1 (in fact the converse of Corollary 3.4). Theorem 4.5.
Let
f: ]Rm
> 9
be continuous and
38
I u
*~ u
f(u E)
in
*~ s
L~(~) in
L~(~)
then
(i)
In order that
(HI)
f
s = f(u), f
must satisfy
has the following form
f(Y) = I ce(Yd+l ..... Ym)Pe(Yl ..... Yd ) where
d
is the dimension of the subspace
E = {y s inf{n,d} (H2)
Yd+l . . . . . . Ym
0}"' P
E
generated by
A
and
are polynomials of degree at most
which are homogeneous and whose coefficients are constants, Each of the
P
verify if its degree is
r > 2.
[ V(%l,~l),...,(Ar,~ r) 6 ~ < with rank(~l,...,~r) ~ r - i P~r)xI% 2-e (ii)
Reciprocally if
constant for all then
... ~ r = O.
~ # 0
f
satisfies (HI) and (H2) and rank B(~) is n where (B(~))ij = [ aijk~k, 1 < i < q, i <_ J < m, k=l
s = f(u).
Remarks.
(i)
The proof of the above theorem is very much in the spirit
of Theorem 4.1; although in this case there is no assumption rank on the matrix (ii)
of constant
B(~).
The hypothesis of constant rank is satisfied by all the hypothe-
ses of the type
curl or
div
in
L2
but not by the following one
39
u
=
u(x I ..... x n)
8u i ~-~4 E L 2, i = l,...,n.
with stant.
However,
=
(u I ..... u n)
In this case the rank of
B(~)
is not con-
it is still possible to prove Theorem 4.5 in the above
case (see [Mu3]).
w
An Important Example:
the Variational
Case.
As we have mentioned earlier all the results of the above sections were known for the variational proofs
case for a long time.
Furthermore all the
for this case have guided those given above in a more general
context. Let us summarize the problem in the variational
*~ u
u
where
~
wl,~(~; IRm)
in
is a bounded open set of
derivatives
are given by
]Rn
as
be quasiconvex
of
~ §
(here all the conditions on the
curl(Vu ~) = 0).
In Section 2 we proved that a necessary lower semicontinuity
case, we have
f
and sufficient condition for
with respect to weak * convergence
(in the variational
case, we will omit the
A
is that
f
of A-quasiconvex),
i.e., [ ;
for every
f(~ + V~(x))dx ~ f(~) meas D
(5.1)
D
~ 6]R m, for every
and for every 2.3 (here since
D
~ 6 W i '~ ( ~ ; ~ m )
a hypercube of u
~Rn.
with
IDV~(x)d x ~ 0
This results from Theorem 2.1 and
is a gradient we have trivially that
Vu ~ - Vu 6 Ker A).
This condition corresponds up to minor changes to the quasiconvexity dition given by Morrey
([Mol],[Mo2];
see also
con-
[Mel] and [Sil]) which is
40
fG
f(Z + V~(x))dx ~ I
for every
G c ~n
E W~'~(G; ~m)
f(~)dx
(5.2)
G
bounded domain, for every (i.e., ~ C W I'~
and
~ = 0
~ E ~ TM on
and for every
SG).
The difference between (5.1) and (5.2) comes from the fact that when we established (5.1) we did it under the following convergence Vu ~ in
L=
while now we have
u~ - * % u
in
*% Vu
W I'~.
We give now the proof of Morrey's result.
Theorem 5.1.
Let
ent condition for convergence
f
in W I'~
f:IR TM
) ~
be continuous.
A necessary and suffici-
to be lower somicontinuous with respect to weak * i.e.
lJm inf I f(VuV(x))dx _> I f(Vu(x))dx ~->oo is that
f
is quasiconvex, i.e., satisfies (5.2). (i) Necessity.
Proof:
(5.3)
The necessity follows directly from (5.1) which
was established in Theorem 2.1. to be a hypercube containing
Fix
~ E W 0' (G; IRTM)
G; defining
~ = 0
on
and then choose D - G
D
and using
(5.1) we get (5.2). (ii)
Sufficiency.
The sufficiency of (5.2) (for (5.3)) although very
similar to the proof of Theorem 2.3 has to be done again, but we will omit all parts which are similar to that of Theorem 2.3. Step i.
First we will consider a hypercube
As in Theorem 2.3 we define for
VZ(x)
Dk
of edge length
I ~.
x 6 Dk
1
[ Vu(x)dx.
meas D k ~D k
(5.4)
41
Observe that f(Vug(x)) - f ( V u ( x ) )
=
+ f(~
Step 2.
f(Vu+(VuV-Vu)) - f ( ~ +
(Vug-Vu))
+ (?uU-Vu)) - f(V~) + f(?~) - f(Vu).
(5.5)
In order to obtain (5.3) from (5.5), the important term to be
estimated in (5.5) is easily estimated.
f(V~ + (VuU-Vu)) - f(Vu)
all the others will be
Let
~
=
uv
-
u
(5.6)
and observe that by definition
~ *~ 0
in
WI,~(Dk; ~m).
(5.7)
We therefore have
R
IICII
>
0
as
u * ~.
(5.8)
L Define
H
a hypercube of edge
(i.e., such that
(~ - 2R~) k
Dk
d(Dk,H ) = R~) and let
NV(x) = ~ 0
[ n~
Observe that
which is contained in
if
x s ~D k
if
x E H~.
(5.9)
~(x)
is Lipschitz (with constant
M~ = max{1,]]?~]]
,))
in
L ~D k U H9
since if
[ n~
x E H~
and
y E ~D k
(x)-n ~ (y)] = ]SV(x)] !
R~
! Ix-y] i
Mv
I~-yl-
So if we use MacShane's lemma (see, for example, Chap. X of [ETI]) we can extend
v
to the whole of
Dk
in such a way that
(5.1o)
42
(i) (ii)
qV(x) = q~(x) q~
if
x 6 ~Dk U H
(5.11)
is Lipschitz with constant
M ~.
(5.12)
We also can conclude that
Vq ~ - V~ ~
>
0
a.e.
as
~ + ~,
(5.13)
and hence lim inf r| if(V~+V~V(x)) _ f(V~+V~(x))idx = 0. v+oo JDk But since
q~ 6 W~'~(Dk )
f
and since
f
f ( ~ + Vqg(x))dx > f Dk
(5.14)
is quasiconvex we get
(5.15)
f(V~)dx. Dk
Therefore combining (5.14) and (5.15) we get
f
(5.16)
lira inf ~ f(V~ + V~ ~(x))dx > | f (V~-)dx. v+oo ~Dk JDk
Step 3.
We then proceed as in Theorem
2.3. Let ~ be approximated by a
union of such hypercubes D k and let us denote by H k this union. Then using (5.16) into
(5.5) we get for every g > o lim
inf
f(Vu~)(x))dx
>~
f(Vu(x))dx
- E ,
(5.17)
Hk
since for every g > o we may find k large enough so that in (5.5) I
( f ( V u + V ~ V ) - f ( V u + V ~ ~))dx and I
want.
(f(Vu)- f(Vu))dx
are as small as we []
43
Remarks. in
The above theorem is still valid if we have weak convergence
W l's, s ~ i
instead of weak * convergence in
W i'~
provided
f
satis-
fies the following hypotheses (i) (ii)
f(F) > m
for some
m E~
and for every
F E~nm
If(Fl)-f(F2)l ! K(l + IFI Is-I + IF21S-I)IFI - F21 and for every
for some
K > 0
FI,F 2 6JR rim.
(The proof is essentially the same as the above one; see Morrey [Moll; see also Meyers [Mel] for weaker conditions on
f
than (i), (ii).)
We have as a consequence of the above theorem and of Theorem 3.1 that Corollary 5.2.
If
f
is quasiconvex then
f
is rank one convex, i.e.,
f(%F + (I-%)G) < %f(F) + (l-%)f(G)
for every
F,G s
Furthermore if
with rank f s C2(]R rim)
(F-G) < 1
(5.18)
and for every
then (5.18) is equivalent
to
~ 6 [0,i]. the Legendre-
Hadamard (or ellipticity) condition
I i,J,~,8 for every
% E~ n
'
~ E]R m
~2f(F) ~Fi ~Fj8 l i l j ~
and
F = (F
> 0 8 _
(5 19)
) i~ l
We now turn our attention to quasiaffine functions (i.e., functions such that
~
and
-$
are quasiconvex).
In the variational context we
will equivalently call such functions (following Ball
[Ba2]), null
Lagrangians, for reasons which will be obvious later. First we start with a lemma which will be important in the remaining part of the section.
44 Le~m~ 5.3.
lao ~ 6 W O' (D; ~Rm)
Let
(with
denotes any subdeterminant of the matrix
I
D c]R n)
then, if
sub det (V~)
V~ 6 ]Rnm,
(5.20)
sub det(V~(x))dx = 0. D
Proof:
This is a well known result (see, for example,
[Mol]).
o
We now state the main properties of quasiaffine functions (or null Lagrangians). Theorem 5.4. (i)
~
(ii)
The following conditions are equivalent:
is a null Lagrangian (or quasiaffine).
For every
F E ~ nm
+ V~(x))dx = [ r
I r D (iii)
and for every
~ f w"1'~ 0 to; ]Rm) = ~(F)meas D.
For every
F,G E ~Rnm, with rank
(F-G) <__ i, and
@(IF + (I-I)G) = I@(F) + (I-I)@(G) and if
@
is
C2(]R nm)
(iv)
A E~n Let
I E [0,I]
(5.22)
then (5.22) is equivalent to
~2~(F) l i l j ~ 8 = 0 i,]~,8 ~ F i ~ F j B for every
(5.21)
jD
(5.23)
~ s
adJsF
denote the matrix of all
subdeterminants of the matrix
~(F) = ~(0) +
s x s
(i < s < inf{n,m})
F 6]Rrim, then inf {n,m} ~ s=l
o(s)
(5.24)
45
where
n) =
m!
o(s) = ( )(s
product in RG(s) Remarks.
and
(i)
If
n!
s~(m-s)~s~(n-s)~ B s E ]Ro(s)
' <';'>o(s).
denotes the scalar
are constants.
m = n = 2, then (5.24) Just means that 2
~(F) = ~(0) +
J~I=IB~jFIj + B2det F. i,
(ii) reduce
The Euler Lagrange e q u a t i o n s f o r the f u n c t i o n a l to an i d e n t i t y
if
Proof of Theorem 5.4: see Ball ([Ba2]).
~
[flr
is a n u l l Lagrangian.
We s k e t c h the p r o o f h e r e , f o r a d e t a i l e d proof
The equivalence between (1) and (ii) is Just the defini-
tion of a null Lagranglan, the implication (ii) ~ (ill) is Corollary 5.2 applied to
~
and
(ill) ~ (iv)
-#.
This is based on a result of Edelen ([Edl]) and of
Erlcksen ([Erl]) which shows that (5.23) implies (5.24) if then by mollifying (iv) ~ (il)
~
one gets (5.24) for continuous
~ 6 C2
and
~.
This is a direct consequence of Lemma 5.3.
m
We now state a more precise result, involving weak convergence in which will be useful in the next chapter. ([Rel],[Re2]) and to Ball ([Bal],[Ba2]).
W I'p,
This result is due to Reshetnyak We will omit the proof (for de-
tails see [Bal],[Ba2]), since it is similar to that of the weak * convergence already given. Theorem 5.5. and let
(i)
We state it for simplicity in the case Let
adj F
p > 2, then the map u
wl,p(~; ~3)
into
(ii)
p > 3
If
wl,p(~; ~3)
into
m - n - 3.
denote the matrix of cofactors of ~
adJ Vu
F s
is weakly continuous from
L~/2(~). then the map u LP/3(~).
>
det Vu
is weakly continuous from
48
We now conclude this section with two useful results which give in some particular cases a pointwise condition for quasiconvexity. Theorem 5.6.
(i)
If
n = 1
or
The first one is
m = I, then convexity and quasicon-
vexity are equivalent. (ii)
If for every
F 6~nm
there exists
g: ~
> 9
continuous so that
f(F) = g(~(F))
where g
~
is a null Lagrangian, then
(5.29)
f
is quasiconvex if and only if
is convex. Proof: (ii)
(i)
This is nothing else than Corollary 3.2.
The proof is divided in two steps.
Step !"
Suppose that
I
g
is convex; we want to show that
f(F + V~(x))dx ~ I
(s.3o)
f(F)dx n
for every Since
c~n g
and every
~ s wO
%~; ~m).
is convex, there exists
A(F) 6 ~
so that
g(~(F + V~)) ~ g(~CF)) + A(F) C~(F + V~) - ~(F)).
If we integrate, over Lagrangian (i.e., Step 2.
~, (5.31) and if we use the fact that
(5.31)
~
is a null
Jr ~(F + V~(x))dx = ~fl[~(F)dx), we obtain (5.30).
We want to show now that if
vex, i.e., for every
v,w 6 ~ ,
% 6 (0,i)
f
is quasiconvex then
g
is con-
we have
g(~v + (l-~)w) ! ~g(v) + (1-~)g(w). We may suppose without loss of generality
that
O(F) ~ A
(5.32) (where
A E ~)
47
for all
F
trivial.
otherwise
f(F)
would be constant and then the theorem is
Suppose now that we can prove
there exists
G,H 6 ~ n m
(see the end of the proof) that
so that
~(G) = v
I
~(H)
(5.33)
w
rank(G - H) ! i,
then it is easy to deduce one convex
(5.32).
Since
f
is quasiconvex,
it is rank
(Corollary 5.2) and therefore
g(%v + (l-%)w) = g(%~(G) + (I-%)~(H)) = g(~(%G + = f(lG +
(I-%)H))
(I-%)H)
< %f(G)+(l-l)f(H)
where we have used (5.33) in the first equality, null Lagrangian and rank(G-H) the fact that
f
< i
= lg(v)+(l-%)g(w)
the fact that
in the second equality,
(5.33).
is a
and finally
is rank one convex.
Therefore in order to end the proof it is sufficient satisfying
#
(5.34)
Since
~
to find
C,H E ~ n m
is a null Lagranglan we deduce by Theorem 5.4
that n
~(F + a @ b) = #(F) +
for every
F, for every
m
[ [ (#'(F))u a b ~=i ~=i
a s ]Rn, b s ]RTM
where
a @ b s ~nm
(5.35)
has coeffici-
ents (a O b)u w = a bw,
i < ~
(5.36)
48 and (it is easy to see from Theorem 5.4 that if then
~
is
~
is a null Lagrangian
C I)
~(F) = ~Fu~
(r
Since we assumed that G 6 ~ nm, ~ (i ! ~ !
n)
i < ~ < n,
,
i < ~ < m.
(5.37)
is not constant we deduce that there exist and
(i < ~ < m)
so that
I ~(G) = v (r (c))
# 0.
(5.38)
v This is easily seen by using Theorem 5.4 (in particular (5.24)). let
s
(i < ~ < inf{n,m})
In fact
be such that (we adopt the notation of (5.24))
I B ~ = (BsI ..... Bo(s)) s # 0 s O(s) i Bj = 0 E ~ O(j)
for every
J < s. Bo(s) s _ # 0, t~en we may choose
Suppose, without loss of generality, that G
as follows, for
Gij =
1 < i < n, i < J < m
I 0
if
i # j
I
if
I < i=
v - ~(0) if s
let
or if_ i =
J >s
J <s
i = j = i.
BO (s) Then it is not difficult to see that
G
satisfies (5.38) with
~ = ~ = i.
With the help of (5.38) we are now able to complete the proof of the theorem by establishing (5.33).
Let
H = G + a ~ b
(5.39)
49 where
a E~n,
have already
b E~m
have to be determined in order to ensure (5.33); we
~(G) = v
by (5.38).
Therefore we want to solve
w ffi ~(H) = r
+ a ~ b).
(5.40)
(~'(G))~ua b = w - v,
(5.41)
Using (5.35) we get
~,~ therefore by choosing
a ffi 6 ~ bu = w - v 6 (~'(G))~ u (where
6ij
(5.42)
is the Kronecker symbol) we have indeed obtained (5.33) and
this concludes the proof. Finally we conclude this section with an important result of Morrey ([Moll, [Mo2]) for minimal hypersufaces in parametric form which gives a pointwise equivalent for quasiconvexity. notation. for
Let
m = n+l
M E Rn(n+l)
and
D: Rn(n+l)
For this let us introduce some > ~n+l
be defined as follows:
we let
D(M) ffi (DI(M) .... ,Dn+l(M))
(5.43)
Dk(M) = (-i) k+l det ~
(5.44)
where
A
where the
~ n(n+l)
is the matrix
For example if
(n x n) matrix obtained by suppressing the k th llne in M. n = 2
and
u: (Xl,X2)
>
(Ul,U2,U3)
50 then ~u 2 8u 3 ~u 3 ~u I ~x 2 8Xl, ~x I 8x 2
~u 2 ~u 3 -(~x I ~x 2
D(Vu) -
Theorem 5.7.
Let
that there exists
m = n+l
g:
n+l
and let > ~
dgree 1 (i.e., g(Ix) = %g(x)
~u I 8u 3 ~u I Bu 2 8x I 8x2, ~x I ~x 2 D
~u I ~u 2 ~x 2 ~Xl). (5.45)
be defined as above.
Suppose
continuous and positively homogeneous
for every
% > 0
and
x E~n+l)
such that
f(F) = g(D(F))
for every
F E ~n(n+l).
Then
f
of
(5.46)
is quasiconvex
if and only if
g
is
convex. Proof: ([Mol], (i) set of
The proof given here is slightly different
[Mo2]) and follows the same pattern as that of Theorem 5.6. Let
g
be convex, then for
~ E w "i'~'0 tG; ~m)
f(F + V~(x))dx = I G
g
G
a bounded open
g(D(F + ?~(x)))dx ~ ~ f(F)dx. 7G
is convex there exist constants
g(D(F + V~(x))~ > g(D(F)) +
Integrating
Ai, 1 < i < n+l
(5.47)
so that
n+l [ AI(F)(D(F + V~(x)) - D(F)). i~l
(5.48)
(5.48) and using Lemma 5.3 we get
[
f(F + V~(x))dx > I G
(ll)
(G
~n)- we want to show that
I
Since
from that of Morrey
Let now
g(D(F))dx - / G
f
be quaslconvex
f(F)dx. G
then
f
is rank one convex (by Corollary
5.2), i.e., f(IMl+(l-~)M 2) ! Xf(M I) + (l-l)f(M 2)
(5.49)
51 for all
~ E [0,i], MI,M 2 EIR n(n+l)
with
rank(Mi-M2) _< i.
We want to prove that
g(%D I + (I-%)D 2) ! %g(D I) + (l-%)g(D 2) for all
% E [0,I], DI,D 2 EIR n+l.
Case i.
If
For this consider two cases:
%D I + (I-%)D 2 = 0, then, since
of degree i, we have
g(0) = 0
(5.50)
and since
g
g
is positively homogeneous
is positive we deduce im-
mediately (5.50). Case 2.
If
%D I + (I-%)D 2 # 0, then we can find
MI,M 2 E ~ n(n+l)
so
that
I D(MI) = DI, D(M 2) = D 2 D(~M I + (I-~)M2) = XD(MI) + (I-~)D(M2) = ~D I + (I-~)D 2
(5.51)
rank(M2-M I) ! 1 this is possible by a result in IDa2] (Proposition 8, [Da2],
which is in
the same spirit as the construction of (5.33) in Theorem 5.6).
Hence
using (5.49) we get
g(%D I + (I-%)D 2) = f(kM I + (I-%)M 2) ! %f(M I) + (l-%)f(M 2) ! %g(D I) + (l-~)g(D2)"
o
52
w
Parametrized Measures We now introduce the notion of parametrized measures which underlies
all the analysis developed here and will be important in the next chapters. We will limit ourselves only to the results we will need in the next chapter.
The main result of this section is due to Tartar ([Ta2]), although
it is based on the notions of generalized curves and surfaces introduced by Young and MacShane ([Yol]-[Yo4];
[Mal], [Ma2]).
For more extensive
results on parametrized measures see Berllocchi and Lasry ([BLI], [BL2]). Theorem 6.1.
Let
K c ~ m, ~ c]R n
be bounded and open and let
f: ~ m
be continuous (i)
Let
> ~m
u :
be such that
u (x) 6 K
a subsequence
a.e.; then there exists
S
S
and a family of probability measures {Vx}xE ~
{Us}~= I
such
that supp ~
c K
(6.1)
X
f(u s)
h f
in
L~(~)
(6.2)
where ~(x) = ~R m ~x(1)f(l)dl. (ii)
is as above then there exists a sequence
Reciprocally if X
{us}~.1 (Us: a continuous
f: K
> A m)
with
u (x) E K S
>
f<us) *~ ~ in L| where
f
satisfies (6.3).
(6.3)
a.e. and such that for all
53 Proof:
(i)
To every measurable function
u : ~----> K
we associate a
S
gs
measure
i n t h e f o l l o w i n g way
= I ~(X,Us(X))dx for every to
u
~ 6 C0(~ • ~m); we will call
Hs
(6.4)
the (Radon) measure associated
(see [Chl], [Bol] for basic properties of Radon measures). S
We may (see [Bn i~,p.31-34)
then extract a weakly convergent subseauen-
ce(without loss of generality we suppose that the whole sequence converges). ~s
~
i.e.,
The limit
(by
H
<~s,~>
>
for all ~ 6 C0(~ x]Rm).
(6.5)
has the following properties:
(&)
~ > 0
(6.6)
(6)
supp(H) c ~ x "K
(6.7)
(y)
proj~H = dx;
(6.8)
supp ~
we denote the support of
the projection on
~
of
H
~
and by (6.8) we just mean that
is the Lebesgue measure).
(6.7) and (6.8) is easy since (~)
For all
~ > 0
we have
<~,~> = lim <~s,~> = lim f ~(X'Us(X))dx > 0 S-~O
S-~O
thus (6.6). (6)
For all
such that
qb -- 0
on
~ x K
<~'~> = s-~=limI ~(X,Us(X))dx = 0, thus (6.7).
we have
To check (6.6),
54 (Y)
If
r
= ~(x)
then
deduce
~
.q
(6.8).
We c o n c l u d e from t h e p r o p e r t i e s ous w i t h r e s p e c t
of
~
to L e b e s g u e m e a s u r e ;
deduce the existence
that
~
is absolutely
continu-
u s i n g t h e Radon-Nykodym t h e o r e m we
of a f a m i l y o f p r o b a b i l i t y
measures
(~), x
with
K, s u c h t h a t
supp X
\
(6.9)
= f~ ~x d x '
that is, (6.10)
<~'#> : II]R~m q b ( x ' X ) ~ x ( l ) d l d x " We now u s e t h e h y p o t h e s i s
(6.2)
to d e d u c e t h a t
<Us,~f> = f ~(x)f(Us(X))dx
for all
> <~,~f> = I ~ m
we have
~(x)f(%)~x(~)dxd%
and therefore T(x) : I~Rm f(1)~x(~)dl. (ii) Reciprocal: Let ~
x
he as in the theorem, then define (6.n)
= I~xdX;
we
want to show that there exists a sequence {Us}s=I
that, for all continuous f: K f(us) - ~
> ~,
f = )( Vx(1)f(%)d%
in L~(~).
(Us: f3
K)
so
55
It is sufficient
M - {m: m
to show that if
associated with a measurable
function
u: ~---->
K}
and N = {~: (a) ~ >__ 0, (8) supp ~ c ~ • K,
then
N c M
since
~
is in
of the proof we have shown that we will show first that closed convex hull of
tions
Let
M
r
Note also that by the first part
M c N.
In order to prove that N c c-~ M
N c (the
M).
ml,...,m p E M, therefore there exist measurable
u I , 9 . .,Up : ~ " >
all
N.
is convex and then that
K
<mi,~> = for
= dx}
(where the closure is taken in the same sense as in (6.5))
defined in (6.11)
Step i.
(y) p r o J ~
func-
so that
Ir
(6.12)
s C0(Q x~Rm).
We want to show that
I
I
%imi E M for every %i > 0 with %i = i. i=l i'l As usual we can find (as in Theorem i.i) characteristic functions Xi so that
X~ (x) = Xi(~)
*% l i
in
L~(~),
i
= 1 . . . . . p.
(6.13)
We then let
re(x) =
and observe that for every
r
I Xi(x)ul(x) i:l
(6.14)
r s Co(~ • m m) = i=iXi(x)r
(6.15)
56
Therefore combining (6.13) and (6.15) we obtain
P I r
P / li@(x,ui(x))dx =
i=l And if we let
me
(6.16)
~ %i<mi,r >i=l
to be defined as follows
(6.17)
<me,C> = I 4)(x,ve(x))dx P we deduce from (6.16) that Ste p 2.
Z ~imi E M. i=l
We want to show now that
As a consequence of the
N c co M.
Hahn-Banach Theorem we have that the closed convex hull of intersection of all the closed half spaces containing
co M =
n
{<m,@o> + a 0 ~ 0
for all
M
is the
M; in other words,
m E M};
(6.18)
r but, by definition of
c--~M(Here
n {1%(x,u(x)) + a0 t 0 r J~
@0 s CO(~ x ~ m)
Let
M, we get from (6.18) that
and
for all meas. funct, u: ~
a0 E ~ . )
~ E N, we want to show that
(6.20)
+ a 0 ~ 0 for every
#0 E C0(~ x ~ m)
and
<m,~o> + a 0 ~ 0
Let
r
> K}.(6.19)
and
a0
a0 E ~
for all
such that
m E M.
be as in (6.21) and define
(6.21)
57
I ~0(x) - inf {r ~EK u
(6.22)
Xo(X,X) = r
(6.23)
- ~o(X) ~ o.
Since (6.21) holds we deduce that (6.24)
I ~ 0 ( x ) d x + a 0 ~ 0. fl
Therefore using the fact that
U E N
and ( 6 . 2 4 ) we deduce t h a t
<~,r > + a0 = + + a0
<~,~0 > + a 0 = I @0(x)dx + a 0 ~ O, which is precisely
(6.20).
From Theorem 6.1 we deduce a criterion for strong convergence Corollary 6.2. (p < ~)
Let
if and only if
Proof:
(i)
*x u
us
in
~x = 6u(x)
Suppose that
u
L~(~) ' then
us
> u
(the Dirac measure at >
u
strongly in
([Ta2]).
strongly in
Lp
u(x)).
L p, then by Theorem
S
6.1 we have f(u(x)) = L m
for every (li)
Ux(A)f(%)d%
f, which is precisely to say that If
9x = 6u(x)
~
x
(6.25)
= 6
u(x)"
we deduce from Theorem 6.1 that
u
2
* h u2
in
L~j'~
S *
and combining this with the assumption that
u
oo
~u
in
L (~), we
S
deduce strong convergence,
o
58 ~t
Examples.
(i)
If
u
s
\
u
from T h e o r e m 6.1 the immediate
in
L ~176 and
conclusion
f(x) = x
then we have
that
u(x) = IR m ~x(R)RdR
(ii)
If
u: [0,i] ----> ~
then we have that, if
is continuous and periodic of period 1
u (x) - u(sx), s
f(Us(X))
~
<9,f(%)> =
f
l
0
so in this case we can choose
v
x
= ~.
f(u(x))dx,
CHAPTER II APPLICATIONS
w
Nonlinear Conservation Laws In this section we will see how the theory developed in the previous
chapter (especially w
and w
can he applied to the existence of solu-
tions (see Theorem 1.2) of the following equation
~fu+
where
f: ~
> ~
(u) - 0,
x c~,
t~0,
(l.1)
is a given smooth function.
Equations of the above type are important in physics and are known as conservation laws.
We will he dealing here only with a single equation of
the type (i.i) and we will present the analysis of Tartar ([Ta2]).
Some
recent results of DiPerna [Dil] indicate that the theory presented in Chapter I can also be applied to systems of equations of the type (I.I). Before starting with the analysis, let us recall very briefly some well known facts about (i.I) (see for details [Lal], [La2]).
I u t + (f(u)) x - 0, u(x,O)
does not generally t h e initial data
u
-
t ~ 0
(1.2)
Uo(X)
have a global
is.
x E~,
The Cauchy problem
smooth solution
no m a t t e r how s m o o t h
Therefore one is lead to search for weak solutions
of (i.I) and by this we mean that
u
is a bounded measurable function
which satisfies
~O/i(u,t + f(U),x)dXdt . 0
(1.3)
60
for every $ E CO(~, x (O,m)).
Similarly
u
is a weak solution of (1.2) if
for every S E C~([R x [0,-)) co
~ rj_~ (USt +f(U)Sx)dXdt
+
f
u^(x)S(x,O)dx = O. _co
(1.4)
U
For physical as well as mathematical
reasons
city of weak solutions of (1.2)), the solution
(in order to ensure uniu
is required to satisfy
the entropy condition, namely that the inequality
q(u) t + q(u) x ! 0 holds in the sense of distributions
(1.5)
for every convex function
(called an entropy for (I.i)) and where
q
q: ~
~>
(called the entropy flux) is
given by q'(u) = f'(u)q'(u).
Remark.
(1.6)
Note that all smooth solutions of (i.I) satisfy
(1.5) and in
this case (1.5) is actually an equality. We now prove the main theorem of this section which was established by Tartar [Ta2]. Theorem i.I. C I. in
Let
Suppose that L~(~)
q
{u E}
be a bounded open set and
f: R
is a sequence of functions such that
and that for each convex function
n(u~)t q(Ue)x where
~ c~ 2
+
is in a compact set of
satisfies
(1.6); then
> R uE
be
*%
u
n -1,2 Wlo c (~)
(i.7)
61 f(u E)
*~
f'(u E)
>
f(u)
in
f'(u)
(1.8)
L~(~)
(strongly) in
LP(~)
for all
Furthermore if there is no interval on which
u
E
>
u
(s~rongly) in
LP(~)
f
for all
p < ~.
(1.9)
is affine, then
p < ~.
(1.1o)
We now give the proof of the theorem and then we will give some hints on how to apply the theorem tO the equation (1.2). Proof:
Step i.
vex function
q
We first want to show (1.8).
and consider the sequence
U C = (uE,f(uE),Q(uE),q(ue))
Since
{u E}
We start by fixing a con-
is bounded in
E L4(~).
(i. II)
L~, we have (after extraction of a subsequence)
that UC
*&
U = (u,v,w,z)
in
~ . L4(~)
(1.12)
We therefore want to show
v(x,t)
By assumption
v
e
= f(u(x,t))
(1.13)
a.e..
(1.7) we have that if
z f(
uE
),
w
E
~ N(
uE
),
z
s
= q(
u~
),
(1.14)
then I Du E
~v ~
~z e +~-i-
is in a compact set of
W-I,2-^. loc (u)
(1.15)
is in a compact set of
-1,2 WIo c (~)
(1.16)
62 where (1.15) is deduced from (1.7) by choosing deduce (using the notation of w
A = {(a,8,u
n(x) - x.
Therefore
we
of Chapter I) that
6m4:
a~ - 8Y = 0}.
(1.17)
By Theorem 4.1 of Chapter I we then obtain immediately that
Q(U e)
=
u~z ~ - vEw E
*~
Q(U)
= uz-vw
in
Lm(~).
(1.18)
Using (1.14) we may rewrite (1.18) as ueq(ue)-f(ue)n(ue)
*x
uz-vw
in
L~(~).
(1.19)
We can rewrite also (1.19) in terms of parametrized measures (see w Using Theorem 6.1 we deduce that there exists a family of probability measures
V
so
that
x,t
u(x,t)
=
v(x,t)
- <9
w(x,t)
- <~
z(x,t)
-
I Using
(1.19)
we g e t
(dropping
x~t X,t x,t X~
;X>
a.e.
;f(X)>
a.e.
;~(l)>
a.e.
(1.2o)
t;q(X)>
the
a.e..
indices
x,t
in
~x,t)
- <9,q(X)> - .
(1.21)
Combining (1.20) and (1.21) we obtain
for
all
convex
functions
-
~
(f(A)-v)n(X)>
and for
(1.22)
- 0
q'(X)
-
f'(X)q'(X).
63
n, namely
We now make a particular choice of
n(x)-
1~-u[
(1.23)
We immediately deduce that
q(~)
J
f(u)-f(A)
if
~ ~ u
t
f(A)-f(u)
if
A ~ u.
(1.24)
Observing that, for this particular choice,
(1.25)
(~-u)q(~) - (f(~)-v)U'(~) = (v-f(u))[~-u[,
and inserting
(1.23),
(1.24) into (1.22) we get
(v-f(u))<~;[~-u[> From (1.26) we deduce that
(i)
if
<9;ll-ul
(li)
if
< ;Ix-ul>
v = f(u)
> # O, t h e n
= O, t h e n
=
(1.26)
o.
since
v = f(u)
V = 6
and thus U
v = f(u).
We have t h e n i n d e e d p r o v e d ( 1 . 8 ) . Step 2.
In o r d e r t o p r o v e ( 1 . 9 ) and (1.10) and hence t o c o n c l u d e t h e
proof, we will show that the support of which
f
V
is contained In an interval on
is afflne.
Without loss of generallty, we may assume that at u = f(u) = O.
Then (1.22) becomes
(x,t)
we have
(using (1.13))
- f(A)~(X)>
-
0
(1.27)
64
for all convex
N.
Using (1.20),
(1.13) and the fact that
u ~ f(u) " 0
we have
t Let
~,B
=
0
(1.28) <~;f(k)>
-
O.
be such that
co(supp v) = [~,8]
where
co M
must have V
(1.29)
denotes the closed convex hull of e < 0 < 8
and also that if
M.
e ~ 0, then
is a Dirac measure and the problem is solved.
consider the case Define
g,h C BV
8 = 0
in which case
Thus it remains only to
u < 0 < 8.
(the set of functions of bounded varfations, i.e.,
functions whose derivatives
I
are measures)
by
g(k) - [
pv du
(1.30)
h(X) *
f ( ~ ) v dU.
(1.31)
We may assume therefore that
g,h
vanish outside
We also see immediately from (1.30) that Equation
In view of (1.28) we
g(%) < 0
[a,8], by using (1.28). on
(a,8).
(1.27) yields
-
= 0
(1.32)
and hence - + = 0.
(1.33)
65
Using the fact that
q'
ffi f'n'
we deduce that
~ 0
for every convex function By linearity
n
(1.34)
and thus for any increasing function
(1.34) will hold for a difference of two increasing functions
and hence for any smooth function.
We therefore deduce
h - gf' = O.
Observe that by (1.30),
(1.35)
(1.31) we have
f(l)g' - ~h' = O.
Combining
(1.36)
(1.35) and (1.36) we get
(fC%)g - %h)' = O.
Using the
n'.
fact
that
g, h
(1.37)
vanish outside
[a,8]
we deduce from (1.37)
that f(~)g
Since (1.35) holds and since
- lh - O.
(1.38)
g(A) < 0
on
(a,B)
we obtain from (1.38)
that
f(A) - Af'(~) - 0
on
(a,~),
(1.39)
and hence f(~) = cA
Thus
f'
is constant on
(a,B)
for all
~ s (a,B).
and we deduce (1.9) from (1.29).
(1.40)
66 Finally if there is no interval on which (1.29) and (1.40) that
Vx,t = 6u(x,t)
f
is affine we obtain from
(the Dirac measure) and hence
D
(Corollary 6.2) we deduce (i.i0). With the help of the above theorem, we are now able to prove an existence theorem for nonlinear conservation laws. Theorem 1.2 (.Existence Theorem).
Let
u0 6 WI'~(~).
xEIR,
I u t + f(u) x = O,
Then
t >0 (1.2)
u(x,0) = u0(x)
oo has a weak solution Proof:
Step i.
u s L . Consider the parabolic approximation of (1.2) I u~ + f(u E)
- eu E
X
=
XX
0
(1.2 E)
u~(x,0) = u0(x). Then, standard results on parabolic equations of the type (1.2 E) ([0s imply the existence of classical solutions every bounded open set
bounded in
X
(1.2 C) by
L~
u e, integrating over
~
Step 2.
s
of (1.2 E) such that, for
fl of ~ x~{+,
u E , eu E
Multiplying
u
uE
X
bounded in
(1.41)
~
L2(fl).
and using (1.41) we get
(1.42)
By (a possible) extraction of a subsequence we deduce from
(1.41) that u
E*
9
u
in
~(~).
(1.43)
67
Our aim is then to show that
u
is a weak solution of (1.2).
For this
purpose we want to use Theorem i.I and therefore we need to show that
n(ue)t~ + q(ue) x
f o r every convex
n
-1,2 Wlo c (~)
is in a compact set of
and where
Observe first that, since
q u
E
(1.44)
satisfies (1.6). is a classical solution of (1.2 e) we have
e = En(u e) - n"(uE)(r n(u ) t + q(Ue)x xx
u~) 2
(1.45)
By a result of Murat (see Lemma 28 in [Ta2]), in order to show (1.44) it is sufficient to prove that
rl(u e ) t + q(u~:)x s (compact s e t o f
W- 1 ' 2 + bounded s e t of ~dr(fl))
fl bounded s e t of where _A~(R) d e n o t e s t h e s e t of m e a s u r e s . i s in a bounded s e t of
W-I'~(~)
W- 1 ' ~
(1.46)
The f a c t t h a t
is trivial
since
uE
q(u C)
+ q(u E) t x i s bounded in L~(~).
Therefore, in order to prove (1.46), it remains to show (by (1.45)) that Cq(u E)
is in a compact set of
W-I'2(~)
and
q"(u~)(v~-ue) 2
XX
is in a
x
bounded set of _~(fl). The second term is easily seen to be in a bounded set of ~ ( ~ )
by (1.42) and the convexity of
n.
Therefore it suffices to
show that
ll n(U )xxllw-l,2
>
O, as
e § O.
(1.47)
By definition (see [Adl]) we have
I I n(USxxl Iw-l,2 ': where
sup ~( cn(u e) xx # ( x , t ) d x d t ~ j,
(1.48)
68
II~llwl,2•
~= {r
Integrating by part and using Schwarz's inequality we get
x
l]En(Ue)xXllw_l, 2 ! ~
(1.49)
r162
II/~ uEIIe211n'(ue) IIe~ sup{I[r
Thus using (1.42) we have indeed obtained (1.47) and hence (1.46). We may then apply Theorem i.I and the arbitrariness of
to get a weak
solution of (1.2).
w
Existence Theorems in Nonlinear Elasticity In this section we show how to apply the results of the previous chapter
to elasticity.
We follow here the presentation of Ball ([Ba2]).
We use material coordinates. tion, a point
x E ~
We let
~ =~3
be the reference configure-
occupies in the deformed configuration the position
u(x) = (Ul,U2,U 3) 6RR 3.
The deformation gradient
F
is defined by
grad u I F = Vu =
grad u 2
(2.1)
grad u 3 We require that
u
is locally invertlble and orientation preserving, i.e.,
det F > 0
for all
x C ~;
(2.2)
if the material is incompressible we will impose that
det F = i
for all
x C ~.
(2.3)
If we suppose ~he material to be hyperelastic, it is then characterized by a strain energy function
W(x,F).
are conservative with potential
~(u)
We assume also that the body forces then the energy to be minimized is
69
f l(u) - | [W(x,Vu(x)) + ~(u(x))]dx. Jfl
(2.4)
We suppose furthermore that the displacement is prescribed on part of the boundary
~fll (= ~fl) with measure
~fl -
i.e.,
8ill,
~i
u = u0
on
We make the following hypotheses on
> 0, and the traction is
0
~i"
on
(2.5)
W, ~
and
fl, which are satisfied
(see the end of this section) by certain elastic materials. (HI)
There exists
g(x,',.,')
g: fl x ~ 9 X iR9 x ~ +
convex for every
x
> ~
continuous and
such that
W(x,F) ffi g(x,F,adJ F,det F)
where
adJ F
denotes the matrix of cofactors of
(2.6)
F.
Ball ([Ba2]) calls
such functions, W, polyconvex. (H2)
There exist
K > 0, c E ~ ,
p ~ 2, q > p--~l' r > i
g(x,F,G,H) ~ c + K(IFI p + IGI q + IH[ r)
for all (H3)
x E fl, (F,G,H) E ~ 9 AS
H § 0
x~9
such that
(2.7)
x~+.
we h a v e
g(x,F,C,H)
> ~
(2.8)
(i.e., infinite energy is required to compress a volume into a point). (H4)
9:~3
> ~+
is continuous.
(HS)
fl is a connected Lipschitz domain.
The set of admissible deformations we will be considering is the following:
70
~-
{u E wI'P(~; ~3): adJ Vu E Lq(~), det Vu > 0
a.e. and
det
Vu E Lr(~),
u = u0
on
Bnl}.
We then have the following theorem established by Ball [Ba2]. Theorem 2.1. there exists Proof:
Suppose that there exists
u E ~F such that
I(~)
uI s
with
l(u I) < ~. then
is minimum.
By a result of Morrey ([Mo2], p. 82) we have ,u(x),Pdx < k[f 'Vu(x) iPdx + (f In
-
fl
f o r some k > O and f o r a l i
(2.9)
'u0Ids) p] @nI
u E wI'P(R; ~3)
with
u = u0
on
BR1.
Therefore using (H2) we deduce that l(u) >__ constant + K{[ [IVu(x) lp + ladJ Vu(x) lq n
+ idet Vu(x) lr]dx}.
(2.1o)
Using (2.9) into (2.10) we obtain (from now on we denote any constant by
K)
/~{
+K
> K + KIIuil --
ladJVu(x)[ q
wl,P
+ Idet Vu(x) Ir}dx. Let
un
be a minimizing sequence of
I
(2.11)
in ~, then from (2.11) it
follows that
Un i adJ
Vu
det Vu n
% u
in
wI,P
%
G
in
Lq
H
in
L r"
(2.12)
71
By Theorem 5.5 of Chapter I we deduce that
G = adJ V~,
(2.13)
H = det V~
and hence
(VUn,ad j VUn,det Vu n)
~
(V~,adJ ~ , d e t
~)
in
(2.14)
L I.
w
Since
g(x,',',')
is convex (and since (2.12) implies that
>
u
u
n
a.e.)
we d e d u c e by t h e
L1
version
o f Theorem 1.1 o f C h a p t e r I t h a t
(2.15)
I(~) ! lira inf l(Un). n-~O But obviously det Vu > 0
u - u0
a.e.; thus
on
~i;
and since
I(~) < ~
we must have
u E ~.
The aame analysis can be carried over for materials which are incompressible (i.e., det F = i).
~i
We now let the set of deformations be
" {u 6 wI'P(~;IR3); adJ Vu s Lq(~), det Vu ~ i and
a.e. in
u = u0
on
~I}.
With the same hypotheses as in the above theorem we have Theorem 2.2. exists
u E~"I
Proof:
If there exists so that
I(~)
is
uI f~ 1
with
l(u I) < =, then there
minimum.
The proof is almost identical with that of Theorem 2.1; for
details see [Ba2]. Before considering specific elastic materials, we need to define a simple criterion in order to check the polyconvexity of a given function Suppose that the material under consideration is isotropic, i.e.,
W.
72 W(x,F) = f(x,vl,v2,v3) where
vi
are the eigenvalues of F ~ F T
and
(2.16)
f
is symmetric in the
v i.
We then have the following theorem (for a proof see [TFI], [Ba2]). Theorem 2.3. fi(x,',-,') and let f
Let
fl,f2: ~ x]R 3
> IR be continuous and such that
is symmetric, convex and nondecreasing for each
f3: ~ x (0,~)
'
> ~
x E
be continuous and convex for each
x.
Let
be such that
f(x,vl,v2,v3) = fl(X,Vl,V2,v3) + f2(x,v2v3,v3vl,VlV 2) + f3(x,vlv2v3) , then
W
(2.17)
(defined by (2.16)) is polyconvex.
With the help of the above theorem it is easy now to see that there are ~dels
in elasticity whose stored e n e r ~
of ~ e o r e m
2.1 and 2.2.
functions satisfy the hypotheses
We give here one ex~ple of inco~ressible ma-
terials (for more details about these ex~ples
and for other models see
[Ba2]). Consider stored e n e r ~ pressible ~terlals.
W(x,F) =
The e n e r ~ has the following f o ~
I (~i ~ ai(x) v I i=l
+ where
functions introduced by Ogden [Ogl] for incom-
~ b (x)[(v2v3)6j + (v3v I) 8j + (VlV2)SJ - 3] J=l j
ai(x) ~ O, bj (x) ~ 0
away from
0
~i ~i ) + v2 + v 3 - 3
for all
(det F = VlV2V 3 = i).
x s ~ ~en
and
aI
and
bI
(2.18)
are bounded
it is easy to check that
satisfies the hypotheses of Theorem 2.2, provided
eI ~ 2
and
W
81 ~ ~i/el-l.
73
The Mooney - Rivlln materials
are included in the above example and
satisfy also the hypotheses of Theorem 2.2.
CHAPTER III DUAL AND RELAXED PROBLEMS
w
Dual Problems In this section and in the next one we will turn our attention to
minimization problems of the form
(P)
inf {F(v,Av)) vEV
where
V
is a topological vector space and
ous linear operator from
V
into
W
A: V
>
W
is a continu-
(a topological vector space).
In fact we will consider often the following type of problems. Example.
Let
~
be a bounded open set of
]Rn
and (P) be of the
following form
therefore in this case if
V - u 0 + w~'P(fl;~ m) u on
~fl
(i.e., u E V
u E wI'P(fl;~ m)
and
u - u0
W ~ LPnm(~)' Au = Vu
and
F(u,Au) = Jl f(x,u(x),Vu(x))dx.
if and only
in the sense of traces),
In this section we will introduce the notion of dual problems, noted by (P*), associated dard results,
to (P).
in convex analysis,
We will then outline some of the stanrelating
(P) and (P*).
The results
we will give, here, rely on the convexity of the functional In the next section we will consider problems weakly lower semicontinuous is convex we mean that if
de-
(P) where
F. F
is not
and hence is not convex (when we say that G(u) = F(u,Au)
then
G
is convex).
F
Then in general,
75
the problem (P) does not have any solution in
V
and one is lead to
introduce the notion of "generalized solutions" of (P).
We will see one
possible approach to this type of problems, using some of the concepts and results of Chapter I. We begin this section by introducing the notions of polar and bipolar of a function. Ekeland and
We follow throughout this section the presentation of
T~mam [ETI] and for more details and for the proofs that
we will omit we refer to this book. Let
V
be a locally convex space and let
f: V
> ~-
]R U {•
we then have the following definition and proposition: Definition.
Let
F(V)
be the set of functions
f
which are point-
wise supremum of continuous affine functions. Proposition i.i. semlcontinuous from
f E r(V) V
if and only if
into ~
and if
f
f
is convex and lower
takes the value
-~
then
f E _~o. We now introduce the definition of the polar and the bipolar functions of
f
which play a crucial role in convex analysis.
Definition.
Let
V
and
pairing
<.;.>.
f*: V *
> ~, is defined by
V*
be placed in duality by a bilinear
The polar (or conjugate) function of
f*(x*) - sup{<x;x*> - f(x)}. xCV Similarly the bipolar of
f**(x)
f
(1.1)
is defined as
- sup
x*s
f, denoted by
{<x;x*>
- f*Cx*)}.
(1.2)
7B
Remarks. (1)
f* E F(v*)
(il)
f*(O) - -inf f(x).
Example.
Let
I
•
-
0
if
xs
+~o if
x ~ A,
then
•
- sup{<•215 xs
which is nothln E else than the support function of
XA where
co A
A
(see [Rol]); and
= X~--oA
denotes the closed convex hull of
A.
The polar and bipolar have the following properties. Proposition 1.2. zation of
f
f E r(v) (li)
then
(1)
(i.e., if
If
f: V
g E F(V)
> ~, and
then
g ~ f
f** then
is the
F-reEularl-
g ~ f**) a n d if
f = f**.
f* - f***.
Remark.
Proposition 1.2 shows that if
the lower convex envelope of
f: V
> ~
then
f**
is
f.
With the help of the notions introduced above, we may now return to our original (or primal) problem
(P)
inf{F(v,Av)}. vEV
(1.3)
77
Let
V*
and
W*
be the dual spaces of
be the adJoint of
A.
V
and
W
and let
sup {-F* (A*w*,-w*) } w*6W* F*
>
V*
We then define the dual problem of (P) by
(P*)
where
A*: W*
is the polar of
(I. 4)
F.
Similarly one defines the bidual problem of (P) by
(P**)
(1.5)
inf{F**(v,Av) }. vs
We then deduce the following: Proposition 1.3. Proof:
(i)
-~ < sup(P*) < inf(P**) < inf(P) < +oo.
By definition of
F*
we have
F*(A*w*,-w*) = sup{<~,A*w*> - - F(~,B)}. ~6V
(1.6)
sew Using the fact that
A*
is the adJoint of
A
we deduce that
F*(A*w*,-w*) = sup{ - F(~,B)}; =EV
(1.7)
Sew hence in particular (choosing
a = v
and
B = Av) we get
F*(A*w*,-w*) > -F(v,Av).
(1.8)
Thus sup(P*) < inf(P).
(ii) that
Applying (1.9) to (P**) and using Proposition 1.2 which shows
P***
and
P*
are the same problems we get
sup(P*) < inf(P**).
(i.i0)
78
Since trivially
inf(P**) < inf(P)
It is of mathematical
interest
we get the result, (see below and Section 2) as well as
from the point of view of applications when
inf P** = inf P.
restrictions
on
o
to know when
sup P* = inf P
In general one cannot expect, without
F, that the inequalities
of Proposition
or
further
1.3 are equalities
(for examples see [Ro2]). In the remaining part of the section we will investigate sup P* - inf P
the case where
while we will leave to the next section the discussion of
the second equality
inf P** = inf P.
We give now a result in this direc-
tion (for a proof see [ETI]). Theorem 1.4.
Assume that
that there exists
v0 C V
w
>
F(v,w) (i)
(ii) (iii)
F
is convex,
such that
is continuous at
F(v0,Av0)
inf (P) < m
is finite and
and the function
Av0, then
inf (P) - sup (P*) (P*) If
has at least one solution (P)
has a solution
~
The above theorem
nonparametric
[Te2],
= 0.
(I.Ii)
(or a similar one) has been an important
many aspects of optimization. ([Tel],
~*
then
F(v,Av) + F*(A*w*,-w*)
T~mam
that
tool in
An interesting application was given by
[ETI]) to the problem of minimal hypersurfaces
form and also to problems in plasticity
giving a simple example of applications tion which allows us to calculate easily for related results see [MS1]).
([TSI]).
in
Before
of Theorem 1.4, we give a proposlF*
(for a proof, see [ETI];
79 Proposition 1.5. Let F(u)
=
I f(x,u(x))dx (~ c~n).
be defined on Le(R), i < = <
(1.12)
Assume that there exists
m
u0 E L~(~)
such that
F(u O) < 4"~176then for all
u* we have
F*(u*) = I f*(x,u*(x))dx. Furthermore if there exists
u~ 6 L~(~)
(1.13)
F*(u~) < 4~
such that
then (1.14)
F**(u) = ] f**(x,u(x))dx. n
We may now give an example (see [ETI]) of application of Theorem 1.4. Example.
Consider the problem f(x)u(x)dx:
(P)
where
~1 c ~ n
i s a bounded open s e t and
6 WI0'2(~) }
f s L2(~).
The E u l e r e q u a t i o n
associated to (P) is
i
-~u = f
in
u = 0
on
~.
Then, in the above notations, we have: I V = W ,2(~) V* = w-l'2(~) A
W = W* = L2(~)
is the gradient operator and
F(u,Au) = ~
~
f
-A*
is the divergence operator
80
Setting
1 f iAvl2
G(Av) = ~
(I.15)
H(v) = -I f(x)v(x)dx
(i.16)
it is easy to see that
G*(w*) = ~1
f
(1.17)
lw, Cx) i2dx
0
if
A'w* + f = 0
(1.18) H*(A*w*) = ~ -~
otherwise.
Therefore the dual problem can be set as follows:
(P*)
sup2 {- 7I
I lw*(x) 12dx}.
w*CL- (a) n dlv w*=f
We thus get the following well known result: Theorem 1.6. (P) has a unique solutlon tlon
w* (1)
(il)
and (P*) has a unique solu-
and inf (e) = sup (P*) w* - -grad ~.
Proof:
The fact that (P) has a unique solution is a well known fact,
the other assertions of the theorem follow directly from Theorem 1.4.
w
Relaxed Variational Problems and Appllcatlons The analysis of the preceding section relied heavily on the convexity
of the functional case where
F
F.
It is the aim of this section to consider the
fails to be weakly lower semlcontlnuous end thus
i.i of Chapter I)
F
is not convex.
our analysis to the following problem
(Theorem
We will, for simplicity, restrict
8~
U inf~J f(Vu(x))dx:
where
~
u C u0
is a bounded open set of ]Rn
and
f: ]Rnm
> ~R
is continu-
ous (we will impose later some growth condition at infinity on
f).
It is well known that if the functional
F(u) = / f(Vu(x))dx
is not weakly lower semicontinuous then in general (P) does not have any solution (in the space
u 0 + w~'P(~; ]Rm)), therefore one is lead to
introduce the notion of "generalized solution" of (P).
We will see later
that this concept is interesting not only from the mathematical point of view but is also useful in some physical models.
In order to define
these solutions, we introduce the so-called relaxed problem
where
Qf
is the lower quasiconvex 9nvelope of
Qf = sup{~: ~ < f
and
#
We will prove below (Theorem 2.2) that minimizing sequences of
(P)
converge
we will then call the solutions of
f, i.e.,
quasiconvex}.
(2.1)
inf (P) = inf (QP) and that
(weakly) to solutions of
(QP);
(QP) "generalized solutions" of
(P).
This way of defining "generalized solutions" is not the only possible; another way of doing so is by using the notion of parametrized measures defined in Section 6 of Chapter I, and this was indeed the idea of Young and MacShane ([Yol]-[Yo4], [Mal], [Ma2]).
However different these two
approaches may be, the ideas of Young and MacShane have lead to the Intro-
82
duction of relaxation,
they have in particular proved the above result
(i.e., inf (P) - inf (QP)) in the case m ffi n = I. extended by Ekeland m ffi i, n > i
and
[Ekl] and by Ekeland and T~mam [ETI] to the cases m > i, n ffi 1
Berliocchi and Lasry in particular
[BLI],
(see also Marcellini and Sbordone
[BL2]).
Theorem 5.6, suggest an interesting conceptual
n ffi i) and the theorem we will obtain here or
difference
(i.e., m - 1
(see also
or
[Da3]); in the cases
we saw that the notions of quasiconvexity
n - 1
[MSI],
However the results of Chapter I,
between the results already obtained in relaxation
m - i
Their result was then
and convexity
coincide and hence in particular
Qf = f**
(f**
being the lower convex envelope of
similarly the relaxed problem
(QP)
f; see the preceding section),
is nothing else in these cases than
the bidual problem (P**), i.e.,
(P**)
inf{laf**(Vu(x))dx:
In higher dimensions
(m > 1
and
u s Uo + Wol'P(fl;]Rm)}.
n > l) however, we only have
f > Qf > f** and therefore inf(P) > inf(QP) > inf(P**). m
In the next proposition we will give an example of strict inequality inf(P**)
< inf(QP)
prove that
and in the main theorem (Theorem 2.2) we will finally
inf(P) = inf(QP).
83 Example. (P)
Let
u: R cl~ 2
> ~2
inf{I (det Vu(x))2dx:
(i.e., m ~ n = 2)
and consider
u = u 0 on a~, u (CI(~; ~2)};
(2.2)
therefore here f(Vu) - (det Vu) 2.
(2.3)
We then claim the following: Proposition 2.1.
if
inf(P**) - 0 < inf(P) = inf(QP) = ~ i
Proof:
(i)
det Vu0(x)dx
.
The fact that (2.4)
inf(P) = inf(QP) is trivial by the results of Chapter I (see Theorem 5.6); in fact and
]2
(P)
(QP) are the same problems. (ii)
Now let us calculate
inf(P).
By Jensen's inequality (see
[Mo2 ]) we have
volI ~ Thus, for every
f
(det Vu(x))2dx >
u C cl(~; ~2)
(
~ 1
such that
( d e t Vu(x))2dx > v ~
f
f2 det Vu(x)dx
u = u0
on
Vu0(x)dx
.
(2.5)
~ , we get from ( 2 . 5 ) '
(2.6)
and hence
inf(P) > ~ Since the problem
(P)
(2.7) is an equality.
VUo(X)dx
.
(2.7)
has actually a solution (see [Dal] for details),
84
(iii)
In order to conclude the proof of the proposition, it remains
to show that, for
F E ~ 2 x 2 = ~4,
f**(F) E O.
(2.7)
We know by Caratheodory's Theorem (see [Rol], Corollary 17.1.5) that
f(F) > f**(F)
inf
=
5 Z %4f (Fi) i=l -
5
~i ->~ [ ~i and
Therefore, since
=
z
i=l
~ XiF i = F . i=l
(2.8)
f(F) > 0, we get
0 < f**(F) < inf{X(det G) 2 + (l-A)(det H)2: I > ~ > 0, XG + (I-X)H = F}.
But it is a simple exercise to show that for every exist
X E (0,1), G
and
H
2 • 2
(2.9)
matrix
F, there
so that
I XG + (I-k)H
=
F
(2.10) det G = det H = 0; hence combining (2.9) and (2.10) we get (2.7). We now state the hypotheses of the main theorem of this section. (HI)
~ c Rn
(H2)
f: R n m
Let
be a bounded open set with Lipschitz boundary. > R
be continuous and satisfy the following coercivity
condition
N u=l
8
N ~i
8
(*)
85 for every d9 ~ b
F 6 ~ nm, for some
> 0
and where
a,c E ~ ,
~ : ~nm
N ~ i
(an integer), 8v > i,
> ~, ~ = I,...,N, are null Lagrangians
(see Theorem 5.4 of Chapter I). Examples.
(i)
The case where
f
satisfies a condition of the
form
a + bIFl 8 ! f(F) ! c + dIFI 8
is a particular case of (H2). d
= d > b
= b > 0
for
I
It suffices to choose
~ = i ..... N
and for
N = nm, 8~ = B > I,
F = (Fij)l
~I(F) = FII ........... ~n(F) = Fnl ~n+l(F) = FI2 ......... ~2n(F) = Fn2 .
.
.
.
.
.
.
e
,
.
.
.
,
.
.
.
.
.
.
.
.
o
.
.
.
.
.
.
.
.
.
.
.
.
.
....................... ~nm(F) = Fnm. (ii)
Another case contained in (H2) is the case where
Proposition 2.1) and
f
n = m
(as in
satisfies
a + bidet FI E ! f(F) ! c + d[det FI E .
Choose
N = 1
Remarks.
and (i)
~ (F) = det F. Hypotheses (HI) and (H2) are equivalent to those con-
sidered by Ekeland and form
T~mam ([ETI], Chap. X), where functionals of the
I f(grad u(x))dx, u: ~ c ~ n
> ~
(i.e., m = i) are studied, and
to those considered in [Dal] where the functional have the form I f($(Vu(x)))dx
where
u: ~ c ~ 3
> R3
and
~
is a null Lagranglan.
However the coercivlty condition (*) is too strong to include the case of parametric integrands, in particular that considered in IDa2], where the functional has the form
~ f(D(Vu(x)))dx, J
u: ~ n
) ~n+l
and
D
is as
86 in Theorem 5.7 (of Chapter I). for
8~ - i, 9 = I,...,N
In these problems (*) is satisfied only
(while in (H2) 89 > i), since
f
in these
f
satisfies (*))
cases is positively homogeneous of degree I. (ii)
Observe that
~
N
h(F) = a +
l bx)[~,~(F) I ~"i
is quasiconvex and hence h<
(ill)
Qf < f.
Note that if
a < f(F) < b + ciFI p
for some
a,b s ~, c ~ 0
and
p ~ i
(in particular if
then the definition of quasiconvexity given in w
of Chapter I is equivalent
to I
f(F)dx
f(F + V~(x))dx ~ I D
D
for every bounded domain
D c R
n
, F C
~nm
and
~ E C ( D ; R m)
(in w
E W ~ ' ~ ( D ; R m) but by approximation we may obtain the above inequality). We now can state the main theorem (proved in [Da3]) of this section. Under the above hypotheses Theorem 2.2.
For
(u0 E wl'~176 ~m)) (i) (ii) (iii)
u
s
= u0
#9(VuS)
(HI)-(H2) we have
every
I
there exists on
with
u = u0
on
8~
{uS}st1, u s { W ' (~; ~m), such that
~R,
% ~u(Vu)
;f(vus(x))dxG
oo
u s W ' (R; l~m)
>
in
L
8v
(~), 9 -
I~ Qf(Vu(x))dx' as
1 ..... N, as s §
S
"~ OO D
87 Remark.
Observe that if
f
satisfies the coerclvlty condition of
Example 1 above, i.e.,
a +
blFI%f(F)
! c + dIFI 8
then the conclusion (ll) of the above theorem means that us
~ u
W I ,8(~; ]Rm).
in
In order to prove the above theorem we will need the following lemma and for this recall that (see (2.1))
Qf - sup{S: S ~ f
L~"-" 2.3.
Let
D cRn
and
S
quasiconvex}.
be a hypercube and for every
F E ~nm
let
(2.11) D
Suppose that
f
satisfies the condition
air(F) !b+cIF[ p for some
a,b C A ,
continuous and Proof: step i. E > 0
c ~ O, p ~ 1
and for all
F s
Then
Q'f
is
Q'f m Qf.
The proof is decomposed in four steps. We show first that
be arbitrary.
~,~ E C~(D; I~m)
Q'f
is continuous.
Let
H E I R nm
and
We then have by definition that there exist
so that (we take
IQ''f(F) -
D
to be the unit hypercuhe)
f(F+V~(x))dxl ! g D
(2.12)
88
IQ'f(F+H) - [Df (F+H+V~ (x))dx [ < ~ . Since
f
is continuous, by choosing
IHI
(2.137
small enough we have
lJDf(F+H+V~(x))dx - JDf (F+~(x) )dx{ < ~e
(2.14)
IIDf(F+H+V*(x))dx- ID f(F+Vr
(2.15)
Using the definition of
Q'f
<2"
we get
Q'f(F) < f f(F+V~(x))dx 2 D
(2.16)
Q'f(F+H) < f f(F+H+?~(x))dx. J D
(2.17)
Using (2.17) and (2.12) we get Q'f(F+H) - Q'f(F) < /D f(F+H+U~(x))dx - ]D f(F+V~(x))dx + 2'
(2.18)
and applying (2.14) to (2.18) we obtain Q'f(F+H) - Q'f(F) < e.
(2.19)
Similarly using (2.16), (2.13) and (2.15) we obtain Q'f(F) - Q'f(F+H) < IDf (F+V~(x))dx - I f(F+H+V~(x))dx + ~e< D Thus
Q'f
(2.20)
is continuous.
Step 2. dent of
s
D.
We next want to show that the definition of Let
D1
rotation) there exist F C ]Rnm, let
and
D2
I > 0
be two hypercubes of ]Rn. and
x0 s ~n
so that
Q'f
is indepen-
Then (up to a
D 2 - x 0 + %D I.
For
89
Q{f(F) = inf
Q~f(F)
i meas D I
ea~1 D 2
= inf
f(F+V~(x))dx: ~ E C0(DI; A m)
(2.21)
f(F+V$(x))dx: ~ E C0(D2; A m) 9
(2.22)
DI
D2 Then, for every
E C0(D2; Am), we have Q~f(F) < 1 -- meas D 2
f
f(F+V~(x))dx. D2
(2.23)
We therefore deduce Q~f(F) < - -
1 f f(F+V~(x))dx %nmeas D I Xo+%D I _ ____!___l I f(F+~(x0+%Y))dy. meas D I DI
Observe that for every
(2.24)
~ 6 C0(DI; A m ) we have that
~(y) ffi %
Therefore applying (2.24) to the function Q~f(F) < - -
(2.25)
s C0(D2;Am). ~
defined in (2.25) yields
i [ f(F+V~(y))dy, meas D I ~D I
and taking the infinimum in (2.26) over every
~ f C0(DI; A m)
(2.26)
we get
Q~f(F) ! Qi f(F)"
Similarly one gets that Ste~ 3. every
Let
F E A nm
D
Qlf ! Q2f"
be the unit hypereube.
and for every
We now want to show that for
~ 6 Co(D; A TM) we have
g0
I
DQ'f(F+V#(y))dy
> fDQ'f(F)dy - Q'f(F).
(2.27)
co
For every
e > 0
affine function
we can approximate
~0 E W 0' (D; ]Rm)
$ 6 C0(D; ~m)
(see Proposition
by a piecewise
2.1, p. 286 in [ETI])
such that 1
Q'f(F+V0(x))dx
'f(F+V0(x))dx I _< -~-.
-
(2.28)
D Since
~
is plecewlse afflne we may decompose
Di, 1 < i < N, and find
A i E]R nm
?~
Ai
=
D
into open subsets
so that
in
Di
and
Q'f(F+VqJ(y))dy = [ meas DIQ'f(F+Ai). D
(2.29)
i=l
We now use the definition of
Q'f
to get
~i 6 Co(Di; ]Rm), 1 < i < N,
such t h a t
~ > measi D i f Di f(F+Ai+V~i(x))dx, Q'f(F+Ai) + ~-We then define
X: D
> ]Rm
(2.30)
by
X(x) " ~(x) + ~i(x)
Observe that
1 < i < N.
1 m X C W O' (D; ~m).
for
Since
x 6 D i.
f
(2.31)
satisfies
the growth condition oO
of the le---- (a < f(F) < b + clFIP), we may find II f(F+Vx(x))dx D Combining
-
fDf(F+vz(x))dx
(2.29) and (2.30) we get
I --<7" ~
Z C Co(D; ~m)
so that
(2.32)
91
Nf
[ i=l
f(F+Vx(x))dx <__[DQ, f (F+V~ (x))dx + ~, e Di
(2.33)
i.e., fDf(F+VX(x))dx < ;DQ'f(F+V*(x))dx + ~. s Using (2.32) and (2.28) in (2.33) we obtain f f(F+VZCx))dx < fDQ'fCF+V*(x))dx + e, D since
e > 0
Step 4.
is arbitrary and
Z s C0(D; ~m)
Q'f
oo
open set of ~n, ~ 6 C0(~; ~RTM)
is quasiconvex. and
f Q'f(F+Vr
0
D In
tion of
we deduce (2.27).
We are now able to complete the proof of the theorem.
(i) We first show that
Let
(2.34)
F s ]Rnm.
Let
~
be any bounded
We want to prove that
>_I Q'f(F)dx.
be a hypercube containing
~
and extend
(2.35)
~
to be identically
D-~. Then apply Step 3 (and Step 2 which shows that the definiQ'f
is independent of
D) to get
I~Q'f(F+V~(x))dx = IDQ'f(F+Vr
- IJD-~Q'f(F)dx
>--fDQ'f(F)dx- JD-fl f Q'f(F)dx. (ii) convex.
It remains only to prove that
Q'f = Qf.
Let
(2.36)
h < f be quasi-
We then deduce from the definition of the quasiconvexity of
h
that Q'h - h.
Since
h < f we also have
(2.37)
92
h = Q'h < Q'f < f for e v e r y
quasiconvex function
(2.38)
h < f, hence
Qf < Q'f.
Since
Q'f
is
also quasiconvex we deduce the result. With the help of the above lenmm we are now able to prove Theorem 2.2. Proof of Theorem 2.2:
Fix
~ > 0
of generality in supposing that wise we may find
u
and observe that there is no loss is piecewise affine in
O c ~, an open set, and
w E WI'~(~; ~m)
~.
Other-
such that
(see Prop. 2.9, Chap. X in [ETI])
meas(~
w
-
O)
(2.39)
n
<
is piecewise affine in
w = u0
Jw(x)-u(x)[ ! n
llvw - vull
on
(2.41)
x E ~
for all
So if we can prove the theorem for
w
and
sets
u
is piecewise affine in
Ai, i < i < I, so that
into small hypercubes
f
Vu
~
(2.42)
p > i.
(2.43)
O, by defining
- O, we will have proved the theorem for every Since
(2.40)
~
for all
< n Lp --
0
S
~ u
in
u E wl'~(~;~m).
we may decompose
is constant in
u
A i.
~
into open
Then decompose
Ai
R~,z 1 _< p --< Pi' so that
Qf(Vu(x))dxl __<"9-31
(2.44)
f(Vu(x))dx I ! 3 ~ "
(2.4s)
Fi
A i- U R p p,,l
f
Pi
A i- U R p p-1
93 n
But on each
R~
we have 1
Qf(Vu) - inf~
| p pf(Vu+V~(x))dx: ~ E C0(RI;
kmeas {~q}q-l' ~q 6 Co(Ri; ~m)
Therefore there exists
so that for
lRm)}. q
large
enough Qf (Vu)dx
Now extend whole of ~n
~q
< n -- 31P i "
be periodicity (in each variable) from
and for
s
Rp
(2.46)
to the
an integer let
q,S
(x) = i ~q(SX). S
(2.47)
We obtain from (2.47) that
Vq,s(X) = 0
if
p.
x E ~R~ (since ~q is periodic and ~q E Co(Ri, ~m)) (2.48) V~q,s(X) = V~q(SX).
We now use the coercivity condition (H2) on we deduce that for every
|
{$v(Vu+V~q)}q= 1
v = I,...,N.
is bounded in
Therefore there exist
(2.49) f
(and
Qf).
From (*)
L
8~(R~)
(and
8V > i)
~v 6 LB~(R~)
so that
(possibly after the extraction of a subsequence) ~v(Vu+V~q(X))
9 ~ ~(x)
Observe also that since we have
in LS~(R~), ~ - i ..... N as q § ~. ~
is a null Lagrangian and
(2.50)
P ~q 6 =C0(Ri; ~m)
94
I
6u(x)dx = lim I
RPi
q-~
~u(Vu+V~q (x))dx
=
Rp
I
r (2.51) I,... ,N.
-
As we extended (in each variable) ~(sx)
~
by periodicity
q from
R Pi
;
i meas R E
(in (2.47)), we do it for
to the whole of 6u(x)dx ffi ~ ( V u )
~n in
~u
and therefore deduce that L U(R~),
(2.52)
RE I,...,N.
=
We then take the diagonal sequence of (2.50) and (2.52), to get
~ (Vu+V~s(SX))
Summarizing
9 ~(Vu)
(2.47),
in
LSU(R~),
~ ffi i ..... N.
(2.48) and (2.53) and for
x s R Pi
(2.53) defining
s
u (x) - u(x) + ~s,s(X) - u(x) + !s ~s(sx)'
(2.54)
we obtain
I uS(x) " u(x) ~(
vs u )
for
x ~u(Vu)
x s ~R p in
(2.55)
B
L ~(RP), ~) - i ..... N.
We therefore have defined
us
on
us = u
on
Ai
(2.55)).
By letting
P and hence on Ri, p
(2.56) P Ui RE p-I
(using
_ Ui Ri, p we have then constructed
p-i I u on A i, i < i < I, and thus on R = U A i. Obviously from the coni=l struction of u s and from (2.56), u s has the required properties (i) s
and (ll); so it only remains to prove (iil)
g5 ff~ I
Q,,udx-r
<_
iffil JA i
l
"A i
< i~l{]f
Qf(Vu)dx
+
f
f (VuS (x))dxl} Pi
Pi
=
I
~ f~ ~u>d~-f~ ~<~u~x i ~-~~-~ We now use (2.44), (2.45) and the fact that
u
s
= u
on
Ai -
(2.57)
Pi U p-i
R~
to get
If Qf(Vu(x))dx- fnf(VuS(x))dxI I
-- -T
Pi
i=l p=l
pQf (Vu) dx Ri
f(VuS(x))dx I 9
(2.58)
RPi
Finally observe that
IR~ f(VuS(x))dx
"
f
R~
fCVu+V~s(SX))dx
"~
s
[sR~ ~(Vu+V~s(Y))dY
ffi fR~f(Vu+ V~s(Y))dy,
(2.59)
the last equality being a consequence of the periodicity of
~s"
Therefore using (2.46) and (2.59) we get
If
ff(vuS(x))dx
n
and combining (2.58) and (2.60) we have indeed obtained the result,
(2.60) o
96
Conclusions.
(i)
We deduce from the theorem that
inf(P) = inf(QP). However, in general, the hypothesis (H2) is not sufficient to guarantee the existence of solutions for (QP), but if
f
satisfies the following
particular form of (*)
a + DIFI ~ ! f(F) ! c +
with
~ > i, then the direct method of the calculus of variations implies
that (QP) possesses a solution (see [Mo2]). and
dlFI 8
f(F) = g(det F)
Similarly if
n = m = 3
then (QP) possesses also a solution (see [Dal] for
details). (ii)
The result obtained in Theorem 2.2 should be extendable to func-
tionals of the form
I f(x,u(x),Vu(x))dx
and growth conditions on Rellich's Theorem, that on
(with the appropriate continuity
f) since the important dependence is, by Vu.
Theorem 2.2 has an interesting physical interpretation.
It is custom-
arily assumed in nonlinear hyperelasticity that the strain energy function is elliptic and that even if elliptlcity failed in certain regions one would not be able to detect it by static experiments.
The above
theorem can be interpreted, somehow, as a mathematical model of this fact, since weak convergence measures some kind of averages and an average is what is usually measured in experiments.
An interesting applica-
tion of the above result (in fact that of [ETI]) is given for an antiplane shear problem in elasticity by Gurtln and
T~mam [GTI].
To con-
97
clude these notes we will give another example of applications of Theorem 2.2 to a problem of equilibrium of gases (see [Dal]). We use here, material (Lagrangian) coordinates and we let a gas occupy a volume
~ =~3
in a given reference configuration.
figuration the particle
x s ~
occupies the position
suppose, further, that the deformation is specified on ~fl).
on
In a deformed conu(x) s ~
(namely
We u - u0
The energy of the deformation, in the absence of external forces,
is t h e n [ I [ det Vu(X)p(V)dV}dx E(u) = Jfl'#l where P -
P
P(V)
state.
i s the pressure and
V
(2.61)
the s p e c i f i c volume. The r e l a t i o n s h i p
at constant temperature (or entropy) is given by an equation of We will assume that the equation of state is the Van der Waal's
equation which at constant temperature has the form 2 an V2
P(V) = nRT V-nb
where
T
denotes the temperature, n
(2.62)
is the number of moles, a, b
and
R
are given constants (see for general references [Cal]). Let f(t)
ffi
?
P(V)dV
(2.63)
1
end l e t
f**
be i t s
lower convex envelope;
f**(t) ffi
?
R(V)dV
then if
(2.64)
i
R
has the following properties, better illustrated in the following diagram:
98 P,R
V
..... Figure i.
Plot of
P
Plot of
R
Plot of P and R as functions of V.
More precisely, in the case illustrated in Figure i, if V A , V B are such that A-
(VA,P(VA)), B = (VB,P(VB)), C -
(Vc,P(Vc))
then I P(t) R(t) -
P(VA)
if
t ~ (VA,VC)
if
t E [VA,Vc]
and Vc Vc IVA P(t)dt = ;VA R(t)dt" (The llne
AC
is usually called the Max-wellRne.)
If
(P) and
inf{I f(det Vu(x))dx:
u = u0 on
3~, u C CI(~; R3)}
and V C
99
(QP)
inf{I f**(det Vu(x))dx:
u = u0
on
3fl, u E cl(~; 13)}
(note that by Proposition 2.1, the problem (QP) is different from (P**)) then it is possible to show the following: Theorem 2.4. det V~(x) > 0
u
(i) (QP) possesses a solution for all
(u s cl(~; m 3)
and
x s ~).
(ii) inf (P) = inf(QP). (ill) More precisely for any solution there exists a minimizing sequence
{u }
u
of
(with det ~
(with det Vu
S
(P)
(QP)
> 0
> O)
in ~) of
S
such that
I iet VU s
' det V~ fin
f(det VUs)dX
Proof:
(i)
>
LI(~)
f**(det ~)dx.
The existence of solutions follows from e result in [Dal].
(ii) and (iii) result from Theorem 2.2. (There are, however, some difficulties in applying the above theorem since
f
satisfies neither the
continuity nor the coercivity condition required in (H2); but these difficulties can be removed by a more careful construction of the sequence {uS } and we refer for details to [Dal].)
u
APPENDIX Since t h e w r i t i n g the results of w
of these notes, the author has improved (see [Da4])
(Chapter I) in some particular cases.
Before describing the results of the Appendix, let us recall the hypotheses of Chapter I l u
~
u in Lm(~ ) . m
(H)
f(u ) " ~ ~ where ~ c ~ n In w
n
~u..
Au ~ = [ ~ ~ a.. ----J-~ bounded in L (~) ~j~l k~l ijk ~Xk]l~i~ q q in L (~)
is a bounded open set and f : ]Rm-->]R is a continuous function.
a necessary condition for weak lower semicontinuity (i.e., ~ f ( u ) )
was isolated and called A-quasiconvexity. This condition turned out to be sufficient in some particular cases. Recall that Definition.
A continous function f : ]R m - >
IR is said to be A-quasiconvex
if f
I
f(~ + ~(x)) dx ~ | D
f(~) dx
(A.I)
jD
for every ~ 6 ]R m, for every hypercube D c ]R n and for every ~ 6 L(D) where L(D) = {~ 6 LI(D); [ ~(x) dx = 0 and ~ E Ker A}. JD The aim of this Appendix is to show that for some special operators A (e.g., A = curl or A = div and hence for the variational case) one may further restrict the set L(D) by including a condition on the support of ~ 6 L(D) (thus answering Remark (ii) p. 13);
which therefore makes more precise the
notion of quasiconvexity. Before doing that we need to isolate a special class of functions which are in Ker A.
101
Notations.
If A is defined as in Hypothesis (H), i.e., m
n
~uj (A.2)
Au = (j[l i = k=l I aijk--~Xk]l~i~ q we will denote by B B : v(x I _ ..... x n) = (v,,: ..., v ) P
> Bv
the operator p n ~v___~_~ 1 ! I b%%/~) ]] i ~=i 3x) ii.
BV-'=
where blp ~ 6 ~ ,
p >. i an integer and where B satisfies ABv 5 0
Remark~
(A.3)
(A. 4)
for all v E C2(~;~P).
(A.4) implies that Bv 6 Ker A.
Examples of operators A and B. (~) Let m = nr (r ~ I an integer) and u = (u I .... ,Ur) , with u. : ~ n 3
> ]R n, i ~ j ~ r.
(A.5)
Let A be the operator (A.6)
Au = (curl u I ..... curl Ur) where, for v : ~ n
curl
v
=(3X~ 3
-
> ]R n,
8Xi)l.i
((~v I
8vj)
(~v 2 _ ~v.)jj
~x. - ~Xl j~2 ; 3
8xj
3x 2
>.3 ;'";
8Vn_l
~v) n
~xn
~x n-I
(A. 7) "
Therefore for A as in (A.6) we have q = n(n-l)r/2. We may then choose p = r and B as follows Bw = Vw = (grad w I ..... grad Wr) where w. : ~ n 3
> JR, i .< j ~ r. Then A and B satisfy (A.2) - (A.4).
(A. 8)
102
(8) Let m = n and > ]R
u(x I ..... Xn) = (u I ..... Un), uj :
I ~ j ~ n,
(A.9)
with ~u I Au = (~Xl,
~u2, Du n " ~x 2 .... ~Xn)e ]Rn
(A.IO)
then the only B satisfying (A.3) and (A.4) is Bv - 0
for all v 9 C2(~Q;]RP).
(A.II)
With the help of the notations above, we may introduce the following definition Definition.
A continuous function f :
~m
> ]R is said to be A-B-quasi-
convex, where A and B satisfy (A.2) - (A.4), if I
f(u + B~(x))dx
>, I
G for every U 9 ]R 6 WoI'O~
TM, for
f(u) dx
(A.12)
G
every bounded domain G c ]R n and for every
P) 9
We then have in~nediately Proposition A.I. Proof:
If f is A-quasiconvex,
then f is A-B-quasiconvex.
Let G be a bounded domain of ~ n
and let K be a hypercube of l~ n
containing G. Let ~ 9 W o' (G;]RP). Extend ~ from G to the whole of K in the following way in K - G.
~-0 We then deduce that B ~ EL(K),
(A.13)
i.e., Co
B ~ 6 L (K) m I
B~(x) dx K
B ~ 6 Ker A.
Using the A-quasleonvexlty of f we obtain
0
(A.14)
103
f
(A.15)
I f(~ + B~(x)) dx ~ | f(~) dx, K ~K and therefore I G
f(P + B~(x)) dx = I f(~ + B~(x)) dx - [ f(~) dx K "K-G [ f(~) dx. )G
Remark.
In the variational case, i.e., m = nr 16j,
u = Vv = (grad Vl,... , grad Vr) , v. : ~ n ... > 3 Au = rot Vv E 0 By = Vv,
the definition of the A-B-quasiconvexity corresponds exactly to that of Morrey ([Mol], [Mo2]) given in w
pp. 39 - 40.
We may now state the main theorem of this Appendix; recall first that [ u~ (H)
*" u
Au ~ f(u ~)
where ~ is a bounded open set of ~ n Theorem A.2.
*~ Au *~ s
in L~(a) in L~(~) q in Lm(~)
and f : ~ m
> ]R is continuous
I) Necessity : If, for every sequence {u9} satisfying (H),
) f(u), then f is A-B-quasiconvex. 2) Sufficiency : If {ug}, u satisfy Hypothesis (H) and if furthermore either (~) f is A-quasiconvex and u 9 and u are such that (H) u o
- u 6 Ker A;
or (B) f is A-B-quasiconvex and satisfies
If(u)
- f(v)[
a > 0, 8 ~ i, u, v 6 ~ m
~ a(1 +
lul B-z + Ivl B-l) l u - v l ,
A and B satisfy (A.2) - (A.4) and
(A.16)
104
For every u
~
0, Au
O, there exist v ~ C W I ' B ( ~ ; ~ P ) o
and w ~ E L~(~) such that
(NAB)
u~
=
vv
Bv ~ + w ~
w~
0 in W I ' 8 ( ~ ; ~ P ) > 0 in L~(~); m
then % ~ f(u). Proof:
I) Necessity : This is just Theorem 2.1 and Proposition A.I above. 2) Sufficiency : The first part (~) is only Theorem 2.3. Part (~),
once Hypothesis (HAB) assumed, follows exactly the pattern of the proofs of Theorem 2.3 and Theorem 5.1 ((HAB) replacing Step 2 of Theorem 5.1; for more []
details see [Da4]). Before proceeding further we need to make some remarks on Theorem A.2 Part (~) Remarks:
(i) Observe first that (A.16) is purely technical and comes from
the fact that in (HAB) we did not assume B = ~. The important condition in the above theorem is obviously (HAB) . We will see below that the following operators satisfy (HAB) i) A = curl, B = grad 2) A = div, B = curl 3) A = (curl, div), B = (grad, curl); while those defined in (A.10), (A.II) do not satisfy (HAB). (ii) The definition of the operator B above and the Hypothesis (HAB) imply that one may decompose u~ into Bv ~ ~ Ker A and w ~, where w ~ is a sum of a boundary term (since v
. is assumed to be 0 on ~ ~, while u ~ is not)
and of a term in (Ker A)"i. For example in Theorem 5.1 (i.e., for the variational case) we have automatically that u
= Bv ~, but in general v ~ # 0 on ~ ~,
105
by the u~e of Mac Shane's Lemma one is able to correct that and therefore to get (HAB). (iii) It is also interesting to compare the Hypothesis
(HAB) with
that of constant rank used by Murat in [Mu3] (Murat's result is mentioned in Theorem 4.5 p. 37 of these notes). Using a theorem of Schulenberger and Wilcox ([SWI],
[Kal]) the hypothesis of constant rank of the operator A implies
a condition very similar to (HAB) (for more details see Lemma 3.6 in [Mu3]). However the method, we will use in Theorem A.4 below, is somehow different. We now want to show that operators of the type div or curl (and hence for the variational case) satisfy Hypothesis
(HAB) . This will result from well
known theorems on the existence and regularity of elliptic operators. We first introduce some notations. Notations.
(i) Let A be the operator defined in (A.2), i.e.,
AB =
We denote by A
the operator defined as f
J~ for
< A*u(x); v ( x ) >
all u E Co(~;~
q
)
~
dx = f
J~
< u(x); Av(x) > dx
m
v C C (~;~). o
(ii) In particular we will denote by curl A
associated to
the operator A
= curl, i.e., curl u = ( ~-i. 3 8u I
8u~ ~xi ) l.i 8u j, 8u 2 8x. J
~u~,
80n_ 1
~x2 j~3
~x
n
therefore here q = n(n-l)/2. If we denote by y(p) = (p-l) n - p(p-l) 2
, I ~ p ~ n
~u n
106
and if v(x I ..... x n) = (Vl,V 2 ..... Vn(n_l)/2 ) we then have
curl
*
v =
(Pi 1 ~vP-9+Y(p) ~x ~=i
~
From the above definitions Lemma A.3.
n
~=p+l
is defined
n
)
~V~-P+Y(v) ~x
it is easy to deduce
(i) If u : ~ c ~ n
curl * curl u = - A where A
n~
-
that
igp~n :
> ]R n, then
u + grad div u, for every u E C 2 '
as ~2u n
nu
=
(J~ ~2Ul'~x~ ~ 3--~--. . . . . ." )'j=l
(Au I ..... bUn)
= ~=i
3 (ii) If u : ~ c ~ n
(iii) If u : ~ C ~ n
n u = An(n_l)/2
* A n (n_l)/2 u
= A n curl
Remark.
* u,
C2
for every u
(A.20) C 3"
theorem
(i) If m = n, A = curl, B = grad,
then A and B satisfy
(HAB) of T h e o r e m A.2.
(ii) If m = n, A = div, B = curl
, then A and B satisfy
(HAB).
If m = nr, s $ r and u(x I ..... x n) = (u I ..... Us,Us+ I ..... u r) Au = (curl u I ..... curl Us,div Bu = (grad u I ..... grad Us,CUrl
then the above
theorem implies
(A.19)
then
u E 0, for every u
We may now prove the following Theorem A . 4 .
3
curl u, for every u 6 C 3.
> ~ n(n-l)/2
div curl curl
(A.i8)
> ]R n, then
curl A
Hypothesis
(A.17)
Us+l ..... div u )r Us+l .... ,curl
that A and B satisfy
(HAB).
u )z
(A.21)
107
Proof:
(i) Let ~ be a bounded domain of ~ n ,
with a sufficiently regular
boundary and let u
:~ 0 in L~(~) n
Au ~ = curl u ~ - *~
w
We w a n t
to
9 LB(~)
such
n
show t h a t ,
given
g ~ 1,
(A.22) 0 in nn(n_l)/2
o n e may f i n d
(~).
(A.23)
v~ 9 WI'~(~) o
and
that u
= grad
v
~
I w~->
v
+ w
(A.24)
0 in wl,~(~)
(A.25)
0 in LB(~). n
(A.26)
For this, let us consider the weak form of Laplace's equation I
< grad v~(x) ; grad ~(x) > dx fI
)
dx, for all ~ E WI'~'(~),
o
(A.27)
where ~' ~ I is given. By the classical results on uniformly elliptic equations (c.f., for example Theorem 7.2 in [Srl]) we dedu6e the existence of a solution v ~ E WI'~(~) of o I I (A.27), with ~ given by ~ + ~, = I, such that [I v~I[
~ Ell div u~ll
WI,~
(A.28)
W-I,~
where K is a constant independent of ~ and W -I'~ denotes the dual of W I'~'. o From (A.22)
(i.e., dlv u ~ E W-I'~), from (A.27) and (A.28) we deduce imme-
diately that v
~ 0 in WI,~;
being arbitrary we have indeed obtained (A.25). We then define w ~ E L ~ ( ~ ) b y m
w
= u
- grad v .
(A.29)
108
Combining (A.22) and (A.25) we obtain w~ ~
0 in L~(~), for all ~ >. i. m
(A.30)
In order to conclude the proof of Part (i) of the theorem, we only need to show that in (A.30) the convergence is strong. Therefore let ~ E C=~ o
n)
and observe that by Lemma A.3 one has <
A
~ > dx = -
n
< w ; curl
+
f
curl ~ > dx
< w ; grad div ~ > dx.
(A.31)
J~ We now use (A.29) in (A.31) and integrate by parts to get I
< wV
A
~ > dx = [ < curl u V ; curl ~ > dx j~
n
+ I
< grad vg; curl* curl ~ > dx
+
< u ; grad
-
< grad
div
v ; grad
~ > dx
div
~ > dx.
(A.32)
From (A.27) we immediately deduce that <
and therefore,
If
< w~; A
A
n
~ > dx =
for every ~ E C o ( ~ ; ~
n
< curl u ; curl ~ > dx;
n
(A.33)
),
~ > dx I ~ ]I curl u~ll L~l[ curl @]]
L~'
Kll curl u~ll Loll ~]I wl,~,, I I with ~ + ~, = i. Recall also that by (A.23) curl u ~ E L
(A.34) .
Using again the local regularity of elliptic operators
(c.f. for example,
[Agl] Theorem 6.2 if ~ = 2 and [Srl] Theorem 9.5 if ~ # 2), we obtain, using (A.30), that
109
II JII W I ,~(~, )
.< K(II curl u"ll
+ II w'~ll
La(~)
),
(A.35)
L~(~)
where ~' is such that ~' c c ~ and K is a constant. Using Rellich's Theorem (see [Adl]) we deduce that w ~) - - >
0 in LC~(~ '), for every ~' c c ~.
(A .36)
m
We finally want to show that (A.30) and (A.36) imply (A.26), thus establishing Part (i) of the theorem. For this, let ~ be large enough so that i .< 13 < e, we therefore want to show w
-->
0 in LS(~). m
(A.26)
Consider
I lw~(x)IS dx--
r lw"(x)l 6 dx
J ~-~
where e > 0 is arbitrary and ~
r lw'~<x)l 6
+
dx,
J~
(A.37)
s
c c ~ is such that
II w"ll B
(meas
(~-,Qg))(~
s
~< "~"
,
(A.38)
Le(~) Using (A.36) we deduce that for 9 sufficiently I
large
{w~Cx) lB dx ~ ~s ,
(A.39)
C and by Holder's inequality we have
I lwV(x) ~-~
~-~ g ~ .
~-~ (A.40)
Combining (A.39) and (A.40) we have indeed obtained (A.26) and this achieves the proof of Part (i) of the theorem. (il) This part is very similar to Part (i) and therefore we will leave out the details.
110
Let l) *
u"
co
~
(A.41)
0 in L (~) m
div u ~ *~
0 in L co(~),
(A.42)
we want to show that, given B ~ I, we may find v ~ 9 wl'8(~,~" n(n-l)/2) 0
and w ~ 9
LB(~) so that n
U
=
v~
curl ~
V
~
(A.43)
+ W
~ 0 in wl'B(~;~n(n-l)/2)
w~
> 0 in LS(~).
(A.44) (A.45)
n
So let v ~ e W I ' ~ ( ~ ; ~ n(n-l)/2), ~ ~ I given, be the weak solution of 0
J( - An(n_l)/2 v ~ = curl u ~ in (A.46)
t
v
= 0
on ~ ,
which satisfies II v~)il
(A.47)
,< Kil curl u~)ll WI,~
W-I,~
where-K is a constant independent of ~. We then deduce from (A.41), (A.46) and (A.47) that ~ 0 in W I '~(~;~ n(n-l)/2);
v
(A.48)
being arbitrary we obtain (A.44). Now let ~) W
*
~) =
U
'0 .
(A.49)
" 0 in L~(~).-
(A.50)
--
curl
V
From (A.41) and (A.48) we deduce that w
l)
n
In order to conclude the proof, it only remains to show that in (A.50) the convergence is strong. A
w ~) = A n
From (A.49) we have that
u~ - A n
curl* v , in the sense of distributions. n
(A.51)
111
Using Lemma A.3 we deduce An w~ = An u = A
n
- curl
An(n_l)/2 v
u ~ + curl* curl u (A.52)
= grad dlv u ~, in the sense of distributions,
where we have used (A.46) in the second equality of (A.52). As in Part (1) from the regularity of elliptic operators, from (A.42) and (A.50) we obtain that w
~
0 in
WI,~ (~;~ n). loc
(A.53)
Since in (A.52) ~ is arbitrary, by choosing ~ so that i ~ 8 < ~, we deduce from Rellich's Theorem,
from (A.50) and (A.53) that w
[]
> 0 in LS(~). m
As in Chapter I w
from the results on weak lower semicontinulty
(i.e.,
s >~ f(u)) we deduce easily some results on weak continuity. Definition.
A continuous function f : ]R TM
affine (resp. A-quasiaffine)
> IR is said to be A-B-quasl-
if f and - f are A-B-quaslconvex
(resp. A-quasi-
convex). We then get immediately as a consequence of Theorem A.2 that if
(H)
u
~ u
Au ~)
" Au in L (~)
f(u ~))
Theorem A.5.
*" s
in L~(~) m
in L=~
Under the hypotheses of Theorem A.2
I) Necessity.
If for every sequence {u ~} satisfying
is A-quasiaffine and thus f is A-B-quaslafflne. 2) Sufficiency.
If ~u ~} and u satisfy (H) and either
(e) f is A-quasia~fine and u ~ and u are such that
(H), ~ = f(u) then f
112
(Ho) Au ~ - Au - 0; or (8) f is A-B-quasiaffine,
A and B satisfy
(HAB) of Theorem A.2;
then s = f(u). Proof: -
The proof is a direct consequence
of Theorem A.2 applied to f and
f.
Corollary A.6:
Let g : ~
S
> ~
be convex and let
f(u) = g(~l(U) ..... ~ s (u)) where ~I .... '~s are A-B-quasiaffine, Proof: Remark.
The proof is identical As seen in Chapter I w
A-B-quasiaffine
functions
then f is A-B-quasiconvex.
to that of Corollary
2.5 of Chapter I.
if A = curl and B = grad,
are just the subdeterminants
then the
of the matrix Vu.
[]
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INDEX
A)
Affine functions:
7,18,24,61,63,66,68,75
Affine in the directions of Anti-plane shear problem:
B)
Bidual problem:
96
77,82
Bipolar function:
75,76
B.V. (bounded variations)
C)
A: 22,26
functions:
64
Calculus of variations:
1,2,4,5,7,12,14,22
Carath~odory's Theorem:
84
Cauchy problem:
59
Characteristic function: Coercivity condition: Compact case:
9,55
84,85,87,93,99
20,24
Compensated Compactness: Conjugate function: F,G-convergence:
7
75
6
Convex functions and Convexity:
7,8,10,11,13,14,15,18,24,26,50,
60,61,62,64,65,67,69,71,72,74,75,78,80,82,112 Convex hull:
55,56,64,76
Convex in the directions of
D)
Dirac measure:
A:
22
57,64,66
Direct methods of the calculus of variations: Distributions: Dual problem: Dual space:
3,12,31,37 6,74,77,80 41,77
1,96
118
E)
Elasticity:
68,72,96
Elliptic e~uation: Ellipticity: Entropy:
11,105,107,108,111
5,22,25,43,96
60,97
Entropy condition: Entropy flux:
60
60
Equilibrium of sas:
97
Euler-Lagrange equations:
F)
Fourier
transform:
G)
Generalized curve:
32
52
Generalized solution:
2,6,75,81
Generalized surface: Growth condition:
H)
I)
18,45,79
52
12,81,90,96
Hahn-Banach Theorem:
56
Homoseneization:
6
Hyperelasticity:
68,96
Incompressible material: Isotropic material:
68,71,72
71
J)
Jensen's inequality:
L)
Lagrangian coordinates:
83 97
Legendre-Hadamard condition: Lipschitz domain:
5,19,22,25,31,43
69,84
Lowe K convex envelope:
76,82,97
Lower quasiconvexenvelope: Lower semicontinuity:
75
81
119
M)
MacShane's Lemma: >~xwell line:
41,105
98
Mazur's Lemmm:
ii
Minimal hypersurface in non-parametric form: Minimal hypersurface in parametric form: MooneyyRivlin materials:
N)
Non-convex problems:
73
6
Nonlinear conservation law: Nonlinear elasticity: Null Lagrangian:
o)
6,68
72
Optimal control theory:
P)
6,59,66
18,43,44,45,46,47,48,85,93
Ogden material:
Optlmization:
2,6
78
Parabolic approximation: Parabolic equation:
66
66
Parametric integrands: Parametrized measure:
85 5,52,62,81
Partial differential equation: Pieeewise affine function: Plancherel formula: Plasticity:
33
78
Polar function: Polyconvexity:
49
75,76 18,69,71,72
1
90,92
78
120
Q)
Quadratic case:
5,31,37
Quasiaffine, A-quasiaffine function:
18,19,24,43,44,111,112
Quasiconvex~ A-quasiconvex function:
5,12,13,14,15,16,17,18,22,
23,24,39,40,42,43,46,47,49,50,81,82,86,87,91,92,100,102,103,11].,112
R)
Radon measure:
53
Radon-Nykodym Theorem: Rank one convexity: r-regularization:
43,47,50 76
Relaxation theorem: Relaxed problem:
6,82
6,74,80,81,82
Rellich's Theorem:
s)
54
96,109,111
Schwarz's inequality:
67
Strain energy function: Strong convergence: Suhdeterminant:
68,96
11,24,57
24,44,112
Support function:
76
u)
Unicity of weak solutions of nonlinear conservation law:
v)
Van der Waal'~ equation of s.tate: Variational case:
w)
97
12,13,18,20,21,24,27,31,36,39,43,100,103
Weak and weak*continuity: Weak and weak*convergence:
2,3,4,5,7,8,17,19,24,25,26,37,111 1,2,3,7,8,12,43,45,53,81,96
Weak and weak*lower semicontinuity:
2,3,4,7,8,15,19,24,25,26,
39,40,74,80,81,100,111 Weak solution:
60
11,59,60,66,68,107,]10
