Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
922 Bernard Dacorogna
Weak Continuity and Weak Lower Sem...
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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
922 Bernard Dacorogna
Weak Continuity and Weak Lower Semicontinuity of NonLinear Functionals
SpringerVerlag 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): 46XX ISBN 3540414882 SpringerVerlag Berlin Heidelberg New York ISBN 0387114882 SpringerVerlag 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, reuse 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 SpringerVerlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140543210
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#NSFEng. CME8023824.
Science
WEAK CONTINUITY AND WEAK LOWER SEMICONTINUITY OF NONLINEAR 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
LegendreHadamard 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 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 ( ~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 wellknown 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 + (lA)w) < Af(v) + (ll)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 + (lXl(VX))W, and observe that by (i.ii) and by the definition of v
X
we have
*
i U
9 Iv + (lI)w,
in
L~(D)
fCu v) = x~f(v) + (lx~)fCw)
*
(1.13) If(v) + (lI)f(w), in
L~.
Therefore, using (1.7), we get lim ~nf
uv
"~
/ f(uV(x))dx = [If(v) + (ll)f(w)] meas D D
I f(~(x))dx = f(tv+(11)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.
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 Aquasiconvexity, 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 Aquasiconvex 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, Aquasiconvexity 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 Aquasiconvexity
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 Aquasi
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 Aquasiconvex.
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 ~ Aquasiconvexity.
As a matter of exercise we will prove in a slightly different way that convexity implies Aquasiconvexity. Proposition 2.2. Proof:
Let
convexity ~
f: R m
> R
Aquasiconvexity.
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 ii 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 Aquasiconvex,
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([2H 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
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~Bx) + ~r~Y(v 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}
WI, 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
WI'2(~)
A
0 in
WI'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)e2~i~.Xdx.
the hypotheses (4.4) we get (since
e2~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
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: ~md
>
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% 2e (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 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+(VuVVu))  f ( ~ +
(VugVu))
+ (?uUVu))  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~ + (VuUVu))  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~
! Ixy] 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 IsI + IF21SI)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)
(FG) < 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 ~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) + (ll)f(M 2)
(5.49)
51 for all
~ E [0,i], MI,M 2 EIR n(n+l)
with
rank(MiM2) _< 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(M2M 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.3134)
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
>
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 = lim f ~(X'Us(X))dx > 0 S~O
S~O
thus (6.6). (6)
For all
such that
qb  0
on
~ x K
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.
HahnBanach 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
+ a0 = + + a0
+ 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))
~
=
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
WI,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)
= uzvw
in
Lm(~).
(1.18)
Using (1.14) we may rewrite (1.18) as ueq(ue)f(ue)n(ue)
*x
uzvw
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)
=
the
a.e..
indices
x,t
in
~x,t)
  .
(1.21)
Combining (1.20) and (1.21) we obtain
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'(~) = (vf(u))[~u[,
and inserting
(1.23),
(1.24) into (1.22) we get
(vf(u)) From (1.26) we deduce that
(i)
if
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

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)

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
WI'~(~)
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
WI'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 )xxllwl,2
>
O, as
e § O.
(1.47)
By definition (see [Adl]) we have
I I n(USxxl Iwl,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/ell.
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{ ~, and
then
g ~ f
f** then
is the
FreEularl
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{   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 < =
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
fDQ'f(F)dx JDfl 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
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 __d~f~ ~ 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 Aquasiconvexity. 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 Aquasiconvex
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. 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 ABquasi
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 Aquasiconvex,
then f is ABquasiconvex.
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 Aquasleonvexlty 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 "KG [ f(~) dx. )G
Remark.
In the variational case, i.e., m = nr 16j, 3 Au = rot Vv E 0 By = Vv,
the definition of the ABquasiconvexity 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 ABquasiconvex. 2) Sufficiency : If {ug}, u satisfy Hypothesis (H) and if furthermore either (~) f is Aquasiconvex and u 9 and u are such that (H) u o
 u 6 Ker A;
or (B) f is ABquasiconvex and satisfies
If(u)
 f(v)[
a > 0, 8 ~ i, u, v 6 ~ m
~ a(1 +
lul Bz + Ivl Bl) 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. ]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(nl)/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)
WI,~
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 WI'~), 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
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
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(nl)/2) 0
and w ~ 9
LB(~) so that n
U
=
v~
curl ~
V
~
(A.43)
+ W
~ 0 in wl'B(~;~n(nl)/2)
w~
> 0 in LS(~).
(A.44) (A.45)
n
So let v ~ e W I ' ~ ( ~ ; ~ n(nl)/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,~
WI,~
whereK is a constant independent of ~. We then deduce from (A.41), (A.46) and (A.47) that ~ 0 in W I '~(~;~ n(nl)/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. Aquasiaffine)
> IR is said to be ABquasl
if f and  f are ABquaslconvex
(resp. Aquasi
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 Aquasiaffine and thus f is ABquaslafflne. 2) Sufficiency.
If ~u ~} and u satisfy (H) and either
(e) f is Aquasia~fine and u ~ and u are such that
(H), ~ = f(u) then f
112
(Ho) Au ~  Au  0; or (8) f is ABquasiaffine,
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 ABquasiaffine, Proof: Remark.
The proof is identical As seen in Chapter I w
ABquasiaffine
functions
then f is ABquasiconvex.
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 Antiplane 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,Gconvergence:
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
EulerLagrange 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
HahnBanach Theorem:
56
Homoseneization:
6
Hyperelasticity:
68,96
Incompressible material: Isotropic material:
68,71,72
71
J)
Jensen's inequality:
L)
Lagrangian coordinates:
83 97
LegendreHadamard 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 nonparametric form: Minimal hypersurface in parametric form: MooneyyRivlin materials:
N)
Nonconvex 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, Aquasiaffine function:
18,19,24,43,44,111,112
Quasiconvex~ Aquasiconvex 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
RadonNykodym Theorem: Rank one convexity: rregularization:
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