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Convex Function of a Measure the Unbounded Case
FERMAT Days 85: Mathematics for Optimization J.-B. Hiriart-Urruty (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1986
1 03
CONVEX FUNCTION OF A MEASURE THE UNBOUNDED CASE
F . DEMENGEL and R. TEMAM Laboratoire d'Analyse NumCrique UniversitC de Paris Sud - BSt. 425 91405 Orsay Cedex (France)
INTRODUCTION Some mechanical problems - in soil mechanics for instance, or for elastoplastic materials obeying to the Prandtl-Reuss Law - lead t o variational problems of the type
where .J, is a convex lower semi-continuous function such that is conjugate 1c,* has a domain B which is unbounded and convex. In order t o try t o solve this type of problems, we are led to study the possibility of extending the convex function .J, to an appropriate convex subset M+ of the space M'(R) of bounded measures on St. More precisely, we will study the case where $* is as follows
and B is a convex unbounded set of the space E of symmetric tensors of order 2
F. Demengel and R. Temam
104
on RN. If do is its asymptotic cone, A the polar cone of do, and if l l ~ ( B( l)) is bounded, we may extend $ to the convex subset of M'(R):
= { p E M1(R),p = h d s
M#)
+ ps,
ll,oh E L2(R,ds),((ps E A))
(2) }
After giving the definition of $(/A), we follow the study carried out in [3], were B was a bounded set. In fact, most of the results proved in [3] are still valid in the unbouded case, under the previous assumptions. The main result is a duality formula, and an approximation result by smooth functions for the metric:
1. ASSUMPTIONS 'AND GEOMETRIC PROPERTIES
Let X be a finite dimensional euclidean space and $ a function from X into R. We assume that $ is convex, 1.s.c. and proper (i.e., its range does not include -00,
and $ is not identically equal t o
+00.
We recall that the conjugate (or
Legendre transform) $* of $ is defined on X' = X by:
$* is also convex, 1.s.c. and proper, and $** coincides with $ (see for instance
J. Moreau [7], R.T. Rockafellar [8], and I. Ekeland
-
R. Temam [4]). Let us
B = {s E x : $*(s) < +w},
(1.1)
recall that the domain B of $* is defined as
We will suppose that
B is closed. The restriction of $* to B is continuous. OEDom$
('I
l l denotes ~ the projection on A in If.
(2) This notation is explained in the text; it means ( p s I p) 5 0, for any cp lying in
CF(n,Ao).
Convex function of a measure
105
(this last condition is equivalent t o saying t h a t $* is bounded from below on
B). Since $ is convex, [ $ ( t f ) - $ ( O ) ] / t is a nondecreasing function of t and then the limit
exists in
R u {+co}; $,is
called the asymptotic function of
and is lower
semi-continuous when $ is, since it may also be written as
Let us observe that under the hypothesis (1.4), $ ,
is nothing else than
the conjugate of the indicator function of B: $,oo(F)
= SUP F * 7 llEB
This result may be obtained by Theorems 8.5 and 13.3 of Rockafellar (cf. [8] .~
p. 66 and 116), or by the following simple argument. Let us assume that q belongs to B ; we have $(tE)
2 t E . rl - $ * ( r l ) ,
and then
q being arbitrary in B , we obtain that:
For the reverse inequality let us suppose t h a t [ belongs t o the domain of
4,.
Then $ ( t < ) / t is bounded when t goes to infinity. In particular, for all c > 0 and for all t , there exists qt E B such that
Dividing by t , we obtain
F. Demengel and R. Temam
106
for some constant c such that $* 2 c, which implies, letting t tend to infinity:
This proves the reverse inequality when
E belongs t o the domain of $m.
Now
if $m(() = $00, then for all A > 0, there exists T such that for t 2 T
Then there exists qt in B such t h a t
which implies: qt * t t 2 At
+c
and then Xk(€)
L
rlt€>
A+
C
t
*
This shows, since A is arbitrary, t h a t
X x 9 = +0O. According to the definition of
I
4,, we see t h a t $m is convex and positively
homogeneous (and 1.s.c. since $ is 1.s.c.). Its domain, denoted by A, is then a convex cone with vertex 0. We denote by A' its polar cone, or equivalently
Let us make more precise the link between A' and the set of asymptotic directions of B , which is defined as the set of t h e [ 's for which there exists q E B such that q
+ R+( C B.
(l)
~~
( l ) We recall that this condition does not depend on q
p. 6 3 ) .
EB
(see Rockafellar 181, Th. 8 . 3 ,
Convex function o f a measure
107
We now state
Proposition 1.1. Under the above assumptions Ao is the asymptotic cone of B , i.e., AO
Proof. Let
E
= { E :\d4 E B , 4 + ~ + (c B } .
be in A’, 4 in B and X in R’,
and let us assume t h a t 4
+ A(
is
not in B. Then using the geometric form of t h e Hahn-Banach Theorem, there exists (40,P) in X x R such that
and 4o.b L P for every b in B . Then 40 E A; in fact 4 * 40
I SUP 40 b€B
*
b = 1clo0(40) 5
P < +w.
We then have A v o . [ 5 0, and
P < 40 (X + 4 ) *
< 40 4, which contradicts (1.8). For the converse conclusion, let us suppose t h a t
Q
+ R+E C B (with 4 in
B ) , and let us show t h a t ( belongs t o A’. Taking f’ in A, and writing t h a t
llcl0(f’)< +a, we obtain:
and the supremum on the right-hand side cannot be finite unless equivalently
E E A’.
I
(‘ . ( 5 0, or
F. Demengel and R. Temam
108
Examples We now give some examples of convex sets appearing in mechanics, and which will be studied in this paper. For each of them, we will give the asymptotic cone and the polar cone A' (see further examples after Proposition 1.2): a) The bounded convex sets: when B is bounded, and using for instance
Proposition 1.1, we see that A' = {0}, A = X. The study which follows has already be made in that case (cf. Demengel-Temam [3]). b) Let E = X denote the space of symmetric tensors of order two on R", E D be the space of symmetric tensors with null trace, and let B = B D x RId, with B D bounded in E D . Convex sets of this type are studied in perfect plasticity (cf. Temam [S]). We have in such a case:
A' = 0 x RId,
A = E D x 0.
c) When B is a cone with vertex 0, then B = A' and
+w
= 0 on A. This case
will often occur in the following, and will be used for the Prandtl Reuss law in plasticity. We now deduce from the properties of B some results for
$w.
Proposition 1.2. The following statements (i), (ii) and (iii) are
equivalent: (i) There exists r < 00 such that
B c B(0,r) + A'
(1.9)
(ii) There exists r > 0 such that, for every ( in A, (1.10)
(This implies that A is closed.)
(iii)
+ is at most linear at infinity on A:
Convex function of a measure
109
Proof. Let us first show that (i) is equivalent t o (ii). Suppose that (i) holds, then for ( in A
Using the lower semi-continuity of
4w, it is then easy t o see that A is closed.
On the other hand let us suppose that
$m
is at most linear at infinity. For all
q in B we write q = I I A ~ + I I A o ~where , we denote by IIAand IIAo respectively
the projection operators in X on A and A'. Then we have, using Proposition 2, p. 24, of Aubin-Cellina [l]:
which implies that
5 r,
InAql
and consequently that
B
C
+
B ( 0 , r ) A'.
Let us now see that (iii) is equivalent to (i) or (ii). Using the definition of
4,and the fact that $I*
is bounded from below on B (by (1.4)) we have:
5 sup
tl EX
= $(,) which implies that
€
0
7
7
-c
- c for € in X ,
4 is at most linear at infinity on A when (i) or (ii) is fulfilled.
For the reverse implication, when 1c, is at most linear on A, we get for every in A:
E
110
or
F. Dernengel and R. Ternarn
$w
is at most linear at infinity on A. I
Let us give two other examples of convex sets; the first one appears in mechanics and satisfies (i), the second does not. d) Let B be the convex set of R2 (cf. Fremond-Friaii [5]), defined as follows: (z,y)ER2 : - l < y < + l , z > - -
1 - y2
2}.
(1.11)
B contains 0 in its interior. Moreover, it is easy t o calculate A' and A; we obtain
A'={Y=O:X>O}
and n ^ B C (-2 5 X 5 0, -1 5 Y 5 +1} is bounded. e) Let us now consider a convex set for which (i) is not fulfilled:
B = { (X, Y ) : Y 2 X 2
+ m}
A' = { X = 0,Y A = { Y < 0) U (0,O) (A is not closed)
(with m E R+)
> 0)
n = A''
(1.12) (1.13)
= {Y 5 0}
IIxB = (anB ) U {Y = 0) which is not bounded. I
(1.14) (1.15)
The following Proposition 1.3 gives us further informations on the link between the behaviour of qh and its restrictions to A and A' respectively. We assume t h a t A is closed. Then: Proposition 1.3.
(i) The following inequality is valid for all ( in X:
Convex function of a measure
111
(ii) Let us assume that 0 belongs to B and that (1.17) where g is an even convex continuous function defined on
R,g* its conjugate. Then there exist
c1,
c3
2 0,
> 0, such that for every
cg
in X :
'$(El 2 and
c1
{cl161
+ c24(nA0E) - c3},
(1.18)
is strictly positive if 0 is in the interior of B .
(i) Proposition 1.1 gives that A' c B whenever 0 belongs to B and then :
W) = sup{E sEB
rl - $ * ( r l ) }
*
2 SUP { E rl - s*(lrll)} *
11EAO
= sup{A sup x>o
( *
7 - g*(A)}
qEAo 111151
= suP{A X>O
lnAoEI
- g*(A)}
= g((nAO[() since g = g**. Now, using (1.17), we note that
and we have obtained (1.18) in this case:
F. Demengel and R . Temarn
112
Now if 0 lies in the interior of B, assume t h a t B(0,r) C B for some r > 0. We then have:
Icl(<) 2
SUP
{E
*
vEB ( 0 ~ )
7 - 1cI*(rl))
= SUP { E V A - g * ( W O
2 rlFl - c. Adding this inequality t o $(() 2 $ ( l l , ~ E ) , which holds if 0 E B, we obtain
which is exactly (1.18). I
To end this Section, let us give a n example of conjugate function $* which fulfills the conditions of (ii) and will often be used in the following: we denote by +* the convex functional
$*(El
=
;A(. { +oo
( when
E belongs to B,
if not,
where A is a linear positive operator on X;
(1.19)
+ is easy to calculate and we obtain
When B is a cone of vertex 0, B = ho and
Moreover, when 0 is in the interior of B, we are in the conditions of (ii) in Proposition 1.3 (with +*(()
=
g*(IfIA)
where we denote by
I E ~ Athe norm
(A[. <)'/'). Using the orthogonal decomposition
we get the inequality:
(1.21)
Convex function of a measure
113
2. THE MEASURE $ ( p ) 3.1. Measures with values in a convex cone Let R be a n open bounded set of RN, and X be a real euclidean finite dimensional space. We denote by M'(R,X) t h e space of bounded measures on R, with values in X . If p belongs t o M'(R,X), we write in this section
the Lebesgue decomposition of p , h being in L'(R,X),(') ps being singular with respect t o the Lebesgue measure dx. We suppose on the other hand t h a t
A is a closed convex cone in X , with vertex 0, and A' is its polar cone. We will say t h a t p takes its values in A (and we will sometimes write p E A) if
where Co(R, A') denotes t h e space of compactly supported continuous functions on R, with values in A'. Let us remark first that when N = 1 and A = R+ (respectively R - ) , the notion of measure with values in A as above coincides with t h e notion of positive (respectively negative) measure. We now give some properties or equivalent definitions.
Proposition 2.1. (i) The assertion (2.2) is equivalent to the following
I
( p p) 5
(ii) If p = ul
0 , V p E L 1 ( R , d p , A o )or C r ( R , A 0 ) . ( 2 )
+ u2 with ul
orthogonal to
values in A, then ul and
u2
u2,
(2.3)
and if p takes its
take their values in A too.
( l ) This is the space of functions from R into X that are L' for the Lebesgue measure dx.
(2) For K C X , we denote by
Cr(f2K , ) (resp. Co(R, K ) ) the space of Cm (resp. continu-
ous) functions from R into K with compact support in f2. Similarly, if u E M'(R, X ) and
15 p 5 00, LP(R,u,K ) is the space of Lp functions from R into X that take their values in K ,u almost everywhere.
F. Demengel and R. Temam
114
(iii) If p = hu with u real valued a n d 2 0, h a u-integrable function with values in X, it is equivalent to say that p takes its values in A or that h does,
Y
almost everywhere.
Proof. (i) The equivalence of (2.2) and (2.3) follows easily from Lemma 2.1 below.
It is enough to show that if ( p I p) 5 0 for all p E C r ( R , A O ) , then we also have ( p 19) 5 0 for all p E L1(R,p,Ao).But if p E L1(R,p,Ao), we consider the sequence p j E C r ( R , A o ) provided by Lemma 2.1 below (used with u,B , v , replaced by p, A', p: p j converges t o p in L1(R, p , A') aa j
-+
00,
and the inequalities ( p I pj) 5 0 for all j , give ( p I p) .< 0 at
the limit. (ii) Let A l and A 2 be two disjoint measurable sets such that ui is supported by A i , for i = 1, 2, and let p be in Co(R,A'). X
A
Then
X
A E L'(R,p) ~ ~ and
takes ~ its ~ values in A'; therefore
which implies t h a t
or equivalently
(vi I p) 5 0 since vr is supported by Ai. I We now state Lemma 2.1 used above and whose proof is given at the end of Section 2.
Lemma 2.1. The sole assumptions are that B is a closed convex set of an euclidean space X a n d that v belongs to L ' ( R , u , B ) , where u E M'(R,X) a n d R is an open bounded set ofRN. Then there exists a sequence v j E Cr(R,B) such that, as j vj
-, v in L'(R, v ) and u-almost
everywhere,
-+
00,
Convex function of a measure
115
The proposition 2.2 below gives a definition of the "projections" of p E
M1(0,X ) onto A and A" respectively. Proposition 2.2. Each measure p E M1(0,X ) may be written as the s u m of two measures taking respectively their values in A and A". Moreover, there exists a minimal decomposition - in the sense of modulus - that we write
Ifp = hlpl, then n ~ =pI I ~ h l p lnAop , = nAohlpl and when p is a function, l l ~ and p n ~ o pcoincide with the projections onto A and A" in the ordinary sense (defined almost everywhere).
Proof. Let h be p/lpI. T h e projections of h onto A and A" respectively are defined p everywhere, and they are p measurable (because the projection onto a convex set is continuous). We then write
and then p = hlpl = nAhlpI
+ nAOhlp1.
But (n.kh)lpl (respectively (n,oh)[p1) is a measure with values in A (respectively A"), and then the announced decomposition exists. Let us now suppose that p
with p1 E A, p2 E A'.
= p1+ p2
We can write
with vi orthogonal to IpI. Let us remark that necessarily
v1
= -v2. According
to Proposition 2.1, assertions (ii) and (iii), we get hllpl E A, hz(p1 E A', and then hl E A, h2 E A' and
F. Demengel and R. Temam
116
which implies that h = hl
+ hz
11-11-almost everywhere and
which is the minimal property mentioned of rI~1-1and
n~01-1. 3
2.2. Definition of $ ( p )
We suppose in this subsection t h a t B is a closed convex set which contains 0, that $* is a convex function on X with domain B , satisfying (1.2) and
$* 1 0, +*(O) = 0. We recall t h a t under these hypotheses, the conjugate
(2.5)
+ of $* verifies also:
Moreover if $* is the asymptotic function of 11, (or equivalently the conjugate of
x ~ )with , domain A, we will suppose that 3r < +co such t h a t B c B ( 0 , r )+ A’ (which implies that A is closed and nhbounded.)
(2.7)
We are now given a bounded measure 1-1 on an open set R of R N , and we write
it,s Lesbegue decomposition, where we assume that 1-1’ (we note 1-1’
takes its values in A
= hSIpSI). Using N. Bourbaki [2] and Proposition 1.2 (ii) of
Section 1, we see that $ = ( h S ) is 11-1’1
integrable. Moreover if we suppose t h a t
11, o h is Lesbegue integrable we will define $ ( p ) as follows
117
Convex function of a measure
Remark 2.1. (i) Assume that p = hlvl = h2u2 with
v1
and
v2
two nonnegative measures
such t h a t p is absolutely continuous with respect t o each of them. T h e proof of Bourbaki [2] shows, by using t h e homogeneity of
$m
that
$ ~ ~ ( his ,v,) integrable for i = 1, 2,
(2.9)
We will then write with no possible confusion
(ii) In [2] we defined t h e measure + ( p ) with formula (2.8) by supposing t h a t $ (which was not necessarily convex) is at most linear at infinity and t h a t
existed. It has been shown in [2] t h a t $ ( p ) is always defined in t h a t case (i.e., t h a t $ o h is ds integrable and $ ( p ) is a bounded measure). Moreover when
4 is convex and the domain B
of $* is bounded, q!~enjoys
remarkable properties. We seek here to extend these properties t o the case where B is unbounded. We will see t h a t even under some restrictive assumptions on t h e geometry of B , infinity and
+m
4 is not in general at most linear at
is not everywhere finite so t h a t the right-hand side of (2.8)
does not make sense automatically.
(2.11)
Proposition 2.3. Under the assumptions (2.5) t o (2.7) the set
M+ (0)is convex,
118
F. Demengel and R. Temarn
and the equality holds true above when 0 is in the interior of B and +* satisfies (1.1 7). Proof. Because of (1.8) and (1.16), we have almost everywhere
and since In~hland +(nAoh) are in L'(n,ds), t h e inclusion (2.12) follows. When 0 is in the interior of B, we apply the assertion (ii) of Proposition 1.3,
cl,
c2
> 0, and hence + ( l l ~ o hE) L'(R,ds).
R e m a r k 2.2. Following the steps in t h e proof of the Krasnoselskii Theorem [6], we obtain
4,
that
is continuous from L1(R,A) into L'(R) if and only if it maps L ' ( Q A)
into L'(R); see the appendix. I We now give the main results of this section. To this end, let us define the sets:
D$(C,")
= {w E
DQ(C0) = {TI
D&)
E
C,"(Sz,X) : v(x) E B , vx E n}
(2.13)
Co(R,X) : w(2) E B , vx E n}
(2.14)
+ w(x) E B , ([PI + dx) a.e., +* o w E ~ l ( ~ , d s ) } ,
= {w E L'(R, IpJ ds,X):
(we easily have
(2.15)
D$(Cr)c D$(Lk)),and we define for p in C(fi),p 2 0: (2.16)
where A will be
('1
D$(Co), D+(Cr) or D@(LE).We have the following result.
This supremum being allowed to be equal t o
+GO
119
Convex function of a measure
Theorem 2.1. The value of $(p,p) defined in (2.16) does not change when A is equal to
D+(Co),D+(CF)or D@(Li).In
addi-
tion, (i) The restriction of $ ( p , - ) to
C$(n)may
be extended as a
measure when $(p,p) is finite for every cp 2 0 in CO(n), and we write for cp in Co(n):
This measure is nonnegative and it is bounded if $ ( p , 1) < +oo.
(ii)
$(p,
.) is a bounded measure on 0 if and only if $ ( p ) also
is, and in this case, we have
Remark N
More precisely + ( p ) is a measure if and only if + ( p ,
a)
is
.
Proof. We begin t o show that the supremum on the right side of (2.16) does
not, change if we take for A ,
D$(Co),
D+(Cr)or D+(LL).It is obvious that
Now, let us suppose first that the supremum on the right-hand side is finite and let S > 0 be given; there exists
210
in D@(Lh)such that
We now use the following lemma whose proof is given at the end of Section 2.
F. Demengel and R. Temam
120
Lemma 2.2. The sole assumptions are that B is a closed convex set of the euclidean space X, 0 E B and that $* is a convex
continuous function from B into R; we assume also t h a t
00
E
L1(R,IpI+ d z , B ) , $* 0210 E L'(R,ds) where P E M ' ( R ; X ) . Then there exists a sequence vj E
CF(n,B) such
that, as
j -,00, vJ converges to v o in L1(R,Ipl+ ds, B ) and IpI + ds a.e., and $* o v j
$* o vo in L'(R, d z ) .
--f
We successively show assertions (i) and (ii). (i) For t h a t purpose, let us remark that
y3
$(p,p) defined by (2.16) is
H
additive on the cone
Indeed let
p1
and 0
p 2
D?
be two functions in
I N P , P1 + 9 2 ) 5 &P,
. We obviously have @(@,,I
N
(01)
+ & ( P >P Z ) ,
which shows that D ( $ ( p , . ) ) is a convex cone with vertex 0. Now, given
6 > 0, let us take vi in D@(Lb)for i = 1, 2 such that $(P,Pi)
5
S,
PviPi -
S,
and let v=-
where cp = (01
+
p 2
# 0, and
P*(Vi)Pi
c
+ 6,
i = 192,
ViPi
P
= 0 elsewhere.
It is clear that v belongs to L ' ( Q , lpl+ d s , X) and v(z) E B for almost everywhere on R ; Moreover 2
2 . "
and using the convexity of +* we get for p ( z ) # 0:
\PI+
dz
Convex function of a measure
or equivalently
121
2
(which remains true when ~ ( z =) 0). We finally obtain
which implies t h a t d ( p , * )is additive, since 6 When
'p
may be written in the form
let us define
d b ,P) = &P,
with P = (01 - 9 2 , P1 1 0,
P2
9 1
> 0 is arbitrary.
- 'p2, with
cpi
in D + ( $ ( p ? . ) ) >
(2.19)
Pl)- d ( p ,(02)
1 0.
The additivity property allows us t o show that (2.19) does not depend on the choice of
p1
and
'p2
such that p =
'p1
- p 2 . T h e positivity of
& ( p , - ) easily implies t h a t J ( p , . ) is a bounded measure when $ ( p , 1) is
finite. Let us suppose secondly that the last term on the right of (2.18) is
+w; then for every A > 0 there exists vo in D,,j(L1p) such t h a t
Using once more Lemma 2.2 there exists
vj
-,
VO, vj
E
CF(R,B) such
t h a t for j sufficiently large, we have:
which implies t h a t the first term on the left-hand side of (2.18) is +oo too. (ii) Let us now suppose t h a t p belongs t o M+(R). Then $ ( p ) is a bounded measure on R,
11 o h is in L'(R, d z ) and we have for all v
that v(z) E B almost everywhere and for all
(o
in L'(R, dp) such
2 0 in &(R):
F. Demengel and R. Temam
122
and ps almost everywhere, ps = hS Ips 1,
This inequality shows t h a t $ ( p , ’p) is finite for all
’p
2 0 in Co(n), and
N
implies t h a t $ ( p , .) is a bounded measure when $ ( p ) also is, i.e., when p belongs to M+(R). We want now t o show t h a t q ( p , .) and $ ( p ) coincide in that case. We will show in a first step t h a t
and then t h a t
+ o h is d s integrable when $ ( h d z ,1) is finite, $ w ( h s ) is
ps integrable when & w ( h S I p S / , l ) is finite and that we have in addition
the inequalities
J,$oh 5 4(hdz,l)
(2.21)
J, $M(hS)dIPSl 5 J w ( hs IPs I, 1).
(2.22)
Let us show t o begin with equality (2.20). We proceed like in (31. Let 6
> 0 be given and
(v1,vZ)E
Co(R,B) such t h a t
123
Convex function of a measure
L $m(PS,P) -
hSV2PIPSI
Let now 0 be a set of Lebesgue measure 6, Ips I that ps is supported by 0. Since 11.’ such that
s,
IP(V1
+ h dx measurable, such
is singular, we may choose 6
>0
- vz)h - ( + * ( V l ) - +*(w2))1 dx c
Let us then consider the function on n\0, on 0 ;
v = { v1 212
v is Ipl
+ dx integrable, with values in B ; moreover:
The left-hand side may be written in the form
We finally have obtained:
5 4 ( P , P)+ 3€, Y
which ends the proof of equality (2.20). Let us now suppose that $ ( p , - ) is a bounded measure; then there exists a constant c
> 0 such that
124
F. Demengel and R. Temarn
We then show that t,b o h E L 1 ( R , d x ) . For
E
> 0 and given 6 > 0, we
consider a disjointed covering of R by universally measurable sets A:, mes(Af) < 6, and we define t h e vectors X f :
Since II, = $I**, there exists
vE
ef
in B , such that
L'(R,dz), v(z) E B for almost every z in R and using $ ( O ) = 0:
11,(h6)ds 5
1 R
h6v - $*(v)
+ E mes R
5 & ( h d z , l )+ c mes R. Since h6
4
h in L'(R) when 6
--f
0, there exists a subsequence 6'
+
0
such that h 6 ' ( z )-, h ( s )everywhere on R. Using t h e lower semi-continuity of 11, and Fatou's Lemma, we may write $ o h ( z )dz 5
lirninf 4 0 h 6 ( z )dz, 6+0
5 lim inf 6-0
$ 0 h6(z)ds,
5 & ( h d z ,1)+ E mes R , which implies t h a t 11, 0 h is Lebesgue integrable. We could use the same argument of approximation by piecewise linear functions, replacing t,b by
L+m
t,bw, d x by lpsl and h by h S , t o obtain the analagous inequality o h s IPs I
5 i ~ (s IP h s 1,1)+ E mes 0.
Convex function of a measure
125
And this ends the proof of Theorem 2.1. I We conclude this section by the proof of Lemmas 2.1 and 2.2:
Proof of Lemma 2.1. From the definition of L ' ( R , v , B ) , there exists a sequence wj E Co(R,X) which converges to v in
L1(R,v , X ) . Then, since the projector IIB in X onto
B is Lipschitzian, the sequence I I B w ~ belongs t o
CO(L?, B ) and
converges to v
in L ' ( R , v , B ) . Now assume that v E Co(R,B). By a mollification procedure we can contruct a sequence v j E Cr(R,B)which converges to v uniformly as j Such a sequence is obtained by setting only satisfy the conditions
pj
1 0 and
vj
--* 00.
= p j * v where the mollifiers p i must
JRN
pj(s)
dz = 1.
By combining these two remarks, we obtain for every v E L1(R,p,X),a sequence vi E Cr(R,B) which converges t o v p-a.e., and the result follows. Proof of Lemma 2.2.
(i) In a first step we approximate v by functions in L"(R, v,B ) , v = lpl+ dx (note that B is not necessarily bounded). For that purpose we set for every integer p,
We easily have Ivp(s)l5 p, Ivp(x)l 5 Iv(z)I and up(.) E B v-a.e. From the Lebesgue dominated convergence theorem, u p converges to v in L'(Q, v,B ) as p
--f
00.
Then, due to the convexity of $*, and since + * ( O ) = 0, $* 2 0,
Since $* is continuous on B , $*(v,(x)) p
--f
00,
+
$*(v(z)),ds-a.e.
in R , as.
and it follows from the Lebesgue dominated convergence theorem
that Jn $* o u p -+
sn $*
0
v.
F. Demengel and R. Temam
126
(ii) We are now reduced to the case where furthermore v takes its values in
B, for some p. Lemma 2.1, applied with
Y,B
replaced by 1p1
+ dx, B,,
provides a sequence v j E Cr(n,B,) which converges t o v in L1(R,v,B,) and u-a.e. continuity of
Then v j converges t o v dx-almost everywhere and by the
4*,
We also observe that
and then the Lebesgue dominated convergence theorem applies and shows that
ln 4* o v j converges to ln +* o v
The proof is complete. I 3. APPROXIMATION RESULTS
We recall the hypothesis on $ of Section 2:
$J
is supposed to be convex,
B = dom $* is closed,
11 1 0 , WJ)= 0,
(34
A is closed and IIAB is bounded.
(3.2)
We may endow M + ( n ) with the distance defined by:
and Lemma 3.1 below proves that from every bounded sequence in M+(R) we can extract a subsequence which is vaguely convergent towards a measure in
M+ (0). L e m m a 3.1. Let pj be a sequence in MJ,(R) weakly convergent towards a bounded measure p . Then if $ ( p j ) is bounded in
M'(R), p belongs to MJ,(R)and we have
Convex function of a measure
Proof. Let us suppose by contradiction that p
127
4 M+(R). By
($(/A,1)) = +w,and for every real A > 0 there exists w in that
Theorem 2.1,
Cr(n,B ) such
n
Since pj is vaguely convergent to p , there exists J > 0 such that for j > J :
Therefore for all j > J :
and
sn $ ( p j ) cannot be bounded; this contradicts the hypothesis.
We thus
have shown that p E M+(R),and we may conclude for every 6 > 0 and p 2 0 in Co(n) the existence of w in C F ( 0 , B )such that
P
5 liminf
-
($(pj)
J-+W
s,
ll*(v)p
+
I $4+ 6,
which ends the proof of Lemma 3.1. I Let us now define a topology on M$(Rj which is less finer than that defined by the metric (3.3). This second topology may be defined by the family of semi-dist ances: (3.5)
( l ) This means
s,
+ ( p ) p 5 liminfj,=
s,
+(/A,) p, V p E C o ( n ) , p 2 0.
F. Demengel and R. Temam
128
for all v in
C(fi),
v 2 0. It is immediate that this topology is stronger than
the weak topology which can be defined by the family of semi-norms:
where v is in
C,(n)
We are now going t o show the following approximation result:
Lemma 3.2. For every p in M @ ( n )there , exists a sequence o f functions z c j in
C?(n),such that
Proof. let O j be an increasing sequence of functions in
being pointwise convergent t o l n ; let
~j
Cr(n),0 I 9 j I 1, O j
be a sequence of positive real numbers
converging t o zero, p be a function in D(RN),p being even,
and uj = p E j * (Ojp). We are going t o show t h a t
By the convexity of +* we have, for all v in +*
and then, using
JRN p
This implies that
= 1,
(PCj
* v) I
PEj
C?(n),v ( z ) E B :
* +*(v)
Convex function of a measure
for pcj
* V
= pcj * V E
129
P(G,B) c a&v(ejP)).(l)
Using the Lemma 3.3 just below we have
On the other hand, since .II, ( p c j * ( e j p ) ) is bounded in M 1 , we may extract from it a subsequence which is vaguely convergent towards Y E M'(R). According t o Lemma 3.1, we have
and therefore
Lemma 3.2. Let
p1
and
p2
be in
C(G), with 0 I
p1
I
p2.
Then for every p in M'(R, X )
The proof of this lemma follows that of the lemma 2.4 established in [3], and consists in the use of the duality formula (2.16).
Remark 3.1. The proof of Lemma 3.2 contains the inequality in the sense of measures:
F. Demengel and R. Temam
130
for all 0 in C,(R), Indeed, let on
uj
E
be in
< d(suptO, an), and
p a regularizing function as before.
Cr(n);u j tends to p vaguely on R,$J(uj)-, $ ( p ) vaguely
R. We have by the convexity of $ and using $(O) = 0,
$J
2 0:
Now the convergence of u j towards p implies that of pr * 8uj towards pt
* 9p;
using Lemma 3.1, we then have
5 liminf $ ( p t * O G ( u j ) ) . 3-M
It is clear that the convergence of $J(uj) towards $ ( p ) implies that of pc*8$(u3) towards pt
* 9 $ ( p ) . We finally have obtained
APPENDIX To begin with, let us recall the definition of the Caratheodory function:
Definition A . l . A function f ( s , u ) of two variables +m, s E
-00
R (where R is a bounded open set of RN) is said to
satisfy the Caratheodory conditions (cf.161) if it is continuous with respect to u for a.e. x E R and measurable with respect to s for all values of u. We denote by F the operator defined on the set of real functions on R by
where f satisfies the Caratheodory conditions. Let u s suppose that h is a convex cone of vertex 0 in a Banach space E, and suppose that p, r are two real numbers such that 1 < p, r < +oo. We then have the following result:
131
Convex function of a measure
Theorem A . l . If F maps L*(R,A) in L’(R,G) (where G is a Banach space), then F is continuous from LP(R,A) into Lr(R, G) for the norm topology. Proof. We proceed as in the theorem 2.1 of Ref. 6, and begin t o recall the definition of the convergence in measure:
Definition A.2. We will say that un --t u in measure if Vc
> 0, 3 N , V n 2 N
mes{z : I(un - .)(.)I
> E } < E.
In fact, it is easy to see that Definition A.2 is equivalent to the following:
Definition A.3.
u, + u
in measure if
Y E > 0, V6 > 0 3 N , V n 2 N mes{z : I(u, - .)(.)I
+
> 6) < c.
Let us recall the lemma 2.1 of IS]:
Lemma A.l. let-R be a set of finite measure. Then the operator F transforms every sequence of functions that converges in measure into a sequence of functions that also converges in measure . We can now prove the theorem; we may suppose that F ( 0 ) = 0 and show that F is continuuous at zero. Indeed, let us introduced for a fixed u in
LP(R,A) the operator H defined on the convex cone A - u:(l) H(v)= F(v
+ u)- F ( u )
Suppose by contradiction that there exists a sequence of functions Pn E
LP(R,A - u) such that lPnlLP(n,E)
(l) h - u = { w - u : : € E } .
+ 0,
F. Demengel and R. Temam
132
and such that We may assume, by extracting a subsequence, t h a t
n
and suppose for a while t h a t we have proved the existence of sequences Ek p n k ,and sets
Gk
c R, such that the following conditions are fulfilled:
Let us then consider t h e sets
u m
Dk= G k \
Gi.
i=k+l
We have: r=k+1
a=k+l
Let us now define the function II, by
We then have
> -2a- - =a 3
4 lies in LP and
3
a
3 '
so H $ E L'. On the other hand we have
+ 0,
133
Convex function of a measure
which is a contradiction. It remains t o prove conditions (a), (b), (c) and (d) by induction. Suppose t h a t e l = mes R, pnl = p 1 7G1 = G , and suppose t h a t Ek, G k , ( o a k
have been constructed. Since H (pnk)E L' we may choose
such t h a t for all
then
,,s
~ k + l
IHp,,I'
<
D,
Ek/2
<
Ek+l
(if not
5 , which
2Ek+1
2
€k
and would have mes
G k
5
2Ek+l
and
contradict condition (c). Now pnk converges in
measure t o zero, so by Lemma A . l , IHpnkI converges too in measure towards 0. Then there exists
nk+l
such t h a t
Let us define the set
Then mes R\mes
Fk+l
< E k + l . Let
Gk+1
a
>a--=3
= R\Fk+l.
We then obtain
2a 3
This ends the proof of Theorem A . l .
REFERENCES [l] AUBIN-CELLINA, Differential inclusions, Springer Verlag, 264 (1984). [2] N. BOURBAKI, Elements de Mathbmatiques, livre VI, Inthgration,
Chapitre 5, Paris (1965). [3] F. DEMENGEL and R. TEMAM, Convex function o j a measure and ap-
plications, Indiana J. Math., 33 (5) (1984), 673-709. [4] 1. EKELAND and R. TEMAN, Analyse conveze et probltmes variation-
nels, Dunod, Paris, (1974).
F. Demengel and R. Temam
134
[5] M. FREMONT and A. FRIAA, Analyse limite.
Comparaison des
mkthodes statique et cinkmatique, C. R. Acad. Sci. Paris, 286, sCrie A
(1978), 107-110. 161 M. A. KRASNOSELSKII, Topological methods in the theory of non linear integral equations, Pergamon Press, London (1964). [7] J.-J. MOREAU, Fonctionnelles convexes, Skminaires Equations aux de'rivkes partielles, Collkge de France (1966). [8] R.T. ROCKAFELLAR, Convex Analysis, Princeton University ,Press
(1970). [9] R. TEMAM, Probltmes Mathe'matiques e n Plasticitk, Dunod (1983).
1 03
CONVEX FUNCTION OF A MEASURE THE UNBOUNDED CASE
F . DEMENGEL and R. TEMAM Laboratoire d'Analyse NumCrique UniversitC de Paris Sud - BSt. 425 91405 Orsay Cedex (France)
INTRODUCTION Some mechanical problems - in soil mechanics for instance, or for elastoplastic materials obeying to the Prandtl-Reuss Law - lead t o variational problems of the type
where .J, is a convex lower semi-continuous function such that is conjugate 1c,* has a domain B which is unbounded and convex. In order t o try t o solve this type of problems, we are led to study the possibility of extending the convex function .J, to an appropriate convex subset M+ of the space M'(R) of bounded measures on St. More precisely, we will study the case where $* is as follows
and B is a convex unbounded set of the space E of symmetric tensors of order 2
F. Demengel and R. Temam
104
on RN. If do is its asymptotic cone, A the polar cone of do, and if l l ~ ( B( l)) is bounded, we may extend $ to the convex subset of M'(R):
= { p E M1(R),p = h d s
M#)
+ ps,
ll,oh E L2(R,ds),((ps E A))
(2) }
After giving the definition of $(/A), we follow the study carried out in [3], were B was a bounded set. In fact, most of the results proved in [3] are still valid in the unbouded case, under the previous assumptions. The main result is a duality formula, and an approximation result by smooth functions for the metric:
1. ASSUMPTIONS 'AND GEOMETRIC PROPERTIES
Let X be a finite dimensional euclidean space and $ a function from X into R. We assume that $ is convex, 1.s.c. and proper (i.e., its range does not include -00,
and $ is not identically equal t o
+00.
We recall that the conjugate (or
Legendre transform) $* of $ is defined on X' = X by:
$* is also convex, 1.s.c. and proper, and $** coincides with $ (see for instance
J. Moreau [7], R.T. Rockafellar [8], and I. Ekeland
-
R. Temam [4]). Let us
B = {s E x : $*(s) < +w},
(1.1)
recall that the domain B of $* is defined as
We will suppose that
B is closed. The restriction of $* to B is continuous. OEDom$
('I
l l denotes ~ the projection on A in If.
(2) This notation is explained in the text; it means ( p s I p) 5 0, for any cp lying in
CF(n,Ao).
Convex function of a measure
105
(this last condition is equivalent t o saying t h a t $* is bounded from below on
B). Since $ is convex, [ $ ( t f ) - $ ( O ) ] / t is a nondecreasing function of t and then the limit
exists in
R u {+co}; $,is
called the asymptotic function of
and is lower
semi-continuous when $ is, since it may also be written as
Let us observe that under the hypothesis (1.4), $ ,
is nothing else than
the conjugate of the indicator function of B: $,oo(F)
= SUP F * 7 llEB
This result may be obtained by Theorems 8.5 and 13.3 of Rockafellar (cf. [8] .~
p. 66 and 116), or by the following simple argument. Let us assume that q belongs to B ; we have $(tE)
2 t E . rl - $ * ( r l ) ,
and then
q being arbitrary in B , we obtain that:
For the reverse inequality let us suppose t h a t [ belongs t o the domain of
4,.
Then $ ( t < ) / t is bounded when t goes to infinity. In particular, for all c > 0 and for all t , there exists qt E B such that
Dividing by t , we obtain
F. Demengel and R. Temam
106
for some constant c such that $* 2 c, which implies, letting t tend to infinity:
This proves the reverse inequality when
E belongs t o the domain of $m.
Now
if $m(() = $00, then for all A > 0, there exists T such that for t 2 T
Then there exists qt in B such t h a t
which implies: qt * t t 2 At
+c
and then Xk(€)
L
rlt€>
A+
C
t
*
This shows, since A is arbitrary, t h a t
X x 9 = +0O. According to the definition of
I
4,, we see t h a t $m is convex and positively
homogeneous (and 1.s.c. since $ is 1.s.c.). Its domain, denoted by A, is then a convex cone with vertex 0. We denote by A' its polar cone, or equivalently
Let us make more precise the link between A' and the set of asymptotic directions of B , which is defined as the set of t h e [ 's for which there exists q E B such that q
+ R+( C B.
(l)
~~
( l ) We recall that this condition does not depend on q
p. 6 3 ) .
EB
(see Rockafellar 181, Th. 8 . 3 ,
Convex function o f a measure
107
We now state
Proposition 1.1. Under the above assumptions Ao is the asymptotic cone of B , i.e., AO
Proof. Let
E
= { E :\d4 E B , 4 + ~ + (c B } .
be in A’, 4 in B and X in R’,
and let us assume t h a t 4
+ A(
is
not in B. Then using the geometric form of t h e Hahn-Banach Theorem, there exists (40,P) in X x R such that
and 4o.b L P for every b in B . Then 40 E A; in fact 4 * 40
I SUP 40 b€B
*
b = 1clo0(40) 5
P < +w.
We then have A v o . [ 5 0, and
P < 40 (X + 4 ) *
< 40 4, which contradicts (1.8). For the converse conclusion, let us suppose t h a t
Q
+ R+E C B (with 4 in
B ) , and let us show t h a t ( belongs t o A’. Taking f’ in A, and writing t h a t
llcl0(f’)< +a, we obtain:
and the supremum on the right-hand side cannot be finite unless equivalently
E E A’.
I
(‘ . ( 5 0, or
F. Demengel and R. Temam
108
Examples We now give some examples of convex sets appearing in mechanics, and which will be studied in this paper. For each of them, we will give the asymptotic cone and the polar cone A' (see further examples after Proposition 1.2): a) The bounded convex sets: when B is bounded, and using for instance
Proposition 1.1, we see that A' = {0}, A = X. The study which follows has already be made in that case (cf. Demengel-Temam [3]). b) Let E = X denote the space of symmetric tensors of order two on R", E D be the space of symmetric tensors with null trace, and let B = B D x RId, with B D bounded in E D . Convex sets of this type are studied in perfect plasticity (cf. Temam [S]). We have in such a case:
A' = 0 x RId,
A = E D x 0.
c) When B is a cone with vertex 0, then B = A' and
+w
= 0 on A. This case
will often occur in the following, and will be used for the Prandtl Reuss law in plasticity. We now deduce from the properties of B some results for
$w.
Proposition 1.2. The following statements (i), (ii) and (iii) are
equivalent: (i) There exists r < 00 such that
B c B(0,r) + A'
(1.9)
(ii) There exists r > 0 such that, for every ( in A, (1.10)
(This implies that A is closed.)
(iii)
+ is at most linear at infinity on A:
Convex function of a measure
109
Proof. Let us first show that (i) is equivalent t o (ii). Suppose that (i) holds, then for ( in A
Using the lower semi-continuity of
4w, it is then easy t o see that A is closed.
On the other hand let us suppose that
$m
is at most linear at infinity. For all
q in B we write q = I I A ~ + I I A o ~where , we denote by IIAand IIAo respectively
the projection operators in X on A and A'. Then we have, using Proposition 2, p. 24, of Aubin-Cellina [l]:
which implies that
5 r,
InAql
and consequently that
B
C
+
B ( 0 , r ) A'.
Let us now see that (iii) is equivalent to (i) or (ii). Using the definition of
4,and the fact that $I*
is bounded from below on B (by (1.4)) we have:
5 sup
tl EX
= $(,) which implies that
€
0
7
7
-c
- c for € in X ,
4 is at most linear at infinity on A when (i) or (ii) is fulfilled.
For the reverse implication, when 1c, is at most linear on A, we get for every in A:
E
110
or
F. Dernengel and R. Ternarn
$w
is at most linear at infinity on A. I
Let us give two other examples of convex sets; the first one appears in mechanics and satisfies (i), the second does not. d) Let B be the convex set of R2 (cf. Fremond-Friaii [5]), defined as follows: (z,y)ER2 : - l < y < + l , z > - -
1 - y2
2}.
(1.11)
B contains 0 in its interior. Moreover, it is easy t o calculate A' and A; we obtain
A'={Y=O:X>O}
and n ^ B C (-2 5 X 5 0, -1 5 Y 5 +1} is bounded. e) Let us now consider a convex set for which (i) is not fulfilled:
B = { (X, Y ) : Y 2 X 2
+ m}
A' = { X = 0,Y A = { Y < 0) U (0,O) (A is not closed)
(with m E R+)
> 0)
n = A''
(1.12) (1.13)
= {Y 5 0}
IIxB = (anB ) U {Y = 0) which is not bounded. I
(1.14) (1.15)
The following Proposition 1.3 gives us further informations on the link between the behaviour of qh and its restrictions to A and A' respectively. We assume t h a t A is closed. Then: Proposition 1.3.
(i) The following inequality is valid for all ( in X:
Convex function of a measure
111
(ii) Let us assume that 0 belongs to B and that (1.17) where g is an even convex continuous function defined on
R,g* its conjugate. Then there exist
c1,
c3
2 0,
> 0, such that for every
cg
in X :
'$(El 2 and
c1
{cl161
+ c24(nA0E) - c3},
(1.18)
is strictly positive if 0 is in the interior of B .
(i) Proposition 1.1 gives that A' c B whenever 0 belongs to B and then :
W) = sup{E sEB
rl - $ * ( r l ) }
*
2 SUP { E rl - s*(lrll)} *
11EAO
= sup{A sup x>o
( *
7 - g*(A)}
qEAo 111151
= suP{A X>O
lnAoEI
- g*(A)}
= g((nAO[() since g = g**. Now, using (1.17), we note that
and we have obtained (1.18) in this case:
F. Demengel and R . Temarn
112
Now if 0 lies in the interior of B, assume t h a t B(0,r) C B for some r > 0. We then have:
Icl(<) 2
SUP
{E
*
vEB ( 0 ~ )
7 - 1cI*(rl))
= SUP { E V A - g * ( W O
2 rlFl - c. Adding this inequality t o $(() 2 $ ( l l , ~ E ) , which holds if 0 E B, we obtain
which is exactly (1.18). I
To end this Section, let us give a n example of conjugate function $* which fulfills the conditions of (ii) and will often be used in the following: we denote by +* the convex functional
$*(El
=
;A(. { +oo
( when
E belongs to B,
if not,
where A is a linear positive operator on X;
(1.19)
+ is easy to calculate and we obtain
When B is a cone of vertex 0, B = ho and
Moreover, when 0 is in the interior of B, we are in the conditions of (ii) in Proposition 1.3 (with +*(()
=
g*(IfIA)
where we denote by
I E ~ Athe norm
(A[. <)'/'). Using the orthogonal decomposition
we get the inequality:
(1.21)
Convex function of a measure
113
2. THE MEASURE $ ( p ) 3.1. Measures with values in a convex cone Let R be a n open bounded set of RN, and X be a real euclidean finite dimensional space. We denote by M'(R,X) t h e space of bounded measures on R, with values in X . If p belongs t o M'(R,X), we write in this section
the Lebesgue decomposition of p , h being in L'(R,X),(') ps being singular with respect t o the Lebesgue measure dx. We suppose on the other hand t h a t
A is a closed convex cone in X , with vertex 0, and A' is its polar cone. We will say t h a t p takes its values in A (and we will sometimes write p E A) if
where Co(R, A') denotes t h e space of compactly supported continuous functions on R, with values in A'. Let us remark first that when N = 1 and A = R+ (respectively R - ) , the notion of measure with values in A as above coincides with t h e notion of positive (respectively negative) measure. We now give some properties or equivalent definitions.
Proposition 2.1. (i) The assertion (2.2) is equivalent to the following
I
( p p) 5
(ii) If p = ul
0 , V p E L 1 ( R , d p , A o )or C r ( R , A 0 ) . ( 2 )
+ u2 with ul
orthogonal to
values in A, then ul and
u2
u2,
(2.3)
and if p takes its
take their values in A too.
( l ) This is the space of functions from R into X that are L' for the Lebesgue measure dx.
(2) For K C X , we denote by
Cr(f2K , ) (resp. Co(R, K ) ) the space of Cm (resp. continu-
ous) functions from R into K with compact support in f2. Similarly, if u E M'(R, X ) and
15 p 5 00, LP(R,u,K ) is the space of Lp functions from R into X that take their values in K ,u almost everywhere.
F. Demengel and R. Temam
114
(iii) If p = hu with u real valued a n d 2 0, h a u-integrable function with values in X, it is equivalent to say that p takes its values in A or that h does,
Y
almost everywhere.
Proof. (i) The equivalence of (2.2) and (2.3) follows easily from Lemma 2.1 below.
It is enough to show that if ( p I p) 5 0 for all p E C r ( R , A O ) , then we also have ( p 19) 5 0 for all p E L1(R,p,Ao).But if p E L1(R,p,Ao), we consider the sequence p j E C r ( R , A o ) provided by Lemma 2.1 below (used with u,B , v , replaced by p, A', p: p j converges t o p in L1(R, p , A') aa j
-+
00,
and the inequalities ( p I pj) 5 0 for all j , give ( p I p) .< 0 at
the limit. (ii) Let A l and A 2 be two disjoint measurable sets such that ui is supported by A i , for i = 1, 2, and let p be in Co(R,A'). X
A
Then
X
A E L'(R,p) ~ ~ and
takes ~ its ~ values in A'; therefore
which implies t h a t
or equivalently
(vi I p) 5 0 since vr is supported by Ai. I We now state Lemma 2.1 used above and whose proof is given at the end of Section 2.
Lemma 2.1. The sole assumptions are that B is a closed convex set of an euclidean space X a n d that v belongs to L ' ( R , u , B ) , where u E M'(R,X) a n d R is an open bounded set ofRN. Then there exists a sequence v j E Cr(R,B) such that, as j vj
-, v in L'(R, v ) and u-almost
everywhere,
-+
00,
Convex function of a measure
115
The proposition 2.2 below gives a definition of the "projections" of p E
M1(0,X ) onto A and A" respectively. Proposition 2.2. Each measure p E M1(0,X ) may be written as the s u m of two measures taking respectively their values in A and A". Moreover, there exists a minimal decomposition - in the sense of modulus - that we write
Ifp = hlpl, then n ~ =pI I ~ h l p lnAop , = nAohlpl and when p is a function, l l ~ and p n ~ o pcoincide with the projections onto A and A" in the ordinary sense (defined almost everywhere).
Proof. Let h be p/lpI. T h e projections of h onto A and A" respectively are defined p everywhere, and they are p measurable (because the projection onto a convex set is continuous). We then write
and then p = hlpl = nAhlpI
+ nAOhlp1.
But (n.kh)lpl (respectively (n,oh)[p1) is a measure with values in A (respectively A"), and then the announced decomposition exists. Let us now suppose that p
with p1 E A, p2 E A'.
= p1+ p2
We can write
with vi orthogonal to IpI. Let us remark that necessarily
v1
= -v2. According
to Proposition 2.1, assertions (ii) and (iii), we get hllpl E A, hz(p1 E A', and then hl E A, h2 E A' and
F. Demengel and R. Temam
116
which implies that h = hl
+ hz
11-11-almost everywhere and
which is the minimal property mentioned of rI~1-1and
n~01-1. 3
2.2. Definition of $ ( p )
We suppose in this subsection t h a t B is a closed convex set which contains 0, that $* is a convex function on X with domain B , satisfying (1.2) and
$* 1 0, +*(O) = 0. We recall t h a t under these hypotheses, the conjugate
(2.5)
+ of $* verifies also:
Moreover if $* is the asymptotic function of 11, (or equivalently the conjugate of
x ~ )with , domain A, we will suppose that 3r < +co such t h a t B c B ( 0 , r )+ A’ (which implies that A is closed and nhbounded.)
(2.7)
We are now given a bounded measure 1-1 on an open set R of R N , and we write
it,s Lesbegue decomposition, where we assume that 1-1’ (we note 1-1’
takes its values in A
= hSIpSI). Using N. Bourbaki [2] and Proposition 1.2 (ii) of
Section 1, we see that $ = ( h S ) is 11-1’1
integrable. Moreover if we suppose t h a t
11, o h is Lesbegue integrable we will define $ ( p ) as follows
117
Convex function of a measure
Remark 2.1. (i) Assume that p = hlvl = h2u2 with
v1
and
v2
two nonnegative measures
such t h a t p is absolutely continuous with respect t o each of them. T h e proof of Bourbaki [2] shows, by using t h e homogeneity of
$m
that
$ ~ ~ ( his ,v,) integrable for i = 1, 2,
(2.9)
We will then write with no possible confusion
(ii) In [2] we defined t h e measure + ( p ) with formula (2.8) by supposing t h a t $ (which was not necessarily convex) is at most linear at infinity and t h a t
existed. It has been shown in [2] t h a t $ ( p ) is always defined in t h a t case (i.e., t h a t $ o h is ds integrable and $ ( p ) is a bounded measure). Moreover when
4 is convex and the domain B
of $* is bounded, q!~enjoys
remarkable properties. We seek here to extend these properties t o the case where B is unbounded. We will see t h a t even under some restrictive assumptions on t h e geometry of B , infinity and
+m
4 is not in general at most linear at
is not everywhere finite so t h a t the right-hand side of (2.8)
does not make sense automatically.
(2.11)
Proposition 2.3. Under the assumptions (2.5) t o (2.7) the set
M+ (0)is convex,
118
F. Demengel and R. Temarn
and the equality holds true above when 0 is in the interior of B and +* satisfies (1.1 7). Proof. Because of (1.8) and (1.16), we have almost everywhere
and since In~hland +(nAoh) are in L'(n,ds), t h e inclusion (2.12) follows. When 0 is in the interior of B, we apply the assertion (ii) of Proposition 1.3,
cl,
c2
> 0, and hence + ( l l ~ o hE) L'(R,ds).
R e m a r k 2.2. Following the steps in t h e proof of the Krasnoselskii Theorem [6], we obtain
4,
that
is continuous from L1(R,A) into L'(R) if and only if it maps L ' ( Q A)
into L'(R); see the appendix. I We now give the main results of this section. To this end, let us define the sets:
D$(C,")
= {w E
DQ(C0) = {TI
D&)
E
C,"(Sz,X) : v(x) E B , vx E n}
(2.13)
Co(R,X) : w(2) E B , vx E n}
(2.14)
+ w(x) E B , ([PI + dx) a.e., +* o w E ~ l ( ~ , d s ) } ,
= {w E L'(R, IpJ ds,X):
(we easily have
(2.15)
D$(Cr)c D$(Lk)),and we define for p in C(fi),p 2 0: (2.16)
where A will be
('1
D$(Co), D+(Cr) or D@(LE).We have the following result.
This supremum being allowed to be equal t o
+GO
119
Convex function of a measure
Theorem 2.1. The value of $(p,p) defined in (2.16) does not change when A is equal to
D+(Co),D+(CF)or D@(Li).In
addi-
tion, (i) The restriction of $ ( p , - ) to
C$(n)may
be extended as a
measure when $(p,p) is finite for every cp 2 0 in CO(n), and we write for cp in Co(n):
This measure is nonnegative and it is bounded if $ ( p , 1) < +oo.
(ii)
$(p,
.) is a bounded measure on 0 if and only if $ ( p ) also
is, and in this case, we have
Remark N
More precisely + ( p ) is a measure if and only if + ( p ,
a)
is
.
Proof. We begin t o show that the supremum on the right side of (2.16) does
not, change if we take for A ,
D$(Co),
D+(Cr)or D+(LL).It is obvious that
Now, let us suppose first that the supremum on the right-hand side is finite and let S > 0 be given; there exists
210
in D@(Lh)such that
We now use the following lemma whose proof is given at the end of Section 2.
F. Demengel and R. Temam
120
Lemma 2.2. The sole assumptions are that B is a closed convex set of the euclidean space X, 0 E B and that $* is a convex
continuous function from B into R; we assume also t h a t
00
E
L1(R,IpI+ d z , B ) , $* 0210 E L'(R,ds) where P E M ' ( R ; X ) . Then there exists a sequence vj E
CF(n,B) such
that, as
j -,00, vJ converges to v o in L1(R,Ipl+ ds, B ) and IpI + ds a.e., and $* o v j
$* o vo in L'(R, d z ) .
--f
We successively show assertions (i) and (ii). (i) For t h a t purpose, let us remark that
y3
$(p,p) defined by (2.16) is
H
additive on the cone
Indeed let
p1
and 0
p 2
D?
be two functions in
I N P , P1 + 9 2 ) 5 &P,
. We obviously have @(@,,I
N
(01)
+ & ( P >P Z ) ,
which shows that D ( $ ( p , . ) ) is a convex cone with vertex 0. Now, given
6 > 0, let us take vi in D@(Lb)for i = 1, 2 such that $(P,Pi)
5
S,
PviPi -
S,
and let v=-
where cp = (01
+
p 2
# 0, and
P*(Vi)Pi
c
+ 6,
i = 192,
ViPi
P
= 0 elsewhere.
It is clear that v belongs to L ' ( Q , lpl+ d s , X) and v(z) E B for almost everywhere on R ; Moreover 2
2 . "
and using the convexity of +* we get for p ( z ) # 0:
\PI+
dz
Convex function of a measure
or equivalently
121
2
(which remains true when ~ ( z =) 0). We finally obtain
which implies t h a t d ( p , * )is additive, since 6 When
'p
may be written in the form
let us define
d b ,P) = &P,
with P = (01 - 9 2 , P1 1 0,
P2
9 1
> 0 is arbitrary.
- 'p2, with
cpi
in D + ( $ ( p ? . ) ) >
(2.19)
Pl)- d ( p ,(02)
1 0.
The additivity property allows us t o show that (2.19) does not depend on the choice of
p1
and
'p2
such that p =
'p1
- p 2 . T h e positivity of
& ( p , - ) easily implies t h a t J ( p , . ) is a bounded measure when $ ( p , 1) is
finite. Let us suppose secondly that the last term on the right of (2.18) is
+w; then for every A > 0 there exists vo in D,,j(L1p) such t h a t
Using once more Lemma 2.2 there exists
vj
-,
VO, vj
E
CF(R,B) such
t h a t for j sufficiently large, we have:
which implies t h a t the first term on the left-hand side of (2.18) is +oo too. (ii) Let us now suppose t h a t p belongs t o M+(R). Then $ ( p ) is a bounded measure on R,
11 o h is in L'(R, d z ) and we have for all v
that v(z) E B almost everywhere and for all
(o
in L'(R, dp) such
2 0 in &(R):
F. Demengel and R. Temam
122
and ps almost everywhere, ps = hS Ips 1,
This inequality shows t h a t $ ( p , ’p) is finite for all
’p
2 0 in Co(n), and
N
implies t h a t $ ( p , .) is a bounded measure when $ ( p ) also is, i.e., when p belongs to M+(R). We want now t o show t h a t q ( p , .) and $ ( p ) coincide in that case. We will show in a first step t h a t
and then t h a t
+ o h is d s integrable when $ ( h d z ,1) is finite, $ w ( h s ) is
ps integrable when & w ( h S I p S / , l ) is finite and that we have in addition
the inequalities
J,$oh 5 4(hdz,l)
(2.21)
J, $M(hS)dIPSl 5 J w ( hs IPs I, 1).
(2.22)
Let us show t o begin with equality (2.20). We proceed like in (31. Let 6
> 0 be given and
(v1,vZ)E
Co(R,B) such t h a t
123
Convex function of a measure
L $m(PS,P) -
hSV2PIPSI
Let now 0 be a set of Lebesgue measure 6, Ips I that ps is supported by 0. Since 11.’ such that
s,
IP(V1
+ h dx measurable, such
is singular, we may choose 6
>0
- vz)h - ( + * ( V l ) - +*(w2))1 dx c
Let us then consider the function on n\0, on 0 ;
v = { v1 212
v is Ipl
+ dx integrable, with values in B ; moreover:
The left-hand side may be written in the form
We finally have obtained:
5 4 ( P , P)+ 3€, Y
which ends the proof of equality (2.20). Let us now suppose that $ ( p , - ) is a bounded measure; then there exists a constant c
> 0 such that
124
F. Demengel and R. Temarn
We then show that t,b o h E L 1 ( R , d x ) . For
E
> 0 and given 6 > 0, we
consider a disjointed covering of R by universally measurable sets A:, mes(Af) < 6, and we define t h e vectors X f :
Since II, = $I**, there exists
vE
ef
in B , such that
L'(R,dz), v(z) E B for almost every z in R and using $ ( O ) = 0:
11,(h6)ds 5
1 R
h6v - $*(v)
+ E mes R
5 & ( h d z , l )+ c mes R. Since h6
4
h in L'(R) when 6
--f
0, there exists a subsequence 6'
+
0
such that h 6 ' ( z )-, h ( s )everywhere on R. Using t h e lower semi-continuity of 11, and Fatou's Lemma, we may write $ o h ( z )dz 5
lirninf 4 0 h 6 ( z )dz, 6+0
5 lim inf 6-0
$ 0 h6(z)ds,
5 & ( h d z ,1)+ E mes R , which implies t h a t 11, 0 h is Lebesgue integrable. We could use the same argument of approximation by piecewise linear functions, replacing t,b by
L+m
t,bw, d x by lpsl and h by h S , t o obtain the analagous inequality o h s IPs I
5 i ~ (s IP h s 1,1)+ E mes 0.
Convex function of a measure
125
And this ends the proof of Theorem 2.1. I We conclude this section by the proof of Lemmas 2.1 and 2.2:
Proof of Lemma 2.1. From the definition of L ' ( R , v , B ) , there exists a sequence wj E Co(R,X) which converges to v in
L1(R,v , X ) . Then, since the projector IIB in X onto
B is Lipschitzian, the sequence I I B w ~ belongs t o
CO(L?, B ) and
converges to v
in L ' ( R , v , B ) . Now assume that v E Co(R,B). By a mollification procedure we can contruct a sequence v j E Cr(R,B)which converges to v uniformly as j Such a sequence is obtained by setting only satisfy the conditions
pj
1 0 and
vj
--* 00.
= p j * v where the mollifiers p i must
JRN
pj(s)
dz = 1.
By combining these two remarks, we obtain for every v E L1(R,p,X),a sequence vi E Cr(R,B) which converges t o v p-a.e., and the result follows. Proof of Lemma 2.2.
(i) In a first step we approximate v by functions in L"(R, v,B ) , v = lpl+ dx (note that B is not necessarily bounded). For that purpose we set for every integer p,
We easily have Ivp(s)l5 p, Ivp(x)l 5 Iv(z)I and up(.) E B v-a.e. From the Lebesgue dominated convergence theorem, u p converges to v in L'(Q, v,B ) as p
--f
00.
Then, due to the convexity of $*, and since + * ( O ) = 0, $* 2 0,
Since $* is continuous on B , $*(v,(x)) p
--f
00,
+
$*(v(z)),ds-a.e.
in R , as.
and it follows from the Lebesgue dominated convergence theorem
that Jn $* o u p -+
sn $*
0
v.
F. Demengel and R. Temam
126
(ii) We are now reduced to the case where furthermore v takes its values in
B, for some p. Lemma 2.1, applied with
Y,B
replaced by 1p1
+ dx, B,,
provides a sequence v j E Cr(n,B,) which converges t o v in L1(R,v,B,) and u-a.e. continuity of
Then v j converges t o v dx-almost everywhere and by the
4*,
We also observe that
and then the Lebesgue dominated convergence theorem applies and shows that
ln 4* o v j converges to ln +* o v
The proof is complete. I 3. APPROXIMATION RESULTS
We recall the hypothesis on $ of Section 2:
$J
is supposed to be convex,
B = dom $* is closed,
11 1 0 , WJ)= 0,
(34
A is closed and IIAB is bounded.
(3.2)
We may endow M + ( n ) with the distance defined by:
and Lemma 3.1 below proves that from every bounded sequence in M+(R) we can extract a subsequence which is vaguely convergent towards a measure in
M+ (0). L e m m a 3.1. Let pj be a sequence in MJ,(R) weakly convergent towards a bounded measure p . Then if $ ( p j ) is bounded in
M'(R), p belongs to MJ,(R)and we have
Convex function of a measure
Proof. Let us suppose by contradiction that p
127
4 M+(R). By
($(/A,1)) = +w,and for every real A > 0 there exists w in that
Theorem 2.1,
Cr(n,B ) such
n
Since pj is vaguely convergent to p , there exists J > 0 such that for j > J :
Therefore for all j > J :
and
sn $ ( p j ) cannot be bounded; this contradicts the hypothesis.
We thus
have shown that p E M+(R),and we may conclude for every 6 > 0 and p 2 0 in Co(n) the existence of w in C F ( 0 , B )such that
P
5 liminf
-
($(pj)
J-+W
s,
ll*(v)p
+
I $4+ 6,
which ends the proof of Lemma 3.1. I Let us now define a topology on M$(Rj which is less finer than that defined by the metric (3.3). This second topology may be defined by the family of semi-dist ances: (3.5)
( l ) This means
s,
+ ( p ) p 5 liminfj,=
s,
+(/A,) p, V p E C o ( n ) , p 2 0.
F. Demengel and R. Temam
128
for all v in
C(fi),
v 2 0. It is immediate that this topology is stronger than
the weak topology which can be defined by the family of semi-norms:
where v is in
C,(n)
We are now going t o show the following approximation result:
Lemma 3.2. For every p in M @ ( n )there , exists a sequence o f functions z c j in
C?(n),such that
Proof. let O j be an increasing sequence of functions in
being pointwise convergent t o l n ; let
~j
Cr(n),0 I 9 j I 1, O j
be a sequence of positive real numbers
converging t o zero, p be a function in D(RN),p being even,
and uj = p E j * (Ojp). We are going t o show t h a t
By the convexity of +* we have, for all v in +*
and then, using
JRN p
This implies that
= 1,
(PCj
* v) I
PEj
C?(n),v ( z ) E B :
* +*(v)
Convex function of a measure
for pcj
* V
= pcj * V E
129
P(G,B) c a&v(ejP)).(l)
Using the Lemma 3.3 just below we have
On the other hand, since .II, ( p c j * ( e j p ) ) is bounded in M 1 , we may extract from it a subsequence which is vaguely convergent towards Y E M'(R). According t o Lemma 3.1, we have
and therefore
Lemma 3.2. Let
p1
and
p2
be in
C(G), with 0 I
p1
I
p2.
Then for every p in M'(R, X )
The proof of this lemma follows that of the lemma 2.4 established in [3], and consists in the use of the duality formula (2.16).
Remark 3.1. The proof of Lemma 3.2 contains the inequality in the sense of measures:
F. Demengel and R. Temam
130
for all 0 in C,(R), Indeed, let on
uj
E
be in
< d(suptO, an), and
p a regularizing function as before.
Cr(n);u j tends to p vaguely on R,$J(uj)-, $ ( p ) vaguely
R. We have by the convexity of $ and using $(O) = 0,
$J
2 0:
Now the convergence of u j towards p implies that of pr * 8uj towards pt
* 9p;
using Lemma 3.1, we then have
5 liminf $ ( p t * O G ( u j ) ) . 3-M
It is clear that the convergence of $J(uj) towards $ ( p ) implies that of pc*8$(u3) towards pt
* 9 $ ( p ) . We finally have obtained
APPENDIX To begin with, let us recall the definition of the Caratheodory function:
Definition A . l . A function f ( s , u ) of two variables +m, s E
-00
R (where R is a bounded open set of RN) is said to
satisfy the Caratheodory conditions (cf.161) if it is continuous with respect to u for a.e. x E R and measurable with respect to s for all values of u. We denote by F the operator defined on the set of real functions on R by
where f satisfies the Caratheodory conditions. Let u s suppose that h is a convex cone of vertex 0 in a Banach space E, and suppose that p, r are two real numbers such that 1 < p, r < +oo. We then have the following result:
131
Convex function of a measure
Theorem A . l . If F maps L*(R,A) in L’(R,G) (where G is a Banach space), then F is continuous from LP(R,A) into Lr(R, G) for the norm topology. Proof. We proceed as in the theorem 2.1 of Ref. 6, and begin t o recall the definition of the convergence in measure:
Definition A.2. We will say that un --t u in measure if Vc
> 0, 3 N , V n 2 N
mes{z : I(un - .)(.)I
> E } < E.
In fact, it is easy to see that Definition A.2 is equivalent to the following:
Definition A.3.
u, + u
in measure if
Y E > 0, V6 > 0 3 N , V n 2 N mes{z : I(u, - .)(.)I
+
> 6) < c.
Let us recall the lemma 2.1 of IS]:
Lemma A.l. let-R be a set of finite measure. Then the operator F transforms every sequence of functions that converges in measure into a sequence of functions that also converges in measure . We can now prove the theorem; we may suppose that F ( 0 ) = 0 and show that F is continuuous at zero. Indeed, let us introduced for a fixed u in
LP(R,A) the operator H defined on the convex cone A - u:(l) H(v)= F(v
+ u)- F ( u )
Suppose by contradiction that there exists a sequence of functions Pn E
LP(R,A - u) such that lPnlLP(n,E)
(l) h - u = { w - u : : € E } .
+ 0,
F. Demengel and R. Temam
132
and such that We may assume, by extracting a subsequence, t h a t
n
and suppose for a while t h a t we have proved the existence of sequences Ek p n k ,and sets
Gk
c R, such that the following conditions are fulfilled:
Let us then consider t h e sets
u m
Dk= G k \
Gi.
i=k+l
We have: r=k+1
a=k+l
Let us now define the function II, by
We then have
> -2a- - =a 3
4 lies in LP and
3
a
3 '
so H $ E L'. On the other hand we have
+ 0,
133
Convex function of a measure
which is a contradiction. It remains t o prove conditions (a), (b), (c) and (d) by induction. Suppose t h a t e l = mes R, pnl = p 1 7G1 = G , and suppose t h a t Ek, G k , ( o a k
have been constructed. Since H (pnk)E L' we may choose
such t h a t for all
then
,,s
~ k + l
IHp,,I'
<
D,
Ek/2
<
Ek+l
(if not
5 , which
2Ek+1
2
€k
and would have mes
G k
5
2Ek+l
and
contradict condition (c). Now pnk converges in
measure t o zero, so by Lemma A . l , IHpnkI converges too in measure towards 0. Then there exists
nk+l
such t h a t
Let us define the set
Then mes R\mes
Fk+l
< E k + l . Let
Gk+1
a
>a--=3
= R\Fk+l.
We then obtain
2a 3
This ends the proof of Theorem A . l .
REFERENCES [l] AUBIN-CELLINA, Differential inclusions, Springer Verlag, 264 (1984). [2] N. BOURBAKI, Elements de Mathbmatiques, livre VI, Inthgration,
Chapitre 5, Paris (1965). [3] F. DEMENGEL and R. TEMAM, Convex function o j a measure and ap-
plications, Indiana J. Math., 33 (5) (1984), 673-709. [4] 1. EKELAND and R. TEMAN, Analyse conveze et probltmes variation-
nels, Dunod, Paris, (1974).
F. Demengel and R. Temam
134
[5] M. FREMONT and A. FRIAA, Analyse limite.
Comparaison des
mkthodes statique et cinkmatique, C. R. Acad. Sci. Paris, 286, sCrie A
(1978), 107-110. 161 M. A. KRASNOSELSKII, Topological methods in the theory of non linear integral equations, Pergamon Press, London (1964). [7] J.-J. MOREAU, Fonctionnelles convexes, Skminaires Equations aux de'rivkes partielles, Collkge de France (1966). [8] R.T. ROCKAFELLAR, Convex Analysis, Princeton University ,Press
(1970). [9] R. TEMAM, Probltmes Mathe'matiques e n Plasticitk, Dunod (1983).