Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

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by  Adolfo Arroyo-Rabasa

LOWER SEMICONTINUITY AND RELAXATION OF LINEAR-GROWTH
INTEGRAL FUNCTIONALS UNDER PDE CONSTRAINTS
ADOLFOARROYO-RABASA,GUIDODEPHILIPPIS,ANDFILIPRINDLER
ABSTRACT. We show general lower semicontinuity and relaxation theorems for
linear-growth integral functionals defined on vector measures that satisfy linear
PDEsideconstraints(ofarbitraryorder). Theseresultsgeneralizeseveralknown
7 lower semicontinuity and relaxation theorems for BV, BD, and for more general
1 first-orderlinearPDEsideconstrains. Ourproofsarebasedonrecentprogressin
0 theunderstandingofsingularitiesinmeasuresolutionstolinearPDE’sandofthe
2
correspondinggeneralizedconvexityclasses.

n KEYWORDS: Lower semicontinuity, functional on measures, A-quasiconvexity,
a
generalizedYoungmeasure.
J
9 DATE:January10,2017.

]
P
A
h. 1. INTRODUCTION
t
a The theory of linear-growth integral functionals defined on vector-valued mea-
m
sures satisfying PDEconstraints iscentral tomanyquestions ofthecalculus ofvari-
[ ations. In particular, their relaxation and lower semicontinuity properties have at-

1 tracted a lot of attention, see for instance [AD92, FM93, FM99, FLM04, KR10b,
v Rin11, BCMS13]. In the present work weunify and extend a large number of these
0
results by proving general lower semicontinuity and relaxation theorems for such
3
2 functionals. Our proofs are based on recent advances in the understanding of the
2 singularities thatmayoccurinmeasuressatisfying (under-determined) linearPDEs.
0 Concretely, let W ⊂ Rd be an open and bounded subset with Ld(¶ W ) = 0 and
.
1 consider for a vector Radon measure m ∈ M(W ;RN) on W with values in RN the
0
functional
7
1 dm dm s
: F#[m ]:= f x, (x) dx+ f# x, (x) d|m s|(x). (1.1)
v ZW (cid:18) dLd (cid:19) ZW (cid:18) d|m s| (cid:19)
i
X
Here, f: W ×RN →RisaBorelintegrand thathaslineargrowthatinfinity, i.e.,
r
a
|f(x,A)|≤M(1+|A|) forall(x,A)∈W ×RN,
wherebythe(generalized) recessionfunction
f(x′,tA′)
f#(x,A):=limsup , (x,A)∈W ×RN,
t
x′→x
A′→A
t→¥
takesonlyfinitevalues. Furthermore, onthecandidate measures m ∈M(W ;RN)we
imposethek’th-order linearPDEsideconstraint
Am := (cid:229) Aa ¶ a m =0 inthesenseofdistributions.
|a |≤k
1
2 A.ARROYO-RABASA,G.DEPHILIPPIS,ANDF.RINDLER
ThecoefficientsAa ∈Rn×N areassumedtobeconstantandwewrite¶ a =¶ a 1...¶ a d
1 d
foreverymulti-indexa =(a ,...,a )∈(N∪{0})d with|a |:=|a |+···+|a |≤k.
1 d 1 d
Wecallmeasures m ∈M(W ;RN)withAm =0inthesenseofdistributions A-free.
WewillalsoassumethatA satisfiesMurat’sconstantrankcondition(see[Mur81,
FM99]),thatis,wesuppose thatthereexistsr∈Nsuchthat
rank(kerAk(x ))=r forallx ∈Sd−1, (1.2)
where
Ak(x ):=(2p i)k (cid:229) x a Aa , x a =x a 1···x a d
1 d
|a |=k
istheprincipalsymbolofA. WealsorecallthenotionofwaveconeassociatedtoA,
which plays a fundamental role in the study of A-free fields and first originated in
theTartar–Murattheoryofcompensatedcompactness [Tar79,Tar83,Mur78,Mur79,
Mur81,DiP85].
Definition1.1. LetA bek’th-orderlinearPDEoperatorasabove,A :=(cid:229) |a |≤kAa ¶ a .
Thewaveconeassociated toA istheset
L A := kerAk(x )⊂RN.
|x[|=1
Note that the wave cone contains those amplitudes along which it is possible to
constructhighlyoscillatingA-freefields. MorepreciselyifA ishomogeneous, i.e.,
A =(cid:229) |a |=kAa ¶ a ,thenP∈L A ifandonlyifthereexistsx 6=0suchthat
A(Ph(x·x ))=0 forallh∈Ck(R).
Ourfirstmaintheoremconcerns thecasewhen f isAk-quasiconvex initssecond
argument, where
Ak := (cid:229) Aa ¶ a
|a |=k
isthe principal part of A. Recall from [FM99]that aBorel function h: RN →Ris
calledAk-quasiconvex if
h(A)≤ h(A+w(y))dy
ZQ
forallA∈RN andallQ-periodicw∈C¥ (Q;RN)suchthatAkw=0and wdy=0,
Q
whereQ:=(−1/2,1/2)d istheunitcubeinRd. R
Theorem 1.2 (lower semicontinuity). Let f: W ×RN → [0,¥ ) be a continuous
integrand. Assume that f has linear growth at infinity and is Lipschitz in its second
argument, uniformlyinx. Assumefurtherthatthereexistsamodulusofcontinuity w
suchthat
|f(x,A)− f(y,A)|≤w (|x−y|)(1+|A|) forallx,y∈W ,A∈RN. (1.3)
andthatthestrongrecession function
f(x,tA)
f¥ (x,A):= lim existsforall (x,A)∈W ×spanL A. (1.4)
t→¥ t
Then,thefunctional
dm dm s
F[m ]:= f x, (x) dx+ f¥ x, (x) d|m s|(x)
ZW (cid:18) dLd (cid:19) ZW (cid:18) d|m s| (cid:19)
LOWERSEMICONTINUITYANDRELAXATIONOFINTEGRALFUNCTIONALS 3
issequentially weakly*lowersemicontinuous formeasuresinthespace
M(W ;RN)∩kerA := m ∈M(W ;RN) : Am =0
(cid:8) (cid:9)
ifandonlyif f(x,q)isAk-quasiconvex foreveryx∈W .
Note that according to (1.7) below, F[m ] is well defined. Since the strong reces-
sionfunction iscomputed onlyatamplitudes thatbelongtospanL A.
Remark1.3. TheconclusionofTheorem1.2extendstosequencessuchthatAm →
j
0strongly inW−k,q(W ;Rn)forsome1<q<d/(d−1).
Notice that f¥ in (1.4) is a limit, and differently from f#, it may fail to exist for
A ∈ (spanL A)\L A (for A ∈ L A the existence of f¥ (x,A) follows from the Ak-
¥
quasiconvexity, see Corollary 2.19). If we remove the assumption that f exists for
points in the subspace generated by the wave cone L A, we still have the following
partiallowersemicontinuity result(cf.[FLM04]).
Theorem 1.4 (partial lower semicontinuity). Let f: W ×RN →[0,¥ ) be a con-
tinuous integrand. Assume that f has linear growth at infinity and is Lipschitz in
its second argument, uniformly in x. Assume further that there exists a modulus of
continuity w suchthat
|f(x,A)− f(y,A)|≤w (|x−y|)(1+|A|) forallx,y∈W ,A∈RN. (1.5)
Then,
dm
f x, (x) dx≤ liminfF#[m ] : m ⇀∗ m andAm →0inW−k,q ,
ZW (cid:18) dLd (cid:19) n j→¥ j j j o
where
dm dm s
F#[m ]:= f x, (x) dx+ f# x, (x) d|m s|(x).
ZW (cid:18) dLd (cid:19) ZW (cid:18) d|m s| (cid:19)
Remark1.5. Asspecial casesofTheorem1.2weget,amongothers, thefollowing
well-knownresults:
(i) For A = curl, one obtains BV-lower semicontinuity results in the spirit of
Ambrosio–Dal Maso [AD92] and Fonseca–Mu¨ller [FM93], also see [KR10b]
forthecaseofsignedintegrands.
(ii) ForA =curlcurl,where
d
curlcurlm := (cid:229) ¶ m j+¶ m k−¶ m i−¶ m k
(cid:18) ik i ij i jk i ii j(cid:19)
i=1 j,k=1,...,d
is the second order operator expressing the Saint-Venant compatibility condi-
tions (see [FM99, Example 3.10(e)]), we re-prove the lower semicontinuity
andrelaxation theorem inthespaceoffunctions ofbounded deformation (BD)
from[Rin11].
(iii) Forfirst-orderoperators A,asimilarresultwasprovedin[BCMS13].
(iv) Earlier work in this direction is in [FM99, FLM04], but did not consider the
singular part(concentration ofmeasure).
If we dispense with the assumption of Ak-quasiconvexity on the integrand, we
havethefollowingtworelaxation results:
4 A.ARROYO-RABASA,G.DEPHILIPPIS,ANDF.RINDLER
Theorem1.6(relaxation). Let f: W ×RN →[0,¥ )beacontinuous integrandthat
isLipschitzinitssecondargument(uniformlyinx),haslineargrowthatinfinity,and
is such that there exists a modulus of continuity w as in (1.5). Further we assume
that A is a homogeneous partial differential operator and that the strong recession
function
f¥ (x,A) existsforall (x,A)∈W ×spanL A.
Then,forthefunctional
G[u]:= f(x,u(x))dx, u∈L1(W ;RN),
ZW
the(sequentially) weakly*lowersemicontinuous envelope
G[m ]:=inf liminfG[u ] : u Ld ⇀∗ m andAu →0inW−k,q
j j j
n j→¥ o
isgivenby
dm dm s
G[m ]=ZW QA f(cid:18)x,d|m |(x)(cid:19)dx+ZW (QA f)#(cid:18)x,d|m s|(x)(cid:19) d|m s|(x),
where QA f(x,q) denotes the A-quasiconvex envelope of f(x, q) with respect to the
secondargument(seeDefinition2.16below).
If we want to relax in the space M(W ;RN)∩kerA we need to assume that
L1(W ;RN)∩kerA is dense in M(W ;RN)∩kerA with respect to a finer topology
thanthenaturalweak*topology (inthiscontextalsosee[AR16]).
Theorem 1.7. Let f: W ×RN →[0,¥ ) be a continuous integrand that is Lipschitz
in its second argument (uniformly in x), has linear growth at infinity, and is such
that there exists a modulus of continuity w as in (1.5). Further assume that A is a
homogeneous partialdifferential operatorandthatthestrongrecession function
f¥ (x,A) existsforall (x,A)∈W ×spanL A.
If for all m ∈M(W ;RN) with Am =0 there exists a sequence (u )⊂L1(W ;RN)∩
j
kerA suchthat
u Ld ⇀∗ m inM(W ;RN) and hu Ldi(W )→hm i(W ), (1.6)
j j
wherehqiis thearea functional defined in (2.2), then the weakly* lower semicontin-
uousenvelope ofthefunctional
G[u]:= f(x,u(x))dx, u∈L1(W ;RN)∩kerA,
ZW
withrespecttoweak*-convergence inthespaceM(W ;RN)∩kerA,isgivenby
dm dm s
G[m ]=ZW QA f(cid:18)x,d|m |(x)(cid:19)dx+ZW (QA f)#(cid:18)x,d|m s|(x)(cid:19)d|m s|(x).
Remark 1.8 (density assumptions). Condition (1.6) is automatically fulfilled in
thefollowingcases:
(i) For A =curl, the approximation property (for general domains) is proved
in the appendix of [KR10a] (also see Lemma B.1 of [Bil03] for Lipschitz
domains). The same argument further shows the area-strict approximation
property in the BD-case (also see Lemma 2.2 in [BFT00] for a result which
coversthestrictconvergence).
LOWERSEMICONTINUITYANDRELAXATIONOFINTEGRALFUNCTIONALS 5
(ii) IfW isastrictlystar-shaped domain,i.e.,thereexistsx ∈W suchthat
0
(W −x )⊂t(W −x ) forallt >1,
0 0
then (1.6) holds for every homogeneous operator A. Indeed, for t >1 we
can consider the dilation of m defined on t(W −x ) and then mollify it at a
0
sufficiently smallscale. Wereferforinstance to[Mu¨l87]fordetails.
As a consequence of Theorem 1.7 and of Remark 1.8 we explicitly state the fol-
lowingcorollary,whichextendsthelowersemicontinuityresultof[Rin11]intoafull
relaxation result. The only other relaxation result in this direction, albeit for special
functions ofbounded deformation, seemstobein[BFT00],other results inthisarea
arediscussed in[Rin11]andthereferences therein.
Corollary 1.9. Let f: W ×Rd×d → [0,¥ ) be a continuous integrand that is uni-
sym
formlyLipschitzinitssecondargument,haslineargrowthatinfinity,andissuchthat
there exists a modulus of continuity w as in (1.5). Further assume that the strong
recession function
f¥ (x,A) existsforall (x,A)∈W ×Rd×d.
sym
Letusconsider thefunctional
G[u]:= f(x,Eu(x))dx,
ZW
defined for u∈LD(W ):={u∈BD(W ) : Esu=0}, where Eu:=(Du+DuT)/2∈
M(W ;Rd×d)isthesymmetrizeddistributional derivative ofu∈BD(W )andwhere
sym
dEsu
Eu=EuLd W + |Esu|,
d|Esu|
isitsRadon–Nikody´m decomposition withrespecttoLd.
Then,thelowersemicontinuousenvelopeofG[u]withrespecttoweak*-convergence
inBD(W ),isgivenbythefunctional
dEsu
G[u]:= SQf(x,Eu(x))dx+ (SQf)# x, (x) d|Esu|(x),
ZW ZW (cid:18) d|Esu| (cid:19)
where SQf denotes the symmetric-quasiconvex envelope of f with respect to the
second argument (i.e., the curlcurl-quasiconvex envelope of f(x, q) in the sense of
Definition2.16).
OurproofsarefairlyconciseandbasedonnewtoolstostudysingularitiesinPDE-
constrained measures. Concretely, we exploit the recent developments on the struc-
tureofA-freemeasuresobtainedin[DR16b]. Inparticular, thestudyofthesingular
part – up to now the most complicated argument in the proof – now only requires a
fairly straightforward (classical) convexity argument. More precisely, the main the-
orem of [KK16] establishes that the restriction of f# to the linear space spanned by
the wave cone is in fact convex at all points of L A (in the sense that a supporting
hyperplane exists). Moreover, by[DR16b],
dm s
d|m s|(x)∈L A for|m s|-a.e.x∈W . (1.7)
Thus, combining these twoassertions, wegain classical convexity for f# at singular
points, which can be exploited via the theory of generalized Young measures devel-
opedin[DM87,AB97,KR10a].
6 A.ARROYO-RABASA,G.DEPHILIPPIS,ANDF.RINDLER
Remark 1.10 (different notions of recession function). Note that both in Theo-
¥
rem 1.2 and Theorem 1.6 the existence of the strong recession function f is as-
sumed, in contrast with the results in [AD92, FM93, BCMS13] where this is not
imposed.
The need for this assumption comes from the use of Young measure techniques
which seem to be better suited to deal with the singular part of the measure, as we
alreadydiscussedabove. Intheaforementionedreferencesadirectblowupapproach
isinstead performedandthisallowstodealdirectlywiththefunctional in(1.1). The
blow up techniques, however, rely strongly on the fact that A is a homogeneous
first-order operator. Indeed, it is not hard to check that for all “elementary” A-free
measuresoftheform
m =P0l , where P0∈L A, l ∈M+(Rd),
the scalar measure l is necessarily translation invariant along orthogonal directions
tothecharacteristic set
X (P ):= x ∈Rd : P ∈kerA(x ) ,
0 0
(cid:8) (cid:9)
whichturnsouttobeasubspaceofRd wheneverA isafirst-orderoperator. Thesub-
spacestructureandtheaforementioned translationinvarianceisthenusedtoperform
homogenization-type arguments. Duetothelackoflinearity ofthemap
x 7→Ak(x ) fork>1,
the structure of elementary A-free measures for general operators is more com-
plicated and not yet fully understood (see however [Rin11, DR16a] for the case
A =curlcurl).Thisprevents, atthemoment,theuseofa“pure”blow-uptechniques
andforces ustopass through thecombination oftheresults of[DR16b,KK16]with
theYoungmeasureapproach.
Thispaperisorganized asfollows: First,inSection2,weintroduce alltheneces-
sary notation and prove a few auxiliary results. Then, in Section 3, we establish the
central Jensen-type inequalities, whichimmediately yieldtheproofofTheorems1.2
and1.4inSection4. TheproofsofTheorems1.6and1.7aregiveninSection5.
Acknowledgments. A.A.-R.issupportedbyascholarshipfromtheHausdorffCen-
terofMathematics and theUniversity ofBonn; the research conducted inthis paper
formspartofthefirstauthor’sPh.D.thesisattheUniversityofBonn. G.D.P.issup-
ported by the MIUR SIR-grant “Geometric Variational Problems” (RBSI14RVEZ).
F.R.acknowledges thesupport fromanEPSRCResearch Fellowshipon“Singulari-
tiesinNonlinear PDEs”(EP/L018934/1).
2. NOTATION AND PRELIMINARIES
We write M(W ;RN) and M (W ;RN) to denote the space of finite and locally
loc
finite vector Radon measures on W ⊂RN. Wewritethe Radon–Nikody´m decompo-
sitionofm ∈M(W ;RN)as
dm
m = Ld W +m s, (2.1)
dLd
where dm ∈L1(W ;RN)and m s∈M(W ;RN)issingular withrespecttoLd.
dLd
Inorder tokeep asimple presentation, wewill often identify u∈L1(W ;RN)with
themeasureuLd ∈M(W ;RN).
LOWERSEMICONTINUITYANDRELAXATIONOFINTEGRALFUNCTIONALS 7
2.1. Integrands and Young measures. For f ∈C(W ×RN) define the transforma-
tion
Aˆ
(Sf)(x,Aˆ):=(1−|Aˆ|)f x, , x∈W ,Aˆ ∈BN,
(cid:18) 1−|Aˆ|(cid:19)
whereBN denotes theopenunitballinRN. Then,Sf ∈C(W ×BN). Weset
E(W ;RN):= f ∈C(W ×RN) : Sf extendstoC(W ×BN) .
(cid:8) (cid:9)
Inparticular, all f ∈E(W ;RN)havelinear growthatinfinity, i.e.,thereexists aposi-
tive constant M such that |f(x,A)|≤M(1+|A|)for all x∈W and all A∈RN. With
thenorm
kfkE(W ;RN):=kSfk¥ , f ∈E(W ;RN),
the space E(W ;RN) turns out to be a Banach space. Also, by definition, for each
f ∈E(W ;RN)thelimit
f(x′,tA′)
f¥ (x,A):= lim , x∈W ,A∈RN,
x′→x t
A′→A
t→¥
exists and defines a positively 1-homogeneous function called the strong recession
¥
functionof f. Evenifonedropsthedependenceonx,therecessionfunctionh might
not exist for h ∈ C(Rd). Instead, one can always define the generalized recession
functions
f(x′,tA′)
f#(x,A):=limsup ,
t
x′→x
A′→A
t→¥
f(x′,tA′)
f (x,A):=liminf ,
#
x′→x t
A′→A
t→¥
which again turn out to be positively 1-homogeneous. If f is x-uniformly Lipschitz
continuous in the A-variable and there exists a modulus of continuity w : [0,¥ ) →
[0,¥ )(increasing, continuous, andw (0)=0)suchthat
|f(x,A)− f(y,A)|≤w (|x−y|)(1+|A|), x,y∈W ,A∈RN,
thenthedefinitions of f¥ and f# (and f )simplifyto
#
f(x,tA)
¥
f (x,A):= lim ,
t→¥ t
f(x,tA)
f#(x,A):=limsup .
t→¥ t
AnaturalactionofE(W ;RN)onthespaceM(W ;RN)isgivenby
dm dm s
m 7→ f x, (x) dx+ f¥ x, (x) d|m s|(x).
ZW (cid:18) dLN (cid:19) ZW (cid:18) d|m s| (cid:19)
Inparticular,for f(x,A)= 1+|A|2∈E(W ;RN)–forwhich f¥ (A)=|A|,wedefine
theareafunctional p
dm 2
hm i(W ):= 1+ dx+|m s|(W ), m ∈M(W ;RN). (2.2)
ZW r (cid:12)dLN(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
8 A.ARROYO-RABASA,G.DEPHILIPPIS,ANDF.RINDLER
In addition to the well-known weak* convergence of measures, we say that a se-
quence(m )converges area-strictly to m inM(W ;RN)if
j
m ⇀∗ m inM(W ;RN) and hm i(W )→hm i(W ),
j j
inwhichcasewewrite“m →m area-strictly”.
j
This notion of convergence turns out to be stronger than the conventional strict
convergence ofmeasures,whichmeansthat
m ⇀∗ m inM(W ;RN) and |m |(W )→|m |(W ).
j j
Indeed, the area-strict convergence, as opposed to the usual strict convergence, pro-
hibitsone-dimensionaloscillations. Themeaningofarea-strictconvergencebecomes
clear when considering the following version of Reshetnyak’s continuity theorem,
which entails that the topology generated by area-strict convergence is the coarsest
topology underwhichthenatural actionofE(W ;RN)onM(W ;RN)iscontinuous.
Theorem 2.1 (Theorem 5 in [KR10b]). For every integrand f ∈ E(W ;RN), the
functional
dm dm s
m 7→ f x, (x) dx+ f¥ x, (x) d|m s|(x),
ZW (cid:18) dLN (cid:19) ZW (cid:18) d|m s| (cid:19)
isarea-strictly continuous onM(W ;RN).
Remark2.2. Noticethatifm ∈M(Rd;RN),thenm e →m area-strictly, wherem e is
themollificationofm withafamilyofstandardconvolution kernels, m e :=m ∗r e and
r e (x):=e −dr (x/e )forr ∈C¥c(B1)positiveandevenfunctionsatisfying r dx=1.
R
GeneralizedYoungmeasuresformasetofdualobjectstotheintegrandsinE(W ;RN).
Werecall briefly some aspects of this theory, which was introduced by DiPerna and
Majdain[DM87]andlaterextended in[AB97,KR10a].
Definition2.3(generalizedYoungmeasure). AgeneralizedYoungmeasure,parametrized
byanopensetW ⊂Rd,andwithvaluesinRN isatriple(n x,l n ,n x¥ ),where
(i) (n x)x∈W ⊂M(RN)isaparametrized familyofprobability measuresonRN,
(ii) l n ∈M+(W )isapositive finiteRadonmeasureonW ,and
(iii) (n ¥ ) ⊂ M(SN−1) is a parametrized family of probability measures on
x x∈W
theunitsphereSN−1.
Additionally, werequirethat
(iv) themapx7→n isweakly*measurable withrespecttoLd,
x
(v) themapx7→n x¥ isweakly*measurablewithrespecttol n ,and
(vi) x7→h|q|,n i∈L1(W ).
x
ThesetofallsuchYoungmeasuresisdenotedbyY(W ;RN).
Here,weak*measurability meansthatthefunctionsx7→hf(x, q),n i(respectively
x
x 7→ hf¥ (x, q),n x¥ i) are Lebesgue measurable (respectively l n -measurable) for all
Carathe´odory integrands f: W ×RN → R (measurable in their first argument and
continuous intheirsecond argument).
LOWERSEMICONTINUITYANDRELAXATIONOFINTEGRALFUNCTIONALS 9
Foranintegrand f ∈E(W ;RN)andaYoungmeasuren ∈Y(W ;RN),wedefinethe
dualityparingbetween f andn asfollows:
f,n := hf(x, q),n xidx+ hf¥ (x,q),n x¥ idl n (x).
ZW ZW
(cid:10)(cid:10) (cid:11)(cid:11)
In many cases it will be sufficient to work with functions f ∈ E(W ;RN) which
are Lipschitz continuous. The following density lemma can be found in [KR10a,
Lemma3]:
Lemma2.4. There exists acountable set of functions {f }={j ⊗h ∈C(W )×
m m m
C(RN):m∈N}⊂E(W ;RN) such that for two Young measures n ,n ∈Y(W ;RN)
1 2
theimplication
hhf ,n ii=hhf ,n ii ∀m∈N =⇒ n =n
m 1 m 2 1 2
holds. Moreover,alltheh canbechosentobeLipschitzcontinuous.
m
SinceY(W ;RN)iscontained inthedualspaceofE(W ;RN)viathedualitypairing
hhq,qii,wesaythatasequenceofYoungmeasures(n )⊂Y(W ;RN)convergesweakly*
j
ton ∈Y(W ;RN),insymbolsn ⇀∗ n ,if
j
f,n → f,n forall f ∈E(W ;RN).
j
(cid:10)(cid:10) (cid:11)(cid:11) (cid:10)(cid:10) (cid:11)(cid:11)
Fundamental for all Young measure theory is the following compactness result,
see[KR10a,Section3.1]foraproof:
Lemma2.5(compactness). Let(n )⊂Y(W ;RN)beasequenceofYoungmeasures
j
satisfying
(i) thefunctions x7→h|·|,n iareuniformlybounded inL1(W ),
j
(ii) supjl n j(W )<¥ .
Then,thereexistsasubsequence (notrelabeled)andn ∈Y(W ;RN)suchthatn ⇀∗ n
j
inY(W ;RN).
Young measures generated by means of periodic homogenization can be easily
computed, see[BM84].
Lemma2.6(oscillationmeasures). Letw∈L1 (Rd;RN)beaQ-periodicfunction
loc
andletm∈N. Definethe(Q/m)-periodic functions
w (x):=w(mx).
m
Then,
w ⇀w(x):= w(y)dy inL1(W ;RN),
m
ZQ
foreverymeasurableW ⊂Rd.
Moreover, thesequence (w ) ⊂L1 (Rd;RN)generates thehomogeneous (local)
m loc
Young measure n =(d ,0,d )∈Y (Rd;RN) (a Young measure restricted to every
w 0 loc
compactsubsetofRd),where
hh,d i:= h(w(y))dy forallh∈C(Rd)withlineargrowthatinfinity.
w
ZQ
10 A.ARROYO-RABASA,G.DEPHILIPPIS,ANDF.RINDLER
TheRadon–Nikody´mdecomposition(2.1)inducesanaturalembeddingofM(W ;RN)
intoY(W ;RN):m 7→d [m ],viatheidentification
(d [m ])x :=d ddLmd(x), l d [m ]:=|m s|, (d [m ])¥x :=d dd|mmss|(x).
Inthissense,wesaythatthesequenceofmeasures(m )generatestheYoungmeasure
j
n ifd [m ]⇀∗ n inY(W ;RN),insymbols
h
m →Y n .
j
Thebarycenter [n ]∈M(W ;RN)ofaYoungmeasuren ∈Y(W ;RN)isdefinedas
[n ]:= id,n =hid,n xiLd W +hid,n x¥ il n .
(cid:10)(cid:10) (cid:11)(cid:11)
Usingthenotation aboveitisclear thatfor(m )⊂M(W ;RN)itholds that m ⇀∗ [n ]
j j
asmeasuresonW ifm →Y n .
j
Remark2.7. Forasequence(m )⊂M(W ;RN)thatarea-strictlyconvergestosome
j
limitm ∈M(W ;RN),itisrelativelyeasytocharacterizethe(unique)Youngmeasure
it generates. Indeed, an immediate consequence of the Separation Lemma 2.4 and
Theorem2.1isthat
m →m area-strictly inW ⇐⇒ m →Y d [m ]∈Y(W ;RN).
j j
Insomecasesitwillbenecessarytodeterminethesmallestlinearspacecontaining
thesupportofaYoungmeasure. Withthisaiminmind,westatethefollowingversion
ofTheorem2.5in[AB97]:
Lemma 2.8. Let (u ) be a sequence in L1(W ;RN) generating a Young measure
j
n ∈Y(W ;RN)andletV beasubspace ofRN suchthatu (x)∈V forLd-a.e. x∈W .
j
Then,
(i) suppn ⊂V forLd-a.e.x∈W ,
x
(ii) suppn x¥ ⊂V∩SN−1 forl n -a.e.x∈W .
Finally,wehavethefollowingapproximation lemma,see[AB97,Lemma2.3]for
aproof.
Lemma2.9. ForeveryuppersemicontinuousBorelintegrand f: W ×RN →Rwith
linear growth at infinity, there exists a decreasing sequence (f )⊂E(W ;RN) of the
m
form f =(cid:229) l(m)j ⊗h (withthechoicesofj ,h depending onm)with
m j=1 j j j j
inf f = lim f = f, inf f¥ = lim f¥ = f# (pointwise).
m∈N m m→¥ m m∈N m m→¥ m
Furthermore, the linear growth constants of the h can be chosen to be bounded by
j
thelineargrowthconstant of f.
Byapproximation, wethusget:
Corollary 2.10. Let f: W ×RN →Rbeanuppersemicontinuous Borelintegrand.
Thenthefunctional
n 7→ hf(x, q),n xidx+ hf#(x,q),n x¥ idl n (x)
ZW ZW
issequentially weakly*uppersemicontinuous onY(W ;RN).