https://doi.org/10.1007/s00526-019-1544-x
Calculus of Variations
Potentials for
A
-quasiconvexity
Bogdan Rait,˘a1
Received: 15 October 2018 / Accepted: 29 March 2019 / Published online: 3 June 2019 © The Author(s) 2019
Abstract
We show that each constant rank operatorAadmits an exact potentialBin frequency space. We use this fact to show that the notion ofA-quasiconvexity can be tested against compactly supported vector fields. We also show thatA-free Young measures are generated by sequences Buj, modulo shifts by the barycentre.
Mathematics Subject Classification 49J45·35G05
1 Introduction
A challenging question in the study of non-linear partial differential differential equations is to find which non-linear functionals are well-behaved with respect to weak convergence, which represents the typical topology consistent with physical measurements and has satisfactory compactness properties. In the context of the Calculus of Variations, answering this question amounts, roughly speaking, to describing semi-continuity properties of functionals
E[w] =
ˆ
Ω f(w(x))dx (1)
with respect to weak convergence in certain weakly closed, convex subsets C, say, of Lp-spaces, 1< p<∞), under growth conditions
0 f c(| · |p+1) (2) on the integrandsf. Such subsetsCcan account for differential constraints and boundary con-ditions. Modulo terms removed for simplicity of exposition, such functionals could model, for instance, the energy arising from the deformation of a solid bodyΩ, viewed as a suffi-ciently regular open subset ofRn, where f is a continuous energy density map characterized by the constitutive properties of the material. In accordance with the Direct Method in the
Communicated by J. Ball.
This work was supported by Engineering and Physical Sciences Research Council Award EP/L015811/1.
B
Bogdan Rait,˘aCalculus of Variations, imposing a suitable bound from below on f ensures existence and weak compactness of minimizing sequenceswj. The appropriate continuity property ofE
in this case is that of lower semi-continuity with respect to weak convergence in Lp
wjw ⇒ lim inf
j→∞ E[wj] ≥E[w],
which, if satisfied, implies existence of a minimizerw∈C.
It is well-known that ifCconsists of the whole of Lp, then E is weakly sequentially lower semi-continuous if and only if f satisfying (2) is convex. Of course, convexity of f is sufficient for lower semi-continuity (always understood as weakly sequential throughout this note) in any reasonable classC, but it is hardly necessary in general. For instance, ifCis the space of weak gradients in L2andfis a quadratic form, then one can easily show that fbeing positive on rank-one matrices implies lower semi-continuity. This example, that we will later come back to in more generality, is of particular relevance, as it provides the insight for a second convexity condition, which is necessary for lower semi-continuity with the constraint w= ∇u: ifEis lower semi-continuous, then f is convex along rank-one lines. In particular, for integrandsf of class C2, this is equivalent to the so-called Legendre–Hadamard ellipticity condition
∂2F(X) ∂Xi j∂Xαβ
aiaαbjbβ≥0 for allX,a,b,
where summation over repeated indices is adopted. From this point of view, lower semi-continuity ofE acting on gradients reflects a semi-convexity condition on f. Indeed, it was shown byMorreyin [22] that lower semi-continuity ofEis equivalent withquasiconvexity of f, i.e., the Jensen-type inequality
f(η)
Q
f(η+ ∇u(x))dx
holds for allηand all smooth mapsu with compact support in the open cube Q. On one hand, the quasiconvexity assumption is a plausible constitutive relation for energy functionals arising in solid mechanics [5]; on the other hand, it is but a minor improvement of the lower semi-continuity concept, which makes it particularly difficult to check in applications. The counterexample ofŠverák[32] rules out the possibility of quasiconvexity being a type of directional convexity (see also [7, Ex. 3.5] for the case of higher order gradients). A tractable sufficient condition for quasiconvexity ispolyconvexity, i.e., f is a convex functions of the minors, also introduced byMorreyin [22] in connection with lower semi-continuity and used byBallto obtain existence theorems under very mild growth conditions, giving very satisfactory existence results in non-linear elasticity [4]. The fact that quasiconvexity does not imply polyconvexity is much easier to see, at least in higher dimensions, and follows from an old observation ofTerpstraconcerning quadratic forms [37] (see also [2,6] and the references therein).
continuous with respect to weak convergence inC[24]. The latter question was also studied in generality byBall,Currie, andOlverin [7], leading to the generalization of polyconvexity to the case where energy functionals depend on higher order derivatives. In this case, the definition of quasiconvexity extends mutatis mutandis [20]. As to the question of lower semi-continuity, the analysis of the case when fis a quadratic form (see, e.g., [36, Ch. 17] or [35, Thm. 2]) reveals a different necessary condition of directional convexity, namely with respect to the so-called wave cone ofA. It was shown byDacorognain [12, Thm. I.2.3] that, in order to have lower semi-continuity, it is sufficient to assume the following generalization of quasiconvexity, namely that
f(η)
Q
f(η+w(x))dx
for allη and all boundedw such that´Qw = 0 andAw = 0. However, it is not clear whether this condition is necessary. More recently,FonsecaandMüllershowed in [16] that if one assumes in addition that the fieldsware periodic, in which case f is called A-quasiconvex, then one indeed obtains a necessary and sufficient condition1(under suitable growth assumptions on f). Their result holds under the assumption that the symbol map A(·)ofAis a constant rank matrix-valued field away from 0. This condition, introduced in [30, Def. 1.5] to prove coerciveness inequalities for non-elliptic systems, was first used in the context of compensated compactness byMuratand ensures, as noted on [23, p.502], the continuity of the map
0 =ξ→ProjkerA(ξ), (3)
making tools from pseudo-differential calculus available. In the absence of the constant rank assumption, little is known about the lower semi-continuity problem. One of the few results in this direction was proved byMüllerin [25], answering a long standing question ofTartar (see also [18] for a generalization).
In the proof of the main result of [16], considerable difficulty is encountered when proving sufficiency ofA-quasiconvexity. One reason for this is the absence of potential functions for A, which, if available, should allow one to test with compactly supported functions in the definition ofA-quasiconvexity and, perhaps, use more standard methods.
The main result of the present work is to show that the existence of such a potential in Fourier space is equivalent with the constant rank condition.
Theorem 1 LetAbe a linear, homogeneous differential operator with constant coefficients onRn. ThenAhas constant rank if and only if there exists a linear, homogeneous differential operatorBwith constant coefficients onRnsuch that
kerA(ξ)=imB(ξ) (4)
for allξ ∈Rn\ {0}.
HereA(·),B(·)denote the (tensor-valued) symbol maps of, respectively,A,B. We say that Ahasconstant rankif the map 0 =ξ → rank A(ξ)is constant (see Sect.2for detailed notation). We will regardBas thepotentialandAas theannihilator, although this terminology is not standard.
It is important to mention that the algebraic relation (4) doesnot, in general, imply for vector fieldswthat
Aw=0 ⇒ w=Bu for someu. (5)
To see this, simply takeA= ∇k. In turn, if we impose restrictions onwthat allow for usage
of the Fourier transform, (5) can be shown to hold (Lemma2). As a consequence, standard arguments in the Calculus of Variations lead to the fact that a map f isA-quasiconvex if and only if
f(η)
Q
f(η+Bu(x))dx
for allηand all smooth vector fieldsusupported in an open cubeQ(Corollary1). It is also the case that, under the constant rank condition, the notions ofA-quasiconvexity [16, Def. 3.1]
andDacorogna’sA-B-quasconvexity [12, Eq. (A.12)] coincide. In particular, one can define
A-quasiconvexity via integration over arbitrary domains (Lemma3). As a consequence, the lower semi-continuity properties of functionals (1) in the (asymptoticallyA-free) topologies considered in [3,16], which are natural from the point of view of compensated compactness theory, rely only on the structure of the potentialB.
In fact, we will show that theA-quasiconvex relaxation of a continuous integrand can be described in terms ofBonly. From this point of view, it is natural to investigate the Young measures generated by sequences satisfying differential constraints [16, Sec. 4], as they efficiently describe the minimization of energies that are not lower semi-continuous. We recall that the role of parametrized measures for non-convex problems in the Calculus of Variations was first recognized byYoungin the pioneering works [39–41]. See the monographs [26,27] for a modern, detailed exposition.
Roughly speaking, for 1< p < ∞, we consider a sequencewj converging weakly in
Lpwhich is asymptoticallyA-free and generates a Young measureν. Technically speaking, it suffices to takeAwj to be strongly compact in W−lock,p, wherek is the order ofA. This is (slightly more general than) the topology considered in [16, Rk. 4.2(i)] and is consistent with the topology considered in compensated compactness (see, e.g., [36, Thm. 17.3], which essentially deals with the case of linear Euler–Lagrange equations). In this setting, we will show that the Young measureνis generated by a sequence of smooth mapsBuj, modulo a
shift by the barycentre (Proposition1).
To sum up, under the constant rank condition on the annihilatorA, the objects characteriz-ing the lower semi-continuous relaxation of functionals defined onA-free vector fields (i.e., A-quasiconvex envelopes andA-free Young measures) can be described only in terms of the potentialBconstructed in Theorem1. From this point of view, it is the author’s opinion that the study of functionals
E[w] =
ˆ
Ω f(x, w(x))dxforAw=0 and F[u] =
ˆ
Ω f(x,Bu(x))dx is essentially dual (strictlyunder the constant rank condition). See also [13] and the Appendix of [12].
Since testing with the appropriate quantity is fundamental in the study of partial differential equations, we hope that the observations made in this work will increase the flexibility of analyzing functionals in either class described above. On the other hand, the functionalF seems better suited for incorporating boundary conditions, which will be pursued elsewhere. This paper is organized as follows: In Sect.2we prove the main Theorem1, in Sect.3
(Corollary1), and in Sect. 4we prove that A-free Young measures are shifts of Young measures generated by sequencesBuj.
2 Proof of Theorem
1
We take a moment to clarify notation. By ak-homogeneous, linear differential operatorA onRnfromW toXwe mean
Aw:=
|α|=k
∂αA
αw forw:Rn→W, (6)
whereAα ∈Lin(W,X)for all multi-indicesαsuch that|α| =k, for finite dimensional inner product spacesW,X. We also define the (Fourier) symbol map
A(ξ):= |α|=k
ξαAα∈Lin(W,X) forξ∈Rn.
We also recall the condition mentioned above thatAis ofconstant rankif there exists a natural numberrsuch that
rankA(ξ)=r for allξ ∈Rn\ {0}.
As to the resolution of Theorem1, we recall the notion of (Moore–Penrose)generalized inverse, introduced independently in [8,21,28], to which we refer plainly as the pseudo-inverse, although the terminology is not standard. For a matrixM∈RN×m, its pseudo-inverse M†is the uniquem×N matrix defined by the relations
M M†M=M, M†M M†=M†, (M M†)∗=M M†, (M†M)∗=M†M,
whereM∗denotes the adjoint (transpose) of M. Equivalently, the pseudo-inverse is deter-mined by the geometric property thatM M†andM†Mare orthogonal projections onto imM and(kerM)⊥ respectively. We refer the reader to the monograph [10] for more detail on generalized inverses.
With these considerations in mind, it is easy to see that the projection mapP∈C∞(Rn\
{0},Lin(W,W))defined in (3) can be represented as
P(ξ)=IdW−A†(ξ)A(ξ) forξ ∈Rn\ {0}. (7)
The smoothness ofPis well-known [16, Prop. 2.7]; for a proof using pseudo-inverses see [29, Sec. 4]. By the basic properties of pseudo-inverses, it is easy to see that, with the choice B=P, we have that (4) holds; however, the tensor-valued mapPis 0-homogeneous, hence not polynomial in general. In particular,Pcannot define a differential operator.
On the other hand, motivated by a similar construction in [38, Rk. 4.1], one can speculate thatPand, in fact,A†(·)are rational functions. This is indeed the case, as a consequence of the main result ofDecellin [15], building on the fundamental result of Penrose[28, Thm. 2] and the Cayley–Hamilton Theorem.
Theorem 2 (Decell [15, Thm. 3])Let M∈RN×mand denote by
p(λ):=(−1)N
a0λN +a1λN−1+ · · · +aN
forλ∈R
the characteristic polynomial of M M∗, where a0=1. Define
Then, if r=0, we have that M†=0; else
M†= −ar−1M∗a0(M M∗)r−1+a1(M M∗)r−2+ · · · +ar−1IdN×N
.
Proof (Proof of Theorem1, sufficiency) Suppose thatAhas constant rank. We putM:=A(ξ) in the above Theorem forξ∈Rn\{0}, and abbreviateH(ξ):=A(ξ)A∗(ξ). The first, perhaps most crucial, observation is thatr(ξ), as defined by (8), equals the number of non-zero eigen-values ofM M∗, which equals the number of singular values ofM. This is, in turn, equal to rank M, which is independent ofξby the constant rank assumption onA.
Therefore, ifr(ξ) = r = 0, we have thatA(ξ) = 0N×m,A†(ξ) = 0m×N, so we can
simply chooseB(ξ) = IdW, which satisfies (4) and gives rise to a linear, 0-homogeneous
differential operator. Otherwise, ifr(ξ)=r>0, we obtain
A†(ξ)= −a
r(ξ)−1A∗(ξ)
a0(ξ)H(ξ)r−1+a1(ξ)H(ξ)r−2+ · · · +ar−1(ξ)IdX
.
It is easy to see thatH(·)is a tensor-valued polynomial inξ. The scalar fieldsaj,j=1. . .r,
are such thataj(ξ)is a coefficient of the characteristic polynomial ofH(ξ), hence a linear
combination of minors. In particular,aj are scalar-valued polynomials inξ.
It then follows that, withPas in (3),
B(ξ):=ar(ξ)P(ξ)=ar(ξ)IdW−ar(ξ)A†(ξ)A(ξ) forξ ∈Rn (9)
defines a tensor-valued polynomial that satisfies (4). In particular, (9) gives rise to a linear differential operator. To check that it is homogeneous, it suffices to see thatar(·)is a linear
combination of minors of the same order ofH(·), which is homogeneous sinceA(·)is.
The necessity of the constant rank condition in Theorem1follows from the following Lemma and the Rank–Nullity Theorem.
Lemma 1 Let S⊂Rnbe a set of positive Lebesgue measure and P,Q be two matrix-valued polynomials onRn. Suppose that there exists s such that
rank P(ξ)+rank Q(ξ)=s forξ ∈S.
Then both P and Q have constant rank in S.
Proof We abbreviateRP:=rank P,RQ:=rank Qand assume for contradiction thatRPis
not constant inS. SayRP(S) = {r1,r1+1. . . ,r2}for natural numbersr1 <r2. We also write Md for the map that has input a matrix and returns (a vector of) all its minors of order
d. In particular, MdP, MdQare vector-valued polynomials onRn. We then have that
R−P1({r1,r1+1. . .r2−1})⊂ {ξ∈Rn:Mr2P(ξ)=0},
so that either Mr2P≡0 (which is not the case by definition ofr2) orR− 1
P ({r1,r1+1. . .r2−1}) is Lebesgue-null.2On the other hand,
R−P1({r2})∩S= R−Q1({s−r2})∩S
⊂ R−Q1({s−r2,s−r2+1, . . .s−r1−1})
⊂ {ξ∈Rn:M
s−r1Q(ξ)=0},
which is Lebesgue-null by the same argument. Since
S= [R−P1({r1,r1+1, . . .r2−1})∩S] ∪ [R−P1({r2})∩S],
it follows thatSis Lebesgue-null and we arrive at a contradiction.
It is natural to ask the reversed question, whether a constant rank operatorBadmits an exact annihilatorA. This is indeed the case, as can be shown by a simple modification of the argument above:
Remark 1 LetBbe a linear, homogeneous, differential operator ofconstant rankonRnfrom V toW. Then, we can chooseM:=B(ξ)forξ ∈Rn\ {0}in Theorem2, so that
A(ξ):=ar(ξ)
IdW−B(ξ)B†(ξ)
forξ∈Rn
satisfies (4) and gives rise to a differential operator. In particular, the formula is consistent with [38, Eq. (4.3)]. This fact can be used to extend the L1-estimates in [9,38] from elliptic to constant rank operators.
We conclude the discussion of algebraic properties with two remarks: Firstly, it is quite convenient that the two constructions presented are explicitly computable. On the other hand, performing the computations on simple examples, e.g., involving only div, grad, curl, one easily notices that the operators constructed via our formulas are often over complicated. Perhaps more computationally efficient methods, e.g., in the spirit of [38, Sec. 4.2] can be developed.
3
A
-quasiconvexity
The relevance of Theorem1for analysis can be seen, for instance, from the fact that periodic A-free fields have differential structure:
Lemma 2 Let A,B be linear, homogeneous, differential operators of constant rank with constant coefficients onRn from W to X , and from V to W , respectively. Assume that(4) holds. Then for allw∈C∞(Tn,W)such thatAw=0and
´
Tnw(x)dx =0, there exists u ∈C∞(Tn,V)such thatw =Bu. Similarly, for allw ∈S(Rn,W)such thatAw =0,
there exists u∈S(Rn,V)such thatw=Bu.
HereTndenotes then-dimensional torus, identified in an obvious way with (a quotient of) [0,1]n. The Fourier transform is defined as
ˆ
u(ξ):= ˆ
Tnu(x)e
−2πix·ξdx, (10)
forξ ∈ Zn andu ∈ C∞(Tn). In addition,S(Rn)denotes the Schwartz class of rapidly
Proof Letw∈C∞(Tn,W)have zero average and satisfyAw=0, so that
w(x)= ξ∈Zn\{0}
ˆ
w(ξ)e2πix·ξ,
forx ∈ Tn, where the coefficientsw(ξ)ˆ ∈ kerA(ξ)decay faster than any polynomial as |ξ| → ∞. We define
u(x):= ξ∈Zn\{0}
B†(ξ)w(ξ)ˆ e2πix·ξ,
forx ∈ Tn, which is smooth by homogeneity ofB†(·): sayB has orderl, then B†(·)is
(−l)-homogeneous. We can thus differentiate the sum term by term to obtain Bu(x)=(2πi)l
ξ∈Zn\{0}
B(ξ)B†(ξ)w(ξ)ˆ e2πix·ξ
=(2πi)l
ξ∈Zn\{0}
ˆ
w(ξ)e2πix·ξ
=(2πi)lw(x),
where the exactness relation (4) is used in the second equality, along with the geometric properties of the pseudo-inverse. The proof of the first case is complete.
We give an analogous argument for the case whenw∈S(Rn,W)isA-free. We have the pointwise relationA(ξ)w(ξ)ˆ =0, so that (4) implies thatw∈imB(ξ)and we can define
ˆ
u(ξ):=B†(ξ)w(ξ),ˆ
which satisfies the required properties.
We conclude this Section by showing that one can test with compactly supported smooth maps in the definition ofA-quasiconvexity.
Corollary 1 LetA,Bbe as in Lemma2and f:W →Rbe Borel measurable and locally bounded. Then
QAf(η):=inf ˆ
Tn f(η+w(x))dx:w∈C
∞(T
n,W),Aw=0,
ˆ
Tnw(x)dx =0
,
QBf(η):=inf ˆ
[0,1]n f(η+Bu(x))dx:u∈C
∞
c ((0,1)n,V)
are equal for allη∈W . Moreover, ifBhas order l andα∈ [0,1), we have
QAf(η)=inf ˆ
[0,1]n
f(η+Bu(x))dx:u∈C∞c ((0,1)n,V),uCl−1,α< ε
(11)
for anyη∈W andε >0.
The proof follows standard arguments; in particular we follow [14, Prop. 5.13] and [17, Thm. 4.2] and include the proof for completeness of the present work.
Proof It is obvious thatQAf QBf. To prove the opposite inequality, letε >0,η∈W, and wbe a periodic field as in the definition ofQAf(η). We will constructv∈C∞c ((0,1)n,V) such that
ˆ
[0,1]n
f(η+Bv(x))dx ˆ
[0,1]n
By Lemma2, we have thatw=Bufor a periodic fieldu∈C∞(Tn,V). Say, as before, that
Bhas orderland defineuN(x):=N−lu(N x)forN sufficiently large. This does not change
the value of the integral over the cube. Next, letδ >0 be sufficiently small and truncate to obtainuδN:=ρδuN, whereρδ∈C∞c ([0,1]n)is such thatρδ(x)=1 if dist(x, ∂[0,1]n) > δ
and|∇jρδ|Cδ−jforj=0. . .land some constantC >0. We imposeδN ≥1 and leave
δto be determined. It follows, forc1 ≥1 depending onBonly, that
|BuδN||ρδBuN| +c1
l
j=1
|∇jρδ||∇l−ju N|
c1C ⎛
⎝BuL∞+
l
j=1
(δN)−j∇l−juL∞ ⎞ ⎠
c1C ⎛
⎝BuL∞+
l−1
j=0
∇ju
L∞ ⎞
⎠=:c1CuWB,∞.
Say f is bounded byM >0 on B(0,|η| +c1CuWB,∞). Hence, if we chooseδsuch that Ln({x ∈ [0,1]n:dist(x, ∂[0,1]n)δ})M−1ε, we obtain
ˆ
[0,1]n f(η+Bu δ
N(x))dx
ˆ
dist(x,∂[0,1]n)<δMdx+ ˆ
[0,1]n f(η+BuN(x))dx
M×M−1ε+ ˆ
[0,1]n
f(η+w(x))dx,
which implies (12) withv:=uδN. To prove the equality of the two envelopes, we distinguish two cases: IfQAf(η) >−∞, we can choosewsuch that
ˆ
[0,1]n
f(η+w(x))dxQAf(η)+ε,
and we conclude thatQAf(η) = QBf(η)by (12) sinceε >0 is arbitrary. IfQAf(η)=
−∞, we choosewsuch that
ˆ
[0,1]n f(η+w(x))dx−ε
−1,
so that we can conclude by (12) thatQBf(η)= −∞.
To prove (11), we need only show that the infimum is smaller than the envelope. Firstly, note as above that by replacinguwithuN(x)=N−lu(N x), whereuis extended by
period-icity toRn, the value of the integral does not change. It suffices to chooseN large enough so thatuN has small Cl−1,α-norm. Note that for j=0. . .l−1 we have
∇ju
N∞=Nj−l∇ju∞,
which can clearly be made arbitrarily small.
Finally, to check the Hölder bound, say that{zi+ [0,N−1]n}N
n
|∇l−1u
N(x)− ∇l−1uN(y)| =N−1|∇l−1u(N x−zi)− ∇l−1u(N y−zi)|
∇lu∞|x−y|
(√n N−1)1−α∇lu∞|x−y|α,
which can be made small since 1−α >0. Ifx,y lie in different cubes, which we label Qx,Qy. Letx¯ ∈∂Qx ∩(x,y),y¯ ∈ ∂Qy∩(x,y), so that|x −y| ≥ |x − ¯x| + |y− ¯y|, |x− ¯x|,|y− ¯y|√n N−1, and all derivatives ofuNvanish nearx¯,y¯. Using these facts and
the previous step we get
|∇l−1u
N(x)− ∇l−1uN(y)||∇l−1uN(x)− ∇l−1uN(x¯)| + |∇l−1u
N(y)− ∇l−1uN(y¯)|
(√n N−1)1−α∇lu∞|x− ¯x|α+ |y− ¯y|α
(√n N−1)1−α∇lu∞2−α|x−y|α,
where the last inequality follows by concavity and monotonicity of 0t →tα. The proof
is complete.
Remark 2 Using the argument in Corollary1, one can show for constant rank operatorsA thatA-quasiconvexity, as defined byFonsecaandMüllerin [16, Def. 3.1], coincides with A-B-quasiconvexity, as introduced byDacorognain [12,13] (to be precise, in the original definition ofA-B-quasiconvexity, the operatorBis assumed to be of first order, but this is only a minor technical restriction). In this case, it is not difficult to prove that [13, Thm. 4] is essentially unconditional. A proof of this fact will be given elsewhere.
We also have thatA-quasiconvexity can be defined by integrals over arbitrary domains, instead of cubes.
Lemma 3 LetA,Bbe as in Lemma2and f:W →Rbe Borel measurable, locally bounded, andA-quasiconvex, andΩbe a bounded open set. Then
f(η)
Ω f(η+Bv(y))dy
for allη∈W andv∈C∞c (Ω,V).
The proof follows from a simple argument in the Calculus of Variations [14, Prop. 5.11].
Proof Fixη∈W,v∈C∞c (Ω,V), extended by zero toRn. By the argument in the proof of Corollary1, we writeC:=(0,1)n and have that
f(η) ˆ
C
f(η+Bu(x))dx
for allu∈C∞c (C,V). For sufficiently smallε >0, we can findx0∈Rnsuch thatx0+εΩ⊂ C. We define
u(x):=εlv
x−x0 ε
so that
f(η) ˆ
C
f(η+Bu(x))dx= |C\(x0+εΩ)|f(η)+
ˆ
x0+εΩ
f(η+Bu(x))dx
=(1−εn|Ω|)f(η)+
ˆ
Ω f(η+Bv(y))ε
ndy.
Rearranging the terms we obtain the conclusion.
4
A
-free Young measures
We recall the definition of oscillation Young measures, while also giving a simplified variant of the Fundamental Theorem of Young measures.
Theorem 3 (FTYM, [26,27])LetΩ ⊂Rnbe a bounded, open set and zj ∈L1(Ω,Rd)be
a bounded sequence inL1. Then there exists a subsequence (not relabeled) and a weakly-* measurable mapν:Ω→P(Rd)(orparametrized measureν=(νx)x∈Ω) such that for all
f ∈C(Ω×Rd)we have that
lim inf
j→∞
ˆ
Ω f(x,zj(x))dx ≥
ˆ
Ωf(x,·), νxdx Moreover,
lim
j→∞
ˆ
Ω f(x,zj(x))dx=
ˆ
Ωf(x,·), νxdx if and only if the sequence f(·,zj)is uniformly integrable.
Above,P(Rd)denotes the space of probability measures onRd. In the notation of Theorem3,
we say thatzj generates the Young measureν(in symbols,zj
Y
→ν). We also recall that a sequencezj is said to be uniformly integrable if and only if for allε >0, there existsδ >0
such that for all Borel setsE⊂Ω, we have that
Ln(E) < δ ⇒ sup j
ˆ
E|
zj|dx < ε,
or, equivalently, if
lim α→∞supj
ˆ
{|zj|>α}
|zj|dx =0.
If|zj|pis uniformly integrable, we say thatzjis p-uniformly integrable.
Lemma 4 [16, Prop. 2.4]Let zjgenerate a Young measureνandz˜j → ˜z in measure. Then
zj+ ˜zj generates the Young measureμgiven byμx =νxδz˜(x)forLna.e. x, i.e., ϕ, μx = ϕ(· + ˜z(x), νx
for anyϕ∈C0.
Proposition 1 LetA,Bbe as in Lemma2and have orders k, l, respectively,Ω ⊂Rnbe a bounded Lipschitz domain, and1< p<∞. Letwj, w∈Lp(Ω,W)be such that
wjwin Lp(Ω,W),
Awj →Awin W−kloc,p(Ω,X),
wj
Y
→ν.
Then there exists a sequence uj∈C∞c (Ω,V)such that
Buj0in Lp(Ω,W),
Buj+w→Y ν.
Moreover, uj can be chosen such that(Buj)jis p-uniformly integrable.
A Young measureνsatisfying the assumptions of Proposition1is said to be anA-free Young measure.
Proof By Lemma4and linearity we can assume thatw=0. We will identify maps defined onΩ with their extensions by zero to full-space without mention. Uniform integrability considerations strictly refer to sequences defined onΩ.
Step I. We construct p-uniformly integrable w˜j ∈ Cc∞(Ω,W) such that w˜j0 in
Lp(Ω,W),Aw˜j →0 in W−k,q(Rn,X)for some 1<q< p, andw˜j generatesν.
Recall the truncation operators, defined forα >0 by
ταA:=
A if|A|α αA/|A| if|A|> α,
which are clearly Carathéodory integrands. By Theorem3, we have that
lim α→∞jlim→∞
ˆ
Ω|ταwj|
pdx = lim
α→∞
ˆ
Ω
ˆ
W
|ταA|pdνx(A)dx
=
ˆ
Ω
ˆ
W
|A|pdνx(A)dx<∞,
so that we can choose a diagonal subsequenceαj ↑ ∞such that
´
Ω|ταjwj|
pdxequals the
pth moment ofν. It also follows from Theorem3that(ταjwj)j isp-uniformly integrable.
We now show thatταjwj generatesν. Sincewjconverges weakly in Lp(Ω,W), it
con-verges weakly in L1, hence is uniformly integrable, so thatταjwj−wj→0 in measure. It also follows by elementary manipulations thatταjwj−wj0 in Lp, so that, indeed,ταjwj generatesνby Lemma4.
Let 1<q< p. We have that
ταjwj−wjLq(Ω,W) ˆ
{|wj|>αj}
2q|wj|qdx 2qαqj−p
ˆ
{|wj|>αj}
|wj|pdx→0,
so that Aταjwj → 0 in W
−k,q
loc (Ω,X). We also record that ταjwj is precompact in W−1,q(Ω,W), so thatDβταjwj →0 in W−k,q(Ω,X)for|β|<k.
We can therefore choose a sequence of cut-off functionsρj ∈C∞c (Ω,[0,1])such that
ρj ↑1 inΩandρjAταjwjW−k,q(Rn,X)→0 and
A(ρjταjwj)=ρjAταjwj+
k
m=1
Bm[Dmρj,Dk−mταjwj] →0 in W−k,
whereBmare fixed bi-linear pairings given by the Leibniz rule. To see that this is possible,
considerΩj:={x∈Ω:dist(x, ∂Ω) < j}, wheresj ↓0 will be determined. We require that
ρj =1 inΩ\Ωsj,ρj =0 inΩ2sj and|D
mρ
j|cs−jm,m=1, . . . ,k. It is easy to see that
the sum above is controlled in W−k,qby
k
m=1
DmρjL∞Dk−mταjwjW−k,q c
k
m=1
s−jmDk−mταjwjW−k,q,
so that it suffices to choose anysj ≥maxm=1,...,kDk−mταjwj 1/(2m)
W−k,q ↓0 asj→ ∞. Alter-natively, one can consider a different cut-off sequenceρi ↑1 and employ a diagonalization
argument. We define
˜
wj:=(ρjταjwj)ηε(j),
whereηε(j)denotes a standard sequence of (radial, positive) mollifiers andε(j)↓0 is such
thatw˜j ∈ C∞c (Ω,W)and, therefore,Aw˜j → 0 in W−k,q(Rn,X). The latter inequality
follows since, for allϕ∈C∞c (Rn,W)withϕWk,q 1,
Aw˜j, ϕ = A(ρjταjwj), ϕηε(j)A(ρjταjwj)W−k,qϕηε(j)Wk,q
A(ρjταjwj)W−k,q →0.
It is also clear that ˜wj−ταjwjLp →0, so thatw˜j is p-uniformly integrable, converges weakly to 0 in Lp, and generatesν.
Step II. We projectw˜j on the kernel ofAinRn and show thatPw˜j are p-uniformly
integrable inΩ, converge weakly to zero in Lp, and generateν. Here the L2-orthogonal projection operatorPis given by the multiplier in (7),
Pw(ξ):=P(ξ)w(ξ)ˆ = [IdW−A†(ξ)A(ξ)] ˆw(ξ) forw∈S(Rn,W).
Since the symbolP(·)is homogeneous of degree zero,Pis a singular integral operator of convolution type; in particularPmaps Schwartz functions to Schwartz functions. Moreover, we have that
Fw˜j−Pw˜j
(ξ)=B†(ξ)B(ξ)Fw˜
j(ξ)=A†
ξ
|ξ|
Aw˜
j(ξ) |ξ|k ,
so that, by boundedness of singular integrals on Lq
˜wj−Pw˜jLq(Rn,W)c F−1
Aw˜j | · |k
Lq(Rn,X)
=cAw˜jW−k,q(Rn,X)→0.
It immediately follows by Lemma4thatPw˜jgeneratesν. To see thatPw˜j0 in Lp(Ω,W),
we note that, sincePis (pointwisely) self-adjoint, we have, for anyg∈Lp/(p−1)(Ω,W), ˆ
Ωg,Pw˜jdx =
ˆ
ΩPg,w˜jdx→0,
sincePg∈Lp/(p−1)(Ω,W)by boundedness of singular integrals.
To see thatPw˜j is p-uniformly integrable, we use the idea in [16, Lem. 2.14.(iv)]. We
first note, by boundedness ofPon Lp, that
sup
j
Pw˜j−Pταw˜jLp(Rn,W)csup
j
byp-uniform integrability ofw˜j. Note that for each fixedα,Pταw˜jis bounded in Lrfor any
p<r<∞, hence isp-uniformly integrable. Letε >0. We chooseα >0 such that
sup
j Pw˜j−Pταw˜jL
p(Rn,W)< ε
and also chooseδ >0 such that for each Borel setE⊂ΩwithLn(Ω) < δ, we have that
´
E|Pταw˜j|pdx < εfor all j. It follows that for all suchE,
ˆ
E
|Pw˜j|pdx2p−1
sup
j
ˆ
E
|Pw˜j−Pταw˜j|pdx+sup j
ˆ
E
|Pταw˜j|pdx
< (2ε)p,
where the right hand side is independent of j. The second step is concluded.
Step III. Using Lemma2, we can writePw˜j=Buj, whereuˆj(ξ):=B†(ξ)Pw˜j(ξ), so that
uj ∈S(Rn,V). It remains to cut-offujsuitably.
SinceBhas orderl, we first note that
Dlu(ξ)=B†(ξ)Bu(ξ)⊗ξ⊗l,
so thatBu→Dluis a singular integral operator of convolution type. It follows thatDlu jis
bounded in Lp(Rn)(recall here thatBuj=Pw˜j is bounded in Lpasw˜j ∈C∞c (Ω,W)is a
weakly convergent sequence), soujis bounded in Wl,p(Ω,V).
By compactness of the embedding Wl,p(Ω) → Wl−1,p(Ω), we have uj → u
in Wl−1,p(Ω,V). Since Buj0, we have that Bu = 0. On the other hand, u = F−1[B†(·)](Bu)=0, so thatDl−mu
j →0 in Lp(Ω)form=1, . . . ,l.
We now proceed similarly to Step I. Letρ∈C∞c (Rn)be such thatρj =1 inΩ\Ωsj and
|Dmρj|cs−jm,m=1, . . . ,l, where
sj:= max m=1,...,lD
l−mu
j1L/(p(Ω)2m)→0.
We can then estimate
Buj−B(ρjuj)Lp(Ω)(1−ρj)BujLp(Ω)+
l
m=1
Bm[Dmρj,Dl−muj]Lp(Ω)
BujLp(Ω s j)+c
l
m=1
s−mj Dl−mujLp(Ω),
which tends to zero by p-uniform integrability ofBuj and the choice ofsj. Here Bm is
another collection of bi-linear pairings given by the product rule. It remains to conclude that B(ρjuj)converges weakly to zero in Lp(Ω,W), is p-uniformly integrable, and generates
ν. The proof is complete.
Acknowledgements The author is grateful to Jan Kristensen for introducing him to the problem and for offering insightful comments and helpful suggestions.
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