warwick.ac.uk/lib-publications
Original citation:
de Philippis, Guido and Rindler, Filip. (2017) Characterization of generalized young measures
generated by symmetric gradients. Archive for Rational Mechanics and Analysis
Permanent WRAP URL:
http://wrap.warwick.ac.uk/86324
Copyright and reuse:
The Warwick Research Archive Portal (WRAP) makes this work of researchers of the
University of Warwick available open access under the following conditions.
This article is made available under the Creative Commons Attribution 4.0 International
license (CC BY 4.0) and may be reused according to the conditions of the license. For more
details see:
http://creativecommons.org/licenses/by/4.0/
A note on versions:
The version presented in WRAP is the published version, or, version of record, and may be
cited as it appears here.
Digital Object Identifier (DOI) 10.1007/s00205-017-1096-1
Characterization of Generalized Young
Measures Generated by Symmetric Gradients
Guido De Philippis & Filip Rindler
Communicated byI. Fonseca
Abstract
This work establishes a characterization theorem for (generalized) Young mea-sures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer–Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The “local” proof strategy combines blow-up arguments with the singu-lar structure theorem in BD (the analogue of Alberti’s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences.
1. Introduction
Young measures quantitatively describe the asymptotic oscillations in Lp -weak-ly converging sequences. They were introduced in [48–50] and later developed into an important tool in modern PDE theory and the calculus of variations in [8,9,44,45] and many other works. In order to deal with concentration effects as well, DiPerna and Majda extended the framework to so-called “generalized” Young measures, see [2,17,21,28,30,43]. In the following we will refer also to these objects simply as “Young measures”.
When considering generating sequences that satisfy a differential constraint like curl-freeness (i.e. the generating sequence is a sequence of gradients), the problem immediately arises to characterize the resulting class of Young measures. In applications, these results provide very valuable information on the allowed oscillations and concentrations that are possible under this differential constraint, which usually constitutes a strong restriction.
by gradients of W1,p-bounded sequences, 1< p ≤ ∞. Their theorems put such
gradient Young measures in duality with quasiconvex functions as introduced by Morrey [34]. For generalized Young measures the corresponding result was proved in [21] (also see [22]) and numerous other characterization results in the spirit of the Kinderlehrer–Pedregal theorems have since appeared, see for instance [11,19, 20,29].
Characterization theorems are of particular use in the relaxation of minimization problems for non-convex integral functionals, where one passes from a functional defined on functions to one defined on Young measures. A Kinderlehrer–Pedregal-type theorem allows one to restrict the class of Young measures over which to minimize. This is explained in detail (for classical Young measures) in [36]. A similar application is possible for generalized Young measures.
The characterization of generalized BV-Young measures was first achieved in [28]. A different, “local” proof was given in [40], another improvement is in [25, Theorem 6.2]. All of these arguments crucially use Alberti’s rank-one theorem [1] (see [31] for a short and elegant new proof) and thus, since this theorem is specific to BV, extensions to further BV-like spaces have been prohibited so far. The only partial result for a characterization beyond BV seems to be in [7], but that result is limited to first-order operators (which does not cover BD) and also additional technical conditions have to be assumed.
We now explain briefly the framework underlying this work and introduce some notation to state our main result; precise definitions are given in Section2. Given an L1-bounded sequence of mapsvj:→RN(⊂Rd), the Fundamental Theorem of (generalized) Young measure theory states that there exists a subsequence of the vj’s (which we do not relabel) such that for all continuous f:×RN →Rwith the property that therecession function
f∞(x,A):= lim x→x A→A
t→∞
f(x,t A)
t , x∈, A∈R
N
exists, it holds that
f(x, vj(x))dx→
f, ν:=
f(x, q), νx
dx+
f∞(x,q), νx∞dλν(x),
where (νx)x∈, (νx∞)x∈ are parametrized families of probability measures on RNand∂BN(the unit sphere inRN), respectively, andλ
νis a positive, finite Borel measure on. Together, we callν=(νx, λν, ν∞x )the(generalized) Young measure generated by the (subsequence of the)vj’s.
In plasticity theory [41,42,47], one often deals with sequences of uniformly L1-boundedsymmetric gradients
Euj := 1 2
∇uj+ ∇uTj
.
It is an important problem to characterize the (generalized) Young measures ν generated by such sequences(Euj). We call suchνBD-Young measuresand write
of sequences (uj) as above. Recall that a function u ∈ L1(;Rd)lies in the space BD()of functions of bounded deformation if its distributional symmetrized derivative Euis a bounded Radon measure ontaking values inRd×d
sym.
Our main result is the following:
Theorem 1.1.Letν ∈Y(;Rdsym×d)be a (generalized) Young measure. Then,νis
a BD-Young measure, ν ∈ BDY(), if and only if there exists u ∈ BD()with
[ν] = Eu and for all symmetric-quasiconvex h ∈C(Rdsym×d)with linear growth at
infinity, the Jensen-type inequality
hid, νx
+id, ν∞x dλν
dLd(x)
≤h, νx
+h#, νx∞ dλν
dLd(x)
holds atLd-almost every x ∈.
Here, thegeneralized recession function h#:RN →Rof a maph:RN →R
withlinear growth at infinity, i.e.|h(A)| ≤C(1+ |A|)for some constantC>0, is given as
h#(A):=lim sup A→A
t→∞
h(t A)
t , A∈R
N.
We remark that the use of the generalized recession function can in general not be avoided since not every quasiconvex function with linear growth at infinity has a (strong) recession function (and one needs to test with all those, but see [25, Theorem 6.2]). Further, a bounded Borel function f:Rdsym×d → Ris called
symmetric-quasiconvexif withEψ:=(∇ψ+ ∇ψT)/2,
f(A)≤ −
D
f(A+Eψ(y))dy for allψ∈W01,∞(D;Rd)and all A∈Rdsym×d.
For a suitable integrand f:×Rdsym×d →R, the minimum principle
f, ν →min, ν∈BDY() (1.1)
can be seen as theextension-relaxationof the minimum principle
f(x,Eu(x))dx+
f
∞x, dEsu d|Esu|(x)
d|Esu| →min, u ∈BD().
(1.2) The point is that (1.2) may not be solvable if f is not symmetric-quasiconvex, whereas (1.1) always has a solution. In this situation, our main Theorem1.1then gives (abstract) restrictions on the Young measures to be considered in (1.1). An-other type of relaxation involving the symmetric-quasiconvex envelope of f is investigated in [5] within the framework of general linear PDE side-constraints.
measures that can be generated by sequences in BD, see Section 3. These “spe-cial” Young measures originate from a blow-up procedure and are calledtangent Young measures. There are two types: regular and singular tangent Young measures, depending on whether regular (Lebesgue measure-like) effects or singular effects dominate around the blow-up point.
For the regular tangent Young measures the classical methods of Kinderlehrer & Pedregal [23,24] are applicable. In order to deal with singular tangent measures, we first need to strengthen the result on “good blow-ups” for Young measures with a BD-barycenter from [38], see Lemma 2.14, which is also interesting in its own right. We combine this lemma with the analogue of Alberti’s rank-one theorem in BD from [16], which imposes strong constraints on the underlying BD-deformation (discussed in Remark3.6). Glueing tangent BD-Young measures together, see Lemma4.2, we then prove Theorem1.1in Section4.
We stress that our argument crucially rests on the BD-analogue of Alberti’s rank-one theorem recently proved by the authors in [16], see Theorem2.12. The reason is that this result explains the local structure of singularities that can occur in BD-functions (more precisely, in the singular part of the symmetric derivative). A weaker version of this argument was already pivotal in the work [38]. However, to prove Theorem1.1, the strong version of [16] is needed. Technically, in one of the proof steps to establish Theorem1.1we need to create “artificial concentrations” by compressing symmetric gradients in one direction. This is only possible if we know precisely what these singularities look like, see Lemma3.5and also Remark3.6 for details. It is not clear to us if the use of Theorem2.12can be avoided to prove Theorem1.1.
As another very useful technical tool, we utilize the BD-analogue of the sur-prising observation by Kirchheim and Kristensen [25] that the singular part of a BV-Young measure isunconstrained. Without this observation, a weaker character-ization result could be established, where, however a second,singularJensen-type inequality needs to be required. Indeed, it was shown in Theorem 4 of [38] that in the situation of our theorem automatically also thesingularJensen inequality
h#id, νx∞
≤h#, ν∞x
forλsν-almost everyx∈
holds. It is remarkable (and due to the observations in [25] alluded to above) that this is, however, not needed to prove the characterization result.
The third central new ingredient in the proof is an argument yielding “very good” blow-ups at singular points, which are not only two-dimensional, but even one-dimensional (plus an affine part). This is achieved by iterating the blow-up construction (using the observation that “blow-ups of blow-ups are blow-ups”); see Lemma2.14for details.
2. Setup and Preliminary Results
In this section we recall all the notation and technical tools that will be employed in the subsequent sections. In particular, we collect many results from the framework of generalized Young measure, usually specialized to the BD-case.
2.1. Functions of Bounded Deformation
The space BD was introduced in [32,41,42,47] for applications in plasticity theory, much of the theory relevant to this work is developed in [3,6,13,26,46,47]. As a standing assumption throughout this whole work, let⊂Rdbe an open domain with Lipschitz boundary; in the following proofs we implicitly assumed ≥
2, but the main results are (trivially) true also ford =1 since then BV and BD agree and (symmetric-)quasiconvexity is just convexity. The space BD()offunctions of bounded deformationis defined as the space of functionsu∈L1(;Rd)such that the distributionalsymmetric derivative
Eu:= Du+Du
T
2
is (representable as) a finite Radon measure Eu∈M(;Rdsym×d). Clearly, BD() is a Banach space under the normuBD():= uL1(;Rd)+ |Eu|().
We splitEuaccording to the Lebesgue–Radon–Nikodým decomposition as
Eu=EuLd+Esu, Eu:= dEu
dLd ∈L
1(;Rd×d
sym),
where theapproximate symmetrized gradientEuis the Radon–Nikodým deriva-tive of Euwith respect to Lebesgue measure andEsu is thesingular partofEu
(with respect toLd).
Since there is no Korn inequality in L1, see [12,25,35], it can be shown that BV(;Rd) is a proper subspace of BD(). See [13] for further results in this direction.
Arigid deformationis a skew-symmetric affine mapr:Rd→Rd, i.e.uis of the form
r(x)=u0+Rx, whereu0∈Rd,R∈Rdskew×d.
We have the following Poincaré inequality: for eachu∈BD()there exists a rigid deformationrsuch that
u+rLd/(d−1)(;Rd)≤C|Eu|(), (2.1)
whereC = C()only depends on the domain. This is shown for example in [47] or see [46, Remark II.2.5].
2.2. Symmetric-Quasiconvexity
The appropriate generalized convexity notion related to symmetrized gradients is the following: we call a bounded Borel function f: Rdsym×d → R symmetric-quasiconvexif
f(A)≤ −
D
f(A+Eψ(y))dy for allψ∈W01,∞(D;Rd)and all A∈Rdsym×d,
where D ⊂Rd is any bounded Lipschitz domain. Similar assertions to the ones for quasiconvex functions hold, cf. [18] and [10]. In particular, if f has linear growth at infinity, we may replace the space W10,∞(D;Rd)in the above formula by W01,1(D;Rd). It can further be shown, see [20, Proposition 3.4], that any symmetric-quasiconvex f isconvexin the directionsR(ab)for anya,b∈Rd\ {0}.
Thesymmetric-quasiconvex envelopeS Q f:Rdsym×d →Rof a Borel function
f:Rdsym×d→Ris
S Q f(A)=sup g(A) : gsymmetric-quasiconvex andg ≤ f .
This expression is either identically −∞ or finite and symmetric-quasiconvex. Analogously to the case for usual quasiconvexity (cf. [15,23]), for continuous f, the symmetric-quasiconvex envelope can be written as
S Q f(A)=inf
−
D
fA+Eψ(z)dz : ψ∈W10,∞(D;Rd)
.
2.3. Generalized Young Measures
The following theory is mostly from [2,28,38], where also proofs and examples can be found.
Let again⊂Rdbe a bounded Lipschitz domain. For f ∈C(×RN)and
g∈C(×BN), whereBN denotes the open unit ball inRN, we let
(R f)(x,Aˆ):=(1− | ˆA|)f
x, Aˆ
1− | ˆA|
, x∈, Aˆ ∈BN, and (2.2)
(R−1g)(x,A):=(1+ |A|)g
x, A
1+ |A|
, x∈, A∈RN.
Clearly,R−1R f = f andR R−1g =g. Define
E(;RN):= f ∈C(×RN) : R f extends continuously onto×BN.
In particular, f ∈ E(;RN)has linear growth at infinity, i.e. there exists a constantM ≥0 (in fact,M = R fL∞(×BN)) with
Furthermore, for all f ∈ E(;RN), the(strong) recession function f∞: ×
RN →R, defined as
f∞(x,A):= lim x→x A→A
t→∞
f(x,t A)
t , x∈, A∈R
N, (2.3)
exists and takes finite values. Clearly, f∞is positively 1-homogeneous inA, that is
f∞(x, αA)=αf∞(x,A)for allα≥0. It can be shown that in fact f ∈C(;RN) is in the classE(;RN)if and only if f∞exists in the sense (2.3).
For f ∈C(×RN)with linear growth at infinity, f∞may not exist, but we can always define thegeneralized recession function f#:×RN →Rvia
f#(x,A):=lim sup x→x A→A
t→∞
f(x,t A)
t , x∈, A∈R
N.
It is easy to see that f#is always positively 1-homogeneous and upper semicontin-uous in its second argument. In many other works, f#is just called the “recession function”, but here the distinction to our (strong) recession function f∞is impor-tant.
A(generalized) Young measureν ∈ Y(;RN)⊂E(;RN)∗ on the open set⊂Rdwith values inRNis a tripleν=(νx, λν, νx∞)consisting of:
(i) a parametrized family of probability measures(νx)x∈⊂M1(RN), called
theoscillation measure;
(ii) a positive finite measureλν ∈M+(), called theconcentration measure; and
(iii) a parametrized family of probability measures (ν∞x )x∈ ⊂ M1(SN−1),
called theconcentration-direction measure,
for which we require that
(iv) the mapx →νx is weakly* measurable with respect toLd, i.e. the function
x→ f(x, q), νxisLd-measurable for all bounded Borel functions f:× RN →R;
(v) the mapx→ν∞x is weakly* measurable with respect toλν; and (vi) x→ |q|, νx ∈L1().
Equivalently to (i)–(vi), one may require
(νx)∈L∞w∗(;M1(RN)), λν ∈M+(),
(ν∞x )∈L∞w∗(, λν;M1(SN−1)), x→
|q|, νx
∈L1().
Theduality pairingbetween f ∈E(;RN)andν∈Y(;RN)is given as
f, ν:=
f(x, q), νx
dx+
f∞(x, q), νx∞dλν(x)
:=
RN f(x,A)dνx(A)dx+
∂BN f
∞(x,A)dν∞
The weak* convergenceνj ∗
νinY(;RN)⊂E(;RN)∗is then defined with respect to this duality pairing. If(γj)⊂M(;RN)is a sequence of measures with supj|γj|() <∞, then we say that the sequence(γj)generatesa Young measure
ν∈Y(;RN), in symbolsγj Y
→ν, if for all f ∈E(;RN)it holds that
f
x, dγj
dLd(x)
Ld + f∞
x, dγ
s j d|γs j|
(x)
|γs j|
∗
f(x, q), νx
Ld +f∞(x,q), ν∞ x
λν inM().
Here,γsj is the singular part ofγjwith respect to Lebesgue measure. Equivalently,
we could have definedγj Y
→νby requiring thatδγj ν∗ , whereδγjare “elementary Young measures” that are naturally associated with theγj.
Also, forν ∈Y(;RN)we define thebarycenteras the measure
[ν] :=id, νx
Ld + id, ν∞x
λν ∈M(;RN).
The following is the central compactness result inY(;RN):
Lemma 2.1.(Compactness)Let(νj)⊂Y(;RN)be such that
supj1⊗ |q|, νj
<∞.
Then,(νj)is weakly* sequentially relatively compact inY(;RN), i.e. there exists
a subsequence (not relabeled) such thatνj ν∗ andν∈Y(;RN).
In particular, if(γj)⊂M(;RN)is a sequence of measures with supj|γj|()
<∞as above, then there exists a subsequence (not relabeled) andν ∈Y(;RN) such thatγj
Y
→ν.
By a standard density argument it suffices to check weak*-convergence of Y-oung measures by testing with a countable set of integrands only, which is equivalent to theseparabilityof the spaceE(;RN):
Lemma 2.2.There exists a countable family{ϕ⊗h}∈N ⊂ E(;RN), where ϕ ∈ C()and h ∈ C(RN)such that for ν1, ν2 ∈ Y(;RN)the following
implication holds:
ϕ⊗h, ν1
=ϕ⊗h, ν2
for all∈N ⇒ ν1=ν2.
2.4. BD-Young Measures
A Young measure inY(;Rsymd×d)is called aBD-Young measure,ν ∈BDY(), if it can be generated by a sequence of BD-symmetric derivatives. That is, for all ν∈BDY(), there exists a (necessarily norm-bounded) sequence(uj)⊂BD() withEuj
Y
→ν. When working withBDY(), the appropriate space of integrands isE(;Rdsym×d), since it is clear that bothνxandνx∞only take values inRdsym×d
when-everν∈BDY(). It is easy to see that for a BD-Young measureν∈BDY()there existsu ∈ BD()satisfying Eu = [ν] ; any suchuis called anunderlying deformationofν.
The following results about BD-Young measures constitute a “calculus” for BD-Young measures, which will be used frequently in the sequel see [28,37,38] for proofs (the first reference treats BV-Young measures, but the proofs adapt line-by-line).
Lemma 2.3.(Good generating sequences)Letν∈BDY(). Then:
(i)There exists a generating sequence(uj)⊂BD()∩C∞(;Rd)with Euj Y → ν;
(ii)If, additionally, λν(∂) = 0, then the uj from (i) can be chosen to satisfy
uj|∂=u|∂, where u∈BD()is any underlying deformation ofν.
The proof of this result can be found in [28, Lemma 4].
Lemma 2.4.(Averaging) Letν ∈ BDY() satisfy λν(∂) = 0. Also, assume
[ν] =Eu for some u∈BD()satisfying on e of the following two properties:
(i)u agrees with an affine map on the boundary∂; or
(ii)is a cuboid with one face normalξ ∈Sd−1 =∂Bdand u isξ-directional, that is u(x)=ηh(x·ξ)with h∈BV(R)for someη∈Rd.
Then, there exists a Young measureν¯ ∈BDY()acting on f ∈E(;Rdsym×d)as
f,ν¯=
−
f(x, q), νy
dydx
+
−
f∞(x, q), ν∞y dλν(y)· λν()
|| dx. (2.4)
More precisely:
(1)The oscillation measure(ν¯x)x isLd-essentially constant in x and for all h ∈ C(Rdsym×d)with linear growth at infinity it holds that
h,ν¯x
= −
h, νy
dy a.e.;
(2)The concentration measureλν¯is a multiple of Lebesgue measure,λν¯ =αLd
(3)The concentration-direction measure(ν¯∞x )x isLd-essentially (λν¯-essentially)
constant and for all h∞∈C(∂Bdsym×d)it holds that
h∞,ν¯x∞= −
h∞, ν∞y dλν(y) a.e..
Remark 2.5.We remark that one may consider any averaged Young measure as in the preceding lemma to be defined onanybounded Lipschitz domain D⊂Rd, so thatν¯∈BDY(D)and (2.4) is replaced by
f,ν¯=
D −
f(x, q), νy
dydx
+
D −
f∞(x, q), ν∞y dλν(y)· λν()
|| dx
for any f ∈E(D;Rdsym×d). This can be achieved by a covering argument analogous to the proof of Lemma2.4in [28, Proposition 7] (coveringDwith rescaled copies of).
The proof of case (i) is contained in [28, Proposition 7], the proof of (ii) is similar, but requires an additional standard staircase (piecewise affine) construction to glue the rescaled versions ofu together without incurring an additional jump part. Let and(uj)be as in (ii). First, by Lemma2.3we may assume that there exists a sequence(uj) ⊂ BD()∩C∞(;Rd)with Euj
Y
→ νanduj|∂ = u|∂. For every j ∈ Nletaj k ∈ Rd be defined such that the similar rescaled setsj kl :=
aj kl+j−1,k =1, . . . , j,l=1, . . . ,jd−1, form a cover of. We furthermore assume that thej kl are arranged in a regular grid withj kl (l = 1, . . . ,jd−1) lying in thek’th slice inξ-direction.
Furthermore, denote byγ ∈ Rd the difference between the trace ofu on the face of u in the positiveξ-direction andu in the negativeξ-direction; note that because of the assumption thatuhas the shapeu(x)=ηh(x·ξ),γ is aconstant vector. Then define
wj(x):= ⎧ ⎨ ⎩
1
juj
j(x−aj kl)
+γk
j ifx ∈aj kl+ j
−1,
0 otherwise.
It is easy to see thatwj ∈W1,1(;Rm)(recall thatuj|∂=u|∂). For the weak derivative we get
∇wj(x)=
∇uj
j(x−aj k)
ifx∈aj k+ j−1, 0 otherwise.
Note that the staircase termγk/j is chosen precisely to annihilate the jumps over the slice boundaries in directionξ; over the other boundaries there are no jumps by the assumption on the shape ofu. It can now be checked, following line-by-line the proof of [28, Proposition 7], that thewj generateν¯as required in the lemma.
Corollary 2.6.(Generalized Riemann–Lebesgue lemma)Let u∈BD()that sat-isfied (i) or (ii) from the previous lemma. Then, for every open bounded Lipschitz domain D⊂Rdthere existsν∈BDY(D)that acts on f ∈E(D;Rd×d
sym)as
f, ν=
D −
f(x,Eu(y))dydx
+|Esu|()
||
D −
f
∞x, dEsu d|Esu|(y)
d|Esu|(y)dx.
Moreover,λν(∂)=0.
We will also need the following approximation result, see [28, Proposition 8].
Lemma 2.7.(Approximation)Letν∈BDY()satisfyλν(∂)=0. Also, assume that[ν] = Eu for u ∈BD()satisfying one of the conditions (i), (ii) from Lem-ma 2.4. Then, for all k ∈ N, there exists a partition(Ckl)l of(Ld+λν)-almost
all of into open sets Ckl, l = 1, . . . ,N(k), with diameters at most 1/k and
(Ld+λ
ν)(∂Ckl)=0, and a sequence of Young measures(νk)⊂BDY()such
that
νk ∗
ν inY(;Rdsym×d) as k→ ∞
and for every f ∈E(;Rdsym×d)
f, νk
= N(k)
l=1
f, ν Ckl
,
whereν Ckl designates the averaged Young measure (as in Lemma2.4) of the
restrictionν Cklofνto Ckl.
2.5. Localization of Young Measures
The paper [38] proved two localization principles for BD-Young measures, leading to so-called “tangent Young measures”; here and in the following, for ease of notation, we leave out the dependence of the spaces onRdsym×d.
We first define the following regular, or homogeneous, spaces of (tangent) Young measures forA0∈Rdsym×d(Qbeing the standard unit cube).
Ereg:= 1⊗h : 1⊗h∈E(Q;Rdsym×d)
, Yreg(A0):= σ =(σx, λσ, σx∞)∈Y(Q;R
d×d
sym) : [σ] = A0Ld Q,
σy, σy∞constant iny, λσ =αLd Qfor someα≥0
, BDYreg(A0):=Yreg(A0)∩BDY(Q)
= σ ∈Yreg(A0) : ∃(vj)⊂BD(Q)withEuj
Y
→σ.
Proposition 2.8.(Localization at regular points)Letν ∈Y(;Rdsym×d)be a Young
measure. Then, forLd-almost every x0∈there exists aregular tangent Young
measure
σ ∈Yreg(A0), where A0:=
id, νx0
+id, ν∞x0
dλν dLd(x0),
which satisfies
[σ] =A0Ld Q∈TanQ([ν],x0), σy =νx0 a.e., λσ = dλν
dLd(x0)L d
Q∈TanQ(λν,x0), σy∞=νx∞0 a.e..
Additionally, ifν∈BDY(), thenσ ∈BDYreg.
Here, TanQ(μ,x0)contains all (restricted)tangent measuresofμ∈M(;RN)
atx0∈ , i.e. those measuresσ ∈ M(Q;RN)such that there existsrn ↓0 and
cn>0 withcnT#x0,rnμ
∗
σ, whereTx0,rn(x):=(x−x
0)/rnand
Tx0,rn
# μ:=μ◦(T
x0,rn)−1=μ(x
0+rnq)
is the push-forward ofμunderTx0,rn. A proof for the preceding proposition can
be found in [38, Proposition 1].
We furthermore remark thatσ in Proposition2.8is such that for all open sets
U ⊂ QwithLd(∂U)=0, and allh ∈C(Rd×d)such that the recession function
h∞exists in the sense of (2.3), it holds that
1U ⊗h, σ=h, νx0
+h∞, νx∞0
dλν dLd(x0)
|U|.
For the singular counterpart to Proposition2.8, we first introduce the following spaces for any bounded Lipschitz domainD⊂Rd(we again omit mention ofRdsym×d for ease of notation):
Esing(D):= f ∈E(D;Rsymd×d) : f(x,q)positively 1-homogeneous, Ysing(D):= ν=(νx, λν, ν∞x ) : νx =δ0a.e.
, BDYsing(D):=Ysing(D;Rdsym×d)∩BDY(D).
The duality pairing betweenEsing(D)andYsing(D)is
f, ν:=
D
∂Bd×d
sym
f(x,A)dνx∞(A)dλν(x).
Furthermore, we say that the sequence of measures(μj)⊂M(D;Rdsym×d)generates
the singular Young measureν∈Ysing(D), in symbolsμj Y
→ν, if
f
x, dμj
d|μj|
(x)
d|μj|(x)→
Proposition 2.9.(Localization at singular points)Letν∈Y(;Rdsym×d)be a Young
measure. Then, forλsν-almost every x0∈ and every bounded Lipschitz domain
D⊂Rd, there exists asingular tangent Young measure
σ ∈Ysing(D)
satisfying
[σ] ∈TanD([ν],x0), σy=δ0 a.e.,
λσ ∈TanD(λsν,x0)\ {0}, σy∞=νx∞0 λσ-a.e..
Additionally, ifν∈BDY(), thenσ ∈BDYsing(D).
A proof of this fact can be found in [38, Proposition 2].
2.6. Good Singular Blow-Ups
In this section, as a preparation for the singular analogue of Proposition3.1, we establish a result about good blow-ups for BD-Young measures in Lemma2.14 below.
First, we recall from [38, Theorem 3] the following result. We here state it in a slightly different fashion, namely for Young measures with a BD-barycenter instead ofBD-generatedYoung measures.
Lemma 2.10.(Good singular blow-ups)Letν∈Y()be a Young measure with
[ν] =Eu for some u∈BD().
Forλsν-almost every x0∈and every bounded Lipschitz domain D ⊂Rd, there
exists a singular tangent Young measureσ ∈Ysing(D)as in Proposition2.9with
[σ] = [σ] D=Evfor somev ∈BD(D)and the appropriate assertion among
the following holds:
(i)If id, ν∞x0∈ {/ ab : a,b∈Rd\ {0} }(this includes the case id, νx∞0 =0), thenvis equal to an affine function almost everywhere;
(ii)If id, ν∞x0 =ab (a,b ∈Rd\ {0}) with a =b, then there exist functions
g1,g2∈BV(R),v0∈Rd, and a skew-symmetric matrix R∈Rdskew×d such that
v(x)=v0+g1(x·a)b+g2(x·b)a+Rx, x∈Rda.e.;
(iii)If id, ν∞x0 = aa (a ∈R
d\ {0}), then there exists a function g ∈ BV(R),
v0∈Rdand a skew-symmetric matrix R∈Rdskew×d such that
v(x)=v0+g(x·a)a+Rx, x∈Rda.e..
The proof is the same as in [38, Theorem 3], where the whole argument is only concerned with the barycenter and not the generating sequence. Moreover, we remark that[σ](∂D)can be achieved by a rescaling of the blow-up sequence
rn↓0 intoαrn↓0 for someα∈(0,1)such that[σ](∂(αD))=0 (assuming that 0∈ D).
The next ingredient we will need is the theorem on the singular structure of BD-functions, proved in [16]:
Theorem 2.12.Let u ∈ BD(). Then, for|Esu|-almost every x ∈, there exist
a(x),b(x)∈Rd\ {0}such that
dEsu
d|Esu|(x)=a(x)b(x).
This is the BD-analogue of the following celebrated Alberti rank-one theorem [1]:
Theorem 2.13.(Alberti’s rank-one theorem) Let u ∈ BV(). Then, for |Dsu| -almost every x∈, there exist a(x),b(x)∈Rd\ {0}such that
dDsu
d|Dsu|(x)=a(x)⊗b(x).
We will now state and prove a strengthened version of Lemma2.10. For this, we first define suitable spaces.
In all of the following, letξ ∈ Sd−1and denote byQξ the rotated unit cube (|Qξ| = 1) with one face normalξ. We first define one-directional versions of the spaces Esing,Ysing,BDYsing for A
0 ∈ Rdsym×d\ {0},ξ ∈ Sd−1; as before, we
henceforth leave out the dependence of the spaces onRdsym×d:
Esing(ξ):= f ∈Esing(Qξ;Rsymd×d) : f(y, q)= f(y·ξ,q)
, Ysing(A0, ξ):= σ =(σx, λσ, σx∞)∈Ysing(Qξ;Rdsym×d) : [σ] =A0λσ,
σy∞=σy∞·ξ, λσ isξ-directional
, BDYsing(A0, ξ):=Ysing(A0, ξ)∩BDYsing(Qξ).
Here, theξ-directionality ofλσ means that for all Borel setsB ⊂Qξ it holds that λσ(B+v)=λσ(B)for allv⊥ξ such thatB+v⊂ Qξ. Notice that we require
A0=0 here (the case A0=0 is treated in the next subsection).
Finally, the spacesYsing0 (A0, ξ),BDYsing0 (A0, ξ)are defined to incorporate the
additional constraintλσ(∂)=0.
Lemma 2.14.(Very good singular blow-ups)Letν ∈ Y()be a Young measure with
[ν] =Eu for some u∈BD().
Forλsν-almost every x0∈, there exists a singular tangent Young measure
such that[σ] = [σ] Qξ =Evfor somev∈BD(Qξ)of the form
v(x)=v0+g(x·ξ)η+β(ξ⊗η)x+Rx, x∈ Qξ a.e.,
wherev0∈Rd,β ∈R, g∈BV(R),ξ, η∈Rd\ {0}, and R∈Rdskew×d.
Remark 2.15.We note in passing that this improvement in fact allows one to slightly simplify the proof of the lower semicontinuity result in [38] as well.
Proof. Letu ∈BD()with[ν] =Eu. Then,Esu = id, ν∞x λsν and id, ν∞x =a(x)b(x) forλsν-a.e.x∈, (2.5)
and somea(x),b(x) ∈ Rd. Indeed, denoting byλ∗
ν the singular part ofλν with respect to|Esu|, the above holds at|Esu|-almost everyx∈witha(x),b(x)=0 by Theorem2.12and atλ∗ν-almost everyx∈witha(x)=b(x)=0.
From the previous Lemma2.10we get that there exists a singular tangent Young measureτ ∈Ysing(D)toνatλsν-almost everyx0∈that satisfies (i), (ii) or (iii).
We have that for anyw∈BD(Rd)withEw= [τ],
Ew= [τ] = id, ν∞x0λτ =(ab)λτ
for somea,b ∈ Rd. Indeed, by Proposition2.9we getτy∞ =ν∞x0 forλτ-almost
everyy∈ D, which implies the first equality. Further, ifx0is chosen such that (2.5)
holds, we infer id, τy∞ = id, νx∞0 =a(x0)⊗b(x0)forλτ-almost everyy∈ D. Step 1.Ifa,bare parallel, saya=bafter rescaling, then the statement of the present lemma follows immediately withσ :=τ,ξ :=a/|a| = b/|b|,D = Qξ
(note that we can chooseDin Lemma2.10as we like and we can decide beforehand whethera,bare parallel, see (2.5)). In this case,σ =τ ∈Ysing(aa,a), as follows
directly from Proposition2.9and Lemma2.10(iii).
Step 2.Only in the case id, ν∞x0 =abwitha =bthere is something left to prove. Without loss of generality we assume thata =e1andb =e2, which is
possible after a change of variables. In this case, by Lemma2.10we have that for anyw∈BD(Q)withEw= [τ]there exist functionsg1,g2∈BV(R),w0∈Rd,
and a skew-symmetric matrixR∈Rdskew×d such that
w(y)=w0+g1(y1)e2+g2(y2)e1+R y for a.e.y=(y1, . . . ,yd)∈Rd.
Moreover,
Dw=(e2⊗e1)Dg1⊗Ld−1+(e1⊗e2)L1⊗Dg2⊗Ld−2+RLd, (2.6)
andDsg1⊗Ld−1is singular toL1⊗Dsg2⊗Ld−2.
Case (I). If either Dg1 or Dg2 are the zero measure, the conclusion of the
theorem is trivially true with σ := τ,ξ := e1, η := e2, D the standard open
unit cube. So, henceforth assume that bothDg1,Dg2are not the zero measure. In
the following we denote by g1,g2theapproximate derivativesof g1,g2, i.e. the
densities of Dg1,Dg2with respect to Lebesgue measure.
Case (II).IfDsg
1,Dsg2=0, then we may use the regular localization principle,
ofτ at y0 ∈ Rd. We will argue below that in factσ is a singular tangent Young
measure to our originalνatx0and that[σ] ∈BD(), see Step 3 of the proof.
Case (III).On the other hand, if Dsg
1 = 0 (without loss of generality), then
we claim we can findy0=(s0,t0,y3, . . . ,yd)∈Rdwith the property that there
exists a non-zero singular tangent Young measureσ ∈Ysing(Q)toτ aty0and that
lim r↓0
1
r
t0+r
t0−r
|g2(t)−α|dt=0 and lim
r↓0
1
r|D
sg
2|((t0−r,t0+r))=0
with a constantα∈R.
Indeed, notice that second and third conditions hold forL1-almost everyt0
because the set of Lebesgue points ofg2 has full Lebesgue measure inRand the Radon–Nikodým derivative of Dsg2byL1is zeroL1-almost everywhere. Hence,
they hold for(|Dsg1| ⊗Ld−1)-almost everyy0=(s0,t0,y3, . . . ,yd)∈Rdby Fu-bini’s theorem. By the singular localization principle for Young measures, Propo-sition 2.9, we know that the first property holds for almost every y ∈ Rd with respect toλsτ. By (2.6), and the fact that|Dsg1| ⊗Ld−1is singular with respect to
L1⊗ |Dsg
2| ⊗Ld−2, we have
|Esw| = |e1e2|
|Dsg1| ⊗Ld−1+L1⊗ |Dsg2| ⊗Ld−2
.
Thus,|Dsg1| ⊗Ld−1=0 is absolutely continuous with respect to|Esw|.
Furthermore,|Esw|is absolutely continuous with respect toλsτ because
λs
τ =
√
2| id, ν∞x0|λ
s
τ =
√ 2|Esw|,
since | id, νx∞0| = |e1 e2| = 1/
√
2. Thus, the first condition also holds at (|Dsg1|⊗Ld−1)-almost everyy0and we find at least oney0=(s0,t0,y3, . . . ,yd)
with the claimed properties.
Step 3.We observe thatσ is still a singular tangent Young measure toνat the original pointx0if the latter was chosen suitably. Indeed, it can be easily checked
that the property of being a singular tangent Young measure is preserved when passing to another “inner” (regular or singular) tangent Young measure; the only non-obvious fact here is that “tangent measures of tangent measures are tangent measures”, but this is well-known and proved for example in Theorem 14.16 of [33].
The “inner” blow-up sequence of the BD-primitives of the barycenters has the form
w(n)(
z):=rnd−1cnw(y0+rnz), z∈Rd, withrn↓0 and constantscn=rn−dif we are in case (II) and
cn=
1
1Q(y0,rn)⊗ |q|, τ
= √ 1
2·λτ(Q(y0,rn))
(2.7)
Then,w(n) v∗ in BD(Q)andvhas the property thatEv = [σ]. For Ew(n)
we get
Ew(n)=(e1e2)
rndcn
g1(s0+rnz1)+g2(t0+rnz2)
Ld(dz)
+cnT#y0,rn
Dsg1⊗Ld−1+L1⊗Dsg2⊗Ld−2
.
Letϕ∈C∞c (Rd)and chooseR>0 so large that suppϕ⊂⊂ Q(0,R)=(−R,R)d. Using the special properties of our choice ofy0together with the fact thatcnrn−d
asn→ ∞, we observe that
rndcn
ϕ(z)|g2(t0+rnz2)−α|dz=cn
ϕy−y0
rn
|g2(y2)−α|dy
≤Cϕ∞ 1
rd n
Q(y0,rnR)
|g2(y2)−α|dy
and this goes to zero asn→ ∞. Also,
cn
ϕdTy0,rn
#
L1⊗Dsg
2⊗Ld−2
=cn
ϕy−y0
rn
dDsg2(y2)d(y1,y3, . . . ,yd)
≤Cϕ∞|D
sg
2|(t0+(−rnR,rnR))
rn
·Ld−1((s0,y03, . . . ,y0d)+(−rnR,rnR)d−1)
rnd−1
→0 asn→ ∞.
Hence, the e2-directional parts of Ew(n) in the limit converge to thefixedmatrix
β(e1e2), whereβ =αlimn→∞rndcn(this limit exists after taking a subsequence). More precisely, in case (II),#rn−dandσ is a regular tangent Young measure toτ, thus potentiallyβ =0; otherwise, in case (III) it must hold thatβ =0 (sincecn∼
λs
τ(Q(y0,rn))−1). The e1-directional parts of Ew(n) clearly stay e1-directional
under the operation of taking weak* limits. Thus,vis of the required form, with (ξ, η)=(e1,e2).
2.7. Functional Analytic Properties ofBDYreg,BDYsing
In this section we prove the following lemma about our tangent Young measure spaces:
Lemma 2.16.The sets BDYreg and BDYsing(ab, ξ) are convex and weakly* closed (with respect to the topology induced as a subset of(Ereg)∗andEsing(ξ)∗) for all a,b∈Rd\ {0},ξ ∈ {a,b}.
deformation of the homogeneous Young measure is even affine). We furthermore assume without loss of generality thatξ =a.
Step 1: Weak*-closedness ofBDYsing(ab,a). Let{fn}n∈N= {ϕn⊗hn}n∈N⊂ Esing(a)be a countable set of integrands that determines Young measure conver-gence inYsing(ab,a), this can be achieved by a reasoning analogous to Lem-ma2.2(only takeξ-directionalϕn and positively 1-homogeneoushn). Letσ be in the weak* closure of BDYsing(ab,a). Then, for every j ∈ Nthere exists σj ∈BDYsing(ab,a)with
fn, σj
−fn, σ+1⊗ |q|, σj
−1⊗ |q|, σ≤ 1
j for alln≤ j.
In particular,σj σ∗ inEsing(a)∗and also inY(Qa;Rdsym×d)since the sequence
(σj)is compact in that space. Indeed, 1⊗ |q|, σjis uniformly inj bounded, so we may use the compactness result from Lemma2.1. It is not hard to check that the defining properties ofYsing(ab,a)(such as thea-directionality ofλσ,y→νy∞) are preserved under weak* limits, henceσ ∈Ysing(ab,a).
We need to show also thatσis generated by a sequence of symmetric gradients, which follows by a diagonal argument: select for each j ∈ Na functionuj ∈ BD(Qa)∩C∞(Qa;Rd)(see Lemma2.3) with the property that
Qa
fn(x,Euj(x))dx−
fn, σj+EujL1−
1⊗ |q|, σj≤ 1
j
for alln ≤ j. Then, adding a rigid deformation to theuj’s, Poincaré’s inequality in BD, see (2.1), yields that there exists a (non-relabeled) subsequence such that
Euj Y
→μ∈BDY(Qa). Clearly,μ=σ by construction.
Step 2: Convexity of BDYsing(ab,a)assumingλμ(∂Qa)=λν(∂Qa)=0). Letμ, ν ∈ BDYsing(ab,a)be such thatλμ(∂Qa) = λν(∂Qa) = 0 and let
θ ∈ (0,1). By the approximation principle, Lemma2.7, we have that bothμ, ν are weak* limits of piecewise homogeneous and averaged Young measures. The partition with respect to which the approximations are piecewise constant can be chosen the same for bothμandν(this can be seen from the proof of the averaging principle in [28, Section 5.3]). Thus, by the weak* closedness proved in the first step, it suffices to show the result for homogeneous, one-directional BD-Young measures.
Assume now that we have two homogeneous, one-directional BD-Young mea-suresμ,¯ ν¯ ∈BDYsing(ab,a)withλμ(∂Qa)=λν(∂Qa)=0, which we assume to be defined on a cube with one face orthogonal toa. Indeed, by Remark2.5we can always assume that the averaged Young measuresμ,¯ ν¯are defined onQa(one can also argue by inspecting the proof of Lemma2.7to conclude that the subdivision may be chosen to consist of cubes only).
Without loss of generality we further assumea =e1and said cube to be the
unit cube Q = (−1/2,1/2)d. Thus,[ ¯μ] = Eu,[¯ν] = Ev for u(x) = η1bx1,
v(x) = η2bx1 (we can remove any additional rigid deformation). Indeed, since
let(uj), (vj)⊂BD(Q)∩C∞(Q;Rd)be bounded sequences such thatEuj Y
→ ¯μ,
Evj Y
→ ¯νanduj =u,vj =von∂Q, see Lemma2.3.
Now splitQinto a sliceS1=(−1/2, θ−1/2)×(1/2,1/2)d−1with volume
θand a sliceS2=(θ−1/2,1/2)×(1/2,1/2)d−1with volume 1−θ. Then cover
S1,S2with disjoint re-scaled copies ofQof side length at most 1/j (for instance
arranged in strips), namely
S1=
k∈N
pj k+εjQ
∪N1, S2=
k∈N
qj k+δjQ
∪N2,
whereεj, δj ≤1/jand|N1| = |N2| =0. Set
wj := ⎧ ⎨ ⎩
εjuj
x−pj k εj
+u(pj k) ifx∈ pj k+εjQ,k∈N,
δjvj
x−qj k δj
+v(qj k)+βb ifx∈qj k+δjQ,k∈N,
withβ ∈Rsuch that there is no jump betweenS1andS2, i.e.β =(η1−η2)(θ−1/2).
Then,
Ewj := ⎧ ⎨ ⎩
Euj
x−pj k εj
ifx∈ pj k+εjQ,k∈N,
Evj
x−qj k δj
ifx∈qj k+δjQ,k∈N.
Moreover,(wj)is uniformly bounded in BD(Q)andEwj Y
→ γ ∈ BDYsing(a
b,a). Effectively, inS1,S2we are repeating the averaging construction also
under-lying Lemma2.4and so, similar conclusions to the ones in that lemma hold. In particular, we get forϕ⊗h∈Esing(Q)withϕ∈C(Q),h ∈C(Rdsym×d)that
ϕ⊗h, γ=
1⊗h,μ¯
|Q|
S1
ϕ(x)dx+
1⊗h,ν¯
|Q|
S2
ϕ(x)dx.
Finally, apply the averaging principle, Lemma2.4, toγ to getγ¯ ∈ BDYsing(a b,a), which by (2.4) has the property
ϕ⊗h,γ¯=1⊗h, γ−
Q
ϕ(x)dx
=θ1⊗h,μ¯+(1−θ)1⊗h,ν¯−
Q
ϕ(x)dx
=θϕ⊗h,μ¯+(1−θ)ϕ⊗h,ν¯
forϕ⊗has above. This shows the claim for homogeneous, one-directional BD-Young measures.
Step 3: Convexity of BDYsing(ab,a). To conclude the proof we show that the set ofμ ∈ B DYsing(ab,a)such thatλμ(∂Qa) = 0 is weakly* dense in BDYsing(ab,a). Indeed, assume that a = e1,σ ∈ BDYsing(e1b,e1)and
BV(−1/2,1/2), we haveEuj Y
→ σ. We considerg to be extended continuously to all ofR. Then, define forα >1,
uαj(x):=uj(αx1,x2, . . . ,xd).
It is not hard to see that Euαj →Y σα for someσα ∈ BDYsing(e1b,e1)with
λσ(∂Qa)=0 andσα σ∗ asα↓1. This and the previous step easily implies the convexity ofBDYsing(ab,a).
3. Local Characterization
We first show the characterization result for tangent Young measures, i.e. those Young measures originating from Propositions2.8and2.9above.
3.1. Characterization for Regular Blow-Ups
With the definition ofEreg,Yreg(A0),BDYreg(A0)from Section2.5, we have
the following result about the characterization at regular points:
Proposition 3.1.Letσ ∈Yreg(A
0)for A0∈Rdsym×dand assume that
h(A0)≤
h, σy
+h#, σy∞ dλσ
dLd(y)
for all symmetric-quasiconvex h ∈ C(Rdsym×d)with linear growth at infinity. Then,
σ ∈BDYreg(A0).
Our proof here is quite concise since it is very close to Kinderlehrer & Pedregal’s original argument [23,24] and also essentially the same as the one given for [40, Proposition 3.2].
Proof. Step 1.First, by Lemma2.16, the setBDYreg(A0)is weakly*-closed and
convex (considered as a subset of(Ereg)∗).
We will show below that for every weakly*-closed affine half-spaceHin(Ereg)∗ withBDYreg(A0) ⊂ H, we haveσ ∈ H. Then the Hahn–Banach theorem will
imply thatσ ∈BDYreg(A0). Fix such a weakly* closed half-spaceH. By standard
arguments from functional analysis, see for example [14, Theorem V.1.3], there exists fH ∈Eregsuch that
H= e∗∈(Ereg)∗ : e∗(fH)≥κ
.
In particular,
fH, μ
≥κ for allμ∈BDYreg(A0).
Step 2.For the the symmetric-quasiconvex envelopeS Q fH of fH it holds that
S Q fH(A0) >−∞. Indeed, otherwise we could findw ∈W1A,0∞x(Bd;Rm), that is
w∈W1,∞(Bd;Rm)andw(x)= A0xfor allx∈∂Bd, such that
Bd fH(Ew(y))dy< κ.
Then, using the generalized Riemann–Lebesgue Lemma, Corollary2.6, there exists μ∈BDYreg(A0)with
fH, μ
=
Bd fH(Ew(y))dy< κ,
which is a contradiction.
For a fixedε >0, the function
gε(A):= fH(A)+ε|A|, A∈Rm×d,
lies inEreg. It holds that
S Qgε(A0)≥ S Q fH(A0)+ε|A0|>−∞.
Consequently, the function S Qgε is symmetric-quasiconvex, see the appendix of [23]. Moreover S Qg(A) ≤ (M +1)(1 + |A|). By Lemma 2.5 in [27], even
|S Qg(A)| ≤ ˜M(1+ |A|)for some M˜ = ˜M(d,m,M) >0. Usinggε ≥ S Qgε,
gε∞≥(S Qgε)#, and the assumption,
gε, σ≥
Bd
S Qgε, σy
+(S Qgε)#, σy∞ dλσ
dLd(y)dy≥S Qgε(A0)|B d|.
(3.1)
Next, take a sequence(wj)⊂W1A,0∞x(B
d;Rm)with
−
Bdgε(Ewj(y))dy→S Qgε(A0).
Moreover, possibly discarding leading elements of the sequence(wj),
S Qgε(A0)+1≥ −
Bd
gε(Ewj(y))dy≥ −
Bd
S Q fH(Ewj(y))+ε|Ewj(y)|dy
≥S Q fH(A0)+ ε
|Bd| · EwjL1.
Hence, the sequence (wj)is uniformly bounded in BD(Bd)and, up to a
subse-quence,Ewj Y
→μ ∈ BDY(Bd). Apply Lemma2.4to replaceμby its averaged versionμ¯ ∈BDYreg(A0). Then,
gε,μ¯=gε, μ= lim j→∞
Bd gε(Ewj(y))dy=S Qgε(A0)|B
Combining with (3.1), we get
fH, σ
=gε, σ−ε1⊗ |q|, σ
≥S Qgε(A0)|Bd| −ε
1⊗ |q|, σ
=gε,μ¯−ε1⊗ |q|, σ
≥fH,μ¯
−ε1⊗ |q|, σ
≥κ−ε1⊗ |q|, σ,
sinceμ¯ ∈BDYreg(A0)⊂H. Now letε↓0 to get fH, σ ≥κ. Thus,σ ∈ H.
3.2. Characterization for Singular Blow-Ups
Here we will prove the singular analogue of Proposition3.1.
Proposition 3.2. Ysing0 (ab, ξ) = BDYsing0 (ab, ξ) for all a,b ∈ Rd \ {0}, ξ ∈ {a,b}.
The preceding proposition is surprising since it says thateverysingular Young measure inYsing0 (ab, ξ)is generated by a sequence of symmetric derivatives of BD-functions.
We first record the following lemma, which is a direct consequence of the main result in [25]:
Lemma 3.3.Letμ ∈ M1(X;Rdsym×d)be a probability measure with barycenter
[μ] = id, μ = ab for some a,b ∈ Rd, and let h ∈ C(Rd×d
sym)be positively
1-homogeneous and symmetric-quasiconvex. Then, the Jensen-inequality
h(ab)=h( id, μ)≤ h, μ
holds.
To prove this result we recall a version of the main result of [25]:
Theorem 3.4.Let h∞:Rdsym×d →Rbe positively one-homogeneous and symmetric quasiconvex. Then, h∞is convex at every matrix ab for a,b∈Rd, that is, there exists an affine function g:Rdsym×d →Rwith
h∞(ab)=g(ab) and h∞≥g.
Proof of Lemma3.3. By the preceding theorem,h is actuallyconvexat matrices
ab, that is, the Jensen inequality holds for measures with barycenterab, such as ourμ.
Lemma 3.5.Letν∈BDY(S), where for some z0∈Qa, R>0, a∈Sd−1,
S =S(z0,R):= x∈ Qa : |(x−z0)·a|<R
.
Assume further that there exists a sequence (vj) ⊂ BD(S)with Evj Y
→ ν and
vj(x) = bg(x·a)on∂S for some b ∈ Rd and g ∈ BV(R). Then, there exists ˆ
ν∈BDYsing(ab,a)such that
1⊗h, ν=1⊗h,νˆ (3.2)
for all positively1-homogeneous h ∈ C(Rdsym×d). The condition on the generating
sequence is in particular satisfied if [ν] = Eu for some u ∈ BD(S)with the property that u(x)=bg(x·a)on∂S.
Proof. Note that by Lemma2.3if[ν] =Eufor someu ∈BD(S)with the property thatu(x)=bg(x·a)on∂Sthere always exists(uj)⊂BD(S)∩C∞(S;Rd)with
Euj Y
→ ν anduj(x)=bg(x·a)on∂S. Hence the second part of the statement follows from the first.
Up to a translation and a one-dimensional scaling, we can assume thatx0=0
andR=1. Let
S(z0,r):= x∈ Qa : |(x−z0)·a|<r
, z0∈ Qa, r>0,
and definewj ∈ B D(S)by
wj(x):= ⎧ ⎪ ⎨ ⎪ ⎩
uj(j x) ifx∈S(0,1/j),
bg(−1) ifx·a≤ −1/j, bg(+1) ifx·a≥1/j,
x∈Qa,
where g(±1)is defined in the sense of trace. It can be seen thatEwj generates a Young measureμ∈BDY(Qa)withμx =δ0almost everywhere sinceEwj →0 in measure. Furthermore, for all positively 1-homogeneoush ∈ C(Rdsym×d), we get
from thea-directionality ofgthat
1⊗h, μ= lim j→∞
S(0,1/j)
j h(Euj(j x))dx
= lim
j→∞
Qa
h(Euj)dx
=1⊗h, ν.
Now apply the averaging principle, Lemma2.4, toμto getνˆ ∈BDYsing(ab,a)
