Embed Size (px)
Citation preview
arX
iv:1
505.
0465
4v1
[mat
h.A
P]
18 M
ay 2
015
ON RANK ONE CONVEX FUNCTIONS THATARE HOMOGENEOUS OF DEGREE ONE∗
Bernd Kirchheim and Jan Kristensen
Abstract
We show that positively1–homogeneous rank one convex functions are convex at0 andat matrices of rank one. The result is a special case of an abstract convexity result that weestablish for positively1–homogeneous directionally convex functions defined on an openconvex cone in a finite dimensional vector space. From these results we derive a number ofconsequences including various generalizations of the OrnsteinL1 non inequalities. Mostof the results were announced in (C. R. Acad. Sci. Paris, Ser. I 349 (2011), 407–409).
AMS Classifications.49N15; 49N60; 49N99.
1 Introduction and Statement of Results
It is often possible to reformulate questions about sharp integral estimates for the derivativesof mappings as questions about certain semiconvexity properties of associated integrands (werefer the reader to [Dac89] for a survey of the relevant convexity notions and their roles in thecalculus of variations). Particularly fascinating examples of the utility of this viewpoint arepresented in [I02], where the fact that rank one convexity isa necessary condition for quasicon-vexity that is possible to check in concrete cases, leads to along list of tempting conjectures, allof which – if proven – would have significant impact on the foundations of Geometric FunctionTheory in higher dimensions. The obstacle to success is thatrank one convexity in generaldoes not imply quasiconvexity. This negative result, knownas Morrey’s conjecture [Mo52],was established in [Sv92]. It does, however, not exclude the possibility that some of thesesemiconvexity notions agree within more restricted classes of integrands having natural homo-geneity properties. A very interesting case being the positively 1–homogeneous integrands.Their semiconvexity properties correspond toL1–estimates, and hence are difficult to establishusing interpolation or other harmonic analysis tools. In this paper we investigate the convex-ity properties of such integrands. In particular it is shownin Corollary 1.2 that a positively
∗Version: April 2015
1
1–homogeneous and rank one convex integrand must be convex at0 and at all matrices of rankone. While this class of integrands has been investigated several times before, see in particular[Sv91, Mu92, DH96, DM07], the surprising automatically improved convexity at all matricesof rank at most one remained unnoticed. We remark that [Sv91] (and also [Mu92] and [DH96])yield examples of1–homogeneous rank one convex functions that arenot convexat some matri-ces of rank two so that these results are indeed sharp. The keyconvexity result is best stated inabstract terms, and we take a moment to introduce the requisite terminology (see also Section2 for notation and terminology). LetV be a finite dimensional real vector space,C an openconvex cone inV andD a cone of directions inV. More precisely,V is considered with somenorm‖ · ‖ andC is an open subset ofV with the property thatsx+ ty ∈ C whens, t > 0 andx,y ∈ C. The setD is also a subset ofV, it gives the directions in which we have convexity, andwe assume thattx ∈ D for all x ∈ D, t ∈ R, and that it contains a basis forV. A real–valuedfunction f : C → R is D–convex provided its restrictions to line segments inC in directionsof D are convex. The functionf is positively1–homogeneous providedf(tx) = tf(x) for allt > 0 and allx ∈ C. Finally we say thatf has linear growth at infinity if we can find a constantc > 0 such that|f(x)| ≤ c(‖x‖+ 1) holds for allx ∈ C.
Theorem 1.1. Let C be an open convex cone in a normed finite dimensional real vector spaceV, andD a cone of directions inV such thatD spansV.
If f : C → R is D–convex and positively1–homogeneous, thenf is convex at each point ofC ∩D. More precisely, and in view of the homogeneity, for eachx0 ∈ C∩D there exists a linearfunctionℓ : V → R satisfyingℓ(x0) = f(x0) andf ≥ ℓ onC.
We state separately the special case corresponding to rank one convexity for functions de-fined on the space of realN by n matrices:
Corollary 1.2. A rank–one convex and positively1–homogeneous functionf : RN×n → R isconvex at each point of the rank one conex ∈ RN×n : rank x ≤ 1.
A remarkable result of Ornstein [Or62] states that given a set of linearly independent lin-ear homogeneous constant–coefficient differential operators inn variables of orderk, sayQ0,Q1, . . . , Qm, and any numberK > 0, there is aC∞ smooth functionφ vanishing outside theunit cube,φ ∈ C∞
c ((0, 1)n), such that∫|Q0φ| > K while
∫|Qjφ| < 1 for all 1 ≤ j ≤ m.
This result convincingly manifests the fact that estimatesfor differential operators, usuallybased on Fourier multipliers and Calderon–Zygmund operators, can be obtained for allLp withp ∈ (1,∞) by interpolation and (more directly) even for the weak-L1 spaces but fail to extend tothe limit casep = 1. Ornstein used his result to answer a question by L. Schwarz by construct-ing a distribution in the plane that was not a measure but whose first order partial derivativeswere distributions of first order. He then gave a very technical and rather concise proof of hisstatement for general dimensionn and degreek.
Theorem 1.1, and Corollary 1.2, yield when combined with arguments from the calculus ofvariations, various generalizations of Ornstein’sL1–non–inequality. In particular the approachallows also a streamlined and very elementary proof of an extension of Ornstein’s result to thewider context ofx–dependent and vector–valued operators:
2
Theorem 1.3. Let V, W1, W2 be finite dimensional inner product spaces, and consider twok–th order linear partial differential operators with locally integrable coefficients
aiα ∈ L1loc(R
n,L (V,Wi)) (i = 1, 2)
defined by
Ai(x,D)ϕ :=∑
|α|=k
aiα(x)∂αϕ (i = 1, 2)
for smooth and compactly supported test mapsϕ ∈ C∞c (Rn,V).
Then there exists a constantc such that
‖A2(x,D)ϕ‖L1 ≤ c‖A1(x,D)ϕ‖L1
holds for all ϕ ∈ C∞c (Rn,V) if and only if there existsC ∈ L∞(Rn,L (W1,W2)) with
‖C‖L∞ ≤ c such thata2α(x) = C(x)a1α(x)
for almost allx and each multi–indexα of lengthk.
We derive this result from a more general nonlinear version stated in Theorem 5.1. Thisresult is in turn a consequence of Theorem 1.1 and standard arguments from the calculus ofvariations. The case of constant coefficient homogeneous partial differential operators wasgiven in [KK11].
The link between an Ornstein type result, concerning the failure of theL1–version of Korn’sinequality, and semiconvexity properties of the associated integrand – though expressed in adual formulation – was observed already in [CFM05a]. There it was utilized in an ad-hoc con-struction which required a very sophisticated refinement in[CFMM05b], where it was trans-ferred from an essentially two–dimensional situation intothree dimensions. Our argumentshandle these situations with ease, see Corollary 1.5 below.
It is well–known that the distributional Hessian of a convexfunction f : Rn → R is amatrix–valued Radon measure (see for instance [R68b]). Thenatural question arises if this isvalid also for the semi-convexity notions important in the vectorial calculus of variations. In[CFMM05b] a fairly complicated construction was introduced to show that this is not true forrank one convex functions defined on symmetric2 × 2 matrices. Here we wish to address thequestion of regularity of second derivatives when the function f : V → R is merelyD–convex.First note that whenf is C2, thenD–convexity is equivalent toD2f(x)[e, e] ≥ 0 for all x ∈ Vande ∈ D. Hence using mollification and that a positive distributionis a Radon measure itfollows that the Hessian ofD–convex functionsf : V → R will be a Radon measure providedwe can select a basis(ej) for V consisting of vectors fromD such that additionallyei ± ej ∈ Dfor all 1 ≤ i < j ≤ dimV. Indeed under this assumption onD we can use polarisation to seethat all second order partial derivatives (in the coordinate system defined by(ej)) are measures.It is not difficult to show that the above condition on the coneof directionsD is equivalent tothe statement that the only functionsf : V → R with the property that both±f areD–convex,
3
calledD–affine functions, are the affine functions. This in turn is equivalent to saying that thecone ofD–convex quadratic forms onV,
QD =q : q is aD–convex quadratic form onV
is line–free. It is remarkable, but in accordance with our theme on OrnsteinL1 non–inequalities,that if the cone of directionsD satisfies the technical condition
∃ ℓ ∈ V∗ such thatD ∩ kerℓ = 0, (1.1)
then this simple sufficient condition is also necessary.
Theorem 1.4. Let V be a normed finite dimensional real vector space andD be a cone ofdirections inV satisfying the condition (1.1). Then the distributional Hessian ofD–convexfunctionsf : V → R is a ⊙2V–valued Radon measure onV if and only if the onlyD–affinefunctions are the affine functions. Moreover, when there exists a nontrivialD–affine function,then there also exists aD–convexC1 functionf : V → R with a locally Holder continuous1
first differentialDf : V → V∗, but for which the distributional Hessian fails to be a measure inanyopen nonempty subsetO of V, in the sense that for some unit vectore ∈ V we have
sup
∫
O
f(x)D2ϕ(x)(e, e) dx : ϕ ∈ C∞c (O) and sup |ϕ| ≤ 1
= ∞.
We remark that the condition onℓ ∈ V∗ given in (1.1) is equivalent to stating that the formℓ⊗ℓ is an interior point of the coneQD. Let us record here some examples of cones of directionsfor the case of square matricesV = Rn×n that satisfy condition (1.1):
D(ξ0, ε0) =
a⊗ b : a, b ∈ Rn and|a · ξ0b| ≥ ε0|a||b|
,
whereξ0 ∈ Rn×n andε0 > 0 are fixed and chosen soD(ξ0, ε0) spansRn×n. Indeed, with theusual identifications(Rn×n)∗ ∼= R
n×n, for such cones we have thatξ0⊗ξ0 is an interior point ofQD(ξ0,ε0). Since any2× 2 minor ofn×n matrices defines a nontrivial rank–one affine functiononRn×n we infer in particular the existence ofD(ξ0, ε0)–convexC1,α functions onRn×n whoseHessians are nowhere a measure. When we restrict attention to functions defined onsymmetricrealn× n matricesRn×n
sym the situation becomes more satisfying since there the rank–one coneDn
sym = t ·a⊗a : t ∈ R anda ∈ Rn satisfies the condition (1.1). IndeedId⊗ Id is an interior
point inQDnsym
. We state this result separately:
Corollary 1.5. Let n ≥ 2 be an integer. Then there exists a rank–one convexC1 functionf : Rn×n
sym → R with a locally Holder continuous first derivative, but whose distributional Hes-sianD2f is not a bounded measure in any open nonempty subsetO of Rn×n
sym in the sense thatfor somee ∈ Rn×n
sym we have
sup
∫
O
f(x)D2ϕ(x)(e, e) dx : ϕ ∈ C∞c (O) and sup |ϕ| ≤ 1
= ∞.
In particular, the functionf cannot be locally polyconvex atanyξ ∈ Rn×nsym .
1∀K ⋐ V ∀α < 1 ∃c > 0 ∀ x, y ∈ K: ‖Df(x)−Df(y)‖ ≤ c|x− y|α
4
Corollary 1.5 was announced in [KK11].Finally, we remark that due to concentration effects on rank–1 matrices, see [Al93], our
results also allow us to simplify the characterization of BVGradient–Young measures given in[KR10b], see Theorem 6.2. In fact, this was the original motivation for the work undertaken inthe present paper.
We end this introduction by giving a brief outline of the organization of the paper. In Sec-tion 2 we discuss and introduce our notation and terminology, and also derive some preliminaryresults. The proofs of Theorem 1.1 and Corollary 1.2 follow in Section 3. Section 4 is con-cerned with higher order derivatives of maps between finite dimensional spaces and relaxationof variational integrals defined on them. Previous works in the calculus of variations on re-laxation of multiple integrals depending on higher order derivatives has mostly been based onperiodic test mapswhereas we usecompactly supported tests maps. The two approaches areessentially equivalent, and here we supply the details in the latter case culminating in Theorem4.2. The proofs of Theorems 1.3, 5.1 and 1.4 are presented in Section 5. Section 6 contains adiscussion of BV gradient Young measures, and a characterization, stated as Theorem 6.2, ofthese by duality with a certain subclass of quasiconvex functions.
2 Preliminaries
This section fixes the notation, collects standard definitions and recalls some preliminary resultsthat are all more or less well–known.
We have attempted to use standard or at the least self–explanatory notation and terminology.Thus our notation follows closely that of [Ro] for convex analysis, [Dac89] and [Mu] for cal-culus of variations, and [AFP] for Sobolev functions and functions of bounded variations. Werefer the reader to these sources for further background if necessary.
Let V be a real vector space. A subsetC of V is convex if it is empty or if for all pointsxandy in C also the segment(x, y) between them is contained inC. Letf : V → R := [−∞,∞]be an extended real–valued function. Then its effective domain is the setdom(f) := x ∈ V :f(x) < ∞ and its epigraph is the setepi(f) := (x, t) ∈ V × R : t ≥ f(x). The functionfis convex if its epigraph is a convex subset of the vector spaceV ⊕ R.
By acone of directionsD in V we understand throughout the paper a balanced and spanningcone inV: soD ⊆ V has the properties that for allx ∈ D, t ∈ R we havetx ∈ D, and its linearhull equalsV.
Definition 2.1. LetV be a finite dimensional real vector space andD a cone of directions.(i) A subsetA of V is D–convex provided for any two pointsx, y ∈ A with x− y ∈ D also thesegment[x, y] = (1− λ)x+ λy : λ ∈ [0, 1] is contained inA.(ii) For an arbitrary subsetS of V and an extended real–valued functionf : S → R we saythat f is weaklyD–convexprovided for anyx, y ∈ S such that[x, y] ⊂ S andx − y ∈ D therestrictionf |[x,y] is convex. We say thatf is D–convexif the function
F (x) =
f(x) if x ∈ S∞ if x ∈ V \ S
5
is a weaklyD–convex functionF : V → R.
The reader will notice that this terminology is inspired by [BES63], and that onD–convexsubsetsS of V there is no difference between weakD–convexity andD–convexity ofreal–valuedfunctionsf : S → R.
Lemma 2.2. LetV be a normed finite dimensional real vector space andD a cone of directionsin V. Assumef : S → [−∞,∞) is aD–convex function defined on an arbitrary subsetS of V.If A is a connected component of the interior ofS, then eitherf ≡ −∞ onA or f > −∞ onA.
Proof. The proof relies on two standard observations. First, ifh : (a, b) ⊂ R → [−∞,∞] is aconvex function such thath(t) <∞ for all t ∈ (a, b) andh(t0) > −∞ for somet0 ∈ (a, b), thenh(t) > −∞ for all t ∈ (a, b). This follows easily from the definition of convexity. The secondobservation is that, sinceD spansV, any two points ofA can be connected by a piecewise linearpath inA whose segments are all in directions fromD. This follows by choosing a basis forV consisting of vectors fromD, then declaring it to be an orthonormal basis forV, so that theabove is a well–known property of connected open sets in euclidean spaces.
Now if we havex ∈ A with f(x) > −∞ and ℓ is a line throughx in a direction fromD, then by convexity off |A∩ℓ we infer thatf(y) > −∞ for all y belonging to the connectedcomponent ofA ∩ ℓ containingx. The result follows since we can connect any pointy ∈ Awith x by such line segments contained inA.
The following result and its proof is a variant of a result from [BKK00] (compare Lemma2.2 there) that in turn is a slightly more precise version of awell–known estimate, see [Dac89]Theorem 2.31, but here being adapted to the case of a general coneD of directions.
Lemma 2.3. Let V, ‖ · ‖ be a normed finite dimensional real vector space andD a balancedcone whose linear span isV. If f : B2r(x0) → R is D-convex, thenf is locally Lipschitz inB2r(x0). More precisely we have
|f(x)− f(y)| ≤ L‖x− y‖ (2.1)
for all x, y ∈ Br(x0), where
L =c0rosc(f, B2r(x0))
and the constantc0 depends only on the norm‖ · ‖ and the coneD.
Proof. For the entire proof we fix unit vectorse1, . . . , en ∈ D which form a basis ofV, andnote that due to the equivalence of all norms onRn there is a constantc ∈ (1,∞) such that
1
c
n∑
j=1
|tj| ≤ ‖n∑
j=1
tjej‖ ≤ c
n∑
j=1
|tj| (2.2)
for all tj ∈ R. Now we proceed in three steps.
6
Step 1.The functionf is locally bounded.We start by showing that forx ∈ B2r(x0) and positiveε such that the parallelepipedC with
centerx and sides parallel to the basis vectors and all of length2ε is contained inB2r(x0),
C =
x+
n∑
j=1
tjej : t1, . . . , tn ∈ [−ε, ε]
⊂ B2r(x0),
we havesup f(C) = max f(K), where the setK = ext(C) consists of the vertices of theparallelepipedC. Indeed, by convexity in each of the directionsej ∈ D we find recursively foreachx+
∑n
1 tjej ∈ C,
f(x+n∑
j=1
tjej) ≤ maxε1=±1
f(x+n∑
j=2
tjej + ε1e1)
≤ . . .
≤ max f(K).
Consequently,sup f(C) ≤ max f(K). In order also to get a lower bound we fixy ∈ C, sayy = x +
∑n
1 tjej where each|tj| ≤ ε. PutKy = x +∑n
1 εjtjej : ε1, . . . , εn ∈ −1, 1.Observe that
∑
z∈Ky2−nz = x andKy is contained inC, so again using convexity in each of
the directionsej we find
f(x) ≤∑
z∈Ky
2−nf(z) ≤ 2−nf(y) + (1− 2−n) sup f(C),
hencef(y) ≥ 2nf(x)− (2n − 1)max f(K), and therefore the lower bound
inf f(C) ≥ 2nf(x)− (2n − 1)max f(K)
follows.Step 2.For all x, y ∈ Br(x0) with x − y ∈ D we have|f(x) − f(y)| ≤ m
r‖x − y‖, where
m = osc(f, B2r(x0)).To prove this we can assume thatm < ∞. Now fix s ∈ (r, 2r) and consider the lineL
throughx andy. Selectz ∈ L∩ ∂Bs(x0) such thatx ∈ [y, z], the segment betweeny andz. Byassumption the restriction off toL is convex and since‖y − z‖ ≥ s− r we get
f(x)− f(y)
‖x− y‖ ≤ f(z)− f(y)
‖z − y‖ ≤ m
s− r.
The required conclusion follows sinces < 2r was arbitrary andx versusy can be swapped.Step 3.Now consider anyx ∈ Br(x0) andBε(x) ⊂ Br(x0). If ‖x− y‖ < ε/c2 for the constantc from (2.2), then
y − x =n∑
j=1
tjej wheren∑
j=1
|tj| <ε
c.
7
Hence, the points
yk = x+
k∑
j=1
tjej , k = 1, . . . , n
all belong toBε(x), and so in particular toBr(x0), since, again due to (2.2),
‖yk − x‖ = ‖k∑
j=1
tjej‖ ≤ c
k∑
j=1
|tj| < ε.
Consequently, step 2 and (2.2) ensure that
|f(y)− f(x)| = |f(yn)− f(y0)| ≤n∑
j=1
|f(yj)− f(yj−1)| ≤n∑
j=1
m
r|tj| ≤
m
rc‖x− y‖.
This shows thatf is everywhere insideBr(x0) locally Lipschitz with a constant at mostmc/r.BecauseBr(x0) is convex, the Lipschitz constant off on the entire ball can also not exceedcm/r.
Finally, we observe that this argument also shows thatf is locally Lipschitz everywhere inB2r(x0).
An open convex coneC in V is an open subsetC of V such thatx + ty ∈ C wheneverx,y ∈ C andt > 0.
For a functionf : C → R defined on an open convex coneC in V we define the(upper)recession functionf∞ : C → R as
f∞(x) := lim supt→∞,x′→x
tx′∈C
f(tx′)
t(x ∈ C) (2.3)
The definition implies thatf∞ is positively1-homogeneous:f∞(tx) = tf∞(x) for all x ∈ Candt > 0. It also follows that whenf has linear growth onC, meaning that for some constantcwe have the bound
|f(x)| ≤ c(‖x‖ + 1
)
for all x ∈ C, thenf∞ is real–valued and|f∞(x)| ≤ c‖x‖ for all x ∈ C∞. Whenf is globallyLipschitz onC with Lipschitz constantlip(f, C) = L:
|f(x)− f(y)| ≤ L‖x− y‖
for all x, y ∈ C, then we can simplify the definition of the recession function f∞ to
f∞(x) = lim supt→∞
f(tx)
t(x ∈ C) (2.4)
and of course againlip(f∞, C) ≤ L. Let us also remark that since all norms are equivalent ona finite dimensional vector space and the actual Lipschitz constants play no role here we shall
8
often just state that the function under consideration has linear growth or is Lipschitz withoutspecifying a norm.
There is also a natural notion of lower recession function, but for our purposes it is lessuseful. The reason is that the following result fails for lower recession functions.
Lemma 2.4. Let C be an open convex cone inV and assume thatf : C → R is a D–convexfunction of linear growth onC. Then the recession functionf∞ : C → R is againD–convex.
We omit the straightforward proof. The following simple observation turns out to be crucialin the sequel.
Lemma 2.5. Let V be a normed finite dimensional real vector space andD a balanced coneof directions inV. AssumeC is an open convex cone inV and thatf : C → R is aD–convexfunction. Then
f(x+ y) ≤ f∞(x) + f(y) (2.5)
for all y ∈ C andx ∈ C ∩ D.
Proof. Let y ∈ C andx ∈ C ∩ D. By definition of open convex cone,y + tx ∈ C for all t ≥ 0,and becausef is convex in the direction ofx we get for allt ≥ 1,
f(x+ y)− f(y) ≤ f(y + tx)− f(y)
t=f(t(x+ y
t))
t− f(y)
t,
and in view of (2.3) the conclusion follows by sendingt to infinity.
3 Proof of Theorem 1.1
Throughout this section we assume thatV is a normed finite dimensional real vector space witha cone of directionsD.
Definition 3.1. LetC be a subset ofV andf : C → R a function. We say thatf has atx ∈ C
• a subdifferential if there is a linear functionℓ : V → R such that
f(y) ≥ f(x) + ℓ(y − x) for all y ∈ C.
• a D–subconeif there is aD–convex and positively1–homogeneous functionℓ : V → R
such thatf(y) ≥ f(x) + ℓ(y − x) for all y ∈ C.
As we shall see momentarily the two conditions are in fact equivalent. However, first weestablish the existence ofD–subcones:
9
Lemma 3.2. LetC be a convex open cone inV.If f : C → R is D–convex and positively1–homogeneous, then for anyx ∈ C ∩ D, y ∈ C
andλ ∈ (0, 1) we have
f(y)− f(x) ≥ f(x+ λ(y − x))− f(x)
λ. (3.1)
In particular,f has aD–subcone atx.
Proof. Since adding a linear function tof does not effect the assumptions nor the validity of(3.1) or the existence of aD–subcone, we can suppose in the sequel thatf(x) = 0. By Lemma2.3 the functionf is lipschitz nearx and hence alsof∞(x) = 0. Therefore, using Lemma 2.5we conclude as required
f(y)− f(x) = f(y) = f(x+ (y − x))
≥ f(x
λ+ (y − x)) =
f(x+ λ(y − x))− f(x)
λ.
To see that this implies the existence of aD–subcone atxwe choose anε > 0 soB2ε(x) ⊂ Cand define fors ≥ 1,
gs(y) := sf(x+y
s), y ∈ Bsε(0).
Clearly gs(0) = 0. By Lemma 2.3 we havelip(gs) = lip(f, Bε(x)) and by (3.1) we get fors ≤ t the monotonicity propertygs(y) ≥ gt(y) for all y ∈ Bsε(0). Hence for eachy ∈ V thelimit
g(y) := lims→∞
gs(y) = infs>0
gs(y)
exists inR, and defines a Lipschitz functiong : V → R. As a pointwise limit ofD–convexfunctionsg isD–convex, and since for anyy ∈ V andλ > 0,
g(λy) = lims→∞
sf
(
x+λy
s
)
= λ lims→∞
( s
λ
)
f
(
x+y
s/λ
)
= λg(y),
g is also positively1–homogeneous. Finally, fory ∈ C we get from (3.1) upon taking anyλ ≥ ε/(1 + ‖y − x‖) thatλ(y − x) ∈ Bε(0) so
f(y)− f(x) ≥ 1
λf (x+ λ(y − x)) =
1
λg1(λ(y − x)).
Sinceg1 ≥ g andg is positively one–homogeneous it follows thatg is aD–subcone forf atx.
Proposition 3.3. Let C be an open subset ofV. If the functionf : C → R has aD–subcone atthe pointx0 ∈ C, then it also has a subdifferential atx0.
10
Proof. We will proceed by induction on the dimensionn ≥ 1 of V and suppose that the state-ment is true whenever the dimension of the vector space is less thann.
A moments reflection on Definition 3.1 shows that existence both of aD–subcone and asubdifferential are unaffected by a simultaneous shift of the functionf and the contact pointx0,so we can without loss in generality suppose thatx0 = 0V and, replacingf by itsD–subcone,also thatf isD-convex and positively one-homogeneous on all ofV.
We choose a basise1, . . . , en ∈ D of V and putx = e1 andV = spane2, . . . , en, soV isspanned byD = V ∩D. According to Lemma 3.2 there is aD–subconeg for f in x, and clearlythe restrictiong|V is its ownD–subcone at0V . Therefore, by the induction assumption there is a
subdifferential at the origin: a linear functionl : V → R such thatℓ(y) ≤ g(y) whenevery ∈ V.In particularℓ = 0 if n = 1.
Now we claim that
ℓ(tx+ y) = tf(x) + ℓ(y) for y ∈ V andt ∈ R
is a subdifferential forf at0V . Once this claim is shown, our proof is finished.For this purpose we first note that clearyℓ is linear onV. Sincef(0) = 0, we need only
show thatf(z) ≥ ℓ(z) for eachz = tx+ y, t ∈ R andy ∈ V. But if t = 1 then we have for ally ∈ V that
ℓ(x+ y) = f(x) + ℓ(y) ≤ f(x) + g(y) ≤ f(x+ y),
according to the definition of aD–subconeg of f in x.By positive1–homogeneity off we get now for allt > 0, y ∈ V that
ℓ(tx+ y) = tℓ(x+y
t) ≤ tf(x+
y
t) = f(tx+ y).
Finally, if t ≤ 0, y ∈ V we use Lemma 2.3 to inferf∞(x) = f(x) and now Lemma 2.5 gives
ℓ(x+ y) ≤ f(x+ y) ≤ f∞((1− t)x) + f(tx+ y)
and sof(tx+ y) ≥ ℓ(x+ y)− (1− t)f(x) = ℓ(tx+ y),
which finishes the proof.
Clearly, Theorem 1.1 is now a direct consequence of Proposition 3.3 and Lemma 3.2.
4 Higher order derivatives
We briefly recall some notation and concepts, mainly from multi–linear algebra, that will proveconvenient for dealing with higher order derivatives.
Starting from the standardL (X, Y ) = f : X → Y ; f linear for given finite–dimensionalreal vector spacesX, Y , we setL 0(X, Y ) = Y and inductively
Lk+1(X, Y ) = L (X,L k(X, Y ))
11
for k ∈ N0. As usual (see [Fed69], 1.9–1.10), we identifyL k(X, Y ) with M k(X, Y ), thespace ofY –valuedk–linear maps onX. In the sequel, we are mainly interested in the space ofall Y –valued symmetrick–linear maps
⊙k(X, Y ) = µ ∈ Mk(X ; Y ) : µ(x1, . . . , xk) = µ(xσ(1), . . . , xσ(k)) for all σ ∈ Sym(k),
where Sym(k) is the group of all permutations of the set1, . . . , k. WhenY = R we simplywrite⊙kX instead of⊙k(X,R).
In terms of a basis(ej)nj=1 for X, we can express a mapξ ∈ M k(X, Y ) as aY –valuedhomogeneous polynomial of degreek in kn real variables byfully expanding all brackets:
ξ(x1, . . . , xk) =
n∑
i1=1
. . .
n∑
ik=1
ξi1,...,ikxi11 · . . . · xikk ,
where
ξi1,...,ik = ξ(ei1 , . . . , eik) ∈ Y andxj =n∑
i=1
xijei.
In particular, observe thatξ ∈ M k(X, Y ) is symmetric if and only if for one basis(ej)nj=1 ofX (and then for all)ξi1,...,ik = ξiσ(1),...,iσ(k)
for all σ ∈ Sym(k) and all indicesij ∈ 1, . . . , n.
Obviously, whendimY = N we havedim⊙k (X, Y ) = N(k+n−1
k
).
In view of the standard definition of the (Frechet–)derivative of aCk mappingf : Ω ⊂ X →Y and Schwarz’ theorem on interchangeability of partial derivatives we have for allx ∈ Ω thatDkf(x) ∈ ⊙k(X, Y ).
The following calculations are of particular interest to us: If µ ∈ ⊙k(X, Y ) andfµ(x) =µ(x, . . . , x) for x ∈ X, then
Dfµ(x)(h) = limt→0
1
t
(
µ(x+ th, . . . , x+ th)− µ(x, . . . , x)
)
= µ(h, x, x, . . . , x) + µ(x, h, x, . . . , x) + . . .+ µ(x, . . . , x, h)
= kµ(h, x, . . . , x).
Iterating this, we obtain
Dlfµ(x)(h1, . . . , hl) =k!
(k − l)!µ(h1, h2, . . . , hk, x, . . . , x
︸ ︷︷ ︸
(k−l)−times
),
and finallyDkfµ(x)(h1, . . . , hk) = k!µ(h1, h2, . . . , hk) (4.1)
for all x, h1, . . . , hk ∈ X. In particular, the symmetrick–linear map generating a homo-geneousk–th order polynomialf is unique, the converse representation (of any such polyno-mial by a symmetrick–linear map) is a consequence of Taylor’s Theorem. Thus, we see that
12
⊙k(X, Y ) is precisely the space of allk–th derivatives ofY –valued functions defined on (opensubsetsΩ of) X.
Now we focus, for the sake of notational simplicity, on the only slightly more special situ-ation ofX = R
n with usual scalar product〈·, ·〉 or, equivalently, any finite dimensional innerproduct space. Analogous to before, we find that for allCk functionsg : R → R and eacha ∈ Rn the functionf(x) = g(〈x, a〉) onRn satisfies
Dkf(x)(h1, . . . , hk) = g(k)(〈x, a〉)k∏
i=1
〈a, hi〉,
or, in more convenient tensor notation,
Dkf(x) = g(k)(〈x, a〉)a⊗k, (4.2)
where we identifya ∈ Rn with y 7→ 〈y, a〉 in (Rn)∗ and accordingly
a⊗k = a⊗ . . .⊗ a︸ ︷︷ ︸
k
with the symmetrick–linear forma⊗k(h1, . . . , hk) =∏k
i=1〈a, hi〉. In particular we record thatits coordinates with respect to any orthonormal basis(ej)
nj=1 for Rn are
(a⊗k)i1...ik =k∏
l=1
ail , where ail = 〈a, eil〉.
Next, we assert that the balanced cone
Ds(n, k) := t · a⊗k : t ∈ R anda ∈ Rn
spans⊙k(Rn). Indeed, otherwise we could findµ ∈ ⊙k(Rn) \ 0 which is perpendicular to allof Ds(n, k) with respect to the canonical scalar product on⊙Rn extended fromRn
〈ξ, ζ〉 =n∑
i1=1
. . .n∑
ik=1
ξi1...ikζi1...ik ,
whereξi1...ik , ζi1...ik are the coordinates with respect to the orthonormal basis obtained from(ej)
nj=1. But observing that
〈µ, a⊗k〉 =n∑
i1=1
. . .n∑
ik=1
µi1...ikai1 · . . . · aik = µ(a, . . . , a),
we havefµ ≡ 0 onRn and hence by (4.1) the contradictionµ = 0. As an easy consequence wesee that, using standard notation, also the balanced cone
Ds = Ds(n, k; Y ) = b⊗ a⊗ . . .⊗ a︸ ︷︷ ︸
k
= b⊗ a⊗k ; a ∈ Rn andb ∈ Y
13
spans⊙k(Rn, Y ). Similar observations can also be found in the Appendix A.1 of [SZ04].Finally, we want to clarify why this coneDs(n, k, Y ) plays in the case ofk–th order deriva-
tives the role which the usual cone of rank–one directions has for first order derivatives. So leton a bounded smooth domainΩ of Rn an integrable mapf : Ω → Y be given and assume thatits distributionalk–th derivative satisfies
Dkf(x) ∈ ξ, η a.e., whereξ, η ∈ ⊙k(Rn, Y ).
Theng := Dk−1f : Ω → ⊙k−1(Rn, Y ) is easily seen to belong toW1,∞(Ω,⊙k−1(Rn, Y ))) andtherefore is lipschitz withDg ∈ ξ, η a.e., where, by Rademacher’s theorem, the derivativecan be understood both distributionally and classically a.e. A by now standard argument (see[BJ]) yields thatξ, η as elements ofL (Rn,⊙k−1(Rn, Y )) satisfy rank(ξ − η) ≤ 1. In otherwords, whenξ 6= η the kernel ofξ− η must be(n−1)–dimensional so for a unit vectorν ∈ Rn
we have(ξ − η)(x) = (ξ − η)(〈x, ν〉ν) as elements of⊙k−1(Rn, Y ) for all x ∈ Rn. By thesymmetry ofξ − η we therefore get for allx = x1 ∈ Rn, x2, . . . , xk ∈ Rn
(ξ − η)(x1, x2, . . . , xk) = [(ξ − η)(x)](x2, . . . , xk) = (ξ − η)(〈x1, ν〉ν, x2, . . . , xk)= (ξ − η)(〈x1, ν〉ν, 〈x2, ν〉ν, . . . , 〈xk, ν〉ν)
= (ξ − η)(ν, ν, . . . , ν)k∏
i=1
〈xi, ν〉.
In other words, then necessarilyξ − η = [(ξ − η)(ν, ν, . . . , ν)] ⊗ ν⊗k. Conversely (4.2) showsthat for eachµ ∈ Ds(n, k; Y ) there isf ∈ Ck−1,1(Rn, Y ) such thatDkf ∈ 0, µ a.e., and soDkf is not essentially constant.
Next we present a result that corresponds to the gradient distribution changing techniquesused in the theory of first order partial differential inclusions. Since we are not forced to findexact solutions, as is usually done in this theory, we avoid many technical difficulties and wecan work exclusively within the class ofC∞–maps. First let us recall the Leibniz rule in severalvariables. LetΩ be an open subset ofX. If f : Ω → Y andg : Ω → Z areCk–maps andΨ: Y × Z → V is bilinear, then for eachx ∈ Ω andv = (v1, . . . , vk) ∈ Xk we have
Dk
(
y 7→ Ψ(f(y), g(y))
)
(x)(v) =∑
M⊂1,...,k
Ψ(∇Mf(x)(v), ∇1,...,k\Mg(x)(v)), (4.3)
where we denote∇Mϕ(x)(v) := Dlϕ(x)(vi1 , . . . , vil) if M = i1, . . . , il ⊂ 1, . . . , k hascardinalityl (well–defined due to the symmetry of the higher order derivatives). The rule canbe easily shown by induction onk.
Proposition 4.1. Let Ω be a bounded open subset ofRn, k ∈ N and ξ, η ∈ ⊙k(Rn, Y ) withξ − η ∈ Ds(n, k, Y ) be given (we express this by saying thatξ, η are rank-one connected.Supposeλ ∈ (0, 1) and letu : Ω → Y satisfyDku(x) = C = λξ+ (1−λ)η for all x ∈ Ω (souis in particular a polynomial of degreek). Then for eachε > 0 there are a compactly supportedC∞ mapϕ : Ω → Y and open subsetsΩξ,Ωη of Ω such that
14
(i) Dk(u+ ϕ)(x) = ξ if x ∈ Ωξ,Dk(u+ ϕ)(x) = η if x ∈ Ωη anddist(Dk(u+ ϕ)(x), [ξ, η]) < ε for all x ∈ Ω, where[ξ, η] = tξ + (1− t)η : t ∈ [0, 1].
(ii) |Ωξ| > (1− ε)λ|Ω| and|Ωη| > (1− ε)(1− λ)|Ω|,
(iii) |Dlϕ(x)| < ε for all x ∈ Ω andl < k.
Proof. We start by choosing a1-periodicC∞-functionh : R → [λ − 1, λ] with∫ 1
0h = 0 and
for which there exist intervalsIξ, Iη ⊂ (0, 1) such thath = λ − 1 on Iξ, h = λ on Iη and|Iξ| > (1 − ε/2)λ, |Iη| > (1 − ε/2)(1 − λ). For instance, we could use a (sufficiently ’fine’)mollification of the1–periodic extension of the function(λ− 1)1(0,λ) + λ1(λ,1). Now for eachl ∈ 0, . . . , k we define recursivelyC∞ functionshl by
h0 = h andhl(0) = 0, h′l = hl−1 if l ≥ 1.
SettingH = hk, we notice thath1 is again1–periodic and, by induction,
|H(l)(t)| = |hk−l(t)| ≤‖h1‖∞
(k − l − 1)!|t|k−l−1 (4.4)
for 0 ≤ l < k andt ∈ R. For later reference we also defineχξ to be the1–periodic function thatequals1Iξ −|Iξ| on [0, 1) and similarly forχη. Since these functions integrate to zero over eachinterval of length1, we see that their indefinite integrals (and hence also distributional primitivefunctions)Fξ, Fη are bounded. Hence, partial integration gives for each smooth test functionϕ ∈ C∞
c (R) that
limj→∞
∣∣
∫
R
χξ(jt)ϕ(t) dt∣∣ = lim
j→∞
∣∣
∫
R
Fξ(jt)ϕ′(t)
jdt∣∣ = 0,
and henceχξ,j
∗ |Iξ| and χη,j
∗ |Iη| in L1(R)∗ (4.5)
whereχξ,j(t) = 1(Iξ+Z)(jt) andχη,j(t) = 1(Iη+Z)(jt).Sinceξ andη are rank–one connected, we findb ∈ Y anda ∈ Sn−1 such thatη−ξ = b⊗a⊗k.
Without loss of generality, we can assume thatb 6= 0. Fix a C∞ functionψ : Rn → [0, 1]compactly supported inΩ such that the setΩ+ = x ∈ Ω : ψ(x) = 1 satisfies
|Ω+| > (1− ε/2)|Ω|. (4.6)
Defining nowϕj(x) = bψ(x)j−kH(〈x, ja〉), x ∈ R
n, j ∈ N,
we claim that forj ∈ N sufficiently large the mapϕ = ϕj is the required mapping. The first,and crucial observation is that (4.2) and (4.3) imply that for x ∈ Ω,
‖Dkϕj(x) − b⊗ a⊗kψ(x)H(k)(〈x, ja〉)‖
=∥∥∥
k∑
l=1
∑
#A=l
b⊗(∇Aψ(x)⊗ j−lH(k−l)(j〈x, a〉)a⊗(1,...,k\A)
)∥∥∥
(4.4)≤ ‖b‖
k∑
l=1
cψ,h1,lj−l(j|x|)l−1 ≤ ‖b‖c′ψ,h1,k,diam(Ω)j
−1.
15
Recalling thatH(k) = h and thatµ + b ⊗ a⊗kψ(x)h(〈x, ja〉) ∈ [ξ, η] for all x we see that thelast inequality in (i) is satisfied providedj > ‖b‖c′ψ,h1,k,Ω/ε. SinceDψ ≡ 0 onΩ+ we concludefor the very same reason that the first two statements in (i) hold true for
Ωξ,j = x ∈ Ω+ : 〈jx, a〉 ∈ Iξ + Z,
andΩη,j = x ∈ Ω+ : 〈jx, a〉 ∈ Iη + Z.
Similarly, (4.3), (4.2) and (4.4) implymaxx∈Ω |Dlϕj(x)| ≤ c′′l,ψ,diam(Ω)j−1 if l < k and so (iii)
follows.Finally, to establish (ii) usually a disjoint decomposition of (a large part of)Ω+ into cubes
with a side parallel toa is considered. However, for variety we prefer to give an analyticargument. Denotings(t) = Hn−1(x ∈ Ω+ : 〈x, a〉 = t), we infer from Fubini’s theoremand (4.5) that
limj→∞
|Ωξ,j| = limj→∞
∫
s(t)χξ,j(t) dt =
∫
s(t) dt|Iξ| = |Iξ||Ω+|
> (1− ε
2)2λ|Ω|
> (1− ε)λ|Ω|,
and similarlylimj→∞ |Ωη,j | > (1 − ε)(1 − λ)|Ω|. So, choosingj sufficiently large, also (ii)holds and the proof is finished.
After these preparations we show how functionals with the semiconvexity properties dis-cussed in the previous sections naturally arise when relaxing energy functionals. This phe-nomenon is well known in the Calculus of Variations, but we could not find a concise treatmentof the higher order derivative case that was based on compactly supported test maps in the liter-ature. The preference has been for periodic test maps, see for instance [BCO81] and [FM], andwhile it is possible to derive our results in that context toowe found it useful and worthwhileto here record the approach based on compactly supported test maps. We therefore provide adetailed exposition for the convenience of the reader.
Recall that a subsetS of V isD–convex if for allx, y ∈ S such thatx− y ∈ D we have thatthe segment[x, y] is contained inS.
Theorem 4.2. Let S be an open andDs(n, k; Y )–convex subset ofV = ⊙k(Rn, Y ) and letthe extended real–valued functionF : S → [−∞,∞) be locally bounded above and Borelmeasurable.
Then the relaxation ofF defined as
F(ξ) := inf
∫
(0,1)nF (ξ +Dkϕ(x)) dx :
ϕ ∈ C∞c ((0, 1)n, Y )
ξ +Dkϕ(x) ∈ S for all x
is aDs(n, k, Y )–convex functionF : S → [−∞,∞).
16
Moreover, for any strictly increasing functionω : [0,∞) → [0, 1] with ω(0) = 0 andt/ω(t) → 0 as t ց 0 and anyε > 0 we can without changing the value ofF restrict theinfimum in its definiton to be taken only over thoseϕ which in addition satisfy|Dlϕ(x)| < ε forall x andl < k, and‖Dk−1ϕ(x)−Dk−1ϕ(y)‖ ≤ εω(|x− y|) for all x, y.
Proof. We start by checking that the infimum is unchanged if we restrict the test mapsϕ asdescribed. Since we can replaceω by its concave envelope we can without loss of generality as-sume thatω in addition is concave. Hence fixω, ε as in the statement and letϕ ∈ C∞
c ((0, 1)n, Y )with ξ +Dkϕ(x) ∈ S for all x. We then takej ∈ N such that
2−j max0≤l<k
‖Dlϕ‖L∞ < ε and 4‖Dkϕ‖L∞
√n2−j
ω(√n2−j)
< ε.
Extendϕ to all of Rn by (0, 1)n periodicity and define forx ∈ (0, 1)n, ψ(x) = 2−jkφ(2jx).Thenψ is an admissible test map forF and it is clear that
∫
(0,1)nF (ξ +Dkψ(x)) dx =
∫
(0,1)nF (ξ +Dkϕ(x)) dx and max
0≤l<k‖Dlψ‖L∞ < ε.
To check theω–Holder continuity we letx, y ∈ (0, 1)n. Divide the cube[0, 1]n into 2jn non–overlapping dyadic subcubes each of side length2−j. If x, y belong to the same dyadic subcube,say they both belong to2−jz + [0, 2−j]n, wherez ∈ Zn, then clearly
‖Dk−1ψ(x)−Dk−1ψ(y)‖ = 2−j‖Dk−1ϕ(2jx− z)−Dk−1ϕ(2jy − z)‖≤ ‖Dkϕ‖L∞|x− y|
≤ ‖Dkϕ‖L∞
√n2−j
ω(√n2−j)
ω(|x− y|)
≤ εω(|x− y|).
If x, y belong to distinct dyadic subcubes, sayDx,Dy, then takex ∈ (x, y)∩Dx, y ∈ (x, y)∩Dy
and note that|x − y| ≥ |x − x| + |y − y| and|x − x|, |y − y| ≤ √n2−j . Using thatψ must
vanish in small neighbourhoods ofx, y we estimate as before
‖Dk−1ψ(x)−Dk−1ψ(y)‖ ≤ 2−j‖Dkϕ‖L∞
(
2j|x− x|+ 2j|y − y|)
≤ 2‖Dkϕ‖L∞
√n2−j
ω(√n2−j)
(
ω(|x− x|) + ω(|y − y|))
≤ εω(|x− y|).
Next we turn to theD–convexity. BecauseF <∞ alsoF <∞ and it therefore suffices to showthatF(µ) ≤ λF(ξ)+(1−λ)F(η) if ξ, η ∈ S with η−ξ ∈ D, λ ∈ (0, 1) andµ = λξ+(1−λ)η.Fix real numberss, t sos > F(ξ) andt > F(η) and by the definition of relaxation find mapsϕξ, ϕη ∈ C∞
c ((0, 1)n, Y ) such thatξ +Dkϕξ(x) ∈ S, η +Dkϕη(x) ∈ S for all x, and∫
(0,1)nF (ξ +Dkϕξ(x)) dx < s (4.7)
17
and ∫
(0,1)nF (η +Dkϕη(x)) dx < t. (4.8)
Givenδ > 0 we findε ∈ (0, δ) such that
supF (ζ) : dist(ζ, [ξ, η]) < εε < δ,
andζ ∈ S if dist(ζ, [A,B]) < ε. Next, we setu(x) = µ(x, x, . . . , x)/k! and chooseφ satisfyingthe conditions (i), (ii) and (iii) of Proposition 4.1 forΩ = (0, 1)n, in particular so thatΩξ =intx ∈ Ω : Dk(u + φ)(x) = ξ andΩη = intx ∈ Ω : Dk(u + φ)(x) = η satisfy
Ln(
Ω \ (Ωξ ∪ Ωη)
)
< εLn(Ω). Clearly, we find anl ∈ N such that the familyQξ of all
dyadic cubes of sidelength2−l entirely contained inΩξ satisfiesLn(⋃Qξ) > (1 − ε)Ln(Ωξ)and similarlyLn
(⋃Qη
)
> (1− ε)Ln(Ωη). Hence,
Ωr = Ω \(⋃
Qξ ∪⋃
Qη
)
fulfills Ln(Ωr) < ε and by Proposition 4.1(i), dist(Dk(u + φ)(x), [ξ, η]) < ε for x ∈ Ω.Therefore, ∫
Ωr
F (Dk(u+ φ))(x) dx < δ.
Finally, we define similarly to the first part of the proof the function
ψ(x) =
(u+ φ)(x) if x ∈ Ωr,
(u+ φ)(x) + 2−klϕξ(2l(x− y)) if x ∈ y + [0, 2−l]n ∈ Qξ,
(u+ φ)(x) + 2−klϕη(2l(x− y)) if x ∈ y + [0, 2−l]n ∈ Qη.
AsDkψ(x) = ξ +Dkϕξ(2l(x− y)) in the interior ofy + [0, 2−l]n ⊂ Qξ we have for each such
cube that ∫
y+[0,2−l]nF (Dkψ(x)) dx < sLn
(
y + [0, 2−l]n)
,
see (4.7). Similarly, (4.8) implies∫
y+[0,2−l]nF (Dkψ(x)) dx < tLn
(
y + [0, 2−l]n)
for cubesy + [0, 2−l]n ⊂ Qη. Altogether, in view of Proposition 4.1 (ii),∫
Ω
F (Dkψ) ≤ δ + sLn(⋃
Qξ
)
+ tLn(⋃
Qη
)
≤ δ + sLn(Ωξ) + tLn(Ωη)≤ δ + s(1− Ln(Ωη)) + t(1− Ln(Ωξ))≤ δ + s(1− (1− λ)(1− ε)) + t(1− λ(1− ε))
≤ sλ+ t(1− λ) + (δ + ε(s(1− λ) + tλ))
≤ sλ+ t(1− λ) + δ(1 + s+ t).
18
It is also easy see thatψ − u ∈ C∞c ((0, 1)n, Y ) because it is a finite sum of such functions
(includingφ). Sinceδ > 0 was arbitrary, the proof is complete.
5 A generalization of Ornstein’sL1 Non–Inequality
As was mentioned in the introduction Theorem 1.1 implies theOrnstein non–inequality in thecontext ofx–dependent vector valued operators, and even a version involving certain nonlinearx–dependent differential expressions:
Theorem 5.1.LetF : Rn ×⊙k(Rn,RN) → R be a Caratheodory integrand satisfying
F (x, tξ) = |t|F (x, ξ)
and|F (x, ξ)| ≤ a(x)|ξ|
for almost allx ∈ Rn, all t ∈ R andξ ∈ ⊙k(Rn,RN), wherea ∈ L1loc(R
n) is a given function.Then ∫
Rn
F (x,Dkφ(x)) dx ≥ 0 (5.1)
holds for all φ ∈ C∞c (Rn,RN) if and only if F (x, ξ) ≥ 0 for almost allx ∈ Rn and all
ξ ∈ ⊙k(Rn,RN).
Before presenting the short proof of Theorem 5.1 let us see how it implies the OrnsteinL1
non–inequality stated in Theorem 1.3.
Proof of Theorem 1.3.Only one direction requires proof. To that end we write the differentialoperators in terms of linear mappingsAi(x) : ⊙k (Rn,V) → Wi as follows:
Ai(x)Dkϕ =
∑
|α|=k
aiα(x)∂αϕ
for all ϕ ∈ C∞c (Rn,V). Define
F (x, ξ) := c‖A1(x)ξ‖ − ‖A2(x)ξ‖ for (x, ξ) ∈ Rn ×⊙k(Rn,V),
and note that Theorem 5.1 applies to the functionF . Accordingly,F (x, ·) is a nonnegativefunction for almost allx, and therefore the set inclusion,ker A2(x) ⊃ ker A1(x), must in par-ticular hold for the kernels for all suchx. Fix x such thatF (x, ·) is nonnegative. We define alinear mappingC(x) : W1 → W2 by
C(x) = A2(x) (
A1(x)|(kerA1(x)
)⊥
)−1
projimA1(x).
HerebyC is defined almost everywhere inRn, is measurable, andA2(x) = C(x)A1(x) holdsfor almost allx. Finally, the nonnegativity ofF (x, ·) yields the uniform norm bound‖C‖L∞ ≤c.
19
Proof of Theorem 5.1.We assume that the integral bound (5.1) holds and shall deduce the point-wise boundF ≥ 0. First we reduce to the autonomous case:F = F (ξ). The procedure for do-ing this is a variant of a well–known proof showing that quasiconvexity is a necessary conditionfor sequential weak lower semicontinuity of variational integrals (see [Mo66] Theorem 4.4.2).Let us briefly comment on the details. Denote byC = (−1/2, 1/2)n the open cube centred atthe origin with sides parallel to the coordinate axes and of unit length, and letφ ∈ C∞
c (C,RN).We extendφ to Rn by C periodicity, denote this extension byφ again, and put forx0 ∈ Rn,r > 0, j ∈ N,
ϕj(x) :=
(r
j
)k
φ
(
jx− x0r
)
, x ∈ Rn.
Writing C(x0, r) = x0 + rC we have for allj ∈ N thatϕj |C(x0,r) ∈ C∞c (C(x0, r),R
N), and soextending this restriction toRn \ C(x0, r) by 0 ∈ R
N to get a test map for (5.1) we arrive at
0 ≤∫
C(x0,r)
F(x,Dkφ
(
jx− x0r
))dx.
If we let j tend to infinity, then by a routine argument that uses the Riemann–Lebesgue lemma(see for instance [Dac89] Chapter 1) it follows that
0 ≤∫
C(x0,r)
∫
C
F (x,Dkφ(y)) dy dx.
Sincex0, r were arbitrary we deduce by the regularity of Lebesgue measure that∫
C
F (x,Dkφ(y)) dy ≥ 0 (5.2)
holds for allx ∈ Rn \ Nφ whereNφ is a negligible set. To see that (5.2) in fact holds outsidea negligible setN ⊂ Rn that is independent ofφ one invokes the separability of the spaceCc(C,⊙k(Rn,RN)) in the supremum norm. We leave the precise details of this to the interestedreader and also remark that a straightforward scaling argument shows that we may replaceCby Rn and allow anyφ ∈ C∞
c (Rn,RN) in (5.2). It now remains to prove the pointwise boundin the autonomous caseF = F (ξ). By Theorem 4.2 the relaxationF given by the formula
F(ξ) = infϕ∈C∞
c ((0,1)n,RN )
∫
(0,1)nF (ξ +Dkϕ(x)) dx
is a rank–one convex functionF : ⊙k (Rn,RN) → [−∞,∞). Recall that byrank–one convexfunction in this context we understand a function which is convex in the directions of the coneDs(n, k,R
N). Now the integral bound (5.1) translates toF(0) ≥ 0, and it then follows fromLemma 2.2 thatF is a rank–one convex function which is real–valuedeverywhere. By thehomogeneity ofF and ofC∞
c one also easily checks thatF is 1–homogeneous. Consequently,as the coneDs(n, k;R
N) is balanced and spanning, Theorem 1.1 implies in particularthatFis convex at0 ∈ ⊙k(Rn,RN): there existsζ ∈ ⊙k(Rn,RN) such thatF(ξ) ≥ 〈ξ, ζ〉 for allξ ∈ ⊙k(Rn,RN). Takingξ = ±ζ and using1–homogeneity we see that necessarilyζ = 0. Theconclusion follows becauseF ≥ F .
20
Remark 5.2. The proof given above for Theorem 5.1 can easily be adapted toalso prove thefollowing result. For aC∞ mapφ : Rn → RN we denote byJk−1φ(x) itsk−1 jet atx: the vectorconsisting of all partial derivativesDαφi(x) of coordinate functionsφi corresponding to allmulti–indicesα of length at mostk−1 arranged in some fixed order. HerebyJk−1φ : Rn → R
d
for a d = d(n, k,N).LetF : Rn × Rd ×⊙k(Rn,RN) → R be a Caratheodory integrand (measurable inx ∈ Rn
and jointly continuous in(y, ξ)) satisfying
F (x, y, tξ) = |t|F (x, y, ξ)
and|F (x, y, ξ)| ≤ a(x)(|y|+ |ξ|)
for almost allx ∈ Rn, all y ∈ Rd, t ∈ R andξ ∈ ⊙k(Rn,RN), wherea ∈ L1loc(R
n) is a givenfunction.
Then ∫
Rn
F (x, Jk−1φ(x), Dkφ(x)) dx ≥ 0
holds for all φ ∈ C∞c (Rn,RN) only if F (x, 0, ξ) ≥ 0 for almost allx ∈ Rn and all ξ ∈
⊙k(Rn,RN).
We now turn to:
Proof of Theorem 1.5.Only the necessity part requires a proof. Assume that there exists anontrivialD–affine quadratic formm onV, whereD is a cone of directions satisfying (1.1).
In the setting of Section 4 we consider the spacesY = R andX = V.The second derivatives that we are interested in will belongto the space⊙2(V) ∼= ⊙2(V,R).
The correspondingrank-one coneis the usual one in⊙2(V), namelyt · ℓ⊗ ℓ : t ∈ R andℓ ∈V∗ (indeed, if we choose a basis forV we may identifyV ∼= Rn, ⊙2(V) ∼= Rn×n
sym and therank-one cone is simply the cone of symmetricn× n matrices of rank at most one). This coneis of course balanced and spanning. Next, and more specific toour situation we observe thatfor an open subsetO of V a C2 functionϕ : O → R is D–convex if for allx ∈ O we haveD2ϕ(x) ∈ C where
C :=
µ ∈ ⊙2(V) : µ(e, e) > 0 for all e ∈ D \ 0
.
Clearly,C is an open convex cone, and in the notation from the Introduction we haveC = intQD.The assumption (1.1) therefore amounts toℓ ⊗ ℓ ∈ C for someℓ ∈ V∗. Putµ0 = ℓ ⊗ ℓ so thatµ0 is a rank one form inC. If µm = D2m(0) we have thatµ0 + tµm ∈ C for all t ∈ R. Wenow consider the functionF (µ) := −‖µ‖ on C. By Theorem 4.2 the relaxationF is rank-oneconvex, and sinceF < ∞ alsoF < ∞. We assert thatF ≡ −∞. Assume not, so thatF isfinite somewhere inC. Then by Lemma 2.2F > −∞ everywhere onC, so thatF is a real–valued and rank–one convex function onC. From the definition ofF and the homogeneity of
21
F , C∞c we infer thatF is positively1–homogeneous. By virtue of Theorem 1.1F is therefore
convex atµ0 so that in particular:
F(µ0) ≤ 1
2
(
F(µ0 + tµm) + F(µ0 − tµm)
)
≤ −1
2
(
‖µ0 + tµm‖+ ‖µ0 − tµm‖)
,
which is impossible for large values oft. This contradiction shows thatF ≡ −∞ onC.In view of the definition of the relaxationF we then find for anyµ ∈ C and eachε > 0
anϕ ∈ C∞c (Q,R) with µ + D2ϕ(x) ∈ C for all x ∈ V such that
∫
Q(−‖µ + D2ϕ‖) < −1/ε,
‖ϕ‖L∞(Q) + ‖Dϕ‖L∞(Q) < ε, and
[Dϕ]ω := sup
‖Dϕ(x)−Dϕ(y)‖ω(|x− y|) : x, y ∈ Q, x 6= y
< ε,
whereQ is the unit cube inV with respect to some (from now on fixed) orthonormal basis ofV and the integral is calculated with respect to the corresponding Lebesgue measure. There isnothing special by the open unit cubeQ in the above, and it is possible to replace it by anybounded open subsetO of V. Indeed, we simply replaceϕ by x 7→ λ−2ϕ(λx + y) for λ > 0andy ∈ V and expressO as a disjoint union of closed cubesy + λQ to achieve this.
The existence of the function claimed in the statement of Theorem 1.5 is now an easy con-sequence of the Baire category theorem. We consider the setA of all rank–one convexC1
functionsf : X → R such that
‖f‖ = sup
|f(x)|1 + ‖x‖2 +
‖Df(x)−Df(y)‖ω(|x− y|)(1 + |x|+ |y|) : x, y ∈ X, x 6= y
is finite and equip it with the metricd(f, g) := ‖f − g‖. It is easy to verify that(A, d) is acomplete metric space, and we denote byB the closure of smooth functions inA. With theinherited metric this is clearly also a complete metric space. Now let(Ol)
∞l=1 be a sequence
consisting of all dyadic cubes inV of side length at most one and define, forl, k ≥ 1,
Bl,k =
f ∈ B :
∫
Ol
f∂2ψ
∂xα≤ k if ψ ∈ C∞
c (Ol,R), ‖ψ‖∞ ≤ 1 and|α| = 2
.
Clearly, eachBl,k is a closed set inB and forf ∈ C2 the usual integration by parts argumentapplied to aψ that sufficiently well approximates sign(∂2f/∂xα) shows that we have
f ∈ Bl,k ⇔∫
Ol
∣∣∣∣
∂2f
∂xα
∣∣∣∣≤ k for all |α| = 2.
We finish our proof by showing that eachBl,k has empty interior since thenB \⋃
l,k Bl,k isnonempty and obviously eachf in this set satisfies the conclusion of Theorem 1.5. But other-wise, we would find an open ballB(f, δ) inside someBl,k. The definition ofB as the closure
22
of smooth functions means that we without loss of generalitymay assumef ∈ C∞. Now wefind aC ∈ C considered above such that‖φC‖ < δ/3, hereφC(x) = 1
2C(x, x) and the first
part of our proof ensures the existence of aϕ ∈ C∞c (Ol) such that‖ϕ‖ ≤ ‖ϕ‖∞ < δ/3 but
∫
Ol‖D2(ϕC + ϕ)‖ > 3n4k. In particular, there must exist a multi-indexα of length two such
that∫
Ol|∂2(ϕC + ϕ)/∂xα| > 3k. From this it is clear thatf + ϕC + ϕ ∈ B(f, δ) but
∫
Ol
∣∣∣∣
∂2(f + ϕC + ϕ)
∂xα
∣∣∣∣> 2k,
a contradiction finishing our proof of Theorem 1.5.
6 Gradient Young Measures
A convenient way to describe the one–point statistics in a bounded sequence of vector–valuedmeasures where both oscillation and concentration phenomena must be taken into account isthrough a notion of Young measure that was introduced by DiPerna and Majda in [DM87]. Thisformalism was revisited in [AB97] and a special case of particular importance was developedinto a very convenient and suggestive form. This was the starting point for [KR10b], and weshall briefly pause to describe the relevant set–up here.
Throughout this sectionΩ denotes a fixed bounded Lipschitz domain inRn andH a finitedimensional real inner product spaces.
Let E(Ω,H) denote the space oftest integrands consisting of all continuous functionsf : Ω×H → R for which
f∞(x, z) = limt→∞
f(x, tz)
t
exists locally uniformly in(x, z) ∈ Ω × H. Note that hereby therecession functionf∞ isjointly continuous andz 7→ f∞(x, z) is for each fixedx a positively1–homogeneous function.
The set of Young measures under consideration, denoted byY(Ω,H), is defined to be theset of all triples((νx)x∈Ω, λ, (ν∞x )x∈Ω) such that
• (νx)x∈Ω is a weakly∗Ln measurable family of probability measures onH, and∫
Ω
∫
H
|z| dνx(z) dx <∞,
• λ is a nonnegative finite Radon measure onΩ,
• (ν∞x )x∈Ω is a weakly∗λ measurable family of probability measures onSH, whereSH
is the unit sphere inH.
Here, the mapsx 7→ νx andx 7→ ν∞x are only defined up toLn- andλ–negligible sets, respec-tively. The parametrized measure(νx)x is called theoscillation measure, the measureλ is theconcentration measureand(ν∞x )x is theconcentration-angle measure. As the terms indi-cate, the oscillation measure is the usual Young measure anddescribes the oscillation (or failure
23
of convergence inLn measure), the concentration measure describes the location in Ω and themagnitude of concentration (or failure ofLn equi–integrability) while the concentration–anglemeasure describes its direction inH. Often, we will simply writeν as short-hand for the abovetriple.
To everyH–valued Radon measureγ supported onΩ with Lebesgue–Radon–Nikodym de-compositionγ = aLd + γs, we associate anelementary Young measureǫγ ∈ Y(Ω,H) givenby the definitions
(ǫγ)x := δa(x), λǫγ := |γs|, (ǫγ)∞x := δp(x),
wherep := dγs/d|γs| is the Radon–Nikodym derivative and|γs|(= |γ|s) denotes the totalvariation measure ofγs.
Using the inner product onH we can define a duality pairing off ∈ E(Ω,H) andν ∈Y(Ω,H) by setting
〈〈f, ν〉〉 :=∫
Ω
∫
H
f(x, z) dνx(z) dx+
∫
Ω
∫
SH
f∞(x, z) dν∞x (z) dλ(x).
For a sequence(γj) of H–valued Radon measures withsupj |γj|(Ω) < ∞, we say that(γj)
generatesthe Young measureν ∈ Y(Ω,H), in symbolsγjY→ ν, if for all f ∈ E(Ω,H) we
have〈〈f, ǫγj〉〉 → 〈〈f, ν〉〉, or equivalently,
f(
x,dγjdLn (x)
)
Ln + f∞(
x,dγsjd|γsj |
(x))
|γsj | →⟨f(x, ·), νx
⟩Ln +
⟨f∞(x, ·), ν∞x
⟩λ (6.1)
weakly∗ in C(Ω)∗ as j → ∞. A continuity result due to Reshetnyak [R68] can be used tojustify the formalism (see [KR10b]) in the sense that for measuresγj, γ not charging∂Ω we
haveγj → γ 〈·〉–strictly if and only ifǫγjY→ ǫγ . Here the former is taken to mean thatγj
∗ γ
in C0(Ω,H)∗ and〈γj〉(Ω) → 〈γ〉(Ω), where
〈γ〉(Ω) :=∫
Ω
(
1 + | dγdLn |
2
) 12
dLn + |γs|(Ω).
Standard compactness theorems for measures imply that any bounded sequence(γj) of H–
valued Radon measures onΩ admits a subsequence (not relabelled) such thatγjY→ ν for some
Young measureν ∈ Y(Ω,H). We are particularly interested in the situation where the measuresγj are concentrated onΩ. Making special choices of test integrandsf in (6.1) we deduce thatγj
∗ γ in C0(Ω,H)∗, whereγ = ν⌊Ω and
ν = νxLn + ν∞x λ with νx = 〈id, νx〉 andν∞x = 〈id, ν∞x 〉
is thebarycentre of ν. If λ(∂Ω) = 0, thenγ = ν and the convergence takes place in the strongersense thatγj
∗ γ narrowly onΩ (meaning〈Φ, γj〉 → 〈Φ, γ〉 for all bounded continuous maps
Φ: Ω → H).
24
Given a mapping of bounded variation,u ∈ BV(Ω,RN ), its distributional derivativeDu isanRN×n–valued Radon measure concentrated onΩ and hence we can associate an elementaryYoung measureǫDu ∈ Y(Ω,RN×n). WritingDu = ∇uLn +Dsu for the Lebesgue decompo-sition with respect toLn we have
(ǫDu)x := δ∇u(x), λǫDu:= |Dsu|, (ǫDu)
∞x := δp(x),
wherep := Dsu/|Dsu| is short-hand for the Radon-Nikodym derivative.Any Young measureν ∈ Y(Ω,RN×n) that can be generated by a sequence(Duj), where
(uj) is a bounded sequence inBV(Ω,RN ) that L1–converges to a mapu ∈ BV(Ω,RN) iscalled aBV gradient Young measurewith underlying deformationu. Clearly the underlyingdeformationu is locally unique up to an additive constant vector. If we assume thatλ(∂Ω) = 0,thenDu is the barycentre ofν and identifying terms according to the Lebesgue decompositionλ = aLn+ b|Dsu|+ λ∗ whereλ∗ is a measure which is singular with respect toLn+ |Dsu| wefind
∇u(x) = νx + ν∞xdλ
dLn (x) Ln a.e.x ∈ Ω (6.2)
Dsu
|Dsu|(x) = b(x)ν∞x |Dsu| a.e.x ∈ Ω (6.3)
0 = ν∞x λ∗ a.e.x ∈ Ω. (6.4)
These conditions are clearly necessary forν to be aBV gradient Young measure with barycentreDu. Other necessary conditions follow, loosely speaking, by expressing a relaxation result forsigned integrands of linear growth obtained in [KR10a], in terms of Young measures. It turnsout that these conditions are sufficient forν to be aBV gradient Young measure too. Theproof of this is based on a Hahn–Banach argument similar to that employed by Kinderlehrerand Pedregal in [KP91, KP94]. Hereby a characterization ofBV gradient Young measures isobtained: they are the dual objects to quasiconvex functions in the sense that a set of inequalitiesakin to Jensen’s inequality must hold for the measures and all quasiconvex functions of at mostlinear growth. It can be seen as a nontrivial instance withinthe abstract frame work providedby Choquet’s theory of function spaces and cones (see [LMNS]). In order to state the resultmore precisely we denote byQ the class of all quasiconvex functions of at most linear growth,sof ∈ Q whenf : RN×n → R is quasiconvex and there exists a real constantc = cf such that|f(ξ)| ≤ c
(|ξ|+ 1
)for all ξ.
Theorem 6.1([KR10b]). LetΩ be a bounded Lipschitz domain inRn. Then, a Young measureν ∈ Y(Ω,RN×n) satisfyingλ(∂Ω) = 0 is a BV gradient Young measure if and only if thereexistsu ∈ BV(Ω,RN) such thatDu is the barycentre forν, and, writingλ = dλ
dLnLn + λs forthe Lebesgue–Radon–Nikodym decomposition, the following conditions hold for allf ∈ Q:
(i) f(νx + ν∞x
dλ
dLn (x))≤
⟨f, νx
⟩+⟨f∞, ν∞x
⟩ dλ
dLn (x) Ln a.e.x ∈ Ω,
(ii) f∞(ν∞x ) ≤⟨f∞, ν∞x
⟩λs a.e.x ∈ Ω .
25
Here the symbolf∞ refers to the (upper) recession function off as defined at (2.3). Thecondition (ii) concerning the pointsx seen by the singular part of the measureλ is simplyJensen’s inequality for the probability measureν∞x and the1–homogeneous quasiconvex func-tion f∞. Condition(i) concerning pointsx seen by the absolutely continuous part of the mea-sureλ is reminiscent of both Jensen’s inequality and (2.5). It expresses how oscillation andconcentration must be coupled if created by a sequence of gradients. The question of whetherthis characterization can be simplified, and indeed whethercondition(ii) is necessary at all,was in fact the initial motivation for the work undertaken inthe present paper. Inspection of theabove characterization yields in particular that a nonnegative finite Radon measureλ onΩ withλ(∂Ω) = 0 is the concentration measure for aBV gradient Young measure with underlying de-formationu ∈ BV(Ω,RN) if and only if λs ≥ |Dsu| as measures onΩ. In fact, it is not hard tosee this directly (even without the assumption thatλ(∂Ω) = 0), and the reason for mentioningit at this stage is that we would like to emphasize that the concentration measure is more or lessarbitrary.
In order to state the the main result of this section we introduce the following notation forspecial test integrands:
SQ = SQ(RN×n) =
f ∈ Q :f is Lipschitz,f(ξ) ≥ |ξ| for all ξ, and for somerf > 0 we havef(ξ) = f∞(ξ) when|ξ| ≥ rf
.
In terms of these the result is
Theorem 6.2. Let Ω be a bounded Lipschitz domain inRn. Then a Young measureν ∈Y(Ω;RN×n) satisfyingλ(∂Ω) = 0 is a BV gradient Young measure, if and only ifν hasbarycentreν = Du for someu ∈ BV(Ω,RN) and for allf ∈ SQ we have
f(νx + ν∞x
dλ
dLn (x))≤
⟨f, νx
⟩+⟨f∞, ν∞x
⟩ dλ
dLn (x) (6.5)
for Ln almost allx.
In view of the examples of nonconvex, but quasiconvex positively 1–homogeneous inte-grands given in [Mu92] it seems thatSQ is close to the minimal class of test integrands wecould possibly expect to use in (6.5). For instance we note that it seems to be impossible toreduce the class of test integrands to the smaller one used inthe characterization of ordinaryW1,1 gradient Young measures (i.e., thoseν with λ = 0) given in [Kr99].
Proof. Only the sufficiency part requires a proof, so assumeν =((νx)x∈Ω, λ, (ν
∞x )x∈Ω
)is a
Young measure such thatλ(∂Ω) = 0, ν = Du for someu ∈ BV(Ω,RN) and that (6.5) holds.We will show that conditions(i) and(ii) of Theorem 6.1 are satisfied, and start by Lebesguedecomposing the measureλ asλ = aLn+ b|Dsu|+λ∗, whereλ∗ is a measure which is singularwith respect toLn+ |Dsu|. Consider first condition(ii), noteλs = b|Dsu|+ λ∗ and fixf ∈ Q.Then the recession functionf∞ is a positively1–homogeneous and quasiconvex function. Sincequasiconvexity implies rank-one convexity for real–valued functions we can use Corollary 1.2whereby we infer thatf∞ is convex at all points of the rank one cone. Therefore Jensen’s
26
inequality holds forf∞ andany probability measure with a centre of mass on the rank onecone. According to (6.4) the probability measureν∞x has centre of mass at0 for λ∗ almost allx ∈ Ω, so (ii) holdsλ∗ almost everywhere. For the remaining pointsx ∈ Ω seen byλs weappeal to Alberti’s rank-one theorem [Al93]. Accordingly the matrixDsu/|Dsu| has rank onefor |Dsu| almost allx ∈ Ω, and so by (6.3) the probability measureν∞x has centre of mass onthe rank one cone forb|Dsu| almost allx ∈ Ω. Therefore(ii) holds forb|Dsu| almost allx ∈ Ω,and so we have shown that it holdsλs almost everywhere inΩ. Next we turn to(i), and startby fixing f ∈ Q with the additional property thatf = f∞ outside a large ball in matrix space.Becausef∞ in particular must be rank–one convex and positively1–homogeneous we deducefrom Corollary 1.2 thatf∞ ≥ ℓ for some linear functionℓ onRN×n. Consequently,f ≥ a foran affine functiona onRN×n, and definingg = | · |+(f−a)/ε for ε > 0 we record thatg ∈ SQ
so that(i) holds forg, and thus also forεg. By approximation we deduce that(i) also holds forf . The final step is facilitated by the following approximation result.
Lemma 6.3. Let f ∈ Q andδ > 0. Theng(ξ) = gδ(ξ) := maxf(ξ), f∞(ξ) + δ|ξ| − 1
δ
is
quasiconvex and for somes = s(δ) > 0 we haveg(ξ) = f∞(ξ) + δ|ξ| − 1δ
for all |ξ| ≥ s.Furthermore,(gδ)δ∈(0,1) is equi–Lipschitz, and
gδ(ξ) → f(ξ) and g∞δ (ξ) → f∞(ξ)
pointwise inξ asδ ց 0.
Proof of Lemma 6.3.It is clear thatgδ(ξ) → f(ξ) asδ ց 0 pointwise inξ, thatgδ are quasi-convex and, by Lemma 2.3, that(gδ) is equi–Lipschitz.
It remains to find the numbers = s(δ) with the stated property. The rest then follows easily.Our definition of recession function at (2.3) applied to(f − f∞ + 1
δ)∞ = 0 yields for given
ξ ∈ ∂BN×n andδ > 0 ans = s(ξ, δ) > 0 such that
f(tξ′) < f∞(tξ′) + δ|tξ′| − 1
δ(6.6)
for t ≥ s andξ′ ∈ ∂BN×n with |ξ − ξ′| < 1/s. By compactness of∂BN×n we therefore find ans = s(δ) > 0 such that (6.6) holds for allt ≥ s andξ′ ∈ ∂BN×n. Stated differently we haveshown thatf(ξ) < f∞(ξ) + δ|ξ| − 1
δfor all ξ with |ξ| ≥ s. But theng(ξ) = f∞(ξ) + δ|ξ| − 1
δ
for |ξ| ≥ s, and in particularg∞(ξ) = f∞(ξ) + δ|ξ|.
Fix f ∈ Q. Then by the foregoing lemma and the previous step,(i) holds for the integrandsgδ +
1δ. But then it also holds for eachgδ and so by approximation forf .
References
[Al93] G. Alberti, Rank one property for derivatives of functions with boundedvariation,Proc. Roy. Soc. Edinburgh Sect. A123(2) (1993), 239–274.
27
[AB97] J. J. Alibert and G. Bouchitte,Non-uniform integrability and generalized Young mea-sures, J. Convex Anal.4 (1997), 129–147.
[AFP] L. Ambrosio, N. Fusco and L. Pallara,Functions of bounded variation and free dis-continuity problems, Oxford Mathematical Monographs, The Clarendon Press, OxfordUniversity Press, New York, 2000.
[AIS] K. Astala, T. Iwaniec, I. Prause and E. Saksman,Burkholder integrals, Morrey’s problemand quasiconformal mappings, J. Amer. Math. Soc.25 (2012), no. 2, 507–531.
[BCO81] J.M. Ball, J.C. Currie, and P.J. Olver,Null Lagrangians, Weak Continuity, and Vari-ational Problems of Arbitrary Order, J. Funct. Anal.41 (1981), 135–174.
[BJ] J.M. Ball and R.C. James,Fine phase mixtures as minimizers of energy, Arch. Ra-tion. Mech. Anal.100(1987), no. 1, 13–52.
[BKK00] J.M. Ball, B. Kirchheim, and J. Kristensen,Regularity of quasiconvex envelopes,Calc. Var. Partial Differential Equations11 (2000), 333–359.
[BP87] A. Bonami and S. Poornima,Nonmultipliers of the Sobolev spacesWk,1(Rn),J. Funct. Anal.71 (1987), 175–181.
[BB04] J. Bourgain and H. Brezis,New estimates for the Laplacian, the div-curl, and relatedHodge systems, C.R. Math. Acad. Sci. Paris338(2004), no. 7, 539–543.
[BB07] J. Bourgain and H. Brezis,New estimates for elliptic equations and Hodge type sys-tems, J. Eur. Math. Soc. (JEMS)9 (2007), no. 2, 277–315.
[BES63] H. Busemann, G. Ewald and G.C. Shephard,Convex Bodies and Convexity on Grass-mann Cones. Parts I to IV.Math. Ann.151(1963), 1-41.
[CFM05a] S. Conti, D. Faraco and F. Maggi,A new approach to counterexamples toL1 es-timates: Korn’s inequality, geometric rigidity, and regularity for gradients of separatelyconvex functions, Arch. Ration. Mech. Anal.175(2005), no. 2, 287–300.
[CFMM05b] S. Conti, D. Faraco, F. Maggi and S. Muller,Rank-one convex functions on2 ×2 symmetric matrices and laminates on rank-three lines, Calc. Var. Partial DifferentialEquations24(4) (2005), 479–493.
[Dac89] B. Dacorogna,Direct Methods in the Calculus of Variations, Second Edition, AppliedMathematical Sciences, vol. 78, Springer-Verlag, 2008.
[DH96] B. Dacorogna and J.–P. Haeberly,On convexity properties of homogeneous functionsof degree one, Proc. Roy. Soc. Edinburgh Sect. A126(1996), no. 5, 947–956.
[DM07] B. Dacorogna and P. Marechal,The role of perspective functions in convexity, poly-convexity, rank-one convexity and separate convexity, J. Convex Anal.15 (2008), no. 2,271–284.
28
[DM87] R. J. DiPerna and A. J. Majda,Oscillations and concentrations in weak solutions ofthe incompressible fluid equations, Comm. Math. Phys.108(1987), 667–689.
[Du77] R.M. Dudley,On second derivatives of convex functions, Math. Scand.41(1977), no. 1,159–174. Ibid.46 (1980), no. 1, 61.
[FS08] D. Faraco and L. Szekelyhidi Jr.,Tartar’s conjecture and localization of the quasiconvexhull in R2×2, Acta Math.200(2008), 279–305.
[Fed69] H. Federer,Geometric measure theory, Springer Verlag, 1969.
[FM] I. Fonseca and S. Muller,A-quasiconvexity, lower semicontinuity, and Young measures,SIAM J. Math. Anal.30 (1999), no. 6, 1355–1390.
[I02] T. Iwaniec,Nonlinear Cauchy-Riemann operators inRn, Trans. Amer. Math. Soc.354(2002), 1961–1995.
[IK05] T. Iwaniec and J. Kristensen,A construction of quasiconvex functions, Riv. Mat. Univ.Parma (7)4* (2005), 75–89.
[KP91] D. Kinderlehrer and P. Pedregal,Characterization of Young measures generated bygradients, Arch. Rational Mech. Anal.115(1991), no. 4, 329–365.
[KP94] D. Kinderlehrer and P. Pedregal,Gradient Young measures generated by sequences inSobolev spaces, J. Geom. Anal.4 (1994), 59–90.
[Ki02] B. Kirchheim,Rigidity and geometry of microstructures, Habilitation Thesis, Universityof Leipzig, 2003.
[KS08] B. Kirchheim and L. Szekelyhidi Jr.,On the gradient set of Lipschitz maps, J. ReineAngew. Math.625(2008), 215–229.
[Kr99] J. Kristensen,Lower semicontinuity in spaces of weakly differentiable functions,Math. Ann.313(1999), no. 4, 653–710.
[KK11] B. Kirchheim and J. Kristensen,Automatic convexity of rank–1 convex functions,C. R. Acad. Sci. Paris, Ser. I349(2011), 407–409.
[KR10a] J. Kristensen and F. Rindler,Relaxation of signed integral functionals inBV,Calc. Var.37 (2010), 29–62.
[KR10b] J. Kristensen and F. Rindler,Characterization of generalized gradient Young mea-sures generated by sequences inW1,1 andBV, Arch. Rational Mech. Anal.197 (2010),539–598.Erratum, Ibid. 203(2012), 693–700.
[dLM62] K. de Leeuw and H. Mirkil,Majorations dansL∞ des operateurs differentielsacoefficients constants, C. R. Acad. Sci. Paris254(1962), 2286–2288.
29
[LMNS] J. Lukes, J. Maly, I. Netuka and J. Spurny,Integral representation theory. Applica-tions to convexity, Banach spaces and potential theory, de Gruyter Studies in Maths.35,Walter de Gruyter & Co., Berlin, 2010.
[Ma01] J. Matousek,On directional convexity, Discrete Comput. Geom.,25 (2001), 389–403.
[MP98] J. Matousek and P. Plechac,On functional separately convex hulls, Discrete Com-put. Geom.,19 (1998), 105–130.
[Mo52] C.B. Morrey Jr.,Quasiconvexity and the lower semicontinuity of multiple integrals,Pacific J. Math.2 (1952), 25–53.
[Mo66] C.B. Morrey Jr.,Multiple integrals in the calculus of variations, Springer 1966.
[Mi58] B.S. Mityagin, On the second mixed derivative, Dokl. Akad. Nauk SSSR123 (1958),606–609.
[Mu92] S. Muller,On quasiconvex functions which are homogeneous of degree1, Indiana Univ.Math. J.41 (1992), 295–301.
[Mu99] S. Muller, Rank-one convexity implies quasiconvexity on diagonal matrices, Inter-nat. Math. Res. Notices20 (1999), 1087–1095.
[Mu] S. Muller, Variational models for microstructure and phase transitions, Calculus of Vari-ations and geometric evolution problems (Cetraro, 1996), 85–210, Lecture Notes in Math.1713, Springer, Berlin, 1999.
[Or62] D. Ornstein,A non-inequality for differential operators in theL1-norm, Arch. Ra-tion. Mech. Anal.11 (1962), 40–49.
[R68] Y.G. Reshetnyak,Weak convergence of completely additive vector functions on a set,Siberean Math. J.9 (1968), 1039–1045.
[R68b] Y.G. Reshetnyak,Generalized derivatives and differentiability almost everywhere,Mat. Sb. (N.S.)75(117)(1968), 323–334.
[Ri11] F. Rindler,Lower Semicontinuity and Young Measures for Integral Functionals withLinear Growth, DPhil. Thesis, University of Oxford, 2011.
[Ro] R.T. Rockafellar,Convex Analysis, Princeton University Press, Princeton Landmarks inMathematics, 1997.
[SZ04] P.M. Santos and E. Zappale,Second-order analysis for thin structures, Nonlinear Anal-ysis56 (2004), no. 5, 679–713.
[Sv91] V. Sverak,Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. LondonSer. A433(1991), 723–725.
30
[Sv92] V. Sverak,Rank-one convexity does not imply quasiconvexity, Proc. Roy. Soc. LondonSer. A120(1–2) (1992), 185–189.
[Ta79] L. Tartar, Compensated compactness and applications to partial differential equa-tions, Nonlinear analysis and mechanics: Heriot–Watt Symposium, Vol. IV, pp. 136–212,Res. Notes in Math.39 Pitman, Boston, Mass.–London, 1979.
[vS13] J. Van Schaftingen,Limiting Sobolev inequalities for vector fields and canceling lineardifferential operators, J. Eur. Math. Soc. (JEMS)15 (2013), no. 3, 877–921.
[Zha92] K. Zhang,A construction of quasiconvex functions with linear growthat infinity,Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)19 (1992), 313–326.
Department of Mathematics, University of Leipzig
E-mail address: [email protected]
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe ObservatoryQuarter, Woodstock Road, Oxford OX2 6GG, UK
E-mail address: [email protected]
31
