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Corollary 1.2. A rank–one convex and positively 1–homogeneous function f : RN×n R is convex at each point of the rank one cone x 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 in n variables of order k, say Q0, ∞ Q1,...,Qm, and any number K > 0, there is a C smooth function φ vanishing outside the ∞ n unit cube, φ Cc ((0, 1) ), such that Q0φ >K while Qjφ < 1 for all 1 j m. This result∈ convincingly manifests the| fact| that estimates| for| differential operators,≤ ≤ usually R R based on Fourier multipliers and Calderon–Zygmund operators, can be obtained for all Lp with p (1, ) by interpolation and (more directly) even for the weak-L1 spaces but fail to extend to the∈ limit∞ case p =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 derivatives were distributions of first order. He then gave a very technical and rather concise proof of his statement for general dimension n and degree k. Theorem 1.1, and Corollary 1.2, yield when combined with arguments from the calculus of variations, various generalizations of Ornstein’s L1–non–inequality. In particular the approach allows also a streamlined and very elementary proof of an extension of Ornstein’s result to the wider context of x–dependent and vector–valued operators:
2 Theorem 1.3. Let V, W1, W2 be finite dimensional inner product spaces, and consider two k–th order linear partial differential operators with locally integrable coefficients
ai L1 (Rn, L (V, W )) (i =1, 2) α ∈ loc i defined by i α Ai(x, D)ϕ := aα(x)∂ ϕ (i =1, 2) |Xα|=k for smooth and compactly supported test maps ϕ C∞(Rn, V). ∈ c Then there exists a constant c such that
A (x, D)ϕ 1 c A (x, D)ϕ 1 k 2 kL ≤ k 1 kL holds for all ϕ C∞(Rn, V) if and only if there exists C L∞(Rn, L (W , W )) with ∈ c ∈ 1 2 C L∞ c such that k k ≤ 2 1 aα(x)= C(x)aα(x) for almost all x and each multi–index α of length k.
We derive this result from a more general nonlinear version stated in Theorem 5.1. This result is in turn a consequence of Theorem 1.1 and standard arguments from the calculus of variations. The case of constant coefficient homogeneous partial differential operators was given in [KK11]. The link between an Ornstein type result, concerning the failure of the L1–version of Korn’s inequality, and semiconvexity properties of the associated integrand – though expressed in a dual 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 into three dimensions. Our arguments handle these situations with ease, see Corollary 1.5 below. It is well–known that the distributional Hessian of a convex function f : Rn R is a matrix–valued Radon measure (see for instance [R68b]). The natural question arises→ if this is valid 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 for rank one convex functions defined on symmetric 2 2 matrices. Here we wish to address the × question of regularity of second derivatives when the function f : R is merely –convex. First note that when f is C2, then –convexity is equivalent to D2Vf( →x)[e, e] 0 forD all x D ≥ ∈ V and e . Hence using mollification and that a positive distribution is a Radon measure it follows∈ that D the Hessian of –convex functions f : R will be a Radon measure provided we can select a basis (e ) forD consisting of vectorsV from → such that additionally e e j V D i ± j ∈D for all 1 i < j dim . Indeed under this assumption on we can use polarisation to see ≤ ≤ V D that 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 cone of directions is equivalent to D the statement that the only functions f : R with the property that both f are –convex, V → ± D 3 called –affine functions, are the affine functions. This in turn is equivalent to saying that the cone ofD –convex quadratic forms on , D V = q : q is a –convex quadratic form on QD D V is line–free. It is remarkable, but in accordance with our theme on Ornstein L1 non–inequalities, that if the cone of directions satisfies the technical condition D ℓ ∗ such that kerℓ = 0 , (1.1) ∃ ∈V D ∩ { } then this simple sufficient condition is also necessary. Theorem 1.4. Let be a normed finite dimensional real vector space and be a cone of directions in satisfyingV the condition (1.1). Then the distributional HessianD of –convex functions f : V R is a 2 –valued Radon measure on if and only if the onlyD –affine functions areV the → affine functions.⊙ V Moreover, when there exiVsts a nontrivial –affine function,D D then there also exists a –convex C1 function f : R with a locally Holder¨ continuous 1 first differential Df : D ∗, but for which the distributionalV → Hessian fails to be a measure in any open nonempty subsetV→VO of , in the sense that for some unit vector e we have V ∈V sup f(x)D2ϕ(x)(e, e) dx : ϕ C∞(O) and sup ϕ 1 = . ∈ c | | ≤ ∞ ZO We remark that the condition on ℓ ∗ given in (1.1) is equivalent to stating that the form ∈ V ℓ ℓ is an interior point of the cone D. Let us record here some examples of cones of directions for⊗ the case of square matrices =QRn×n that satisfy condition (1.1): V (ξ ,ε )= a b : a, b Rn and a ξ b ε a b , D 0 0 ⊗ ∈ | · 0 | ≥ 0| || | where ξ Rn×n and ε > 0 are fixed and chosen so (ξ ,ε ) spans Rn×n. Indeed, with the 0 ∈ 0 D 0 0 usual identifications (Rn×n)∗ = Rn×n, for such cones we have that ξ ξ is an interior point of ∼ 0 ⊗ 0 D(ξ0,ε0). Since any 2 2 minor of n n matrices defines a nontrivial rank–one affine function Q n×n × × 1,α n×n on R we infer in particular the existence of (ξ0,ε0)–convex C functions on R whose Hessians are nowhere a measure. When we restrictD attention to functions defined on symmetric real n n matrices Rn×n the situation becomes more satisfying since there the rank–one cone × sym n = t a a : t R and a Rn satisfies the condition (1.1). Indeed Id Id is an interior Dsym { · ⊗ ∈ ∈ } ⊗ point in n . We state this result separately: QDsym Corollary 1.5. Let n 2 be an integer. Then there exists a rank–one convex C1 function Rn×n R ≥ f : sym with a locally Holder¨ continuous first derivative, but whose distributional Hes- 2 → Rn×n sian D f is not a bounded measure in any open nonempty subset O of sym in the sense that for some e Rn×n we have ∈ sym sup f(x)D2ϕ(x)(e, e) dx : ϕ C∞(O) and sup ϕ 1 = . ∈ c | | ≤ ∞ ZO In particular, the function f cannot be locally polyconvex at any ξ Rn×n. ∈ sym 1 K ⋐ α< 1 c> 0 x, y K: Df(x) Df(y) c x y α ∀ V ∀ ∃ ∀ ∈ k − k≤ | − | 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 BV Gradient–Young measures given in [KR10b], see Theorem 6.2. In fact, this was the original motivation for the work undertaken in the 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 preliminary results. 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 relaxation of 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 on periodic test maps whereas we use compactly supported tests maps. The two approaches are essentially equivalent, and here we supply the details in the latter case culminating in Theorem 4.2. The proofs of Theorems 1.3, 5.1 and 1.4 are presented in Section 5. Section 6 contains a discussion of BV gradient Young measures, and a characterization, stated as Theorem 6.2, of these by duality with a certain subclass of quasiconvex functions.
2 Preliminaries
This section fixes the notation, collects standard definitions and recalls some preliminary results that 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 [M¨u] for cal- culus of variations, and [AFP] for Sobolev functions and functions of bounded variations. We refer the reader to these sources for further background if necessary. Let be a real vector space. A subset C of is convex if it is empty or if for all points x V V and y in C also the segment (x, y) between them is contained in C. Let f : R¯ := [ , ] be an extended real–valued function. Then its effective domain is the set dom(V →f) := −∞x ∞: f(x) < and its epigraph is the set epi(f) := (x, t) R : t f(x) . The function{ ∈ Vf is convex∞} if its epigraph is a convex subset of the{ vector spac∈ Ve × R.≥ } By a cone of directions in we understand throughout theV paper⊕ a balanced and spanning cone in : so has theD propertiesV that for all x , t R we have tx , and its linear hull equalsV .D⊆V ∈D ∈ ∈D V Definition 2.1. Let be a finite dimensional real vector space and a cone of directions. (i) A subset of isV –convex provided for any two points x, y D with x y also the segment [x, yA]= V(1 Dλ)x + λy : λ [0, 1] is contained in . ∈A − ∈D { − ∈ } A (ii) For an arbitrary subset of and an extended real–valued function f : R¯ we say that f is weakly –convex providedS V for any x, y such that [x, y] andSx → y the D ∈ S ⊂ S − ∈ D restriction f is convex. We say that f is –convex if the function |[x,y] D f(x) if x F (x)= ∈ S if x ∞ ∈V\S 5 is a weakly –convex function F : R¯. D V → The reader will notice that this terminology is inspired by [BES63], and that on –convex subsets of there is no difference between weak –convexity and –convexityD of real– valued functionsS V f : R. D D S → Lemma 2.2. Let be a normed finite dimensional real vector space and a cone of directions V D in . Assume f : [ , ) is a –convex function defined on an arbitrary subset of . If V is a connectedS → component−∞ ∞ of theD interior of , then either f on or f > S onV .A S ≡ −∞ A −∞ A Proof. The proof relies on two standard observations. First, if h: (a, b) R [ , ] is a ⊂ → −∞ ∞ convex function such that h(t) < for all t (a, b) and h(t0) > for some t0 (a, b), then h(t) > for all t (a, b). This∞ follows easily∈ from the definition−∞ of convexity.∈ The second observation−∞ is that, since∈ spans , any two points of can be connected by a piecewise linear path in whose segmentsD are allV in directions from A. This follows by choosing a basis for consistingA of vectors from , then declaring it to beD an orthonormal basis for , so that the aboveV is a well–known propertyD of connected open sets in euclidean spaces. V Now if we have x with f(x) > and ℓ is a line through x in a direction from ∈ A −∞ , then by convexity of f A∩ℓ we infer that f(y) > for all y belonging to the connected Dcomponent of ℓ containing| x. The result follows−∞ since we can connect any point y A ∩ ∈ A with x by such line segments contained in . A The following result and its proof is a variant of a result from [BKK00] (compare Lemma 2.2 there) that in turn is a slightly more precise version of a well–known estimate, see [Dac89] Theorem 2.31, but here being adapted to the case of a general cone of directions. D Lemma 2.3. Let , be a normed finite dimensional real vector space and a balanced cone whose linearV spank·k is . If f : B (x ) R is -convex, then f is locallyD Lipschitz in V 2r 0 → D B2r(x0). More precisely we have
f(x) f(y) L x y (2.1) | − | ≤ k − k for all x, y Br(x0), where ∈ c L = 0 osc(f, B (x )) r 2r 0 and the constant c depends only on the norm and the cone . 0 k·k D
Proof. For the entire proof we fix unit vectors e1,...,en which form a basis of , and note that due to the equivalence of all norms on Rn there is a∈ constant D c (1, ) such thatV ∈ ∞ n n n 1 t t e c t (2.2) c | j|≤k j jk ≤ | j| j=1 j=1 j=1 X X X for all t R. Now we proceed in three steps. j ∈ 6 Step 1. The function f is locally bounded. We start by showing that for x B (x ) and positive ε such that the parallelepiped C with ∈ 2r 0 center x and sides parallel to the basis vectors and all of length 2ε is contained in B2r(x0),
n C = x + t e : t ,...,t [ ε,ε] B (x ), j j 1 n ∈ − ⊂ 2r 0 ( j=1 ) X we have sup f(C) = max f(K), where the set K = ext(C) consists of the vertices of the parallelepiped C. Indeed, by convexity in each of the directions ej we find recursively for n ∈D each x + t e C, 1 j j ∈ P n n f(x + tjej) max f(x + tjej + ε1e1) ≤ ε1=±1 j=1 j=2 X X ... ≤ max f(K). ≤ Consequently, sup f(C) max f(K). In order also to get a lower bound we fix y C, say n ≤ n ∈ y = x + 1 tjej where each tj ε. Put Ky = x + 1 εjtjej : ε1,...,εn 1, 1 . −n | | ≤ { ∈ {− }} Observe that 2 z = x and Ky is contained in C, so again using convexity in each of P z∈Ky P the directions ej we find P f(x) 2−nf(z) 2−nf(y)+(1 2−n) sup f(C), ≤ ≤ − z∈K Xy hence f(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 B (x ) with x y we have f(x) f(y) m x y , where ∈ r 0 − ∈ D | − | ≤ r k − k m = osc(f, B2r(x0)). To prove this we can assume that m < . Now fix s (r, 2r) and consider the line L through x and y. Select z L ∂B (x ) such∞ that x [y, z],∈ the segment between y and z. By ∈ ∩ s 0 ∈ assumption the restriction of f to L is convex and since y z s r we get k − k ≥ − f(x) f(y) f(z) f(y) m − − . x y ≤ z y ≤ s r k − k k − k − The required conclusion follows since s< 2r was arbitrary and x versus y can be swapped. 2 Step 3. Now consider any x Br(x0) and Bε(x) Br(x0). If x y < ε/c for the constant c from (2.2), then ∈ ⊂ k − k n n ε y x = t e where t < . − j j | j| c j=1 j=1 X X 7 Hence, the points k
yk = x + tjej, k =1,...,n j=1 X all belong to Bε(x), and so in particular to Br(x0), since, again due to (2.2),
k k y x = t e c t < ε. k k − k k j jk ≤ | j| j=1 j=1 X X Consequently, step 2 and (2.2) ensure that
n n m m f(y) f(x) = f(y ) f(y ) f(y ) f(y ) t c x y . | − | | n − 0 | ≤ | j − j−1 | ≤ r | j| ≤ r k − k j=1 j=1 X X
This shows that f is everywhere inside Br(x0) locally Lipschitz with a constant at most mc/r. Because Br(x0) is convex, the Lipschitz constant of f on the entire ball can also not exceed cm/r. Finally, we observe that this argument also shows that f is locally Lipschitz everywhere in B2r(x0). An open convex cone in is an open subset of such that x + ty whenever x, y and t> 0. C V C V ∈ C ∈For C a function f : R defined on an open convex cone in we define the (upper) C → C V recession function f ∞ : R¯ as C → f(tx′) f ∞(x) := lim sup (x ) (2.3) t→∞,x′→x t ∈ C tx′∈C The definition implies that f ∞ is positively 1-homogeneous: f ∞(tx) = tf ∞(x) for all x ∈ C and t> 0. It also follows that when f has linear growth on , meaning that for some constant c we have the bound C f(x) c x +1 | | ≤ k k for all x , then f ∞ is real–valued and f ∞(x ) c x for all x ∞. When f is globally Lipschitz∈ on C with Lipschitz constant lip(|f, )=| ≤L: k k ∈ C C C f(x) f(y) L x y | − | ≤ k − k for all x, y , then we can simplify the definition of the recession function f ∞ to ∈ C f(tx) f ∞(x) = lim sup (x ) (2.4) t→∞ t ∈ C and of course again lip(f ∞, ) L. Let us also remark that since all norms are equivalent on a finite dimensional vector spaceC ≤ 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 without specifying a norm. There is also a natural notion of lower recession function, but for our purposes it is less useful. The reason is that the following result fails for lower recession functions.
Lemma 2.4. Let be an open convex cone in and assume that f : R is a –convex function of linearC growth on . Then the recessionV function f ∞ : RCis → again –convex.D C C → D We omit the straightforward proof. The following simple observation turns out to be crucial in the sequel.
Lemma 2.5. Let be a normed finite dimensional real vector space and a balanced cone of directions in .V Assume is an open convex cone in and that f : D R is a –convex function. Then V C V C → D f(x + y) f ∞(x)+ f(y) (2.5) ≤ for all y and x . ∈ C ∈C∩D Proof. Let y and x . By definition of open convex cone, y + tx for all t 0, and because f∈is C convex∈C∩D in the direction of x we get for all t 1, ∈ C ≥ ≥ f(y + tx) f(y) f(t(x + y )) f(y) f(x + y) f(y) − = t , − ≤ t t − t and in view of (2.3) the conclusion follows by sending t to infinity.
3 Proof of Theorem 1.1
Throughout this section we assume that is a normed finite dimensional real vector space with a cone of directions . V D Definition 3.1. Let be a subset of and f : R a function. We say that f has at x C V C → ∈ C a subdifferential if there is a linear function ℓ: R such that • V → f(y) f(x)+ ℓ(y x) for all y . ≥ − ∈ C
a –subcone if there is a –convex and positively 1–homogeneous function ℓ: R • suchD that D V → f(y) f(x)+ ℓ(y x) for all y . ≥ − ∈ C As we shall see momentarily the two conditions are in fact equivalent. However, first we establish the existence of –subcones: D 9 Lemma 3.2. Let be a convex open cone in . C V If f : R is –convex and positively 1–homogeneous, then for any x , y and λ (0C, →1) we haveD ∈C∩D ∈ C ∈ f(x + λ(y x)) f(x) f(y) f(x) − − . (3.1) − ≥ λ In particular, f has a –subcone at x. D Proof. Since adding a linear function to f does not effect the assumptions nor the validity of (3.1) or the existence of a –subcone, we can suppose in the sequel that f(x)=0. By Lemma 2.3 the function f is lipschitzD near x and hence also f ∞(x)=0. Therefore, using Lemma 2.5 we conclude as required
f(y) f(x)= f(y)= f(x +(y x)) − − x f(x + λ(y x)) f(x) f( +(y x)) = − − . ≥ λ − λ
To see that this implies the existence of a –subcone at x we choose an ε> 0 so B (x) D 2ε ⊂ C and define for s 1, ≥ y g (y) := sf(x + ), y B (0). s s ∈ sε
Clearly gs(0) = 0. By Lemma 2.3 we have lip(gs) = lip(f, Bε(x)) and by (3.1) we get for s t the monotonicity property gs(y) gt(y) for all y Bsε(0). Hence for each y the limit≤ ≥ ∈ ∈ V
g(y) := lim gs(y) = inf gs(y) s→∞ s>0 exists in R, and defines a Lipschitz function g : R. As a pointwise limit of –convex functions g is –convex, and since for any y Vand →λ> 0, D D ∈V λy s y g(λy) = lim sf x + = λ lim f x + = λg(y), s→∞ s s→∞ λ s/λ g is also positively 1–homogeneous. Finally, for y we get from (3.1) upon taking any ∈ C λ ε/(1 + y x ) that λ(y x) B (0) so ≥ k − k − ∈ ε 1 1 f(y) f(x) f (x + λ(y x)) = g (λ(y x)). − ≥ λ − λ 1 − Since g g and g is positively one–homogeneous it follows that g is a –subcone for f at 1 ≥ D x.
Proposition 3.3. Let be an open subset of . If the function f : R has a –subcone at the point x , thenC it also has a subdifferentialV at x . C → D 0 ∈ C 0 10 Proof. We will proceed by induction on the dimension n 1 of and suppose that the state- ≥ V ment is true whenever the dimension of the vector space is less than n. A moments reflection on Definition 3.1 shows that existence both of a –subcone and a D subdifferential are unaffected by a simultaneous shift of the function f and the contact point x0, so we can without loss in generality suppose that x0 = 0V and, replacing f by its –subcone, also that f is -convex and positively one-homogeneous on all of . D D V We choose a basis e1,...,en of and put x = e1 and ˜ = span e2,...,en , so ˜ is spanned by ˜ = ˜ . According∈ to D LemmaV 3.2 there is a –subconeV g for{ f in x, and} clearlyV D V∩D D the restriction g ˜ is its own ˜–subcone at 0 ˜ . Therefore, by the induction assumption there is a |V D V subdifferential at the origin: a linear function ˜l : ˜ R such that ℓ˜(y) g(y) whenever y ˜. In particular ℓ˜ =0 if n =1. V → ≤ ∈ V Now we claim that
ℓ(tx + y)= tf(x)+ ℓ˜(y) for y ˜ and t R ∈ V ∈ is a subdifferential for f at 0V . Once this claim is shown, our proof is finished. For this purpose we first note that cleary ℓ is linear on . Since f(0) = 0, we need only V show that f(z) ℓ(z) for each z = tx + y, t R and y ˜. But if t =1 then we have for all y ˜ that ≥ ∈ ∈ V ∈ V ℓ(x + y)= f(x)+ ℓ(y) f(x)+ g(y) f(x + y), ≤ ≤ according to the definition of a –subcone g of f in x. By positive 1–homogeneityD of f we get now for all t> 0,y ˜ that ∈ V y y ℓ(tx + y)= tℓ(x + ) tf(x + )= f(tx + y). t ≤ t Finally, if t 0, y ˜ we use Lemma 2.3 to infer f ∞(x)= f(x) and now Lemma 2.5 gives ≤ ∈ V ℓ(x + y) f(x + y) f ∞((1 t)x)+ f(tx + y) ≤ ≤ − and so f(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 prove convenient for dealing with higher order derivatives. Starting from the standard L (X,Y )= f : X Y ; f linear for given finite–dimensional real vector spaces X,Y , we set L 0(X,Y )={ Y and→ inductively }
L k+1(X,Y )= L (X, L k(X,Y ))
11 for k N . As usual (see [Fed69], 1.9–1.10), we identify L k(X,Y ) with M k(X,Y ), the ∈ 0 space of Y –valued k–linear maps on X. In the sequel, we are mainly interested in the space of all Y –valued symmetric k–linear maps
k(X,Y )= µ M k(X; Y ): µ(x ,...,x )= µ(x ,...,x ) for all σ Sym(k) , ⊙ { ∈ 1 k σ(1) σ(k) ∈ } where Sym(k) is the group of all permutations of the set 1,...,k . When Y = R we simply { } write kX instead of k(X, R). ⊙ ⊙ n M k In terms of a basis (ej)j=1 for X, we can express a map ξ (X,Y ) as a Y –valued homogeneous polynomial of degree k in kn real variables by fully∈ expanding all brackets:
n n ξ(x ,...,x )= ... ξ xi1 ... xik , 1 k i1,...,ik 1 · · k i =1 i =1 X1 Xk where n ξ = ξ(e ,...,e ) Y and x = xi e . i1,...,ik i1 ik ∈ j j i i=1 X In particular, observe that ξ M k(X,Y ) is symmetric if and only if for one basis (e )n of ∈ j j=1 X (and then for all) ξ = ξ for all σ Sym(k) and all indices i 1,...,n . i1,...,ik iσ(1),...,iσ(k) ∈ j ∈ { } Obviously, when dimY = N we have dim k (X,Y )= N k+n−1 . ⊙ k In view of the standard definition of the (Fr´echet–)derivative of a Ck mapping f : Ω X ⊂ → Y and Schwarz’ theorem on interchangeability of partial derivatives we have for all x Ω that Dkf(x) k(X,Y ). ∈ ∈ ⊙ k The following calculations are of particular interest to us: If µ (X,Y ) and fµ(x) = µ(x,...,x) for x X, then ∈ ⊙ ∈ 1 Dfµ(x)(h) = lim µ(x + th, . . . , x + th) µ(x,...,x) t→0 t − = µ(h, x, x, . . . , x)+ µ(x, h, x, . . . , x)+ ... + µ(x, . . . , x, h) = kµ(h, x, . . . , x).
Iterating this, we obtain
k! Dlf (x)(h ,...,h )= µ(h , h ,...,h , x,...,x ), µ 1 l (k l)! 1 2 k − (k−l)−times and finally | {z } k D fµ(x)(h1,...,hk)= k!µ(h1, h2,...,hk) (4.1) for all x, h1,...,hk X. In particular, the symmetric k–linear map generating a homo- geneous k–th order polynomial∈ f is unique, the converse representation (of any such polyno- mial by a symmetric k–linear map) is a consequence of Taylor’s Theorem. Thus, we see that
12 k(X,Y ) is precisely the space of all k–th derivatives of Y –valued functions defined on (open ⊙ subsets Ω of) X. Now we focus, for the sake of notational simplicity, on the only slightly more special situ- ation of X = Rn with usual scalar product , or, equivalently, any finite dimensional inner product space. Analogous to before, we findh· ·i that for all Ck functions g : R R and each a Rn the function f(x)= g( x, a ) on Rn satisfies → ∈ h i k Dkf(x)(h ,...,h )= g(k)( x, a ) a, h , 1 k h i h ii i=1 Y or, in more convenient tensor notation,
Dkf(x)= g(k)( x, a )a⊗k, (4.2) h i where we identify a Rn with y y, a in (Rn)∗ and accordingly ∈ 7→ h i a⊗k = a ... a ⊗ ⊗ k
⊗k k with the symmetric k–linear form a (h1,...,h| k{z)= } i=1 a, hi . In particular we record that n h in its coordinates with respect to any orthonormal basis (ej) for R are Q j=1 k (a⊗k) = a , where a = a, e . i1...ik il il h il i l Y=1 Next, we assert that the balanced cone
(n, k) := t a⊗k : t R and a Rn Ds { · ∈ ∈ } spans k(Rn). Indeed, otherwise we could find µ k(Rn) 0 which is perpendicular to all ⊙ ∈ ⊙ \{ } of (n, k) with respect to the canonical scalar product on Rn extended from Rn Ds ⊙ n n ξ,ζ = ... ξ ζ , h i i1...ik i1...ik i =1 i =1 X1 Xk where ξi1...ik ,ζi1...ik are the coordinates with respect to the orthonormal basis obtained from n (ej)j=1. But observing that
n n µ, a⊗k = ... µ a ... a = µ(a,...,a), h i i1...ik i1 · · ik i =1 i =1 X1 Xk n we have fµ 0 on R and hence by (4.1) the contradiction µ =0. As an easy consequence we see that, using≡ standard notation, also the balanced cone
= (n, k; Y )= b a ... a = b a⊗k ; a Rn and b Y Ds Ds { ⊗ ⊗ ⊗ ⊗ ∈ ∈ } k
| {z 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 cone s(n,k,Y ) plays in the case of k–th order deriva- tives the role which the usual cone of rank–oneD directions has for first order derivatives. So let on a bounded smooth domain Ω of Rn an integrable map f : Ω Y be given and assume that its distributional k–th derivative satisfies →
Dkf(x) ξ, η a.e., where ξ, η k(Rn,Y ). ∈{ } ∈ ⊙ Then g := Dk−1f : Ω k−1(Rn,Y ) is easily seen to belong to W1,∞(Ω, k−1(Rn,Y ))) and → ⊙ ⊙ therefore is lipschitz with Dg ξ, η a.e., where, by Rademacher’s theorem, the derivative can be understood both distributionally∈ { } and classically a.e. A by now standard argument (see [BJ]) yields that ξ, η as elements of L (Rn, k−1(Rn,Y )) satisfy rank(ξ η) 1. In other words, when ξ = η the kernel of ξ η must be⊙(n 1)–dimensional so for a− unit vector≤ ν Rn we have (ξ η6 )(x)=(ξ η)( x,− ν ν) as elements− of k−1(Rn,Y ) for all x Rn. By∈ the − − h i ⊙ ∈ symmetry of ξ η we therefore get for all x = x Rn, x ,...,x Rn − 1 ∈ 2 k ∈ (ξ η)(x , x ,...,x ) = [(ξ η)(x)](x ,...,x )=(ξ η)( x , ν ν, x ,...,x ) − 1 2 k − 2 k − h 1 i 2 k = (ξ η)( x1, ν ν, x2, ν ν, . . . , xk, ν ν) − h i h k i h i = (ξ η)(ν,ν,...,ν) x , ν . − h i i i=1 Y In other words, then necessarily ξ η = [(ξ η)(ν,ν,...,ν)] ν⊗k. Conversely (4.2) shows − −k−1,1 n ⊗ k that for each µ s(n, k; Y ) there is f C (R ,Y ) such that D f 0,µ a.e., and so Dkf is not essentially∈ D constant. ∈ ∈ { } Next we present a result that corresponds to the gradient distribution changing techniques used in the theory of first order partial differential inclusions. Since we are not forced to find exact solutions, as is usually done in this theory, we avoid many technical difficulties and we can work exclusively within the class of C∞–maps. First let us recall the Leibniz rule in several variables. Let Ω be an open subset of X. If f : Ω Y and g : Ω Z are Ck–maps and → → Ψ: Y Z V is bilinear, then for each x Ω and v =(v ,...,v ) Xk we have × → ∈ 1 k ∈
Dk y Ψ(f(y),g(y)) (x)(v)= Ψ( ˜ M f(x)(v), ˜ {1,...,k}\M g(x)(v)), (4.3) 7→ ∇ ∇ M⊂{X1,...,k} where we denote ˜ M ϕ(x)(v) := Dlϕ(x)(v ,...,v ) if M = i ,...,i 1,...,k has ∇ i1 il { 1 l}⊂{ } cardinality l (well–defined due to the symmetry of the higher order derivatives). The rule can be easily shown by induction on k. Proposition 4.1. Let Ω be a bounded open subset of Rn, k N and ξ, η k(Rn,Y ) with ∈ ∈ ⊙ ξ η s(n,k,Y ) be given (we express this by saying that ξ, η are rank-one connected. Suppose− ∈λ D (0, 1) and let u: Ω Y satisfy Dku(x)= C = λξ +(1 λ)η for all x Ω (so u is in particular∈ a polynomial of degree→ k). Then for each ε> 0 there are− a compactly supported∈ C∞ map ϕ: Ω Y and open subsets Ω , Ω of Ω such that → ξ η 14 (i) Dk(u + ϕ)(x)= ξ if x Ω , Dk(u + ϕ)(x)= η if x Ω and ∈ ξ ∈ η dist(Dk(u + ϕ)(x), [ξ, η]) <ε for all x Ω, where [ξ, η]= tξ + (1 t)η : t [0, 1] . ∈ { − ∈ } (ii) Ω > (1 ε)λ Ω and Ω > (1 ε)(1 λ) Ω , | ξ| − | | | η| − − | | (iii) Dlϕ(x) <ε for all x Ω and l
Ω = x Ω : jx, a I + Z , ξ,j { ∈ + h i ∈ ξ } and Ω = x Ω : jx, a I + Z . η,j { ∈ + h i ∈ η } l ′′ −1 Similarly, (4.3), (4.2) and (4.4) imply maxx∈Ω D ϕj(x) cl,ψ,diam(Ω)j if l lim Ωξ,j = lim s(t)˜χξ,j(t) dt = s(t) dt Iξ = Iξ Ω+ j→∞ | | j→∞ | | | || | Z Z ε > (1 )2λ Ω − 2 | | > (1 ε)λ Ω , − | | and similarly limj→∞ Ωη,j > (1 ε)(1 λ) Ω . So, choosing j 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 treatment of 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], and while it is possible to derive our results in that context too we found it useful and worthwhile to here record the approach based on compactly supported test maps. We therefore provide a detailed exposition for the convenience of the reader. Recall that a subset of is –convex if for all x, y such that x y we have that the segment [x, y] is containedS V inD. ∈ S − ∈D S k n Theorem 4.2. Let be an open and s(n, k; Y )–convex subset of = (R ,Y ) and let the extended real–valuedS function F : D [ , ) be locally boundedV ⊙ above and Borel measurable. S → −∞ ∞ Then the relaxation of F defined as ∞ n k ϕ Cc ((0, 1) ,Y ) (ξ) := inf F (ξ + D ϕ(x)) dx : ∈ k F n ξ + D ϕ(x) for all x Z(0,1) ∈ S is a (n,k,Y )–convex function : [ , ). Ds F S → −∞ ∞ 16 Moreover, for any strictly increasing function ω : [0, ) [0, 1] with ω(0) = 0 and ∞ → t/ω(t) 0 as t 0 and any ε > 0 we can without changing the value of restrict the infimum→ in its definitonց to be taken only over those ϕ which in addition satisfy DlFϕ(x) <ε for | | all x and l k−1 k−1 −j k j j D ψ(x) D ψ(y) 2 D ϕ ∞ 2 x x¯ +2 y y¯ k − k ≤ k kL | − | | − | −j k √n2 2 D ϕ ∞ ω( x x¯ )+ ω( y y¯ ) ≤ k kL ω(√n2−j) | − | | − | εω( x y ). ≤ | − | Next we turn to the –convexity. Because F < also < and it therefore suffices to show that (µ) λ (ξ)+(1D λ) (η) if ξ, η with∞ η ξF ∞, λ (0, 1) and µ = λξ +(1 λ)η. F ≤ F − F ∈ S − ∈D ∈ − Fix real numbers s, t so s > (ξ) and t > (η) and by the definition of relaxation find maps ϕ , ϕ C∞((0, 1)n,Y ) suchF that ξ + DkϕF(x) , η + Dkϕ (x) for all x, and ξ η ∈ c ξ ∈ S η ∈ S k F (ξ + D ϕξ(x)) dx (u + φ)(x) if x Ωr, −kl l ∈ −l n ψ(x)= (u + φ)(x)+2 ϕξ(2 (x y)) if x y + [0, 2 ] ξ, −kl l − ∈ −l n ∈ Q (u + φ)(x)+2 ϕη(2 (x y)) if x y + [0, 2 ] η. − ∈ ∈ Q k k l −l n As D ψ(x)= ξ + Dϕξ(2 (x y)) in the interior of y + [0, 2 ] ξ we have for each such cube that − ⊂ Q F (Dkψ(x)) dx F (Dkψ(x)) dx < t n y + [0, 2−l]n −l n L Zy+[0,2 ] for cubes y + [0, 2−l]n . Altogether, in view of Proposition 4.1 (ii), ⊂ Qη F (Dkψ) δ + s n + t n ≤ L Qξ L Qη ZΩ δ + s n(Ω[)+ t n(Ω ) [ ≤ L ξ L η δ + s(1 n(Ω )) + t(1 n(Ω )) ≤ −L η −L ξ 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 that ψ u C∞((0, 1)n,Y ) because it is a finite sum of such functions − ∈ c (including φ). Since δ > 0 was arbitrary, the proof is complete. 5 A generalization of Ornstein’s L1 Non–Inequality As was mentioned in the introduction Theorem 1.1 implies the Ornstein non–inequality in the context of x–dependent vector valued operators, and even a version involving certain nonlinear x–dependent differential expressions: Theorem 5.1. Let F : Rn k(Rn, RN ) R be a Caratheodory´ integrand satisfying × ⊙ → F (x, tξ)= t F (x, ξ) | | and F (x, ξ) a(x) ξ | | ≤ | | Rn R k Rn RN 1 Rn for almost all x , all t and ξ ( , ), where a Lloc( ) is a given function. Then ∈ ∈ ∈ ⊙ ∈ F (x, Dkφ(x)) dx 0 (5.1) Rn ≥ Z ∞ Rn RN Rn holds for all φ Cc ( , ) if and only if F (x, ξ) 0 for almost all x and all ξ k(Rn, RN )∈. ≥ ∈ ∈ ⊙ Before presenting the short proof of Theorem 5.1 let us see how it implies the Ornstein L1 non–inequality stated in Theorem 1.3. Proof of Theorem 1.3. Only one direction requires proof. To that end we write the differential operators in terms of linear mappings A˜ (x): k (Rn, V) W as follows: i ⊙ → i ˜ k i α Ai(x)D ϕ = aα(x)∂ ϕ |Xα|=k for all ϕ C∞(Rn, V). Define ∈ c F (x, ξ) := c A˜ (x)ξ A˜ (x)ξ for (x, ξ) Rn k(Rn, V), k 1 k−k 2 k ∈ × ⊙ and note that Theorem 5.1 applies to the function F . Accordingly, F (x, ) is a nonnegative · function for almost all x, and therefore the set inclusion, ker A˜2(x) ker A˜1(x), must in par- ticular hold for the kernels for all such x. Fix x such that F (x, ) is⊃ nonnegative. We define a · linear mapping C(x): W W by 1 → 2 −1 C(x)= A˜2(x) A˜1(x) ⊥ proj ˜ . ˜ imA1(x) ◦ | kerA1(x) ◦ n Hereby C is defined almost everywhere in R , is measurable, and A˜2(x) = C(x)A˜1(x) holds for almost all x. Finally, the nonnegativity of F (x, ) yields the uniform norm bound C ∞ · k kL ≤ c. 19 Proof of Theorem 5.1. We assume that the integral bound (5.1) holds and shall deduce the point- wise bound F 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 condition for sequential weak lower semicontinuity of variational integrals (see [Mo66] Theorem 4.4.2). Let us briefly comment on the details. Denote by C = ( 1/2, 1/2)n the open cube centred at − ∞ RN the origin with sides parallel to the coordinate axes and of unit length, and let φ Cc (C, ). n ∈ n We extend φ to R by C periodicity, denote this extension by φ again, and put for x0 R , r> 0, j N, ∈ ∈ k r x x ϕ (x) := φ j − 0 , x Rn. j j r ∈ Writing C(x ,r)= x + rC we have for all j N that ϕ C∞(C(x ,r), RN ), and so 0 0 ∈ j|C(x0,r) ∈ c 0 extending this restriction to Rn C(x ,r) by 0 RN to get a test map for (5.1) we arrive at \ 0 ∈ x x 0 F x, Dkφ j − 0 dx. ≤ r ZC(x0,r) 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 F (x, Dkφ(y)) dy dx. ≤ ZC(x0,r)ZC Since x0, r were arbitrary we deduce by the regularity of Lebesgue measure that F (x, Dkφ(y)) dy 0 (5.2) ≥ ZC n holds for all x R Nφ where Nφ is a negligible set. To see that (5.2) in fact holds outside a negligible set∈N \ Rn that is independent of φ one invokes the separability of the space k n N ⊂ Cc(C, (R , R )) in the supremum norm. We leave the precise details of this to the interested reader⊙ and also remark that a straightforward scaling argument shows that we may replace C by Rn and allow any φ C∞(Rn, RN ) in (5.2). It now remains to prove the pointwise bound ∈ c in the autonomous case F = F (ξ). By Theorem 4.2 the relaxation given by the formula F (ξ) = inf F (ξ + Dkϕ(x)) dx ∞ F ϕ∈C ((0,1)n,RN ) n c Z(0,1) is a rank–one convex function : k (Rn, RN ) [ , ). Recall that by rank–one convex function in this context we understandF ⊙ a function→ which−∞ is∞ convex in the directions of the cone N s(n, k, R ). Now the integral bound (5.1) translates to (0) 0, and it then follows from DLemma 2.2 that is a rank–one convex function whichF is real–valued≥ everywhere. By the F ∞ homogeneity of F and of Cc one also easily checks that is 1–homogeneous. Consequently, N F as the cone s(n, k; R ) is balanced and spanning, Theorem 1.1 implies in particular that is convex atD0 k(Rn, RN ): there exists ζ k(Rn, RN ) such that (ξ) ξ,ζ for allF ξ k(Rn, RN∈). ⊙Taking ξ = ζ and using 1–homogeneity∈ ⊙ we see that necessarilyF ≥ h ζ =0i . The ∈ ⊙ ± conclusion follows because F . ≥F 20 Remark 5.2. The proof given above for Theorem 5.1 can easily be adapted to also prove the following result. For a C∞ map φ: Rn RN we denote by J k−1φ(x) its k 1 jet at x: thevector →α − consisting of all partial derivatives D φi(x) of coordinate functions φi corresponding to all multi–indices α of length at most k 1 arranged in some fixed order. Hereby J k−1φ: Rn Rd for a d = d(n, k, N). − → Let F : Rn Rd k(Rn, RN ) R be a Caratheodory´ integrand (measurable in x 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 ) | | ≤ | | | | Rn Rd R k Rn RN 1 Rn for almost all x , all y , t and ξ ( , ), where a Lloc( ) is a given function. ∈ ∈ ∈ ∈ ⊙ ∈ Then F (x, J k−1φ(x),Dkφ(x)) dx 0 Rn ≥ Z ∞ Rn RN Rn holds for all φ Cc ( , ) only if F (x, 0,ξ) 0 for almost all x 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 a nontrivial –affine quadratic form m on , where is a cone of directions satisfying (1.1). D V D In the setting of Section 4 we consider the spaces Y = R and X = . V 2 2 R The second derivatives that we are interested in will belong to the space ( ) ∼= ( , ). The corresponding rank-one cone is the usual one in 2( ), namely t ℓ ⊙ ℓ V: t ⊙R andV ℓ ⊙ V { · ⊗ ∈ ∈ ∗ (indeed, if we choose a basis for we may identify = Rn, 2( ) = Rn×n and the V } V V ∼ ⊙ V ∼ sym rank-one cone is simply the cone of symmetric n n matrices of rank at most one). This cone is of course balanced and spanning. Next, and more× specific to our situation we observe that for an open subset O of a C2 function ϕ: O R is –convex if for all x O we have D2ϕ(x) where V → D ∈ ∈ C := µ 2( ): µ(e, e) > 0 for all e 0 . C ∈ ⊙ V ∈D\{ } Clearly, is an open convex cone, and in the notation from the Introduction we have = int D. C ∗ C Q The assumption (1.1) therefore amounts to ℓ ℓ for some ℓ . Put µ0 = ℓ ℓ so that µ is a rank one form in . If µ = D2m(0)⊗we∈ have C that µ +∈tµ V for all t⊗ R. We 0 C m 0 m ∈ C ∈ now consider the function F (µ) := µ on . By Theorem 4.2 the relaxation is rank-one convex, and since F < also <−k .k WeC assert that . Assume not,F so that is finite somewhere in . Then∞ byF Lemma∞ 2.2 > everywhereF ≡−∞ on , so that is a real–F valued and rank–oneC convex function on . FromF the−∞ definition of andC the homogeneityF of C F 21 F , C∞ we infer that is positively 1–homogeneous. By virtue of Theorem 1.1 is therefore c F F convex at µ0 so that in particular: 1 (µ ) (µ + tµ )+ (µ tµ ) F 0 ≤ 2 F 0 m F 0 − m 1 µ + tµ + µ tµ , ≤ −2 k 0 mk k 0 − mk which is impossible for large values of t. This contradiction shows that on . In view of the definition of the relaxation we then find for any µF ≡−∞and eachC ε > 0 F ∈ C an ϕ C∞(Q, R) with µ + D2ϕ(x) for all x such that ( µ + D2ϕ ) < 1/ε, ∈ c ∈ C ∈ V Q −k k − ϕ ∞ + Dϕ ∞ <ε, and k kL (Q) k kL (Q) R Dϕ(x) Dϕ(y) [Dϕ] := sup k − k : x, y Q, x = y < ε, ω ω( x y ) ∈ 6 | − | where Q is the unit cube in with respect to some (from now on fixed) orthonormal basis of and the integral is calculatedV with respect to the corresponding Lebesgue measure. There is Vnothing special by the open unit cube Q in the above, and it is possible to replace it by any bounded open subset O of . Indeed, we simply replace ϕ by x λ−2ϕ(λx + y) for λ > 0 and y and express O asV a disjoint union of closed cubes y + λ7→Q¯ to achieve this. The∈V 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 set A of all rank–one convex C1 functions f : X R such that → f(x) Df(x) Df(y) f = sup | | + k − k : x, y X, x = y k k 1+ x 2 ω( x y )(1 + x + y ) ∈ 6 k k | − | | | | | is finite and equip it with the metric d(f,g) := f g . It is easy to verify that (A, d) is a k − k complete metric space, and we denote by B the closure of smooth functions in A. With the ∞ inherited metric this is clearly also a complete metric space. Now let (Ol)l=1 be a sequence consisting of all dyadic cubes in of side length at most one and define, for l,k 1, V ≥ ∂2ψ B = f B : f k if ψ C∞(O , R), ψ 1 and α =2 . l,k ∈ ∂xα ≤ ∈ c l k k∞ ≤ | | ZOl 2 Clearly, each Bl,k is a closed set in B and for f C the usual integration by parts argument applied to a ψ that sufficiently well approximates∈ sign(∂2f/∂xα) shows that we have ∂2f f B k for all α =2. ∈ l,k ⇔ ∂xα ≤ | | ZOl We finish our proof by showing that each B l,k has empty interior since then B l,k Bl,k is nonempty and obviously each f in this set satisfies the conclusion of Theorem 1.5.\ But other- S wise, we would find an open ball B(f, δ) inside some Bl,k. The definition of B as the closure 22 of smooth functions means that we without loss of generality may assume f C∞. Now we 1 ∈ find a C considered above such that φC < δ/3, here φC (x) = 2 C(x, x) and the first ∈ C k k ∞ part of our proof ensures the existence of a ϕ Cc (Ol) such that ϕ ϕ ∞ < δ/3 but 2 4 ∈ k k≤k k D (ϕC + ϕ) > 3n k. In particular, there must exist a multi-index α of length two such Ol k 2 k α that ∂ (ϕC + ϕ)/∂x > 3k. From this it is clear that f + ϕC + ϕ B(f, δ) but R Ol | | ∈ R ∂2(f + ϕ + ϕ) C > 2k, ∂xα ZOl 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–valued measures where both oscillation and concentration phenomena must be taken into account is through a notion of Young measure that was introduced by DiPerna and Majda in [DM87]. This formalism was revisited in [AB97] and a special case of particular importance was developed into a very convenient and suggestive form. This was the starting point for [KR10b], and we shall briefly pause to describe the relevant set–up here. Throughout this section Ω denotes a fixed bounded Lipschitz domain in Rn and H a finite dimensional real inner product spaces. Let E(Ω, H) denote the space of test integrands consisting of all continuous functions f : Ω H R for which × → f(x, tz) f ∞(x, z) = lim t→∞ t exists locally uniformly in (x, z) Ω H. Note that hereby the recession function f ∞ is jointly continuous and z f ∞(x,∈ z) is× for each fixed x a positively 1–homogeneous function. 7→ The set of Young measures under consideration, denoted by Y(Ω, H), is defined to be the ∞ set of all triples ((νx)x∈Ω, λ, (νx )x∈Ω) such that (ν ) is a weakly∗ n measurable family of probability measures on H, and • x x∈Ω L z dνx(z) dx< , H| | ∞ ZΩZ λ is a nonnegative finite Radon measure on Ω, • ∞ ∗ (ν ) is a weakly λ measurable family of probability measures on SH, where SH • x x∈Ω is the unit sphere in H. Here, the maps x ν and x ν∞ are only defined up to n- and λ–negligible sets, respec- 7→ x 7→ x L tively. The parametrized measure (νx)x is called the oscillation measure, the measure λ is the ∞ concentration measure and (νx )x is the concentration-angle measure. As the terms indi- cate, the oscillation measure is the usual Young measure and describes the oscillation (or failure 23 of convergence in n measure), the concentration measure describes the location in Ω and the magnitude of concentrationL (or failure of n equi–integrability) while the concentration–angle measure describes its direction in H. Often,L we will simply write ν as short-hand for the above triple. To every H–valued Radon measure γ supported on Ω with Lebesgue–Radon–Nikod´ym de- d s composition γ = a + γ , we associate an elementary Young measure ǫγ Y(Ω, H) given by the definitions L ∈ (ǫ ) := δ , λ := γs , (ǫ )∞ := δ , γ x a(x) ǫγ | | γ x p(x) where p := dγs/d γs is the Radon–Nikod´ym derivative and γs (= γ s) denotes the total | | | | | | variation measure of γs. Using the inner product on H we can define a duality pairing of f E(Ω, H) and ν Y(Ω, H) by setting ∈ ∈ ∞ ∞ f, ν := f(x, z) dνx(z) dx + f (x, z) dνx (z) dλ(x). hh ii H S ZΩZ ZΩZ H H For a sequence (γj) of –valued Radon measures with supj γj (Ω) < , we say that (γj) Y| | ∞ generates the Young measure ν Y(Ω, H), in symbols γ ν, if for all f E(Ω, H) we ∈ j → ∈ have f, ǫ f, ν , or equivalently, hh γj ii → hh ii dγ dγs f x, j (x) n + f ∞ x, j (x) γs f(x, ), ν n + f ∞(x, ), ν∞ λ (6.1) d n L d γs | j |→ · x L · x L | j | weakly∗ in C(Ω)∗ as j . A continuity result due to Reshetnyak [R68] can be used to → ∞ justify the formalism (see [KR10b]) in the sense that for measures γj, γ not charging ∂Ω we Y ∗ have γj γ –strictly if and only if ǫγj ǫγ . Here the former is taken to mean that γj ⇁γ in C (Ω,→H)∗ h·iand γ (Ω) γ (Ω), where→ 0 h ji →h i 1 dγ 2 γ (Ω) := 1+ 2 d n + γs (Ω). h i |d n | L | | ZΩ L Standard compactness theorems for measures imply that any bounded sequence (γj) of H– Y valued Radon measures on Ω admits a subsequence (not relabelled) such that γ ν for some j → Young measure ν Y(Ω, H). We are particularly interested in the situation where the measures ∈ γj are concentrated on Ω. Making special choices of test integrands f in (6.1) we deduce that γ ⇁γ∗ in C (Ω, H)∗, where γ =ν ¯ Ω and j 0 ⌊ ν¯ =ν ¯ n +¯ν∞λ with ν¯ = id, ν and ν¯∞ = id, ν∞ xL x x h xi x h x i is the barycentre of ν. If λ(∂Ω) =0, then γ =ν ¯ and the convergence takes place in the stronger sense that γ ⇁γ∗ narrowly on Ω (meaning Φ,γ Φ,γ for all bounded continuous maps j h ji→h i Φ: Ω H). → 24 Given a mapping of bounded variation, u BV(Ω, RN ), its distributional derivative Du is ∈ an RN×n–valued Radon measure concentrated on Ω and hence we can associate an elementary N×n n s Young measure ǫDu Y(Ω, R ). Writing Du = u + D u for the Lebesgue decompo- sition with respect to∈ n we have ∇ L L (ǫ ) := δ , λ := Dsu , (ǫ )∞ := δ , Du x ∇u(x) ǫDu | | Du x p(x) where p := Dsu/ Dsu is short-hand for the Radon-Nikod´ym derivative. | | N×n Any Young measure ν Y(Ω, R ) that can be generated by a sequence (Duj), where ∈ N 1 N (uj) is a bounded sequence in BV(Ω, R ) that L –converges to a map u BV(Ω, R ) is called a BV gradient Young measure with underlying deformation u. Clearly∈ the underlying deformation u is locally unique up to an additive constant vector. If we assume that λ(∂Ω) =0, then Du is the barycentre of ν and identifying terms according to the Lebesgue decomposition λ = a n + b Dsu + λ∗ where λ∗ is a measure which is singular with respect to n + Dsu we find L | | L | | dλ u(x) =ν ¯ +¯ν∞ (x) n a.e. x Ω (6.2) ∇ x x d n L ∈ Dsu L (x) = b(x)¯ν∞ Dsu a.e. x Ω (6.3) Dsu x | | ∈ | | 0 =ν ¯∞ λ∗ a.e. x Ω. (6.4) x ∈ These conditions are clearly necessary for ν tobea BV gradient Young measure with barycentre Du. Other necessary conditions follow, loosely speaking, by expressing a relaxation result for signed integrands of linear growth obtained in [KR10a], in terms of Young measures. It turns out that these conditions are sufficient for ν to be a BV gradient Young measure too. The proof of this is based on a Hahn–Banach argument similar to that employed by Kinderlehrer and Pedregal in [KP91, KP94]. Hereby a characterization of BV gradient Young measures is obtained: they are the dual objects to quasiconvex functions in the sense that a set of inequalities akin to Jensen’s inequality must hold for the measures and all quasiconvex functions of at most linear growth. It can be seen as a nontrivial instance within the abstract frame work provided by Choquet’s theory of function spaces and cones (see [LMNS]). In order to state the result more precisely we denote by Q the class of all quasiconvex functions of at most linear growth, N×n so f Q when f : R R is quasiconvex and there exists a real constant c = cf such that f(ξ)∈ c ξ +1 for all→ξ. | | ≤ | | Theorem 6.1 ([KR10b]) . Let Ω be a bounded Lipschitz domain in Rn. Then, a Young measure ν Y(Ω, RN×n) satisfying λ(∂Ω) = 0 is a BV gradient Young measure if and only if there exists∈ u BV(Ω, RN ) such that Du is the barycentre for ν, and, writing λ = dλ n + λs for ∈ dLn L the Lebesgue–Radon–Nikodym´ decomposition, the following conditions hold for all f Q: ∈ dλ dλ (i) f ν¯ +¯ν∞ (x) f, ν + f ∞, ν∞ (x) n a.e. x Ω, x x d n ≤ x x d n L ∈ L L (ii) f ∞(¯ν∞) f ∞, ν∞ λs a.e. x Ω . x ≤ x ∈ 25 Here the symbol f ∞ refers to the (upper) recession function of f as defined at (2.3). The condition (ii) concerning the points x seen by the singular part of the measure λ is simply ∞ Jensen’s inequality for the probability measure νx and the 1–homogeneous quasiconvex func- tion f ∞. Condition (i) concerning points x seen by the absolutely continuous part of the mea- sure λ is reminiscent of both Jensen’s inequality and (2.5). It expresses how oscillation and concentration must be coupled if created by a sequence of gradients. The question of whether this characterization can be simplified, and indeed whether condition (ii) is necessary at all, was in fact the initial motivation for the work undertaken in the present paper. Inspection of the above characterization yields in particular that a nonnegative finite Radon measure λ on Ω with λ(∂Ω)=0 is the concentration measure for a BV gradient Young measure with underlying de- formation u BV(Ω, RN ) if and only if λs Dsu as measures on Ω. In fact, it is not hard to ∈ ≥| | see this directly (even without the assumption that λ(∂Ω) = 0), and the reason for mentioning it at this stage is that we would like to emphasize that the concentration measure is more or less arbitrary. In order to state the the main result of this section we introduce the following notation for special test integrands: RN×n f is Lipschitz, f(ξ) ξ for all ξ, and for some SQ = SQ( )= f Q : ≥| |∞ . ∈ rf > 0 we have f(ξ)= f (ξ) when ξ rf | | ≥ In terms of these the result is Theorem 6.2. Let Ω be a bounded Lipschitz domain in Rn. Then a Young measure ν Y(Ω; RN×n) satisfying λ(∂Ω) = 0 is a BV gradient Young measure, if and only if ν has∈ barycentre ν¯ = Du for some u BV(Ω, RN ) and for all f SQ we have ∈ ∈ dλ dλ f ν¯ +¯ν∞ (x) f, ν + f ∞, ν∞ (x) (6.5) x x d n ≤ x x d n L L for n almost all x. L In view of the examples of nonconvex, but quasiconvex positively 1–homogeneous inte- grands given in [M¨u92] it seems that SQ is close to the minimal class of test integrands we could possibly expect to use in (6.5). For instance we note that it seems to be impossible to reduce the class of test integrands to the smaller one used in the characterization of ordinary W1,1 gradient Young measures (i.e., those ν with λ =0) given in [Kr99]. ∞ Proof. Only the sufficiency part requires a proof, so assume ν = (νx)x∈Ω, λ, (νx )x∈Ω is a Young measure such that λ(∂Ω) = 0, ν¯ = Du for some u 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 Lebesgue decomposing the measure λ as λ = a n + b Dsu + λ∗, where λ∗ is a measure which is singular L | | with respect to n + Dsu . Consider first condition (ii), note λs = b Dsu + λ∗ and fix f Q. Then the recessionL function| | f ∞ is a positively 1–homogeneous and quasiconvex| | function. Since∈ quasiconvexity implies rank-one convexity for real–valued functions we can use Corollary 1.2 whereby we infer that f ∞ is convex at all points of the rank one cone. Therefore Jensen’s 26 inequality holds for f ∞ and any probability measure with a centre of mass on the rank one ∞ ∗ cone. According to (6.4) the probability measure νx has centre of mass at 0 for λ almost all x Ω, so (ii) holds λ∗ almost everywhere. For the remaining points x Ω seen by λs we ∈ ∈ appeal to Alberti’s rank-one theorem [Al93]. Accordingly the matrix Dsu/ Dsu has rank one s ∞ | | for D u almost all x Ω, and so by (6.3) the probability measure νx has centre of mass on the rank| one| cone for b ∈Dsu almost all x Ω. Therefore (ii) holds for b Dsu almost all x Ω, | | ∈ | | ∈ and so we have shown that it holds λs almost everywhere in Ω. Next we turn to (i), and start by fixing f Q with the additional property that f = f ∞ outside a large ball in matrix space. ∈ Because f ∞ in particular must be rank–one convex and positively 1–homogeneous we deduce from Corollary 1.2 that f ∞ ℓ for some linear function ℓ on RN×n. Consequently, f a for an affine function a on RN×n≥, and defining g = +(f a)/ε for ε> 0 we record that g≥ SQ |·| − ∈ so that (i) holds for g, and thus also for εg. By approximation we deduce that (i) also holds for f. The final step is facilitated by the following approximation result. ∞ 1 Lemma 6.3. Let f Q and δ > 0. Then g(ξ) = gδ(ξ) := max f(ξ), f (ξ)+ δ ξ δ is quasiconvex and for∈ some s = s(δ) > 0 we have g(ξ) = f ∞(ξ)+ δ ξ 1 for all| | −ξ s. | | − δ | | ≥ Furthermore, (gδ)δ∈(0,1) is equi–Lipschitz, and g (ξ) f(ξ) and g∞(ξ) f ∞(ξ) δ → δ → pointwise in ξ as δ 0. ց Proof of Lemma 6.3. It is clear that g (ξ) f(ξ) as δ 0 pointwise in ξ, that g are quasi- δ → ց δ convex and, by Lemma 2.3, that (gδ) is equi–Lipschitz. It remains to find the number s = s(δ) with the stated property. The rest then follows easily. Our definition of recession function at (2.3) applied to (f f ∞ + 1 )∞ = 0 yields for given − δ ξ ∂BN×n and δ > 0 an s = s(ξ, δ) > 0 such that ∈ 1 f(tξ′) < f ∞(tξ′)+ δ tξ′ (6.6) | | − δ for t s and ξ′ ∂BN×n with ξ ξ′ < 1/s. By compactness of ∂BN×n we therefore find an ≥ ∈ | − | s = s(δ) > 0 such that (6.6) holds for all t s and ξ′ ∂BN×n. Stated differently we have shown that f(ξ) < f ∞(ξ)+ δ ξ 1 for all ξ≥with ξ ∈s. 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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 Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK E-mail address: [email protected] 31 0 we find ε (0, δ) such that ∈ sup F (ζ): dist(ζ, [ξ, η]) <ε ε < δ, { } and ζ if dist(ζ, [A, B]) <ε. Next, we set u(x)= µ(x, x, . . . , x)/k! and choose φ satisfying ∈ S n the conditions (i), (ii) and (iii) of Proposition 4.1 for Ω = (0, 1) , in particular so that Ωξ = int x Ω: Dk(u + φ)(x) = ξ and Ω = int x Ω: Dk(u + φ)(x) = η satisfy { ∈ } η { ∈ } n Ω (Ω Ω ) < ε n(Ω). Clearly, we find an l N such that the family of all L \ ξ ∪ η L ∈ Qξ dyadic cubes of sidelength 2−l entirely contained in Ω satisfies n( ) > (1 ε) n(Ω ) ξ L Qξ − L ξ n n and similarly η > (1 ε) (Ωη). Hence, S L Q − L S Ω = Ω r \ Qξ ∪ Qη n k fulfills (Ωr) < ε and by Proposition 4.1(i), [ dist([D (u + φ)(x), [ξ, η]) < ε for x Ω. Therefore,L ∈ F (Dk(u + φ))(x) dx < δ. ZΩr Finally, we define similarly to the first part of the proof the function