Liftings, Young Measures, and Lower Semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs (u j , Du j ) j for (u j ) j ⊂ BV( ;Rm) under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional F : u → ∫ f (x, u(x),∇u(x)) dx, u ∈ W1,1( ;Rm), ⊂ Rd open, to the space BV( ;Rm). Lower semicontinuity results of this type were first obtained by Fonseca and Müller (Arch Ration Mech Anal 123:1–49, 1993) and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that f be Carathéodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising F in the x and u variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.


Introduction
Finding an integral representation for the relaxation F * * of the functional To obtain the inequality " " in (3), it suffices to bound the right hand side of (4) from below by f (x 0 , u(x 0 ), ∇u(x 0 )) and to obtain analogous results for dμ |D c u| and dμ dH d−1 J u . The authors of [22] achieve this by noting that the partial coercivity property g(x, y)|A| f (x, y, A) Cg(x, y)(1 + |A|) combined with the rescaling in r of each u j allows for (u j ) j to be replaced by a rescaled and truncated sequence (w j ) j , which is crucially weakly* and L ∞ convergent in BV(B d ; R m ) to a blow-up limit z → ∇u(x 0 )z.
Fonseca and Müller's result confirms that, as in the case for problems posed over W 1, p ( ; R m ) for p > 1, quasiconvexity is still the right qualitative condition to require for variational problems with linear growth. However, it has been an open question as to whether the (x, y)-localisation hypotheses which have been used until now (both for the results in [22] and for the substantial later improvements in [21]) are truly necessary for (3) to hold. Up to an error which grows linearly in A, these assumptions state that f must be such that f (x 0 , y 0 , A) ≈ f (x, y, A) whenever (x, y) is sufficiently close to (x 0 , y 0 ) and that f # (x 0 , y, A) ≈ f # (x, y, A) uniformly in y for |x 0 − x| sufficiently small.
It is known in the case of superlinear growth that for F to be weakly lower semicontinuous over W 1, p ( ; R m ), f need only be quasiconvex in the final variable, Carathéodory, and satisfy the growth bound 0 f (x, y, A) C(1 + |A| p ). For the situation where f has linear growth and the task is to find the relaxation of F to BV( ; R m ), a short overview of available results is as follows: in the scalar valued case where m = 1, Dal Maso [17] obtained the scalar counterpart to (3) (the term over J u admits a simpler form) under the assumption of coercivity for the general case where f = f (x, y, A). If no coercivity is required of f , the task of finding the L 1 -relaxation F 1 * * is highly non-trivial and involves the regularity of f in the x variable, see for instance [5]. In the vector-valued case, Aviles and Giga used the theory of currents to obtain (3) under the assumption that f is continuous, convex in the final variable, coercive, and also satisfies a specific isotropy property, see [10,11]. The situation where f is quasiconvex in the final variable is harder: the case where f is independent of both x and u was settled by Ambrosio and Dal Maso [6], and Kristensen together with the first author obtained (3) under just the assumption that f = f (x, A) is u-independent, Carathéodory and such that f ∞ exists [31].
For the u-dependent case where f = f (x, y, A) and f is quasiconvex in the final variable, the only available results are the original identification in [22] due to Fonseca and Müller, later improvements in [14,15], and finally the most recent results in Fonseca and Leoni [21]. Roughly, these results require (in addition to quasiconvexity) that f is Borel, such that f # exists, and that, for every (x 0 , y 0 ) ∈ × R m and ε > 0, there exists δ > 0 such that |x − x 0 | + |y − y 0 | < δ implies f (x 0 , y 0 , A) − f (x, y, A) ε (1 + f (x, y, A)) for all A ∈ R m×d and that Reasoning by analogy with the cases where f has superlinear growth over W 1, p ( ; R m ) or where f has linear growth and either m = 1 (so that f : × R × R d → R) or f is u-independent (so that f = f (x, A)), the implication is that (3) should hold under just the assumptions that f be quasiconvex in the final variable and possesses sufficient growth and regularity to ensure that the right hand side of (3) is well-defined over BV( ; R m ). This result is, essentially, our Theorem A below.
To avoid the use of extraneous hypotheses, we must pass from lim j f (x, u j (x), ∇u j (x)) to lim j f (x 0 , u(x 0 ), ∇u j (x)) in (4) using only the behaviour of the sequence (u j ) j rather than any special properties of the integrand. In order to do this, we must improve our understanding of the behaviour of weakly* convergent sequences in BV( ; R m ), particularly under the blow-up rescaling x → (x −x 0 )/r . These are much more poorly behaved than weakly convergent sequences in W 1, p ( ; R m ) for p > 1 thanks to interactions between (u j ) j and (Du j ) j in the limit as j → ∞, see Example 3.10. In the reflexive Sobolev case, powerful truncation techniques [23,30] mean that these interactions can be neglected in the sequence under consideration, but no such tools are available in BV( ; R m ) when m > 1.
The main contributions of this paper are the development of a new theory for understanding weakly* convergent sequences and blow-up procedures in BV( ; R m ), together with the use of this theory to provide a new proof for the integral representation of the weak* relaxation F * * of F to BV( ; R m ) under weaker hypotheses. This representation is valid for Carathéodory integrands and does not require the (x, y)-localisation properties of f which have previously been used: f (x, y, A) C(1 + |y| d/(d−1) + |A|), (5) for some C > 0, p ∈ [1, d/(d − 1)), h ∈ C(R m×d ) satisfying h ∞ ≡ 0, and for all (x, y, A) ∈ × R m × R m×d ; (iii) f (x, y, q ) is quasiconvex for every (x, y) ∈ × R m .

Preliminaries
Throughout this work, ⊂ R d will always be assumed to be a bounded open domain with compact Lipschitz boundary ∂ in dimension d 2, and B k , ∂B k will denote the open unit ball in R k and its boundary (the unit sphere) respectively. The open ball of radius r centred at x ∈ R k is B(x, r ), although we will sometimes write B k (x, r ) if the dimension of the ambient space needs to be emphasised for clarity. The volume of the unit ball in R k will be denoted by ω k := L k (B k ), where L k is the usual k-dimensional Lebesgue measure. We will write R m×d for the space of m×d real valued matrices, and id R m for the identity matrix living in R m×m . The map π : × R m → denotes the projection π((x, y)) = x, and T (x 0 ,r ) : R d → R d , T (x 0 ,r ),(y 0 ,s) : R d × R m → R d × R m represent the homotheties x → (x − x 0 )/r and (x, y) → ((x − x 0 )/r, (y − y 0 )/s)). Tensor products a ⊗ b ∈ R m×d and f ⊗ g for vectors a ∈ R m , b ∈ R d , and real valued functions f , g, are defined componentwise by (a ⊗ b) i, j = a i b j and ( f ⊗ g)(x, y) = f (x)g(y) respectively.
The closed subspaces of BV( ; R m ) and C ∞ ( ; R m ) consisting only of the functions satisfying (u) := − u(x) dx = 0 are denoted by BV # ( ; R m ) and C ∞ # ( ; R m ) respectively. We shall use the notation (u) when the domain of integration might not be clear from context, as well as the abbreviation (u) x,r := (u) B(x,r ) . We shall sometimes use subscripts for clarity when taking the gradient with respect to a partial set of variables: that is, if f = f (x, y) ∈ C 1 ( × R m ) then ∇ x f = (∂ x 1 f, ∂ x 2 f, . . . , ∂ x d f ) and ∇ y f = (∂ y 1 f, ∂ y 2 f, . . . , ∂ y m f ).

Measure Theory
For a separable locally convex metric space X , the space of vector-valued Radon measures on X taking values in a normed vector space V will be written as M(X ; V ) or just M(X ) if V = R. The cone of positive Radon measures on X is M + (X ), and the set of elements μ ∈ M(X ; V ) whose total variation |μ| is a probability measure, is M 1 (X ; V ). The notation μ j * μ will denote the usual weak* convergence of measures, and we recall that μ j is said to converge to μ strictly if μ j * μ and in addition |μ j |(X ) → |μ|(X ). Given a map T from X to another separable, locally convex metric space Y , the pushforward operator T # : If T is continuous and proper, then T # is continuous when M(X ; V ) and M(Y ; V ) are equipped with their respective weak* or strict topologies. We omit the proof of the following simple lemma: Lemma 2.1. Let μ ∈ M(X ; V ), ν ∈ M + (X ) satisfy μ ν and let T : X → Y be a continuous injective map. Then it holds that Given a function f : X × V → R which is positively one-homogeneous in the final variable (that is, f (x, t A) = t f (x, A) for all t 0 and A ∈ V ) and a measure μ ∈ M(X ; V ), we shall use the abbreviated notation We note that, if T : X → Y is an injection, then applying Lemma 2.1 to T # μ and |T # μ| lets us deduce If μ is a measure on X × Y then we recall that the Disintegration of Measures Theorem (see Theorem 2.28 in [7]) allows us to decompose μ as the (generalised) product μ = π # |μ| ⊗ ρ, where π # |μ| is the pushforward of |μ| onto X and ρ is a (π # |μ|-almost everywhere defined) parametrised measure. Here, π # |μ| ⊗ ρ is defined (uniquely) via With A assumed to be countably H k -rectifiable, we can define the Radon-Nikodym derivative for any μ ∈ M(R d ) with respect to H k A, given for H kalmost every x ∈ A, by The function dμ dH k A is a Radon-Nikodym Derivative in the sense that dμ dH k A H k A is a k-rectifiable measure and that we can decompose in analogy with the usual Lebesgue-Radon-Nikodym decomposition. A measure μ ∈ M(R d ; V ) is said to be admit a (k-dimensional) approximate tangent space at x 0 if there exists an (unoriented) k-dimensional hyperplane τ ⊂ R d and θ ∈ V such that The existence of approximate tangent spaces characterises the class of rectifiable measures in the sense that μ ∈ M(R d ; V ) possesses a k-dimensional approximate tangent space at |μ|-almost every x 0 ∈ R d if and only if μ is k-rectifiable (see Theorem 2.83 in [7]).
By varying ∇g(a) through R m , we see that μ({a}) = 0, as required.

BV Functions
Given a function u ∈ BV( ; R m ), we recall the mutually singular decomposi- otherwise.
The need to fix a choice of orientation for J u in order to properly define u θ is obviated by the fact that u θ will only appear in expressions of the form 1 0 ϕ(u θ (x)) dθ, which are invariant of our choice of n u .
Given (the precise representative of) a function u ∈ BV( ; R m ), the function associated to its graph is denoted by gr u : x → (x, u(x)). If μ is a measure on satisfying both |μ| H d−1 and |μ|(J u ) = 0 (we will usually take μ = |Du| ( \J u ), its pushforward under gr u then still makes sense as the Radon and Du j → Du strictly in M( ; R m×d ) as j → ∞. We say that u j converges area-strictly to u if u j → u in L 1 ( ; R m ) and, in addition, as j → ∞. It is the case that area-strict convergence implies strict convergence in BV( ; R m ) and that strict convergence implies weak* convergence. That none of these notions of convergence coincide follows from considering the sequence (u j ) j ⊂ BV((−1, 1)) given by u j (x) := x + (a/j) sin( j x) for some a = 0 fixed. This sequence converges weakly* to the function x → x for any a ∈ R\{0}, strictly if and only if |a| 1, but (since the function z → 1 + |z| 2 is strictly convex away from 0) never area-strictly. Smooth functions are area-strictly (and hence strictly) dense in BV( ; R m ); indeed, if u ∈ BV( ; R m ) and (u ρ ) ρ>0 is a family of radially symmetric mollifications of u then it holds that u ρ → u area-strictly as ρ ↓ 0. If ⊂ R d is such that ∂ is Lipschitz and compact, then the trace onto ∂ of a function u ∈ BV( ; R m ) is denoted by u| ∂ ∈ L 1 (∂ ; R m ). The trace map u → u| ∂ is norm-bounded from BV( ; R m ) to L 1 (∂ ; R m ) and is continuous with respect to strict convergence (see Theorem 3.88 in [7]). If u, v ∈ BV( ; R m ) are such that u| ∂ = v| ∂ , then we shall sometimes simply say that "u = v on ∂ ".
The following proposition, a proof for which can be found in the appendix of [32] (or Lemma B.1 of [13] in the case of a Lipschitz domain ), states that we can even require that smooth area-strictly convergent approximating sequences satisfy the trace equality u j | ∂ = u| ∂ : and, if u ∈ L ∞ ( ; R m ), then we can also require that sup j u j L ∞ u L ∞ .
and Du as the mutually singular sum where D u denotes the set of points at which u is approximately differentiable, J u denotes the set of jump points of u, C u denotes the set of points where u is approximately continuous but not approximately differentiable, and N u satisfies Then the following trichotomy relative to L d + |Du| holds: (iii) For |D c u|-almost every x ∈ and for any sequence r n ↓ 0, the sequence (u r n ) r n contains a subsequence which converges weakly* in BV(B d ; R m ) to a non-constant limit function of the form where γ ∈ BV((−1, 1); R) is non-constant and increasing. Moreover, if (u r n ) n is a sequence converging weakly* in this fashion then, for any ε > 0, there exists τ ∈ (1 − ε, 1) such that the sequence (u τr n ) n converges strictly in BV(B d ; R m ) to a limit of the form described by (10).
In all three situations, we denote lim r u r (or lim n u r n ) by u 0 . If the base (blow-up) point x needs to be specified explicitly to avoid ambiguity, then we shall write u r x , u r n x and u 0 x .

Proof.
For points x ∈ D u ∪J u , the conclusion of Theorem 2.4 follows directly from the approximate continuity of ∇u at L d -almost every x ∈ and the existence of the jump triple (u + , u − , n u ) for H d−1 -almost every x ∈ J u . For points x ∈ C u , the weak* precompactness of sequences (u r n ) n follows from the fact that (u r n ) B d = 0 and |Du r n |(B d ) = 1 combined with the weak* compactness of bounded sets in BV(B d ; R m ). The representation of u 0 is non-trivial and can only be obtained through the use of Alberti's Rank One Theorem, see Theorem 3.95 in [2]. It remains for us to show that, given a weakly* convergent sequence u r n * u and ε > 0, we can always find τ ∈ (1 − ε, 1) such that (u τr n ) n is strictly convergent in BV(B d ; R m ).
First note that we can assume that |Du r n | * |Du 0 | in M + (B d ) since (see for instance Theorem 2.44 in [7]) for arbitrary μ ∈ M( ; R m×d ) it is always true that Since as n → ∞, we therefore see that as required.
For x ∈ J u , the function u 0 gives a 'vertically recentered' description of the behaviour of u near x. It will be convenient to have a compact notation for also describing this behaviour when u is not recentered.
If the choice of base point x ∈ J u needs to be emphasised for clarity, we shall write u ± x . This definition is independent of the choice of orientation (u + , u − , n u ) and, for H d−1 -almost every x ∈ J u , the rescaled function u(x + r q ) converges strictly to u ± as r ↓ 0.
The following proposition was first proved in [38]: is continuous when BV( ; R m ) is equipped with the topology of strict convergence.
Theorem 2.7. (The chain rule in BV) Let u ∈ BV( ; R m ) and let f ∈ C 1 (R m ; R k ) be Lipschitz. It follows that v := f • u ∈ BV( ; R k ) and that

Integrands and Compactified Spaces
whenever the right hand side exists for every (x, y, A) ∈ × R m × R m×d independently of the order in which the limits of the individual sequences ( are taken and of the sequences used.
The definition of f ∞ implies that, whenever it exists, it must be continuous.
The following example demonstrates that the assumption in Theorem A that f ∞ 0 cannot be relaxed independently of the other requirements on f : For each fixed k, it is clear that v j,k * 0 in BV((−1, 1); R 2 ) as j → ∞. We can also see, however, that It therefore follows that F * * [0] = −∞. Moreover, for u ∈ (C 1 ∩BV)((−1, 1); R 2 ), we can repeat the procedure above with u j,k = u + v j,k in place of v j,k to obtain F * * ≡ −∞ on BV((−1, 1); R 2 ).
Defining v j,k ∈ BV((−1, 1) 2 ; R 2 ) by v j,k (x, y) = v j,k (x) (since Theorem A is only stated for d > 1) and noting both that v * 0 in BV((−1, 1) 2 ; R 2 ) and we see that the conclusion of Theorem A cannot hold for f and that no general integral formula is possible in this case.
Not every function f : R m×d → R with linear growth possesses a recession function in the sense of Definition 2.8, as simple examples show (this even holds for quasiconvex functions, see [37]). The upper recession function defined by however, always exist in R, and is often denoted by f ∞ in the literature.
is now abstractly defined as the dual of C( × σ R m ), and can also be understood in terms of more familiar spaces of measures as follows: whenever the limit appearing in (11) exists independently of our choice of convergent sequence ((x j , y j , A j )) j ⊂ × R m × R m×d . This method of extension is canonical in the sense that f ∈ C( × R m × R m×d ) occurs as the restriction Given c ∈ R m we extend the addition operator y → y + c continuously from R m to σ R m in this way by setting We see that × σ R m × R m×d . and that the recession function f ∞ exists. In addition, both f and f ∞ admit extensions to C( × σ R m × R m×d ) in the sense of (11). Note however that, as the example f (y, A) = exp(−(|y| − |A|) 2 )|A| demonstrates, the existence of continuous extensions for f and f ∞ does not guarantee that f ∈ E( × R m ). For this to be the case, we must also require that for any sequences ((x j , y j , (11). For f ∈ E( × R m ), this limit always exists by virtue of the continuity of g f at points (x, y, A) ∈ × σ R m × ∂B m×d .
Our interest in integrands which admit extensions to × σ R m × R m×d stems from the fact that, in order to compute limits of the form for weakly* convergent sequences (u j ) j ⊂ W 1,1 ( ; R m ), it is necessary that the extension h of f ∞ exists in some sense. This can be seen by considering sequences of the form u j (x) It turns out (see Example 2.14 below in conjunction with Proposition 4.18) that it is not sufficient to require that this extension exists as the limit (11) (with f = f ∞ ). On the other hand, the requirement that h exist in the sense of the limit (13) is clearly stronger than necessary, since it precludes us from considering any integrands which are unbounded in y such as f (x, y, A) = |y|, for which we can compute representation. Definition 2.11 below provides the optimal existence requirement for an extension of f ∞ from the perspective of computing (14) for as wide a class of integrands f as possible.
whenever the right hand side exists for every (x, y, The definition of σ f ∞ implies that, whenever it exists, σ f ∞ is continuous and also that f ∞ exists and satisfies Thanks to the existence of the limit (13), we see that σ f ∞ exists for all f ∈ E( × R m ). Lemma 2.12. Let f : × R m × R m×d be such that f ∞ exists and f ∞ ≡ 0. Then, for any ε > 0 and K R m , there exists R > 0 such that |A| R implies If f is such that σ f ∞ exists with σ f ∞ ≡ 0 then, for any ε > 0 and k > 0, there for all x ∈ and y ∈ R m satisfying |y| d/(d− 1) k(1 + |A|).

Proof.
To prove the first statement, assume for a contradiction that there exists a sequence of points for some fixed ε > 0. By passing to a subsequence, we can assume that ((x k , y k )) → (x, y) in × K and that A k := A k /(1 + |A k |) converges to some limit B ∈ ∂B m×d and t k : To prove the second statement, we proceed similarly by assuming that there exists a sequence of points Letting t j := 1+|A j | and passing to a subsequence we can assume that ((x j , y j , A j /(1 + |A j |))) j converges to some limit (x, y, B) ∈ ×σ R m ×∂B m×d . Since |y j | d/(d−1) kt j , the definition of σ f ∞ then implies that which gives us the required contradiction.

Definition 2.13. (Representation integrands)
A function f : ×R m ×R m×d → R is said to be a member of R( × R m ) if f is Carathéodory and its recession function f ∞ exists. We shall primarily be interested in the following subsets of R( ×R m ), defined by the growth bounds (which are understood to hold uniformly in their respective parameters for all (x, y, A) ∈ × R m × R m×d ) satisfied by their members: The classes R L ( × R m ) and R w * ( × R m ) are named as such because they represent the largest classes of integrands to which we will refer whilst making statements about liftings and weakly* convergent BV-functions. In particular, the conclusion of Theorem A also holds for all f ∈ R w * ( × R m ). The following example demonstrates that the lower bound which we require for members of , is such that f ∞ exists with f ∞ 0, and is quasiconvex (or even convex) in the final variable: ) be a homeomorphism which can be extended continuously to a map defined on B d by setting u(∂B d ) = 0. For example, we can take u to be the composition As Since By construction, we have that B satisfies Since the function B is zero-homogeneous and u(x) = 0 for x ∈ B d , for every for all s > sufficiently large. We can therefore find s > 1 such that, if we define Now, define f : and is such that f ∞ exists and is given by the formula In particular, We also note that, whilst f ∞ extends continuously to a non-negative function defined on all of σ R d+1 × R (d+1)×d in the sense of (11) (with f = f ∞ ), σ f ∞ does not exist according to Definition 2.11. Finally, we claim that f (y, q ) is convex on R (d+1)×d for every y ∈ R d+1 : this follows from the fact that the function is convex on R for each fixed a 0 and that the map A → A : B(y) is linear. Next, define the functional F : Using the change of variables z = x/r , the fact that lim r ↓0 r log(r −1 ) = 0, and the zero-homogeneity of B, we can compute By virtue of (17), then, we have that lim r ↓0 F[u r ] < 0. It is easy to see that u r * 0 in BV(B d ; R d+1 ) as r → 0 and so we deduce that F * * [0] < 0. In fact, by replacing u with k · u, repeating the procedure above and then letting k → ∞, we can even see that F * * [0] = −∞. On the other hand, since f (y, 0) = 0 for all y ∈ R d+1 and f ∞ 0 on R d+1 × R (d+1)×d , the integral functional given in the statement of Theorem A must be non-negative at u ≡ 0. Hence, the conclusion of Theorem A cannot hold for the integrand f defined above.

Functionals and Surface Energies
where u θ is the jump interpolant defined above by (9). This choice of extension for F to BV( ; R m ) is different to the one discussed in Section 1, where F is extended to BV( ; R m ) by F * * , and is used for technical reasons: whilst the method of extension for F by relaxation is the right choice from the point of view of seeking existence of minimisers, F * * is not continuous with respect to any convergence with respect to which C ∞ ( ; R m ) is dense in BV( ; R m ) (see Example 6.1) and hence not an ideal functional to work with with respect to analysis in BV( ; R m ). By contrast, for integrands f ∈ R w * ( × R m ) (which need not be quasiconvex in the final variable), Theorem 2.15 below, states that F as defined by (18) 3 therefore implies that Theorem A can equivalently be seen as identifying the weak* relaxation of this continuously extended F from BV( ; R m ) to BV( ; R m ), which is the approach that we take in what follows.

Theorem 2.15. Let ⊂ R d be a bounded domain with Lipschitz boundary and let
Then the functional F : BV( ; R m ) → R is area-strictly continuous.
Theorem 2.15 is proved under slightly more general hypotheses in Theorem 5.2 of [38].
Given u ∈ BV( ; R m ) and x ∈ J u , define the class of functions A u (x) by where u ± x is as given in Lemma 2.16 below shows that K f [u] is always H d−1 -measurable and hence that the integral is always well-defined for every u ∈ BV( ; R m ).
such that n u orients J u and u + , u − are the one sided jump limits of u with respect to n u . Fix also ε > 0. The triple (u + , u − , n u ) is Borel and hence |D j u|-measurable, and so Lusin's Theorem implies that there exists a compact set It can easily be seen that μ j converges strictly in M(B d × R m ; R m×d ) to μ as j → ∞. Using Reshetnyak's Continuity Theorem and the positive one-homogeneity of f ∞ , we therefore deduce as j → ∞. By our choice of ϕ and the boundary condition satisfied by each ϕ j , we therefore have that It follows from the arbitrariness of x ∈ K ε and δ > 0 that and note that F := inf ε>0 F ε is equal to K f [u] at |D j u|-almost every x ∈ J u and hence H d−1 J u -almost every x ∈ J u . The conclusion now follows from the fact that the pointwise infimum of a collection of upper semicontinuous functions is upper semicontinuous and hence measurable.

Filip Rindler & Giles Shaw
Proof. This follows directly from Lemma 2.16 combined with the discussion about rectifiability in Section 2.1.

Liftings
In this section, we develop a theory of liftings. In turn we investigate their functional analytic properties, their relationship to BV-functions, a structure theorem, some blow-up results, and a discussion of how integral functionals over BV( ; R m ) can be represented in terms of liftings. Two separate spaces are introduced: the space of liftings L( × R m ), and the space of approximable liftings is larger than strictly necessary for our purposes, but has good compactness properties and is sufficiently well behaved that working in this setting of extra generality allows for a cleaner presentation of most of the results described here and in Sections 4 and 5. The exceptions to this rule is the Jensen inequalities for F, derived in Theorems 5.3 and 5.5, Section 5.2, for which we must work within the more restrictive class AL( × R m ).
The assumption that ⊂ R d with d 2 is used in this paper only in Proposition 2.6 and the Young measure theory developed in Section 4. Consequently, the results presented here are also valid for domains ⊂ R.

Functional Analysis and the Structure Theorem
holds. The space of all liftings is denoted by L( × R m ). Weak* convergence of liftings in L( × R m ) means weak* convergence of the liftings in considered as measures in M( × R m ; R m×d ).
Definition 3.1 was first given by Jung and Jerrard in [28] where the authors initiated the study of elementary liftings (which they refer to as minimal liftings), introduced below in Definition 3.5. This paper also contains the first proofs of Lemma 3.2 and Proposition 3.15 below. Our proofs for these results are new and, in the case of Proposition 3.15, are obtained as a corollary of the Structure Theorem (Theorem 3.11), which does not feature in [28].
The following lemma implies that each γ ∈ L( × R m ) is associated to a unique u ∈ BV # ( ; R m ) satisfying (22). We shall refer to this u as the barycentre of γ , writing [γ ] = u. (22), then it holds that Letting R → ∞ and using the Dominated Convergence Theorem, we therefore obtain Hence, after an integration by parts, which implies the first result. For A ∈ B( ), we can now compute from which it follows that π # |γ | |Du| as desired.
converging weakly* in M( × R m ; R m×d ) to a limit γ and such that [γ j k ] * u for some u ∈ BV # ( ; R m ). Taking the limit in (22) as j → ∞ and using the Dominated Convergence Theorem, it follows that the pair (u, γ ) satisfies (22) and The preceding discussion therefore implies that, upon passing to a further subsequence, [γ j ] * u for some u ∈ BV # ( ; R m ). Passing to the limit again in (22), we find that u = [γ ] and so, since this argument can be applied to any subsequence of (γ j ) j , we reach the desired conclusion.
Using the Dominated Convergence Theorem, we therefore deduce that, for any which is what was to be shown.
where u θ is the jump interpolant defined in Section 2.
where n ∂ is the (inwards pointing) normal orientation vector for ∂ . Applying the chain rule, Theorem 2.7, to ϕ • gr u and writing ∇ϕ = (∇ x ϕ, ∇ y ϕ), we see that Integrating over with respect to x, we deduce Remark 3.6. We note here that (22) and Definition 3.5 both make sense for u ∈ BV( ; R m ) (rather than just BV # ( ; R m )) and γ ∈ M( × R m ; R m×d ) and could be used to define liftings associated to arbitrary BV-functions. The downside for this extra generality is that the barycentre [0] of the zero lifting is no longer unique and, more importantly, that the control sup j |γ j |( × R m ) < ∞ no longer enforces sup j [γ j ] BV < ∞. As a result, the map γ → [γ ] is no longer continuous and Lemma 3.3 is no longer true. Instead, the discussion surrounding (27) in Section 3.4 shows how the behaviour of arbitrary weakly* convergent sequences in BV( ; R m ) can be described in terms of liftings as we have chosen to define them.

Definition 3.7. (Approximable liftings)
A lifting γ is said to be approximable if it arises as the weak* limit of a sequence of elementary liftings. The space of all approximable liftings is denoted by AL( × R m ), Note that, despite being defined as a sequential closure, it is an open question as to whether AL( × R m ) is either sequentially weakly* closed or weakly* closed since weak* topologies are not in general metrizable on unbounded sets. Example 3.8 demonstrates that, in the one-dimensional case at least, the inclusion Letting j → ∞, we deduce B(0, k)) > 0 for every k ∈ N, however, this implies that supp μ ⊂ (−1, 1) × B(0, R) for any R > 0, and so μ ∈ AL((−1, 1) × R 2 ).
The following is a direct corollary of Lemma 3.3 combined with Definition 3.7: Example 3.10 below demonstrates how non-elementary liftings can arise as weak* limits of sequences of elementary liftings, and shows that this phenomenon gives rise to behaviour for integrands which is very different to the u-independent and scalar valued cases.

Filip Rindler & Giles Shaw
We can compute for j 4, Thus, and so, by the construction of ϕ, However, we also see that from which we can conclude that γ = γ [u 0 ].

The Structure Theorem
We now investigate the structure of liftings γ ∈ L( × R m ) and, in particular, how a general lifting γ must relate to the elementary lifting γ satisfies div y μ = 0 (column-wise divergence), since div y μ = 0 implies that f (x)∇ y g(y) dμ(x, y) = 0 for all f ∈ C 0 ( ) and g ∈ C 1 0 (R m ). It turns out that every γ ∈ L( × R m ) can be written in this form: , then γ admits the following decomposition into mutually singular measures: and it is graph-singular with respect to u in the sense that γ gs is singular with respect to all measures of the form gr u # λ where λ ∈ M( ) satisfies both λ H d−1 and λ(J u ) = 0.
Proof. Since γ and γ [u] both satisfy the chain rule (22), we can test with functions of the form ϕ = f ⊗ g for f ∈ C 1 0 ( ), g ∈ C 1 0 (R m ) to obtain and Taking the difference of these two equations then leads us to the identity Since g ∈ C 1 0 (R m ) was arbitrary, it follows that div y ρ x = 0 (column-wise divergence) for η-almost every x ∈ and hence that div y (γ − γ [u]) = 0. Defining we therefore immediately obtain from the definition of γ [u] that Abbreviating and hence that as required.
To show that γ gs is graph-singular with respect to u, we argue as follows: , it follows that γ gs and gr u # λ charge disjoint sets, which suffices to prove the claim. Since is a u-graphical measure, we therefore also deduce that γ [u](( \J u ) × R m ) ⊥ γ gs , as required.
As we have discussed, no more can be said about the structure of γ ∈ L( ×R m ) beyond the conclusion of Theorem 3.11 in general, but a lot more can be said for the special cases where either m = 1 or π # |γ |( ) = |Du|( ) for u = [γ ]. Proof. If m = 1, then the operators div y and ∇ y coincide, and so (24) now states we therefore have that ∇ y ρ x ≡ 0 for π # |γ gs − γ [u] (J u × R)|-almost every x ∈ . Since any distribution whose gradient vanishes must be constant, it must follow that ρ x = c x L 1 for some constant c x ∈ R d . As ρ x ∈ M 1 (R; R d ) is a finite measure, however, we see that c x = 0 and hence that γ gs = γ [u] (J u × R m ). It then follows from Theorem 3.11 that as required.
Lemmas 3.13 and 3.14 below, which are special cases of Theorems 5.3 and D.1 respectively from [16] and for which simplified proofs can be found in [38], show that, since |θ x |(R m ) = 1, the identity (25) in fact forces and hence that γ = γ [u]. Applying Lemma 3.13 to θ x and then applying Lemma 3.14 to the measure we therefore arrive at the following proposition: Proof. It remains only to show the second statement: this follows from the fact that the strict convergence of u j to u implies that lim j |γ [u j ]|( × R m ) = |Du|( ). Thus, for any (non-relabelled) subsequence converging to a limit γ , we can apply Lemma 3.2 and the lower semicontinuity of the total variation on open sets to deduce that |Du|( ) |γ |( × R m ) |Du|( ). It follows that γ = γ [u] and, since this argument can be applied to any subsequence of (u j ) j , that the entire sequence converges γ [u j ] → γ [u].

be the homothety given by
Defining the measure γ r,s ∈ M(B d × R m ; R m×d ) by In the special case where s = c r , we shall use the abbreviations Proof. That γ r,s ∈ L(B d × R m ) follows from the fact that, for ϕ ∈ C 1 0 (B d × R m ), we can compute It is also clear that To show that γ r,s ∈ AL(B d × R m ) if γ ∈ AL( × R m ), it suffices to note that, if (γ [u j ]) j is such that γ [u j ] * γ , then the liftings γ j,r,s ∈ L(B d × R m ) defined by ϕ(z, w) dγ j,r,s (z, w) converge weakly* to γ r,s as j → ∞. Finally, The following result describes how liftings can be blown up around points in a similar fashion to the results presented for BV-functions in Theorem 2.4. Theorem 3.17 will be an indispensable tool for the localisation arguments carried out in Section 5. A similar procedure can be carried out at points x 0 ∈ J u but this is not necessary for the rest of this paper and is therefore omitted for brevity. • if x 0 ∈ D u then u r → u 0 and γ [u r ] → γ [u 0 ] strictly as r → 0, • if x 0 ∈ C u then for any sequence r n ↓ 0 and ε > 0, there exists τ ∈ (1 − ε, 1) such that u τr n → u 0 and γ [u τr n ] → γ [u 0 ] strictly as n → ∞.

Perspective Constructions and Integral Representations
Given an integrand f ∈ R( × R m ), define the corresponding perspective integrand, P f : × R m × R m×d × R → R by Similarly, the perspective measure Pγ ∈ M( × R m ; R m×d × R) of a lifting γ ∈ L( × R m ) is defined by .
Note that P f is positively one-homogeneous in the (A, t) as j → ∞. Note that this holds for (γ [u j ]) j and γ [u] whenever u j → u areastrictly in BV # ( ; R m ).
Recalling the notation introduced in (6) in Section 2, we see that these perspective constructions permit the following computation: we therefore recover For a general element u ∈ BV( ; R m ), we can still write Lemma 3.18 below and Proposition 4.18 in the next section imply that, whenever (u j ) j is a sequence converging weakly* to u in BV( ; R m ), we can expect that for a large class of integrands f , and so for these we can gain a complete understanding of the functional F on all of BV( ; R m ) by considering F L on L( ; R m ).
is clear from the continuity of f , whereas continuity at points (x, y, A, 0) for (x, y, A) ∈ × σ R m × R m×d follows directly from the definition of P f combined with the fact that σ f ∞ exists in the stronger sense that the limit (13) always returns σ f ∞ as noted in Section 2. Consequently, P f must be uniformly continuous on the compact set × σ R m × ∂B m×d+1 and so we can estimate where m : [0, ∞) → [0, ∞) is a modulus of continuity for P f so that for all (x 1 , y 1 , A 1 , t 1 ), (x 2 , y 2 , A 2 , t 2 ) ∈ × σ R m × ∂B m×d+1 and m(r ) → 0 as r → 0.

Young Measures for Liftings
This section considers Young measures associated to liftings and presents the technical machinery required to manipulate them. These Young measures are compactness tools associated to weakly* convergent sequences (γ j ) j ⊂ L( × R m ) whose purpose is to allow us to compute the limiting behaviour of sequences ( P f (x, y, Pγ j )) j for as large a class of integrands f as possible simultaneously. Crucially, every bounded sequence (γ j ) j ⊂ L( × R m ) can (upon passing to a non-relabelled subsequence) be assumed to generate a Young measure ν with the key property that where the duality product f, ν is defined below in Section 4.1. Together with the localisation principles developed in Section 5, the theory developed in this section will lead to a proof of the lower semicontinuity component of Theorem A. The plan for this section is as follows: first, in Section 4.1, Young measures over × R m with target space R m×d are defined abstractly. The elementary Young measures associated to elements of L( × R m ) are then introduced, as are the concepts of Young measure convergence, generation, and the duality product q , q . Section 4.2 is concerned with the proof of the Young measure Compactness Theorem (Theorem 4.13), which shows that any bounded sequence (γ j ) j ⊂ L( × R m ) possesses a subsequence generating a Young measure ν. Section 4.3 then introduces several techniques for manipulating Young measures and, finally, Section 4.4 proves that Young measures can be also used to gain insight into lim j P f (x, y, Pγ j ) for integrands f from the larger class R( × R m ).

Young Measures
In what follows, we will make use of the compactified space σ R m := R m ∞∂B m and the associated spaces of continuous functions and measures introduced and discussed in Section 2.

Definition 4.1. (Generalised Young measures)
An ( × R m ; R m×d )-Young measure is a quadruple ν = (ι ν , ν, λ ν , ν ∞ ) such that: are parametrised measures which are weakly*-measurable with respect to ι ν and λ ν respectively. That is, where M d := inf{|Du|( ) : u ∈ BV ( ; R m ) and u L d/(d−1) = 1}. We denote the set of all ( × R m ; R m×d )-Young measures by Y( × R m ; R m×d ). When there is no risk of confusion, we will usually abbreviate this to Y.
We shall refer to ι ν and λ ν as the reference and concentration measures of ν, respectively, and ν, ν ∞ as the regular and singular oscillation measures of ν.
Recall from Definition 2.10 that E( × R m ) is defined to be the set of functions Under the duality product for f ∈ E( × R m ), Young measures can be seen as continuous linear functionals on E( × R m ). Whilst conditions (1), (2) and (3) # D c [γ ] + γ gs is the Lebesgue singular part of γ .
If γ ∈ L( × R m ), it follows that we have the representation is an elementary lifting and δ[γ [u]] is the elementary Young measure associated to γ [u], then The set of Young measures Y( × R m ; R m×d ) is not a linear space and so q Y cannot be a norm in the technical sense of being positively homogeneous and subadditive. Nevertheless, q Y -bounded sequences in Y( × R m ; R m×d ) share the same compactness properties as norm-bounded sequences in Banach spaces.
We note that the Poincaré condition (28) can now be rephrased as Since the convergence Y → preserves more information than weak* convergence in the space of liftings, we can show that ALY( ×R m ) is sequentially closed under weak* convergence in Y( × R m ; R m×d ), even though it is not known whether AL( × R m ) is sequentially weakly* closed in M( × R m ; R m×d ): is a mass-bounded sequence such that ν j * ν in Y, we can assume that each ν j is the limit of a sequence (δ k j ) k∈N of elementary Young measures (associated either to liftings in the case LY( ×R m ) or elementary liftings in the case ALY( ×R m )) satisfying sup k δ k We can then use a diagonal argument to extract a sequence (δ k j j ) j of elementary Young measures which generates ν.
it follows that γ is determined solely by ν independently of the choice of generating sequence (δ j ) j and that the following definition is coherent: For elements ν ∈ LY( × R m ), if [ν] = γ and [γ ] = u, we will use the notation ν := u.
It follows immediately that The following corollary is now a direct consequence of Corollary 3.4 combined with the identity ϕ ⊗ 1, ν = ϕ(x, y) dι ν (x, y) for ϕ ∈ C 0 ( × R m ).

Duality and Compactness
and note that S admits an inverse S −1 : It follows that S is a linear isomorphism between E( × R m ) and C( × σ R m × B m×d ), and so E( × R m ) can be given the structure of a Banach space by setting We therefore have that (S * ) −1 (hereafter denoted by S − * ) is itself an isometric isomorphism between (E( × R m )) * and M( × σ R m × B m×d ) (via the Riesz Representation Theorem). In particular, Note that if ν ∈ Y( × R m ; R m×d ), then for all ν ∈ Y.
With S and S − * so defined, the duality product between Young measures ν ∈ Y( × R m×d ; R m×d ) and integrands f ∈ E( × R m ) can be understood in terms of a more familiar integral product between measures and continuous functions: So defined, S f is always Carathéodory on × R m × B m×d and is continuous at each point (x, y, B) ∈ ×R m ×∂B m×d . Moreover, if f is continuous and satisfies (where M d is as given in Definition 4.1), and for which there Proof. First, we will show that for any ν ∈ Y( × R m ; R m×d ), S − * ν satisfies the given conditions. Since S f 0 if and only if f 0, and f, ν Every ν x,y satisfies ν x,y (R m×d ) = 1 and so it follows that Since π # ι ν = L d , this condition is verified. conditions (33) and (34) follow directly from the definition of ι ν , λ ν , the construction of S, and the Poincaré inequality (28). Now let μ ∈ M + ( × σ R m × B m×d ) satisfy conditions (33), (34), and (35). It remains to show that S * μ ∈ Y( × R m ; R m×d ). By the Disintegration of Measures Theorem, we can write Let ω = pκ + ω s be the Radon-Nikodym decomposition of ω with respect to κ. (36) is true for arbitrary ϕ ∈ C 0 ( × R m ), this implies that in M( ×R m ). By construction, however, ω s and κ charge disjoint sets. This implies that both sides of (37) must be the zero measure. It follows that 1 − |B|, η x,y = 0 for ω s -almost every (x, y) ∈ × σ R m , and hence that, for ω s -almost every (x, y) ∈ × σ R m , For f ∈ E( × R m ), we may therefore write The above decomposition combined with the fact that η x,y (∂B m×d ) = 1 ω s -almost everywhere suggests that we construct the relevant Young measure by defining (ι ν , ν, λ ν , ν ∞ ) as follows: It is clear that ν x,y , ν ∞ x,y and ι ν , λ ν inherit positivity from κ, η x,y and ω. From (37), we see that p(x, y) (1 − |B|) , η x,y = 1 for κ-almost everywhere (x, y) ∈ × σ R m , and so, since by definition 1, ν x,y = p(x) (1 − |B|) , η x,y , we have that ν x,y ∈ M 1 (R m×d ). Since it is defined as an average, ν ∞ x,y is also a probability measure for λ ν -almost every (x, y) ∈ × σ R m . That (x, y) → | q |, ν x,y ∈ L 1 ( × R m ; ι ν ) follows from the definition of S applied to the integrand f (x, y, A) = 1 + |A|. The desired weak* measurability properties for ν and ν ∞ follow from the definition of p and the fact that η is weakly* ω-measurable. Finally, the Poincaré inequality (28) follows directly from condition (34).
Proof. It suffices to show that conditions (33), and (34), and (35) from Proposition 4.10 are sequentially weakly* closed. This is immediate for condition (35), since the functions ψ(x, y, (σ R m × ∂B m×d )]). This fails to be the case only if there exists ε > 0, δ ∈ [0, 1), and a sequence of radii R j ↑ ∞ such that lim sup The Poincaré inequality (34) for ν j implies We would therefore obtain contradicting the fact that (S − * ν j ) j must be a norm-bounded sequence in M( × σ R m × B m×d ) (by the Uniform Boundedness Theorem). Finally, since the function is lower semicontinuous, it follows (see for instance Proposition 1.62 in [7]) that and that condition (34) is therefore satisfied.
As a consequence of Proposition 4.10 combined with the usual density results for tensor products of continuous functions over a product domain, the following lemma is now immediate:

Lemma 4.12. There exists a countable family of products {ϕ
then ν j * ν.
then there exists a Young measure ν ∈ Y( × R m ; R m×d ) and a subsequence (ν j k ) k ⊂ (ν j ) j such that ν j k * ν as k → ∞.
Proof. Note that by (31) for ν ∈ Y( × R m ; R m×d ) it holds that The result now follows from the sequential weak* closure of combined with the Banach-Alaoglu Theorem.
Then, upon passing to a (non-relabelled) subsequence, there exists a Young measure Proof. Observe that, if δ is the elementary Young measure associated to a lifting γ ∈ L( × R m ), where γ s is the Lebesgue singular part of γ introduced in Definition 4.2. It follows that the sequence of elementary Young measures (δ[γ j ]) j is bounded in Y( × R m ; R m×d ) and so we can combine Theorem 4.13 with Lemma 4.7 to achieve the desired result.

Manipulating Young Measures
The following theorem will be of great importance in the computation of tangent Young measures in Section 5.
Note that the definition of ν ( × R m ) is equivalent to the statement that for all f ∈ E( × R m ), y dλ ν (x, y). for y ∈ B(0, R), ∇η R ∞ is bounded independently of R, and |∇η R (y)| → 0 as |y| → ∞. It suffices, for example, to take For each j, R ∈ N, define and denote by T R j : × R m → × R m the map given by In particular, for every ϕ ∈ C 0 ( × B(0, R)), By the chain rule combined with the fact that ∇ y T R j = ∇η R , we have We therefore see that and hence that For f ∈ E( × R m ), we can therefore compute Since T R 0 can be extended continuously to × σ R m by setting T R 0 (x, ∞e) := (x, 2Re) for (x, ∞e) ∈ × ∞∂B m and ∇η R can be continuously extended to σ R m by setting ∇η R (∞e) = 0 for all ∞e ∈ ∞∂B m , the fact that and c R j → c R 0 for each R > 0 as j → ∞, we can use the fact that γ j Y → ν in combination with Lemma 3.18 and our previous calculation to deduce that Since this limit holds for all f ∈ E( × R m ) it follows from Definition 4.3 and Theorem 4.13 that each sequence (γ R j ) j ⊂ L( × R m ) generates a Young measure It follows then that the family (ν R ) R>0 ⊂ LY( × R m ) converges as R → ∞ to a Young measure ν 0 ∈ LY( × R m ) satisfying It follows that which, upon making use of the density property of H, implies that γ [u j ] generates ν, as required.
where c k j is the constant which ensures that u k j (x) dx = 0. For sufficiently large j we have that for every j, k ∈ N. Taking the limit as j → ∞, we obtain Since π # λ ν (∂ ) = 0, the right hand side of this expression must converge to 0 as k → ∞ and we can therefore use a diagonal argument to find a sequence ( j k ) k such that (γ [u k j k ]) k generates ν and c k j k → 0. Since u k j k | ∂ = ν | ∂ + c k j k , this suffices to prove the lemma.

Extended Representation
All of the results obtained so far in this section have helped us to understand lim j→∞ P f (x, y, Pγ j ) when f ∈ E( ×R m ). However, the integrands f which are of interest for the applications that we have in mind are not members of E( × R m ), only of R( × R m ). In particular, the requirement that both f and f ∞ extend continuously to × σ R m × R m×d is too strong to be satisfied by any integrand which is unbounded in the middle variable.
If ν ∈ LY( × R m ) satisfies λ ν ( × ∞∂B m ) = 0 and we assume just that f ∈ R w * ( × R m ), then the duality product is still well-defined in R. For the singular part this follows from the fact that λ ν is a positive measure and that f ∞ is positive and continuous. For the regular part this follows from ι ν = gr ν # (L d ) and ν ∈ BV( ; R m ). Indeed, the fact that f is Carathéodory and satisfies | f (x, y, A)| C(1 + |y| d/(d−1) + |A|), the ι ν -weak* measurability of the parametrised measure ν, and the Poincaré Condition (28) together imply that the map (x, y) → f (x, y, q ), ν x,y is ι ν -measurable and ι νintegrable. Similarly is also well-defined even if λ ν ( × ∞∂B m ) > 0. Proposition 4.18 and Corollary 4.19 below show that knowledge of ν allows us to say something meaningful about lim j P f (x, y, Pγ j ) for a very large class of integrands f and, at the same time, that this limiting behaviour does not change if f is replaced by a perturbation f ( q , q + c j , q ) where (c j ) j ⊂ R m is a convergent sequence. As a consequence, the limiting behaviour of (F[u j ]) j for any f and any weakly* convergent sequence in BV( ; R m ) can be understood in terms of the Young measures generated by sequences (γ [u j − (u j )]) j ⊂ AL( × R m ).
(ii) If instead λ ν ( × ∞∂B m ) 0 and we assume in addition that σ f ∞ exists according to Definition 2.11, we also have that y). (40) Proof. We shall prove (39) and (40) simultaneously: Step 1: Assume first that f : × σ R m × R m×d → R is Carathéodory and satisfies | f (x, y, A)| C for all (x, y, A) ∈ × R m × R m×d and some fixed C > 0. By the Scorza-Dragoni Theorem, there exists a compact set K ε such that L d ( \K ε ) ε and f (K ε × σ R m × R m×d ) is continuous. By the Tietze Extension Theorem, we can find a continuous function g ∈ C( × σ R m × R m×d ) which restricts to f on K ε × σ R m × R m×d and such that g is bounded and that By the construction of g, however, we see that and that the same estimate holds for ×R m P(g − f )(x, y, Pγ ) . Letting ε → 0, we obtain and so both (39) and (40) hold.
Step 2: Now assume that f : × R m × R m×d → R is Carathéodory and satisfies | f (x, y, A)| C for all (x, y, A) ∈ × R m × R m×d and some fixed C > 0. For K > 0, let ϕ K ∈ C c (R m ; [0, 1]) be such that ϕ K (y) = 1 for |y| K . Define and note that each f K satisfies the hypotheses of Step 1. Since f K → f pointwise in × R m × R m×d as K → ∞ and σ f ∞ ≡ 0, the Dominated Convergence Theorem lets us deduce and, since Step 1 implies Now fix ε > 0 and let R > 0 be large enough that C/(1 + R) < ε. Recalling the map introduced in Section 4.2 together with S − * := (S * ) −1 , the inverse of its adjoint, we see that Proposition 4.10 implies S − * ν( × ∞∂B m × B m×d ) = 0 and so, by the outer regularity of Radon measures, there exists K > 0 such that , we must therefore have that for all j ∈ N sufficiently large. Since our choice of R implies C/(1 + R) < ε, we have that, whenever (x, y, B) ∈ × R m × B m×d and |B| R/(1 + R) so that 1 − |B| 1/(1 + R), Abbreviating δ j := δ[γ j ], we can now use the fact that (1 − ϕ K (y + c j )) = 0 when |y| K − 1 and |c j | < 1, to estimate Since sup j 1 ⊗ | q |, δ j < ∞ and ε > 0 was arbitrary, we therefore deduce and hence that as required.
Step 3: Now assume that f : × R m × R m×d → R is Carathéodory and satisfies | f (x, y, A)| C(1 + |A|) for some C > 0 and is such that (x, y, A) and note that f R satisfies the hypotheses of Step 2. Moreover, since f R → f pointwise in × R m × R m×d as R → ∞, the Dominated Convergence Theorem implies that Splitting and using the fact that Step 2 implies First we assume that λ ν ( × ∞∂B m ) = 0 and prove (39). Since the condition λ ν ( × ∞∂B m ) = 0 is equivalent to S − * ν( × ∞∂B m × ∂B m×d ) = 0 and Proposition 4.10 also forces S − * ν( × ∞∂B m × B m×d ) = 0 we have that For ε > 0 fixed, let K verify (41). By Lemma 2.12, we have that, for all R > 0 sufficiently large, | f (x, y, A)| ε(1 + |A|) whenever (x, y) ∈ × B(0, K + 1) and |A| R. Since ( f − f R )(x, y, A) = 0 whenever |A| < R, and |y| K implies |y + c j | K + 1 once |c j | 1, we therefore have that, once R > 0 is sufficiently large, Since (γ j ) j is a norm-bounded sequence in M( × R m ; R m×d ) and ε > 0 was arbitrary, we therefore deduce that and hence that (39) holds. Next, we prove that (40) holds under the assumption that σ f ∞ ≡ 0. By Proposition 4.10, we have that This is equivalent to the statement Now, if there were ε > 0 such that, for all K ∈ N, we would have that , y, B).
contradicting (42). Taking the converse of (43), we therefore see that for any ε > 0 it holds that, for all K ∈ N sufficiently large, Fix ε > 0 and let K ∈ N verify (44). By Lemma 2.12 we have that, once R > 0 is sufficiently large, |A| R implies that | f (x, y, A)| ε(1 + |A|) for all x ∈ and y ∈ R m such that |y| d/(d−1) Defining A K ⊂ R m × B m×d by and that Since S( f − f R )(x, y, B) = 0 whenever |B| R/(1 + R), we can now estimate, for R > 0 sufficiently large, , y, B) Since ε > 0 was arbitrary, we therefore deduce that as required.
Step 4: Assume now that f : ×R m ×R m×d → R is Carathéodory with f ∞ ≡ 0 as in the previous step, but that f now satisfies the bound | f (x, y, A)| C(1+|y| p + |A|) for some p ∈ [1, d/(d − 1)). For k ∈ N, define f k : × R m × R m×d → R by Each f k satisfies a bound of the form | f k (x, y, A)| It follows that f k falls under the hypotheses of Step 2. From the definition of f k , we can estimate from which it follows that and that an analogous estimate also holds for ν. Since p < d/(d − 1) and c j → 0, the sequence ([γ j ] + c j ) j ⊂ BV( ; R m ) is p-uniformly integrable from which it follows that this estimate is uniform in j. We can therefore let k → ∞ to deduce lim j→∞ ×R m P f (x, y + c j , Pγ j ) = f, ν .
To obtain (40), note that since σ f ∞ ∈ C( × σ R m × R m×d ) and is positively one-homogeneous in the final variable, it follows that f ∞ ∈ E( × R m ). Lemma 3.18 then implies that as required.
Step 6: By Step 4 applied to the integrand f − f ∞ , we have that By Step 5 applied to f ∞ , we also have and so the result follows from adding (45) and (46).
Proof. First assume that f 0. For R > 0 let ϕ R ∈ C c (R m ; [0, 1]) be such that ϕ R (y) = 1 if |y| R and define f R : In addition, σ f ∞ R exists for every R > 0 and satisfies The positivity of f and the Monotone Convergence Theorem then imply that If we only assume that f ∈ R w * ( × R m ) then, letting h ∈ R L ( × R m ) with σ h ∞ = 0 be such that as required, where we used the fact that σ h ∞ = 0 implies h, ν = h, ν ( × R m ) to justify the penultimate equality.

Tangent Young Measures and Jensen Inequalities
Letting λ gs ν := λ ν − dλ ν d|γ [u]| |γ [u]| ( \J u × R m ) denote the graph-singular part of λ ν and using Corollary 4.9 together with the fact that f ∞ 0, we can write Using the equality d|γ [u]| dι ν (x, u(x)) = d|Du| dL d (x) = |∇u(x)|, disintegrating λ gs ν = λ ν = π # λ ν ⊗ ρ on J u × R m and using again the positivity of f ∞ to neglect the (H d−1 J u ) ⊗ ρ-singular part of λ gs , we can then deduce The lower semicontinuity component of Theorem A is now reduced to the task of obtaining the three pointwise inequalities for |D c u|-almost every x ∈ , and for H d−1 -almost every x ∈ J u . In Section 5.1, we will show that the left hand side of each of these inequalities can be computed as a duality product h, σ with σ ∈ ALY( × R m ). In the case of the first two inequalities, h depends only on the A variable, and only on the y and A variables in the third case. From there, the theory in Section 5.2 uses the definitions of quasiconvexity and K f [u] to obtain the desired inequalities.

Tangent Young Measures
Let ν ∈ LY( × R m ) with ν = u ∈ BV # ( ; R m ) and, for x 0 ∈ D u ∪ C u ∪ J u , let u r be as defined in Theorem 2.4. Let (γ j ) j ⊂ L( × R m ) be a sequence which generates ν. Recalling Thanks to the positive one-homogeneity of P f in (A, t) in conjunction with (7) from Section 2, these observations imply that, for f ∈ E( × R m ), We will show that, for L d +|Du|-almost every x 0 ∈ D u ∪C u ∪J u , the family (σ r ) r >0 (or at least a subsequence) converges weakly* to a limit as r → 0. The primary tools in identifying this limit for x 0 ∈ D u ∪ C u are Theorem 3.17 which guarantees the strict convergence of the rescaled liftings (γ [u r ]) r >0 and the graphical Besicovitch Derivation Theorem, Theorem 5.1 introduced below. We note that the usual version of the Besicovitch Derivation Theorem and its generalisation, the Morse Derivation Theorem (see [36] and also Theorems 2.22 and 5.52 in [7]) cannot be used in this situation, since the aspect ratio (c r r −1 )/r −1 = c r corresponding to the scaling present in (50) (which, due to the need to apply Theorem 3.17, cannot be modified) will converge to 0 for x ∈ C u . The following theorem demonstrates that, provided the denominator is a graphical measure, derivatives of measures can be computed using families of sets which are very different to the usual decreasing cylinders B(x, r ) × B(y, r ), so long as the family is sufficiently well behaved according to the differentiating measure: → R m be μ-measurable so that η := gr u # μ ∈ M + ( × R m ). For each x ∈ and r ∈ (0, 1), let c x r > satisfy lim r ↓0 c x r = 0 and y x r ∈ R m satisfy relaxation F * * attains a global minimum at 0, yet which does not possess any weakly* convergent minimising sequences. In the full vector-valued, u-dependent case we can thus only expect to find recovery sequences which converge in the strong L 1 ( ; R m )-topology. We must therefore find a way of constructing approximate recovery sequences (u j ) j ⊂ C ∞ ( ; R m ) such that u j * u in BV( ; R m ) and lim j |F[u j ] − F * * [u]| ε for each ε > 0.
By the Sobolev Embedding Theorem in one dimension it holds that sup j v j ∞ < ∞, which implies that, for some δ > 0, Define the sequence (w j ) j ⊂ W 1,1 ((−1, 1)) by w j (z) = (v j (z)) 2 so that w j * w := 1 [0,1) in BV((−1, 1)). By Lemma 3.18, we have that Since We have therefore shown that lim inf (v j (x))|∇v j (x)| dx > 0 and so we can deduce that, for all r > 0 sufficiently small, (B(x, r )) On the other hand, Lemma 6.2 implies that, for all r sufficiently small, where 1 * := d/(d − 1). It therefore follows that the collection is a fine cover for L ε 0 and so, by the Vitali-Besicovitch Covering Theorem, there exists a countable disjoint set H ε ⊂ G ε whose union covers K f [u]-almost all of L ε 0 . Let B(x 1 , r 1 ), B(x 2 , r 2 ) . . . be a sequence of elements from H ε such that there exists an increasing sequence N 1 , N 2 . . . in N with Let τ ∈ (0, 1) and η τ ∈ C ∞ c (B d ; [0, 1]) be such that η τ ≡ 1 on τ B d . Fix j. For i = 1 . . . N j , let v i ∈ A u (x i ) be such that v i L 1 * 2 u ± x i L 1 * and We can now set v ε,τ j (x) :=