A global method for relaxation in W^{1,p} and in SBV_{p}

Guy Bouchitté^{∗}  Irene Fonseca ^{†}  Giovanni Leoni ^{‡}  Luísa Mascarenhas ^{§}

Abstract

An integral representation formula for a class of functionals defined on W^{1,p} and in SBV_{p} is obtained without requiring the regularity conditions usually imposed in the literature. The approach is based on the general results of [17] and on a Poincaré–Wirtinger’s type inequality introduced by De Giorgi, Carriero and Leaci [34]. Applications to relaxation problems and dimension reduction problems in brittle thin films are presented.

Key words. Relaxation, lower semicontinuity, special functions of bounded variation, brittle thin films

AMS subject classifications. 49J45, 49Q20

Contents

1 Introduction  1
2 Part I: the general method
2.1 Preliminaries  4
2.2 The general method  11
2.3 Proof of Theorem 1.0.1: Lebesgue points  13
2.4 Proof of Theorem 1.0.1: jump points  16
2.5 Proofs of Theorems 1.0.2 and 1.0.3  20
3 Part II: applications
3.1 Preliminary results  22
3.2 Relaxed energy: proof of Theorem 1.0.4  29
3.3 Thin films  31
3.5 Appendix  37
References  45

1 Introduction

In a recent paper Bouchitté, Fonseca and Mascarenhas [17] introduced a general method for the identification of the integral representation of a certain class of functionals defined on the space of functions of bounded variation and with linear growth with respect to the total variation of the derivative. This method has important applications, including the integral representation of the relaxed energy in BV (Ω; ℝ^{d}) corresponding to a functional defined in W^{1,1}(Ω; ℝ^{d}) with a discontinuous integrand and growing linearly at infinity, the homogenization of energy functionals involving bulk and surface terms, the integral representation of relaxed energies when no apriori bounds are available, and the characterization of effective energies under Dirichlet boundary conditions (see [18]; see also [16]).

The purpose of this paper is to extend the work in [17] to functionals defined on SBV_{p}(Ω; ℝ^{d}), p > 1, and with superlinear growth. More precisely, we consider a functional

F : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞]

satisfying for every (u, A) ∈ SBV_{p}(Ω; ℝ^{d}) × A(Ω)

∗Département de Mathématiques, Université de Toulon et du Var-BP 132, La Garde Cedex, France 83957 †Department of Mathematical Sciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.

‡Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy §C.M.A.F., Universidade de Lisboa, Av. Prof. gama Pinto 2, Lisboa Codex, Portugal 1699

(H_{1}) F(u; ·) is the restriction to A(Ω) of a Radon measure; (H_{2}) F(u; A) = F(v; A) whenever u = v L^{N} a.e. on A ∈ A(Ω); (H_{3}) F(·; A) is L^{1}(Ω; ℝ^{d})- lower semicontinuous; (H_{4}) there exists C > 0 such that

\begin{aligned}\frac{1}{C} (∫_{A} |∇u|^{p}dx + ∫_{S(u)∩A}(1 + |u^{+} − u^{−}|)dH^{N−1}) & ≤ F(u; A) \\ & ≤ C(∫_{A}(1 + |∇u|^{p})dx + ∫_{S(u)∩A}(1 + |u^{+} − u^{−}|)dH^{N−1}). \end{aligned}

Here Ω is an open, bounded subset of ℝ^{N}, A(Ω) is the class of all open subsets of Ω, and SBV_{p}(Ω; ℝ^{d}) is the space of functions u ∈ SBV (Ω; ℝ^{d}) such that ∇u ∈ L^{p}(Ω; ℝ^{d×N}) and H^{N−1}(S(u) ∩ Ω) < ∞.

For every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A_{∞}(Ω), where A_{∞}(Ω) is the class of Lipschitz subdomains of Ω, define

m(u; A) := inf{F(w; A) : w ∈ SBV_{p}(A; ℝ^{d}) such that w = u in a neighborhood of ∂A}.  (1.0.1)

The main result of the paper is the following:

Theorem 1.0.1. Under hypotheses (H_{1}) − (H_{4}), for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) we have

F(u; A) = ∫_{A} f(x, u, ∇u) dx + ∫_{S(u)∩A} g(x, u^{+}, u^{−}, ν_{u}) dH^{N−1},

where

f(x_{0}, u_{0}, ξ) := lim sup_{ε→0^{+}} \frac{m(u_{0} + ξ(· − x_{0}); Q_{ν}(x_{0}, ε))}{ε^{N}} ,  (1.0.2)
g(x_{0}, a, b, ν) := lim sup_{ε→0^{+}} \frac{m(u_{x_{0},a,b,ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} ,  (1.0.3)

for all x_{0} ∈ Ω, u_{0}, a, b ∈ ℝ^{d}, ξ ∈ ℝ^{d×N}, ν ∈ S^{d−1}, and where

u_{x_{0},a,b,ν}(x) := \begin{cases}a & if (x − x_{0}) · ν > 0, \\ b & if (x − x_{0}) · ν ≤ 0. \end{cases}

Theorem 1.0.1 was proved by Braides and Chiadó Piat [20] under the additional Lipschitz-type assumptions that there exists a sequence {ω_{k}} of integrable moduli of continuity such that

|F(u + s; A) − F(u; A)| ≤ ∫_{A} ω_{k}(x, |s|) dx  (1.0.4)

for every k ∈ ℕ, A ∈ A(Ω), s ∈ ℝ^{d}, and for all functions u of class C^{1} such that ||u||_{∞} ≤ k, ||u + s||_{∞} ≤ k, ||∇u||_{∞} ≤ k; and that there exists a modulus of continuity ω such that

|F(u; S) − F(v; S)| ≤ ∫_{S} ω(|u^{+} − v^{+}| + |u^{−} − v^{−}|) dH^{N−1}  (1.0.5)

for every u, v ∈ SBV_{p}(Ω; ℝ^{d}) and S ⊂ S(u) ∩ S(v).

Condition (1.0.4) was introduced by Buttazzo and Dal Maso [26] in the Sobolev setting (see Theorem 1.0.2 below) and is equivalent to requiring f to be Carathéodory, while condition (1.0.5) implies, in particular, that g(x, ·, ·, ν) is continuous, uniformly with respect to x and ν. Our approach, which is based on the general results of [17] and on a Poincaré–Wirtinger’s type inequality introduced by De Giorgi, Carriero and Leaci [34], does not require the extra assumptions (1.0.4) and (1.0.5).

Although hypothesis (1.0.4) would slightly simplify the proofs of Lemmas 2.3.1 and 2.3.2 below, we prefer to give Theorem 1.0.1 in its full generality, since it is known that the continuity of f in the u variable is by no means a necessary condition for lower semicontinuity and relaxation results, and we refer the reader to [37] (and the references contained therein) where f(x, ·, ξ) is assumed to be lower semicontinuous, or, in some cases, only measurable (see also Theorem 3.1.2 below). Moreover, conditions of the type (1.0.4) and (1.0.5) are, in general, not easy to verify in the applications.

We remark that Theorem 1.0.1 entails new results even in the Sobolev setting. Indeed, Theorem 1.0.2. Let F_{1} : W^{1,p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞] be a functional satisfying hypotheses (H_{1}) − (H_{3}) and the growth condition

\frac{1}{C} ∫_{A} |∇u|^{p}dx ≤ F_{1}(u; A) ≤ C ∫_{A}(1 + |∇u|^{p})dx,  (1.0.6)

for some C > 0. Then, for every u ∈ W^{1,p}(Ω; ℝ^{d}) and A ∈ A(Ω) we have

F_{1}(u; A) = ∫_{A} f_{1}(x, u, ∇u) dx, where f_{1}(x_{0}, u_{0}, ξ) := lim sup_{ε→0^{+}} \frac{m_{1}(u_{0} + ξ(· − x_{0}); Q_{ν}(x_{0}, ε))}{ε^{N}}

for all x_{0} ∈ Ω, u_{0} ∈ ℝ^{d}, ξ ∈ ℝ^{d×N}, and where, for (v, A) ∈ W^{1,p}(Ω; ℝ^{d}) × A_{∞}(Ω),

m_{1}(v; A) := inf{F_{1}(w; A) : w ∈ W^{1,p}(A; ℝ^{d}) with w = v in a neighborhood of ∂A}.  (1.0.7)

Theorem 1.0.2 is closely related to the result of Buttazzo and Dal Maso [26] mentioned above (see also [24], [25]), where the integral representation formula is obtained under the additional hypothesis (1.0.4).

Theorem 1.0.1 also allows us to recover the integral representation for functionals defined on partitions of sets of finite perimeter previously obtained by Ambrosio and Braides [8] under an additional continuity hypothesis. A similar result was proved by Braides and Piat [20] in the case where the ambiance space BV (Ω; T) is replaced by

SBV_{0}(Ω; ℝ^{d}) := {w ∈ SBV (Ω; ℝ^{d}) : ∇w = 0 a.e. in Ω and H^{N−1}(S(w) ∩ Ω) < ∞}.

Theorem 1.0.3. Let T be a finite subset of ℝ^{d} and let F_{2} : BV (Ω; T) × A(Ω) → [0, ∞] be a functional satisfying hypotheses (H_{1}) − (H_{3}) and the growth condition

\frac{1}{C} H^{N−1}(S(u) ∩ A) ≤ F_{2}(u; A) ≤ C H^{N−1}(S(u) ∩ A).

Then for every u ∈ BV (Ω; T) and A ∈ A(Ω) we have

F_{2}(u; A) = ∫_{S(u)∩A} g_{2}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1},

where

g_{2}(x_{0}, a, b, ν) := lim sup_{ε→0^{+}} \frac{m_{2}(u_{x_{0},a,b,ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} ,

for all x_{0} ∈ Ω, a, b ∈ ℝ^{d}, ν ∈ S^{d−1}, and where, for (v, A) ∈ BV (Ω; T) × A_{∞}(Ω),

m_{2}(v; A) := inf{F_{2}(w; A) : w ∈ BV (A; v(A)) with w = v in a neighborhood of ∂A}.  (1.0.8)

An important example of the kind of functionals considered in Theorem 1.0.1 is given by the relaxed energy

H(u; A) := inf_{{u_{n}}}{lim inf_{n→∞} ∫_{A} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩A} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1} :
u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{1}(A; ℝ^{d})},

where both bulk and surface energies are present.

Theorem 1.0.4. Assume that the densities

f_{0} : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞), g_{0} : Ω × (ℝ^{d})^{2} × S^{N−1} → [0, +∞)

are, respectively, a Carathéodory and a continuous function that satisfy

\begin{matrix}\frac{1}{C} |ξ|^{p} ≤ f_{0} & (x, u, ξ) ≤ C(1 + |ξ|^{p}), \\ \frac{1}{C} (1 + |λ − θ|) ≤ g_{0} & (x, λ, θ, ν) ≤ C (1 + |λ − θ|) and g_{0}(x, λ, θ, ν) & = g_{0}(x, λ, θ, −ν) \end{matrix}  (1.0.9)

for all x ∈ Ω, u, λ, θ ∈ ℝ^{d}, ξ ∈ ℝ^{d×N}, ν ∈ S^{N−1}, and for some C > 0. Then for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) we have

H(u; A) = ∫_{A} Qf_{0}(x, u, ∇u) dx + ∫_{S(u)∩A} Rg_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1},

where Qf_{0} and Rg_{0} are the quasiconvex and the BV-elliptic envelopes of f_{0} and g_{0}, respectively.

The quasiconvex envelope Qf of a function f : ℝ^{d×N} → ℝ is defined as

Qf := sup{G : G is quasiconvex, G ≤ f},

and the BV -elliptic envelope Rg of a function g : ℝ^{2d} × S^{N−1} → ℝ is the greatest BV -elliptic function less than g. We recall that a locally bounded, Borel measurable function f : ℝ^{d×N} → ℝ is said to be quasiconvex if

∫_{(0,1)^{N}} f (ξ + ∇φ(y)) dy ≥ f(ξ)

for all ξ ∈ ℝ^{d×N}, φ ∈ W_{0}^{1,∞}((0, 1)^{N}; ℝ^{d}). Similarly, we say that a locally bounded, Borel measurable function g : ℝ^{2d} × S^{N−1} → ℝ is BV-elliptic if for all (a, b, ν) ∈ ℝ^{2d} × S^{d−1} and for any finite subset T of ℝ^{d} we have

∫_{S(w)∩Q_{ν}} g(w^{+}, w^{−}, ν_{w}) dH^{N−1} ≥ g(a, b, ν)

for all w ∈ BV (Q_{ν}; T) such that w = u_{a,b,ν} on ∂Q_{ν}, where u_{a,b,ν} := u_{0,a,b,ν}, i.e.

u_{a,b,ν}(x) := \begin{cases}a & if x · ν > 0, \\ b & if x · ν ≤ 0. \end{cases}

Theorem 1.0.1 has also important applications in homogenization theory and brittle thin films (see Section 3.3).

As a final remark, it is worth noting that the proofs of Lemma 2.3.1 and 2.3.2 could be simplified if given a function u ∈ SBV_{p}(Ω; ℝ^{d}) one could show that for L^{N} a.e. x_{0} ∈ Ω

lim_{ε→0^{+}} \frac{1}{ε^{N+p}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})|^{p} dx = 0.

This result, which would be of independent interest, is true if p ≤ N/(N − 1) or in the Sobolev space W^{1,p}(Ω; ℝ^{d}), but we have been unable to prove it (or disprove it) in the general case where u ∈ SBV_{p}(Ω; ℝ^{d}).

Similarly, the proofs of Lemma 2.4.1 and 2.4.2 could be significantly shortened if we knew that for H^{N−1} a.e x_{0} ∈ S(u)

lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u_{x_{0},ν}(x)|^{p} dx = 0,

where u_{x_{0},ν} := u_{x_{0},u^{+}(x_{0}),u^{−}(x_{0}),ν(x_{0})}.

To overcome these difficulties we will make use of the approximation procedure described in Lemma 2.1.7 and Lemma 2.1.8, respectively.

2 Part I: The General Method

2.1 Preliminaries.

We start with some notation. Here Ω ⊂ ℝ^{N} is an open, bounded subset, L^{N} and H^{N−1} are, respectively, the N dimensional Lebesgue measure and the N − 1 dimensional Hausdorff measure in ℝ^{N}. For any ν ∈ S^{N−1} := {x ∈ ℝ^{N} : |x| = 1} let {ν_{1}, ⋯, ν_{N−1}, ν} be an orthonormal basis of ℝ^{N} varying continuously with ν, and let Q_{ν} := {x ∈ ℝ^{N} : |x · ν_{i}| < 1/2, |x · ν| < 1/2, i = 1, ⋯, N − 1} be a unit cube centered at the origin with two of its faces orthogonal to the direction ν. We set Q_{ν}(x_{0}, ε) := x_{0} + εQ_{ν}.

We recall briefly some facts about functions of bounded variation which will be useful in the sequel. A function u ∈ L^{1}(Ω; ℝ^{d}) is said to be of bounded variation if for all i = 1, ⋯ d, and j = 1, ⋯ N, there exists a Radon measure μ_{ij} such that

∫_{Ω} u_{i}(x) \frac{∂φ}{∂x_{j}} (x) dx = − ∫_{Ω} φ(x) dμ_{ij}

for every φ ∈ C_{0}^{1}(Ω; ℝ). The distributional derivative Du is the matrix–valued measure with components μ_{ij}. Given u ∈ BV (Ω; ℝ^{d}) the approximate upper and lower limit of each component u_{i}, i = 1, ⋯ d, are given by

u_{i}^{+}(x) := inf \{t ∈ ℝ : lim_{ε→0^{+}} \frac{1}{ε^{N}} L^{N} (\{y ∈ Ω ∩ Q(x, ε) : u_{i}(y) > t\}) = 0\}

and

u_{i}^{−}(x) := sup \{t ∈ ℝ : lim_{ε→0^{+}} \frac{1}{ε^{N}} L^{N} (\{y ∈ Ω ∩ Q(x, ε) : u_{i}(y) < t\}) = 0\},

while the jump set of u, or singular set, is defined by

S(u) := ⋃_{i=1}^{d}{x ∈ Ω : u_{i}^{−}(x) < u_{i}^{+}(x)}.

It is well known that S(u) is N − 1 rectifiable, i.e.

S(u) = ⋃_{n=1}^{∞} K_{n} ∪ E,

where H^{N−1}(E) = 0 and K_{n} is a compact subset of a C^{1} hypersurface. If x ∈ Ω\S(u) then u(x) is taken to be the common value of (u_{1}^{+}(x), ⋯, u_{d}^{+}(x)) and (u_{1}^{−}(x), ⋯, u_{d}^{−}(x)). It can be shown that u(x) ∈ ℝ^{d} for H^{N−1} a.e. x ∈ Ω\S(u) and that H^{N−1} (S(u)△L(u)) = 0, where L(u) stands for the set of Lebesgue points of u. Furthermore, for H^{N−1} a.e. x ∈ S(u) there exist a unit vector ν_{u}(x) ∈ S^{N−1}, normal to S(u) at x, and two vectors u^{−}(x), u^{+}(x) ∈ ℝ^{d} (the traces of u on S(u) at the point x) such that

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{\{y∈Q(x_{0},ε): (y−x)·ν_{u}(x)>0\}} |u(y) − u^{+}(x)|^{N/(N−1)}dy = 0

and

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{\{y∈Q(x_{0},ε): (y−x)·ν_{u}(x)<0\}} |u(y) − u^{−}(x)|^{N/(N−1)}dy = 0.

Note that, in general, (u_{i})^{+} ≠ (u^{+})_{i} and (u_{i})^{−} ≠ (u^{−})_{i}. We denote the jump of u across S(u) by [u] := u^{+} − u^{−}. The distributional derivative Du may be decomposed as

Du = ∇u L^{N} + (u^{+} − u^{−}) ⊗ ν_{u} H^{N−1}⌊ S(u) + C(u),

where ∇u is the density of the absolutely continuous part of Du with respect to the N–dimensional Lebesgue measure L^{N} and C(u) is the Cantor part of Du. These three measures are mutually singular.

As it turns out, the Sobolev inequality

(∫_{Ω} |u(x)|^{N/(N−1)}dx)^{(N−1)/N} ≤ C(N)||u||_{BV}

holds in BV (Ω; ℝ^{d}) when N > 1.

The space SBV (Ω; ℝ^{d}) of special functions of bounded variation, introduced by De Giorgi and Ambrosio [33], is the space of all functions u ∈ BV (Ω; ℝ^{d}) such that

Du = ∇u L^{N} + (u^{+} − u^{−}) ⊗ ν_{u} H^{N−1}⌊ S(u).

The following compactness result for sequences bounded in SBV (Ω; ℝ^{d}) is due to Ambrosio [5].

Theorem 2.1.1. Let Φ : [0, +∞) → [0, +∞), Θ : (0, +∞] → (0, +∞] be two functions, respectively convex and concave, and such that

lim_{t→+∞} \frac{Φ(t)}{t} = +∞, Φ is nondecreasing,
Θ(+∞) = lim_{t→+∞} Θ(t), lim_{t→0^{+}} \frac{Θ(t)}{t} = +∞, Θ is nondecreasing.

Let {u_{n}} be a sequence of functions in SBV (Ω; ℝ^{d}) such that

sup_{n} {∫_{Ω} Φ(|∇u_{n}|) dx + ∫_{S(u_{n})} Θ(|[u_{n}]|) dH^{N−1} + ∫_{Ω} |u_{n}| dx} < +∞.

Then there exists a subsequence {u_{n_{k}}} converging in L^{1}(Ω; ℝ^{d}) to a function u ∈ SBV (Ω; ℝ^{d}), and

∇u_{n_{k}} ⇀ ∇u in L^{1}(Ω; ℝ^{N×d}), [u_{n_{k}}] ⊗ ν_{u_{n_{k}}} H^{N−1}⌊S(u_{n_{k}}) ⇀^{⋆} [u] ⊗ ν_{u}H^{N−1}⌊S(u),

∫_{S(u)∩Ω} Θ(|[u]|) dH^{N−1} ≤ lim inf_{n→+∞} ∫_{S(u_{n})∩Ω} Θ(|[u_{n}]|) dH^{N−1}.

We refer to [19] and to [12] for more details on SBV functions.

For p > 1, we say that a function u ∈ SBV (Ω; ℝ^{d}) belongs to SBV_{p}(Ω; ℝ^{d}) if

∇u ∈ L^{p}(Ω; ℝ^{d×N}) and H^{N−1}(S(u) ∩ Ω) < ∞.

In the remaining of the section we present some theorems due to De Giorgi, Carriero and Leaci [34] (see also [27]) which will be useful in the sequel. We also prove some new related results, Lemma 2.1.7 and 2.1.8, which, we believe, are of independent interest.

Following the notation in [27] and [34], given a = (a_{1}, …, a_{d}), b = (b_{1}, …, b_{d}) ∈ ℝ^{d} we will denote a ∧ b := (min(a_{1}, b_{1}), …, min(a_{d}, b_{d})) and a ∨ b := (max(a_{1}, b_{1}), …, max(a_{d}, b_{d})). Let Q be a cube in ℝ^{N}. For every measurable function u : Q → ℝ^{d}, with u = (u_{1}, …, u_{d}), we set

u_{∗}(s; Q) := ((u_{1})_{∗}(s; Q), …, (u_{d})_{∗}(s; Q)), med(u; Q) := u_{∗}( \frac{1}{2} |Q|; Q),

where, with i = 1, …, d,

(u_{i})_{∗}(s; Q) := inf{t ∈ ℝ : |{u_{i} < t} ∩ Q| ≥ s} for 0 ≤ s ≤ |Q|.

Moreover, for every u ∈ SBV_{p}(B; ℝ^{d}) such that (2γ_{N}H^{N−1}(S(u) ∩ Q))^{N/(N−1)} < \frac{1}{2} |Q| we define

\begin{matrix}τ′(u; Q) := & u_{∗} ((2γ_{N}H^{N−1}(S(u) ∩ Q))^{N/(N−1)}; Q) , \\ τ′′(u; Q) := & u_{∗} (|Q| − (2γ_{N}H^{N−1}(S(u) ∩ Q))^{N/(N−1)}; Q) , \\ ū(x) := & (u(x) ∧ τ′′(u, Q)) ∨ τ′(u, Q), \end{matrix}

where γ_{N} is the isoperimetric constant relative to cubes of ℝ^{N}, that is γ_{N} is the smallest number such that for every Borel set E

min{|E ∩ Q|^{N/(N−1)}, |Q\E|^{N/(N−1)}} ≤ γ_{N}Per_{Q}(E),

where Per_{Q}(E) denotes the perimeter of E in Ω.

Remark 2.1.2. Note that when S(u) = ∅ then τ′(u_{i}; Q) = −∞, τ′′(u_{i}; Q) = +∞ for all i = 1, …, d, and so ū(x) coincides with u(x).

Lemma 2.1.3. ([34], Lemma 2.6) Let p ≥ 1 and let u ∈ SBV (Ω; ℝ) be such that

∫_{K} |∇u(x)|^{p} dx + H^{N−1}(S(u) ∩ K) < ∞

for every compact set K ⊂ Ω. Then

lim_{ε→0^{+}} \frac{∫_{Q(x_{0},ε)} |∇u(x)|^{p} dx + H^{N−1}(S(u) ∩ Q(x_{0}, ε))}{ε^{N−1}} = 0

for H^{N−1} a.e. x_{0} ∈ Ω \ S(u).

Theorem 2.1.4. ([34], Thm. 3.1) Assume that u ∈ SBV (Q; ℝ) and that

(2γ_{N} H^{N−1}(S(u) ∩ Q))^{N/(N−1)} < \frac{|Q|}{2} .  (2.1.1)

If 1 ≤ p < N then

(∫_{Q} |ū − med(u; Q)|^{p^{∗}}dx)^{1/p^{∗}} ≤ \frac{2γ_{N}p(N − 1)}{N − p} (∫_{Q} |∇u|^{p} dx)^{1/p} ,  (2.1.2)
|{ū ≠ u} ∩ Q| ≤ 2 (2γ_{N}H^{N−1}(S(u) ∩ Q))^{N/(N−1)} ,  (2.1.3)

where p^{∗} := pN/(N − p). If p ≥ N then for every q ≥ 1

||ū − med(u; Q)||_{L^{q}(Q;ℝ)} ≤ \frac{2γ_{N}q(N − 1)}{N} |Q|^{\frac{1}{N} + \frac{1}{q} − \frac{1}{p}} ||∇u||_{L^{p}(Q;ℝ)}.  (2.1.4)

Theorem 2.1.5. ([34], Thm. 3.5) Assume that u_{n} ∈ SBV (Q; ℝ), p > 1, and that

sup_{n} ∫_{Q} |∇u_{n}|^{p}dx < ∞,  lim_{n→∞} H^{N−1}(S(u_{n}) ∩ Q) = 0.

Then (up to a subsequence) the sequences {ū_{n} − med(u_{n}; Q)} and {u_{n} − med(u_{n}; Q)} converge to a function u_{0} ∈ W^{1,p}(Q; ℝ) in L^{p} and pointwise almost everywhere, respectively.

Theorem 2.1.6. ([34], Thm. 3.6) Let u ∈ SBV_{loc}(Ω; ℝ) and p > 1, and let x_{0} ∈ Ω. If

lim_{ε→0^{+}} \frac{∫_{Q(x_{0},ε)} |∇u(x)|^{p} dx + H^{N−1}(S(u) ∩ Q(x_{0}, ε))}{ε^{N−1}} = 0

then x_{0} ∉ S(u) and

lim_{ε→0^{+}} med(u, Q(x_{0}, ε)) = aplim_{x→x_{0}} u(x) ∈ ℝ.  (2.1.5)

Moreover there exists ε_{0} > 0 such that for α ∈ ( \frac{3}{4} , 1) and ε, ρ > 0, with 0 < α ε ≤ ρ ≤ ε < ε_{0}, we have

\begin{matrix}τ′ & (u, Q(x_{0}, ρ)) ≤ med(u, Q(x_{0}, ε)) ≤ τ′′(u, Q(x_{0}, ρ)), \\ τ′ & (u, Q(x_{0}, ε)) ≤ med(u, Q(x_{0}, ρ)) ≤ τ′′(u, Q(x_{0}, ε)). \end{matrix}  (2.1.6)

The extensions of Theorems 2.1.4–2.1.6 to the vectorial case may be found in [27].

Lemma 2.1.7. Assume that u ∈ SBV_{p}(Ω; ℝ^{d}), and let ν ∈ S^{N−1}. Then for L^{N} a.e. x_{0} ∈ Ω

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |ū_{ε}(x) − ∇u(x_{0}) \frac{(x − x_{0})}{ε} |^{p} dx = 0,  (2.1.7)

where

u_{ε}(x) := \frac{u(x) − u(x_{0})}{ε} ,  ū_{ε}(x) := (u_{ε}(x) ∧ τ′′(u_{ε}, Q_{ν}(x_{0}, ε))) ∨ τ′(u_{ε}, Q_{ν}(x_{0}, ε)).

Proof. Note that if u ∈ W^{1,p}(Ω; ℝ^{d}) then by Remark 2.1.2 ū_{ε}(x) = u_{ε}(x), and (2.1.7) is a well known fact within the realm of Sobolev spaces.

Given u ∈ SBV_{p}(Ω; ℝ^{d}), it is clearly enough to show the result for d = 1. Consider x_{0} ∈ Ω satisfying

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |∇u(x) − ∇u(x_{0})|^{p} dx = 0,  (2.1.8)
lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)∩S(u)}(1 + |[u]|) dH^{N−1} = 0,  (2.1.9)
lim_{ε→0^{+}} \frac{1}{ε^{N+1}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})| dx = 0.  (2.1.10)

Define v_{ε}(y) := u_{ε}(x_{0} + ε y). For 0 ≤ s ≤ ε^{N} we have (u_{ε})_{∗}(s, Q_{ν}(x_{0}, ε)) = (v_{ε})_{∗}(s/ε^{N}, Q_{ν}) and, in turn,

τ′(v_{ε}, Q_{ν}) = τ′(u_{ε}, Q_{ν}(x_{0}, ε)), τ′′(v_{ε}, Q_{ν}) = τ′′(u_{ε}, Q_{ν}(x_{0}, ε)), med(v_{ε}; Q_{ν}) = med(u_{ε}, Q_{ν}(x_{0}, ε)).

It follows that v̄_{ε}(y) = ū_{ε}(x_{0} + ε y), where v̄_{ε}(y) := (v_{ε}(y) ∧ τ′′(v_{ε}, Q_{ν})) ∨ τ′(v_{ε}(y), Q_{ν}). Thus proving (2.1.7) is equivalent to showing that

{v̄_{ε}(y)} converges to ∇u(x_{0}) · y in L^{p}(Q_{ν}; ℝ^{d}).  (2.1.11)

By (2.1.8) and (2.1.9)

∫_{Q_{ν}} |∇v_{ε}(y)|^{p} dy ≤ C,  lim_{ε→0^{+}} H^{N−1}(Q_{ν} ∩ S(v_{ε})) = 0.

Hence we can apply Theorem 2.1.5 to conclude that there exists a subsequence {v_{ε_{j}} } and a function v_{0} ∈ W^{1,p}(Q_{ν}; ℝ^{d}) such that {v_{ε_{j}} −med(v_{ε_{j}}; Q_{ν})} and {v̄_{ε_{j}} −med(v_{ε_{j}}; Q_{ν})} converge to v_{0} pointwise almost everywhere and in L^{p}, respectively. We claim that

lim_{ε→0^{+}} med(v_{ε}; Q_{ν}) = 0.  (2.1.12)

If (2.1.12) holds then (2.1.11) follows easily. Indeed, (2.1.12) implies that {v_{ε_{j}} } and {v̄_{ε_{j}} } converge to v_{0} pointwise almost everywhere and in L^{p}, respectively. Moreover, by (2.1.10) we have that v_{0}(y) = ∇u(x_{0}) · y. Consequently the whole sequence {v̄_{ε}} converges to ∇u(x_{0}) · y in L^{p} and (2.1.11) is verified.

Thus, to conclude the proof of the lemma it remains to show (2.1.12). Fix t > 0 small. If ∇u(x_{0}) = 0, and since by (2.1.10) we have v_{ε} → 0 in L^{1}(Q), we may find ε̄ > 0 such that if 0 < ε ≤ ε̄ then

|\{y ∈ Q_{ν} : v_{ε}(y) < 2t\}| ≥ \frac{1}{2} .  (2.1.13)

Suppose now that ∇u(x_{0}) ≠ 0. Since

|{y ∈ Q_{ν} : ∇u(x_{0}) · y < t}| ≤ |{y ∈ Q_{ν} : |v_{ε}(y) − ∇u(x_{0}) · y| > t}| + |{y ∈ Q_{ν} : v_{ε}(y) < 2t}|,

with |\{y ∈ Q_{ν} : ∇u(x_{0}) · y < t\}| = \frac{1}{2} + ρ, where ρ = ρ(t) > 0, and as by (2.1.10) \{v_{ε}\} converges to ∇u(x_{0}) · y in L^{1}, and hence in measure, we can find ε̄ > 0 such that |{y ∈ Q_{ν} : |v_{ε}(y) − ∇u(x_{0}) · y| > t}| < ρ for all 0 < ε ≤ ε̄, and so (2.1.13) still holds. We conclude that med(v_{ε}; Q_{ν}) ≤ 2t, and, in turn, this implies that

lim sup_{ε→0^{+}} med(v_{ε}; Q_{ν}) ≤ 2t.

By letting t → 0^{+} we get

lim sup_{ε→0^{+}} med(v_{ε}; Q_{ν}) ≤ 0.

Conversely, for t < 0 small and if ∇u(x_{0}) = 0, we may find ε̄ > 0 such that for ε ≤ ε̄

|\{y ∈ Q_{ν} : v_{ε}(y) < t\}| < \frac{1}{2} .  (2.1.14)

Also, if ∇u(x_{0}) ≠ 0 then we have

\begin{aligned}|\{y ∈ Q_{ν} : v_{ε}(y) < t\}| & ≤ |\{y ∈ Q_{ν} : ∇u(x_{0}) · y < t/2\}| + |\{y ∈ Q_{ν} : |v_{ε}(y) − ∇u(x_{0}) · y| > −t/2\}| \\ & = \frac{1}{2} + ρ + |\{y ∈ Q_{ν} : |v_{ε}(y) − ∇u(x_{0}) · y| > −t/2\}|, \end{aligned}

where ρ = ρ(t) < 0. Again by (2.1.10) there exits ε̃ > 0 such that |{y ∈ Q_{ν} : |v_{ε}(y) − ∇u(x_{0}) y| > −t/2}| < −ρ for all 0 < ε ≤ ε̃. Hence (2.1.14) holds and thus med(v_{ε}; Q_{ν}) ≥ t. By first letting ε → 0^{+} and then t → 0^{−} we obtain (2.1.12).

Lemma 2.1.8. Assume that u ∈ SBV_{p}(Ω; ℝ^{d}). Then for H^{N−1} a.e x_{0} ∈ S(u) and for every ε > 0 there exists a function v_{ε} ∈ SBV_{p}(Q_{ν}(x_{0}, ε)); ℝ^{d}) such that H^{N−1}((S(v_{ε}) \ S(u)) ∩ Q_{ν}(x_{0}, ε)) = 0 with ν = ν_{u}(x_{0}),

lim_{ε→0^{+}} \frac{1}{ε^{N−1}} ∫_{Q_{ν} (x_{0},ε)} |∇v_{ε}(x)|^{p} dx = 0,  lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |v_{ε}(x) − u_{x_{0},ν}(x)|^{p} dx = 0  (2.1.15)

and

lim_{ε→0^{+}} \frac{1}{ε^{N}} |\{x ∈ Q_{ν}(x_{0}, ε) : u ≠ v_{ε}\}| = 0,  lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − v_{ε}(x)| dx = 0.  (2.1.16)

Proof. Clearly it is enough to show the result for d = 1. Since S(u) is a countably (N − 1)-rectifiable set in the sense of Federer, there exists a sequence of compact sets K_{j} such that

S(u) = ∪_{j=1}^{∞}K_{j} ∪ N_{0}, H^{N−1}(N_{0}) = 0,

and each set K_{j} is a subset of a C^{1} hypersurface. For each x = (x_{1}, …, x_{N}) ∈ ℝ^{N} set x′ := (x_{1}, …, x_{N−1}). In a neighborhood Q(y, ε_{0}) ⊂ Ω of each point y ∈ K_{j}, after a rotation, we may find a C^{1} function h_{j} : ℝ^{N−1} → ℝ such that

K_{j} ∩ Q(y, ε_{0}) ⊂ {( x′, x_{N}) ∈ Q(y, ε_{0}) : x_{N} = h_{j}(x′)}.

The function u restricted to the Lipschitz set Ω_{j} := {(x′, x_{N}) ∈ Ω : x_{N} > h_{j}(x′)} has a unique trace on its boundary. Thus, by taking ε_{0} smaller, if necessary, we obtain that the functions

w(x) := \begin{cases}u(x′, x_{N}) & if x_{N} > h_{j}(x′) \\ u(x′, −x_{N} + 2h_{j}(x′)) & if x_{N} < h_{j}(x′) \end{cases} , z(x) := \begin{cases}u(x′, x_{N}) & if x_{N} < h_{j}(x′) \\ u(x′, −x_{N} + 2h_{j}(x′)) & if x_{N} > h_{j}(x′) \end{cases}

belong to the space SBV_{p}(Q(y, ε_{0}); ℝ^{d}) and that S(w)∩Q(y, ε_{0}), S(z)∩Q(y, ε_{0}) ⊂ Q(y, ε_{0})\K_{j}. Thus by Lemma 2.1.3 for H^{N−1} a.e x_{0} ∈ K_{j} ∩ Q(y, ε_{0}) we have that

lim_{ε→0^{+}} \frac{∫_{Q_{ν} (x_{0},ε)} |∇w(x)|^{p} dx + H^{N−1}(S(w) ∩ Q_{ν}(x_{0}, ε))}{ε^{N−1}} = 0.  (2.1.17)

We claim that for H^{N−1} a.e x ∈ K_{j} ∩ Q(y, ε_{0}) we have that w(x) is either u^{+}(x) or u^{−}(x). Indeed fix x_{0} = (x′_{0}, h_{j}(x′_{0})) ∈ K_{j} ∩ Q(y, ε_{0}) for which the following hold:

lim_{ε→0^{+}} \frac{1}{ε^{N−1}} ∫_{Q_{ν} (x_{0},ε)} |∇u(x)|^{p} dx = 0,  lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u_{x_{0},ν}(x)| dx = 0,  (2.1.18)

where

ν = ν_{u}(x_{0}) = ± \frac{1}{\sqrt{1 + |∇_{x′}h_{j}(x′_{0})|^{2}}} (∇_{x′}h_{j}(x′_{0}), −1) ,  (2.1.19)

and such that

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |w(x) − w(x_{0})| dx = 0.  (2.1.20)

Without loss of generality we may assume that (2.1.19) is satisfied with the plus sign (the other case being treated in an analogous way), and, after a change of coordinates, we may take ν := (0, …, 0, −1). Then

\begin{matrix}∫_{Q_{ν} (x_{0},ε)∩Ω_{j}} |w(x) − u^{+}(x_{0})| dx = ∫_{Q_{ν} (x_{0},ε)∩\{x_{N} >h_{j} (x′)\}} |u(x) − u^{+}(x_{0})| dx \\ ≤ ∫_{Q_{ν} (x_{0},ε)} |u(x) − u_{x_{0},ν}(x)| dx + |u^{−}(x_{0}) − u^{+}(x_{0})| |Q_{ν}(x_{0}, ε) ∩ \{h_{j}(x′) < x_{N} ≤ h_{j}(x′_{0})\}| . \end{matrix}  (2.1.21)

Since h_{j} is C^{1} and ∇_{x′}h_{j}(x′_{0}) = 0, we have that |h_{j}(x′) − h_{j}(x′_{0})| ≤ C|x′ − x′_{0}|^{2} for ε > 0 sufficiently small. Thus

lim_{ε→0^{+}} \frac{1}{ε^{N}} |Q_{ν}(x_{0}, ε) ∩ \{h_{j}(x′) < x_{N} ≤ h_{j}(x′_{0})\}| = 0,

and, in turn, dividing (2.1.21) by ε^{N} and letting ε → 0^{+} we get, also by (2.1.18),

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)∩Ω_{j}} |w(x) − u^{+}(x_{0})| dx = 0.

This, together with (2.1.20), implies that w(x_{0}) = u^{+}(x_{0}) and proves the claim.

Fix x_{0} = (x′_{0}, x_{0N}) ∈ K_{j} ∩ Q(y, ε_{0}) for which (2.1.18), (2.1.17) and (2.1.20) hold with, say, w(x_{0}) = u^{+}(x_{0}) and

ν = ν_{u}(x_{0}) = \frac{1}{\sqrt{1 + |∇_{x′}h_{j}(x′_{0})|^{2}}} (∇_{x′}h_{j}(x′_{0}), −1)

and let 0 < η < 1/(2γ_{N}4^{(N−1)/N}). By (2.1.17) there exists ε_{0} > 0 such that

∫_{Q_{ν} (x_{0},ε)} |∇w(x)|^{p} dx + H^{N−1}(S(w) ∩ Q_{ν}(x_{0}, ε)) < η ε^{N−1}  (2.1.22)

for all 0 < ε < ε_{0}, and by Hölder’s inequality, (2.1.2) and (2.1.4), we have

\begin{matrix}∫_{Q_{ν} (x_{0},ε)} |w̄_{ε}(x) − med(w, Q_{ν}(x_{0}, ε))|^{p} dx ≤ & (∫_{Q_{ν} (x_{0},ε)} |w̄_{ε} − med(w, Q_{ν}(x_{0}, ε))|^{p^{∗}} dx)^{1/q} ε^{N/q′} \\ ≤ & Cε^{p} ∫_{Q_{ν} (x_{0},ε)} |∇w(x)|^{p} dx, \end{matrix}

where q := p^{∗}/p, q′ is its Hölder conjugate, with N/q′ = p, and w̄_{ε}(x) := (w(x)∧τ′′(w, Q_{ν}(x_{0}, ε))∨τ′(w, Q_{ν}(x_{0}, ε)). Thus, by (2.1.17), we get

lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |w̄_{ε}(x) − med(w, Q_{ν}(x_{0}, ε))|^{p} dx = 0.  (2.1.23)

We claim that

med(w, Q_{ν}(x_{0}, ε)) = u^{+}(x_{0}) + o (ε^{(p−1)/p})  as ε → 0^{+}.  (2.1.24)

Conditions (2.1.17) and Theorem 2.1.6 imply that w is approximately continuous at x_{0}, with w(x_{0}) = u^{+}(x_{0}), and by (2.1.5) we also have that

lim_{ε→0^{+}} med(w, Q_{ν}(x_{0}, ε)) = u^{+}(x_{0}).  (2.1.25)

Fix α ∈ ( \frac{3}{4} , 1) and ε, ρ > 0 such that α ε ≤ ρ ≤ ε < ε_{0}. Setting w_{ε} := med(w, Q_{ν}(x_{0}, ε)) and

ŵ := (w ∧ (τ′′(w, Q_{ν}(x_{0}, ε)) ∧ τ′′(w, Q_{ν}(x_{0}, ρ)))) ∨ (τ′(w, Q_{ν}(x_{0}, ε)) ∨ τ′(w, Q_{ν}(x_{0}, ρ))),

then, again by Theorem 2.1.4 and (2.1.22),

|w_{ε} − w_{ρ}| = ( \frac{1}{ρ^{N}} ∫_{Q_{ν} (x_{0},ρ)} |w_{ε} − w_{ρ}|^{p^{∗}}dx)^{1/p^{∗}}

≤ Cρ^{−N/p^{∗}} [(∫_{Q_{ν} (x_{0},ρ)} |ŵ − w_{ρ}|^{p^{∗}}dx)^{1/p^{∗}} + (∫_{Q_{ν} (x_{0},ρ)} |ŵ − w_{ε}|^{p^{∗}}dx)^{1/p^{∗}} ]

≤ Cρ^{−N/p^{∗}} [(∫_{Q_{ν} (x_{0},ρ)} |w̄_{ρ} − w_{ρ}|^{p^{∗}}dx)^{1/p^{∗}} + (∫_{Q_{ν} (x_{0},ρ)} |w̄_{ε} − w_{ε}|^{p^{∗}}dx)^{1/p^{∗}} ]

≤ C(αε)^{−N/p^{∗}} [(∫_{Q_{ν} (x_{0},ρ)} |∇u(x)|^{p} dx)^{1/p} + (∫_{Q_{ν} (x_{0},ε)} |∇u(x)|^{p} dx)^{1/p}]

≤ C η^{1/p}α^{−N/p∗} ε^{(p−1)/p},

where we have used the inequalities

|ŵ − w_{ρ}| ≤ |w̄_{ρ} − w_{ρ}|, |ŵ − w_{ϵ}| ≤ |w̄_{ϵ} − w_{ϵ}|,

which follow from (2.1.6). Hence for k ≥ 0 and with ρ_{k} such that α^{k+1}ε ≤ ρ_{k} ≤ α^{k}ε we obtain

|w_{ε} − w_{ρ_{k}}| ≤ |w_{ε α^{k}} − w_{ρ_{k}}| + ∑_{j=0}^{k−1} |w_{ε α^{j+1}} − w_{ε α^{j}} | ≤ C η^{1/p} α^{−N/p∗} ε^{(p−1)/p} ∑_{j=0}^{k}(α^{(p−1)/p})^{j}.

If we now let k → ∞ and use (2.1.25) we get

|w_{ε} − u^{+}(x_{0})| ≤ C η^{1/p} α^{−N/p∗} ε^{(p−1)/p} ∑_{j=0}^{∞}(α^{(p−1)/p})^{j} = C η^{1/p} ε^{(p−1)/p} \frac{1}{1 − α^{(p−1)/p}} .

Hence

lim sup_{ε→0^{+}} \frac{|med(w, Q_{ν}(x_{0}, ε)) − u^{+}(x_{0})|}{ε^{(p−1)/p}} ≤ C η^{1/p}.

To obtain (2.1.24) it now suffices to let η → 0^{+}.

By (2.1.23) and (2.1.24) we obtain that

lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |w̄_{ε}(x) − u^{+}(x_{0})|^{p} dx = 0.  (2.1.26)

With a similar proof we can show that

lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |z̄_{ε}(x) − u^{−}(x_{0})|^{p} dx = 0.  (2.1.27)

We can now define

v_{ε}(x) := \begin{cases}w̄_{ε}(x) & if x_{N} > h_{j}(x′), \\ z̄_{ε}(x) & if x_{N} < h_{j}(x′). \end{cases}

Then (2.1.15) follows from (2.1.17), (2.1.26) and (2.1.27). Note that

H^{N−1}((S(v_{ε}) \ S(u)) ∩ Q_{ν}(x_{0}, ε)) = 0

and, by (2.1.3) and (2.1.17),

\frac{1}{ε^{N}} |\{x ∈ Q_{ν}(x_{0}, ε) : u ≠ v_{ε}\}| = \frac{1}{ε^{N}} |\{x ∈ Q_{ν}(x_{0}, ε) : x_{N} > h_{j}(x′), w(x) ≠ w̄_{ε}(x)\}|

+ \frac{1}{ε^{N}} |\{x ∈ Q_{ν}(x_{0}, ε) : x_{N} < h_{j}(x′), z(x) ≠ z̄_{ε}(x)\}|

≤ C ( \frac{H^{N−1}(S(w) ∩ Q_{ν}(x_{0}, ε))}{ε^{N−1}} )^{N/(N−1)} + C ( \frac{H^{N−1}(S(z) ∩ Q_{ν}(x_{0}, ε))}{ε^{N−1}} )^{N/(N−1)} → 0,

as ε → 0^{+}, so that (2.1.16)_{1} is satisfied. To prove (2.1.16)_{2} we use Hölder’s inequality, (2.1.18) and (2.1.15) to get

\begin{aligned}\frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − v_{ε}(x)| dx ≤ & \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u_{x_{0},ν}(x)| dx \\ & + ( \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |v_{ε}(x) − u_{x_{0},ν}(x)|^{p} dx)^{1/p} → 0 as ε → 0^{+}. \end{aligned}

2.2 The General Method.

In the remaining of this section we assume that hypotheses (H_{1}) − (H_{4}) hold.

Let u ∈ SBV_{p}(Ω; ℝ^{d}), ν ∈ S^{N−1}, and define μ := L^{N}⌊Ω + (1 + |u^{+} − u^{−}|)H^{N−1}⌊(S(u) ∩ Ω). Let A^{∗} := {Q_{ν}(x, ε) : x ∈ Ω, ε > 0}, and for δ > 0, A ∈ A(Ω), set

m^{δ}(u; A) := inf{∑_{i=1}^{∞} m(u; Q_{i}) : Q_{i} ∈ A^{∗}, Q_{i} ∩ Q_{j} = ∅, Q_{i} ⊂ A,

diam(Q_{i}) < δ, μ(A\ ∪_{i=1}^{∞} Q_{i}) = 0}.

Note that the existence of coverings as those needed in the definition of m^{δ} follows from a generalization of Besicovitch’s Covering Theorem (see [43], Th. 5.11) provided we take closed cubes. However, since μ is a Radon measure then μ(∂Q_{ν}(x, ε)) = 0 for a.e. ε > 0, and thus we may consider open cubes instead. Since m^{δ}(u; A) is decreasing as a function of δ, we define

m^{∗}(u; A) := lim_{δ→0^{+}} m^{δ}(u; A).

Lemma 2.2.1.

F(u; A) = m^{∗}(u; A).

Proof. The proof of Lemma 2.2.1 is entirely identical to the proof of Lemma 3.3 of [17], with the exception that the function v^{δ} defined there now belongs to SBV_{p}(Ω; ℝ^{d}) due to the coercitivity hypothesis (H_{4}) and by virtue of Ambrosio’s Compactness Theorem 2.1.1.

Lemma 2.2.2. For μ a.e. x_{0} ∈ Ω and for all ν ∈ S^{N−1}

lim_{ε→0^{+}} \frac{F(u; Q_{ν}(x_{0}, ε))}{μ(Q_{ν}(x_{0}, ε))} = lim_{ε→0^{+}} \frac{m(u; Q_{ν}(x_{0}, ε))}{μ(Q_{ν}(x_{0}, ε))} .

The proof of this result may be carried out exactly as in Lemma 3.5 in [17]. Since the latter relied on Remark 3.2 1) in [17], we now prove that this remark still applies in our present context, precisely, Lemma 2.2.3. If A ∈ A_{∞}(Ω), u ∈ SBV_{p}(Ω; ℝ^{d}), and if

A_{δ} := {x ∈ A : dist(x, ∂A) > δ}

then

lim_{δ→0} m(u; A_{δ}) = m(u; A).

Proof. Given δ, η > 0 choose v ∈ SBV_{p}(Ω; ℝ^{d}) such that v = u on a neighborhood of ∂A_{δ} and

m(u; A_{δ}) ≥ F(v; A_{δ}) − η.

Define

w := \begin{cases}v & in A_{δ}, \\ u & in Ω \ \overline{A_{δ}}. \end{cases}  (2.2.1)

Let ε > 0 be small enough so that v = u on A_{δ} \ A_{δ+ε}. We have

m(u; A) ≤ F(w; A)

≤ F(v; A_{δ}) + C ∫_{A\A_{δ+ε}}(1 + |∇u|^{p}) dx + C ∫_{(A\A_{δ+ε})∩S(u)}(1 + |[u]|) dH^{N−1}

≤ m(u; A_{δ}) + η
+ C ∫_{A\A_{δ+ε}}(1 + |∇u|^{p}) dx + C ∫_{(A\A_{δ+ε})∩S(u)}(1 + |[u]|) dH^{N−1}.

Thus, letting ε → 0^{+} and then η → 0^{+} we conclude that

m(u; A) ≤ m(u; A_{δ}) + C ∫_{A\A_{δ}}(1 + |∇u|^{p}) dx + C ∫_{(A\A_{δ})∩S(u)}(1 + |[u]|) dH^{N−1},  (2.2.2)

and so

lim inf_{δ→0^{+}} m(u; A_{δ}) ≥ m(u; A).

Conversely, fix ε > 0 and choose v ∈ SBV_{p}(Ω; ℝ^{d}) such that v = u on a neighborhood of ∂A and

m(u; A) ≥ F(v; A) − ε.

Let δ_{0} > 0 be small enough so that v = u on A \ A_{δ} for all 0 < δ < δ_{0}. Then

m(u; A_{δ}) ≤ F(v; A_{δ}) ≤ F(v; A) ≤ m(u; A) + ε.

Letting δ → 0^{+} and then ε → 0^{+} we deduce that

lim sup_{δ→0^{+}} m(u; A_{δ}) ≤ m(u; A).

Remark 2.2.4. We can show that if A′ ⊂⊂ A, with A′, A ∈ A(Ω), then

m(u; A)≤ m(u; A′) + C ∫_{A\A′}(1 + |∇u|^{p}) dx + C ∫_{(A\A′)∩S(u)}(1 + |[u]|) dH^{N−1}.

This inequality follows immediately from (2.2.2) with A′ in place of A_{δ}. Similarly, it can be shown that

F(u; A)≤ F(u; A′) + C ∫_{A\A′}(1 + |∇u|^{p}) dx + C ∫_{(A\A′)∩S(u)}(1 + |[u]|) dH^{N−1}.

2.3 Proof of Theorem 1.0.1: Lebesgue Points.

Lemma 2.3.1. For L^{N} a.e. x_{0} ∈ Ω

\frac{dF(u; ·)}{dL^{N}} (x_{0}) ≤ f(x_{0}, u(x_{0}), ∇u(x_{0})),

where f is the function defined in Theorem 1.0.1. Proof. Let u_{ε} and ū_{ε} be defined as in Lemma 2.1.7, and let x_{0} satisfy (2.1.8)–(2.1.10). By Lemma 2.2.2 for L^{N} a.e. x_{0} ∈ Ω

\frac{dF(u; ·)}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{F(u(x_{0}) + ε u_{ε}; Q_{ν}(x_{0}, ε))}{ε^{N}} = lim_{ε→0^{+}} \frac{m(u(x_{0}) + ε u_{ε}; Q_{ν}(x_{0}, ε))}{ε^{N}} < ∞.  (2.3.1)

Thus, by the coercitivity hypothesis (H_{4}), there exists K > 0 such that

\frac{H^{N−1}(S(u_{ε}) ∩ Q_{ν}(x_{0}, ε))}{ε^{N}} ≤ K

for all ε > 0 sufficiently small. Hence (2.1.1) is satisfied for 0 < ε << 1 and by (2.1.3)

\begin{aligned}0 & ≤ \frac{1}{ε^{N}} ∫_{0}^{1} H^{N−1} (\{u_{ε} ≠ ū_{ε}\} ∩ ∂Q_{ν}(x_{0}, λε)) d λ = \frac{1}{ε^{N+1}} |\{x ∈ Q_{ν}(x_{0}, ε) : u_{ε} ≠ ū_{ε}\}| \\ & ≤ \frac{C}{ε^{N+1}} (H^{N−1}(S(u_{ε}) ∩ Q_{ν}(x_{0}, ε)))^{N/(N−1)} ≤ C K^{N/(N−1)} ε^{1/(N−1)} → 0 as ε → 0^{+}, \end{aligned}

and by (2.1.10) and (2.1.7)

\begin{aligned}0 & ≤ \frac{1}{ε^{N}} ∫_{0}^{1} ∫_{∂Q_{ν} (x_{0},λε)} |ε u_{ε} − ε ū_{ε}|d H^{N−1} d λ ≤ \frac{1}{ε^{N+1}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})| dx \\ & + \frac{1}{ε^{N+1}} ∫_{Q_{ν} (x_{0},ε)} |ε ū_{ε}(x) − ∇u(x_{0})(x − x_{0})| dx → 0 as ε → 0^{+}. \end{aligned}

Thus for any sequence ε_{h} → 0^{+} there exists a subsequence (not relabeled) such that for L^{1} a.e λ ∈ (0, 1)

\begin{matrix}μ & (∂Q_{ν}(x_{0}, λε_{h})) = H^{N−1}(∂Q_{ν}(x_{0}, λε_{h}) ∩ S(ū_{ε_{h}})) = 0, \\ lim_{ε_{h}→0^{+}} \frac{1}{ε_{h}^{N}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u_{ε_{h}} ≠ū_{ε_{h}} \}}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1} = 0. \end{matrix}  (2.3.2)

Fix a sequence ε_{h} → 0^{+}, and consider a subsequence (not relabeled) and λ ∈ (0, 1) for which (2.3.2) holds. Given v_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, ε_{h}); ℝ^{d}) such that v_{ε_{h}} = u(x_{0}) + ε_{h} ū_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, λε_{h}) and

F(v_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) ≤ m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N+1},

define

w_{ε_{h}} := \begin{cases}v_{ε_{h}}(x) & in Q_{ν}(x_{0}, λε_{h}), \\ u(x_{0}) + ε_{h} u_{ε_{h}}(x) & in Q_{ν}(x_{0}, ε_{h})\\overline{Q_{ν}(x_{0}, λε_{h})}. \end{cases}

Since w_{ε_{h}} = u(x_{0}) + ε_{h} u_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, ε_{h}), by Remark 2.2.4 we have m(u(x_{0}) + ε_{h} u_{ε_{h}}; Q_{ν}(x_{0}, ε_{h})) ≤ F(w_{ε_{h}}; Q_{ν}(x_{0}, ε_{h}))

≤ F(v_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h})) + C ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h})}(1 + |∇u|^{p}) dx
+ ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u_{ε_{h}} ≠ū_{ε_{h}} }}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1} + C ∫_{Q_{ν} (x_{0},ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}
≤ m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N+1} + C ((1 − λ^{N})ε_{h}^{N} + ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h})} |∇u|^{p} dx)
+ ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u_{ε_{h}} ≠ū_{ε_{h}} }}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1} + C ∫_{Q_{ν} (x_{0},ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}.

Dividing by ε_{h}^{N} yields

\frac{m(u(x_{0}) + ε_{h} u_{ε_{h}}; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N}} ≤ \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N}} + ε_{h} + C(1 − λ^{N})

+ \frac{C}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h})} |∇u|^{p}dx + \frac{C}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}

+ \frac{1}{ε_{h}^{N}} ∫_{∂Q_{ν} (x_{0},λε_{h})∩\{u_{ε_{h}} ≠ū_{ε_{h}} \}}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1}.

We claim that the last three terms on the right hand side approach zero if we first let ε_{h} → 0^{+} and then λ → 1^{−}. Indeed, for the last two integrals this follows from (2.3.2) and from (2.1.9), respectively. Moreover

\frac{1}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h})} |∇u|^{p}dx = \frac{1}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})} |∇u|^{p}dx − λ^{N} \frac{1}{(λ ε_{h})^{N}} ∫_{Q_{ν} (x_{0},λ ε_{h})} |∇u|^{p}dx.  (2.3.3)

Hence by (2.1.8)

lim_{ε_{h}→0^{+}} \frac{1}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h})} |∇u|^{p}dx = |∇u(x_{0})|^{p} − λ^{N} |∇u(x_{0})|^{p}.

The claim is proved if we let λ → 1^{−}.

In conclusion, by (2.3.1), we have shown that

\frac{dF(u; ·)}{dL^{N}} (x_{0}) ≤ lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N}} ,  (2.3.4)

where the liminf in λ is taken only over those λ for which (2.3.2) holds.

To complete the proof of the lemma it remains to show that the right hand side of (2.3.4) is less than or equal to f(x_{0}, u(x_{0}), ∇u(x_{0})). Fix 0 < s < 1 and let v_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s λ ε_{h}); ℝ^{d}) be such that v_{ε_{h}}(x) = u(x_{0}) + ∇u(x_{0}) (x − x_{0}) on a neighborhood of ∂Q_{ν}(x_{0}, sλε_{h}) and

F(v_{ε_{h}}; Q_{ν}(x_{0}, sλε_{h})) ≤ m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, sλε_{h})) + (λε_{h})^{N+1}.

Extend v_{ε_{h}} to be equal to u(x_{0})+∇u(x_{0}) (x−x_{0}) outside Q_{ν}(x_{0}, s λ ε_{h}) and consider a layer L of width (1−s)λε_{h}/2 and a smooth cut-off function φ such that 0 < φ < 1 in L, φ ≡ 1 in Q_{ν}(x_{0}, sλε_{h}), φ = 0 on ∂Q_{ν}(x_{0}, λ ε_{h}) and ||∇φ||_{∞} ≤ 3/(1 − s)λε_{h}. Define w_{ε_{h}}(x) := φ v_{ε_{h}} + (1 − φ)(u(x_{0}) + ε_{h} ū_{ε_{h}}(x)). Since w_{ε_{h}} = u(x_{0}) + ε_{h} ū_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, λ ε_{h}) we have, in view of Remark 2.2.4,

m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h})) ≤ F(w_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h}))
≤F(v_{ε_{h}}; Q_{ν}(x_{0}, sλε_{h})) + C ∫_{Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h})}(1 + |∇u|^{p} + |∇u(x_{0})|^{p}) dx

+ \frac{C}{(1 − s)^{p}(λε_{h})^{p}} ∫_{Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h})} |ε_{h} ū_{ε_{h}}(x) − ∇u(x_{0}) (x − x_{0})|^{p}dx

+ C ∫_{Q_{ν} (x_{0},λ ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}

≤m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, sλε_{h}))

+ (λε_{h})^{N+1} + C(1 + |∇u(x_{0})|^{p})(1 − s^{N})(λε_{h})^{N} + C ∫_{Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h})} |∇u|^{p} dx

+ \frac{C}{(1 − s)^{p}(λε_{h})^{p}} ∫_{Q_{ν} (x_{0},λ ε_{h})} |ε_{h} ū_{ε_{h}}(x) − ∇u(x_{0}) (x − x_{0})|^{p}dx + C ∫_{Q_{ν} (x_{0},λ ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1},

where we have used the facts that ||∇φ||_{∞} ≤ 3/(1 − s)λε_{h}, and that

\begin{matrix}|∇ū_{ε_{h}}| ≤ |∇u_{ε_{h}}| & ≤ |∇u|/ε_{h} L^{N} a.e., \\ H^{N−1}(S(ū_{ε_{h}})\S(u)) = 0, & |[ū_{ε_{h}}]| ≤ |[u_{ε_{h}}]| ≤ |[u]|/ε_{h} H^{N−1} a.e. \end{matrix}  (2.3.5)

Dividing by (λ ε_{h})^{N} yields

\frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h}))}{(λ ε_{h})^{N}} ≤ \frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, sλε_{h}))}{(λ ε_{h})^{N}} + λε_{h}

+ C(1 + |∇u(x_{0})|^{p})(1 − s^{N}) + \frac{C}{(λ ε_{h})^{N}} ∫_{Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h})} |∇u|^{p} dx

+ \frac{C}{(1 − s)^{p}(λε_{h})^{N+p}} ∫_{Q_{ν} (x_{0},λ ε_{h})} |ε_{h} ū_{ε_{h}}(x) − ∇u(x_{0}) (x − x_{0})|^{p}dx

+ \frac{C}{(λ ε_{h})^{N}} ∫_{Q_{ν} (x_{0},λ ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}.

We claim that the last three terms on the right hand side approach zero if we first let ε_{h} → 0^{+}, then s → 1^{−} and finally λ → 1^{−}. Indeed, for the second integral this follows from (2.1.7), while the first and the third integrals may be treated as before (see (2.1.9) and (2.3.3)). Consequently, and also by (2.3.4),

\frac{dF(u; ·)}{dL^{N}} (x_{0}) ≤ lim inf_{λ→1^{−}} lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, sλε_{h}))}{(λ ε_{h})^{N}}

≤ lim sup_{ε→0^{+}} \frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, ε))}{ε^{N}} = f(x_{0}, u(x_{0}), ∇u(x_{0})).

In order to assert the representation (1.0.2) it suffices now to establish the reverse inequality, precisely

Lemma 2.3.2. For L^{N} a.e. x_{0} ∈ Ω

\frac{dF(u; ·)}{dL^{N}} (x_{0}) ≥ f(x_{0}, u(x_{0}), ∇u(x_{0})).

Proof. The proof of this lemma is very similar to the previous one and thus we will indicate only the main changes. Let ε_{h} → 0^{+} be such that

lim sup_{ε→0^{+}} \frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, ε))}{ε^{N}} = lim_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N}} .  (2.3.6)

Fix 0 < s < 1 and let v_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s ε_{h}); ℝ^{d}) be such that v_{ε_{h}}(x) = u(x_{0}) + ε_{h} ū_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, sε_{h}) and

F(v_{ε_{h}}; Q_{ν}(x_{0}, sε_{h})) ≤ m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, sε_{h})) + ε_{h}^{N+1}.

Extend v_{ε_{h}} to be u(x_{0}) + ε_{h} ū_{ε_{h}} outside Q_{ν}(x_{0}, s ε_{h}) and consider a layer L of width (1 − s)ε_{h}/2 and a cut-off function φ such that 0 < φ < 1 in L, φ ≡ 1 on Q_{ν}(x_{0}, sε_{h}), φ = 0 on ∂Q_{ν}(x_{0}, ε_{h}) and ||∇φ||_{∞} ≤ 3/(1 − s)ε_{h}. Define w_{ε_{h}}(x) := φ v_{ε_{h}} +(1−φ)(u(x_{0})+∇u(x_{0}) (x−x_{0})). Since w_{ε_{h}} = u(x_{0})+∇u(x_{0}) (x−x_{0}) on a neighborhood of ∂Q_{ν}(x_{0}, ε_{h}) then, reasoning as in the second part of the proof of Lemma 2.3.1 and by (2.3.5), we have by Remark 2.2.4,

\frac{m(u(x_{0}) + ∇u(x_{0}) (· − x_{0}); Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N}} ≤ \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N}} + ε_{h}

+ C(1 + |∇u(x_{0})|^{p})(1 − s^{N}) + \frac{C}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},sε_{h})} |∇u|^{p} dx

+ \frac{C}{(1 − s)^{p}ε_{h}^{N+p}} ∫_{Q_{ν} (x_{0},ε_{h})} |ε_{h} ū_{ε_{h}}(x) − ∇u(x_{0}) (x − x_{0})|^{p}dx + \frac{C}{ε_{h}^{N}} ∫_{Q_{ν} (x_{0},ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1}.

As before, and invoking (2.3.6), we obtain

f(x_{0}, u(x_{0}), ∇u(x_{0})) ≤ lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N}} .  (2.3.7)

Consider a sequence s_{j} → 1^{−} such that

lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N}} = lim_{s_{j} →1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, s_{j}ε_{h}))}{ε_{h}^{N}} ,  (2.3.8)

and consider a subsequence of {ε_{h}} (not relabeled) and λ ∈ (0, 1) such that, for all j ∈ ℕ, and in light of (2.3.2),

\begin{matrix}μ(∂Q_{ν}(x_{0}, s_{j}λε_{h})) = H^{N−1}(∂Q_{ν}(x_{0}, s_{j}λε_{h}) ∩ S(ū_{ε_{h}})) = 0, \\ lim_{ε_{h}→0^{+}} \frac{1}{ε_{h}^{N}} ∫_{∂Q_{ν} (x_{0},s_{j} λ ε_{h})∩\{u_{ε_{h}} ≠ū_{ε_{h}} \}}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1} = 0. \end{matrix}

Given v_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s_{j}λε_{h}); ℝ^{d}) such that v_{ε_{h}} = u(x_{0}) + ε_{h} u_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, s_{j}λε_{h}) and

F(v_{ε_{h}}; Q_{ν}(x_{0}, s_{j}λε_{h})) ≤ m(u(x_{0}) + ε_{h} u_{ε_{h}}; Q_{ν}(x_{0}, s_{j}λε_{h})) + (s_{j} ε_{h})^{N+1},

define

w_{ε_{h}} := \begin{cases}v_{ε_{h}}(x) & in Q_{ν}(x_{0}, s_{j}λε_{h}), \\ u(x_{0}) + ε_{h} ū_{ε_{h}}(x) & in Q_{ν}(x_{0}, s_{j}ε_{h})\\overline{Q_{ν}(x_{0}, s_{j}λε_{h})}. \end{cases}

Since w_{ε_{h}} = u(x_{0}) + ε_{h} ū_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, s_{j}ε_{h}) we have, as before, and in view of Remark 2.2.4

\begin{matrix}\frac{m(u(x_{0}) + ε_{h} ū_{ε_{h}}; Q_{ν}(x_{0}, s_{j}ε_{h}))}{(s_{j}ε_{h})^{N}} ≤ \frac{m(u(x_{0}) + ε_{h} u_{ε_{h}}; Q_{ν}(x_{0}, s_{j}λε_{h}))}{(s_{j}ε_{h})^{N}} + s_{j}ε_{h} + C(1 − λ^{N}) \\ + \frac{C}{(s_{j}ε_{h})^{N}} ∫_{Q_{ν} (x_{0},s_{j} ε_{h})\Q_{ν} (x_{0},s_{j} λ ε_{h})} |∇u|^{p}dx + \frac{C}{(s_{j}ε_{h})^{N}} ∫_{Q_{ν} (x_{0},s_{j} ε_{h})∩S(u)}(1 + |[u]|) dH^{N−1} \\ + \frac{1}{(s_{j}ε_{h})^{N}} ∫_{∂Q_{ν} (x_{0},s_{j} λε_{h})∩\{u_{ε_{h}} ≠ū_{ε_{h}} \}}(1 + |ε_{h} u_{ε_{h}} − ε_{h} ū_{ε_{h}}|)d H^{N−1}, \end{matrix}  (2.3.9)

where we have used (2.3.5). We can proceed as in the first part of the proof of Lemma 2.3.1 to conclude from (2.3.7)–(2.3.8), (2.3.9), that

f(x_{0}, u(x_{0}), ∇u(x_{0})) ≤ lim inf_{s_{j} →1^{−}} lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u(x_{0}) + ε_{h} u_{ε_{h}}; Q_{ν}(x_{0}, s_{j}λε_{h}))}{(s_{j}ε_{h})^{N}} = \frac{dF(u; ·)}{dL^{N}} (x_{0}).

2.4 Proof of Theorem 1.0.1: Jump Points.

In this section we prove that the jump energy density g is given by (1.0.3).

Lemma 2.4.1.

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) ≤ g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})),  (2.4.1)

where g is the function defined in Theorem 1.0.1.

Proof. Let v_{ε} be the function constructed according to Lemma 2.1.8, let x_{0} satisfy (2.1.18), and set ν := ν_{u}(x_{0}). By Lemma 2.2.1 for H^{N−1} a.e. x_{0} ∈ Ω ∩ S(u)

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) = lim_{ε→0^{+}} \frac{F(u; Q_{ν}(x_{0}, ε))}{ε^{N−1}} = lim_{ε→0^{+}} \frac{m(u; Q_{ν}(x_{0}, ε))}{ε^{N−1}} < ∞.  (2.4.2)

By (2.1.16)

0 ≤ \frac{1}{ε^{N−1}} ∫_{0}^{1} H^{N−1} (\{u ≠ v_{ε}\} ∩ ∂Q_{ν}(x_{0}, λε)) d λ = \frac{1}{ε^{N}} |\{x ∈ Q_{ν}(x_{0}, ε) : u ≠ v_{ε}\}| → 0 as ε → 0^{+},

and

0 ≤ \frac{1}{ε^{N−1}} ∫_{0}^{1} ∫_{∂Q_{ν} (x_{0},λε)} |u − v_{ε}|d H^{N−1} d λ ≤ \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u − v_{ε}| dx → 0 as ε → 0^{+}.

Thus for any sequence ε_{h} → 0^{+} we may find a subsequence (not relabeled) such that for L^{1} a.e λ ∈ (0, 1)

\begin{matrix}μ & (∂Q_{ν}(x_{0}, λε_{h})) = H^{N−1}(∂Q_{ν}(x_{0}, λε_{h}) ∩ S(v_{ε_{h}})) = 0, \\ lim_{ε_{h}→0^{+}} \frac{1}{ε_{h}^{N−1}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u≠v_{ε_{h}} \}}(1 + |u − v_{ε_{h}}|)d H^{N−1} = 0. \end{matrix}  (2.4.3)

Fix a subsequence ε_{h} → 0^{+} and λ ∈ (0, 1) for which (2.4.3) holds. Given z_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, ε_{h}); ℝ^{d}) such that z_{ε_{h}} = v_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, λε_{h}) and

F(z_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) ≤ m(v_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N},

define

w_{ε_{h}} := \begin{cases}z_{ε_{h}} & in Q_{ν}(x_{0}, λε_{h}), \\ u & in Q_{ν}(x_{0}, ε_{h})\\overline{Q_{ν}(x_{0}, λε_{h})}. \end{cases}

Since w_{ε_{h}} = u on ∂Q_{ν}(x_{0}, ε_{h}) we have by Remark 2.2.4,

m(u; Q_{ν}(x_{0}, ε_{h})) ≤ F(w_{ε_{h}}; Q_{ν}(x_{0}, ε_{h}))
≤ F(z_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h})) + C ∫_{Q_{ν} (x_{0},ε_{h})}(1 + |∇v_{ε_{h}}|^{p})dx
+ ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u≠v_{ε_{h}} }}(1 + |u − v_{ε_{h}}|)d H^{N−1} + C ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|)dH^{N−1}
≤ m(v_{ε_{h}}(x); Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N} + C ∫_{Q_{ν} (x_{0},ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx
+ ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u≠v_{ε_{h}} }}(1 + |u − v_{ε_{h}}|)dH^{N−1} + C ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|) dH^{N−1}.

Dividing by ε_{h}^{N−1} yields

\frac{m(u; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ≤ \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} + ε_{h} + \frac{C}{ε_{h}^{N−1}} ∫_{Q_{ν} (x_{0},ε_{h})}(1 + |∇v_{ε_{h}}|^{p})dx

+ \frac{C}{ε_{h}^{N−1}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u≠v_{ε_{h}} \}}(1 + |u − v_{ε_{h}}|)dH^{N−1}

+ \frac{C}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|)dH^{N−1}.

We claim that the last three terms on the right hand side approach zero if we first let ε_{h} → 0^{+} and then λ → 1^{−}. Indeed, for the first two integrals this follows from (2.1.15)_{1} and from (2.4.3), respectively. Moreover

\begin{matrix}\frac{1}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|) dH^{N−1} \\ = \frac{(1 + |[u]|)H^{N−1}⌊S(u)(Q_{ν}(x_{0}, ε_{h}))}{H^{N−1}⌊S(u)(Q_{ν}(x_{0}, ε_{h}))} − λ^{N−1} \frac{(1 + |[u]|)H^{N−1}⌊S(u)(Q_{ν}(x_{0}, λ ε_{h}))}{H^{N−1}⌊S(u)(Q_{ν}(x_{0}, λ ε_{h}))} . \end{matrix}

Hence

\begin{aligned}lim_{ε_{h}→0^{+}} & \frac{1}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|) dH^{N−1} \\ & = \frac{d((1 + |[u]|)H^{N−1}⌊S(u))}{dH^{N−1}⌊S(u)} (x_{0}) − λ^{N−1} \frac{d((1 + |[u]|)H^{N−1}⌊S(u))}{dH^{N−1}⌊S(u)} (x_{0}). \end{aligned}  (2.4.4)

It now suffices to let λ → 1^{−}. In conclusion, by (2.4.2), we have shown that

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) ≤ lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} .  (2.4.5)

To complete the proof of the lemma it remains to show that the right hand side of (2.4.5) is less than or equal to g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})). Fix 0 < s < 1 and let z_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s λ ε_{h}); ℝ^{d}) be such that z_{ε_{h}} = u_{x_{0},ν} on a neighborhood of ∂Q_{ν}(x_{0}, sλε_{h}) and

F(z_{ε_{h}}; Q_{ν}(x_{0}, sλε_{h})) ≤ m(u_{x_{0},ν}; Q_{ν}(x_{0}, sλε_{h})) + (λε_{h})^{N}.

Extend z_{ε_{h}} to be equal to u_{x_{0},ν} outside Q_{ν}(x_{0}, s λ ε_{h}) and consider a layer L of width (1 − s)λε_{h}/2 and a smooth cut-off function φ such that 0 < φ < 1 in L, φ ≡ 1 on Q_{ν}(x_{0}, sλε_{h}), φ = 0 on ∂Q_{ν}(x_{0}, λ ε_{h}) and ||∇φ||_{∞} ≤ 3/(1 − s)λε_{h}. Define w_{ε_{h}} := φ z_{ε_{h}} + (1 − φ)v_{ε_{h}}. Since w_{ε_{h}} = v_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, λ ε_{h}) we have by Remark 2.2.4

m(v_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h})) ≤ F(w_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h})) ≤ F(z_{ε_{h}}; Q_{ν}(x_{0}, sλε_{h}))

+ C ∫_{Q_{ν} (x_{0},λ ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx + \frac{C}{(1 − s)^{p}(λε_{h})^{p}} ∫_{Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h})} |v_{ε_{h}}(x) − u_{x_{0},ν}(x)|^{p}dx

+ C ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(w_{ε_{h}})}(1 + |[w_{ε_{h}}]|) dH^{N−1}

≤ m(u_{x_{0},ν}; Q_{ν}(x_{0}, sλε_{h}))

+ (λε_{h})^{N} + C ∫_{Q_{ν} (x_{0},λ ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx + \frac{C}{(1 − s)^{p}(λε_{h})^{p}} ∫_{Q_{ν} (x_{0},λ ε_{h})} |v_{ε_{h}}(x) − u_{x_{0},ν}(x)|^{p}dx

+ C ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(w_{ε_{h}})}(1 + |[w_{ε_{h}}]|) dH^{N−1},

where we have used the facts that ||∇φ||_{∞} ≤ 3/(1 − s)λε_{h}. Dividing by (λ ε_{h})^{N−1} yields

\frac{m(v_{ε_{h}}(x); Q_{ν}(x_{0}, λ ε_{h}))}{(λ ε_{h})^{N−1}} ≤ \frac{m(u_{x_{0},ν}; Q_{ν}(x_{0}, sλε_{h}))}{(λ ε_{h})^{N−1}}

+ λε_{h} + \frac{C}{(λ ε_{h})^{N−1}} ∫_{Q_{ν} (x_{0},λ ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx

+ \frac{C}{(1 − s)^{p}(λε_{h})^{N−1+p}} ∫_{Q_{ν} (x_{0},λ ε_{h})} |v_{ε_{h}}(x) − u_{x_{0},ν}(x)|^{p}dx

+ \frac{C}{(λ ε_{h})^{N−1}} ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(w_{ε_{h}})}(1 + |[w_{ε_{h}}]|) dH^{N−1}.

We claim that the last three terms on the right hand side approach zero if we first let ε_{h} → 0^{+}, then s → 1^{−} and finally λ → 1^{−}. Indeed, for the first two integrals this follows from (2.1.15). To estimate the third integral, and since

H^{N−1}(S(v_{ε_{h}}) \ S(u)) = 0, |[v_{ε_{h}}]| ≤ |[u]| H^{N−1} a.e.,

we have that if x ∈ (S(w_{ε_{h}}) ∩ S(u_{x_{0},ν}))\S(u) then |[w_{ε_{h}}]| ≤ |[u_{x_{0},ν}]|, and if x ∈ S(w_{ε_{h}}) ∩ S(u) then

|[w_{ε_{h}}]| ≤ |[u_{x_{0},ν}]| + |[u]| ≤ C(1 + |[u]|).

Here we have used the fact that (see (3.8) in [13]) if u, v ∈ BV (Ω) then for H^{N−1} a.e. x ∈ S(u) ∩ S(v) we have ν_{u} = ν_{v} or ν_{u} = −ν_{v}, and, consequently, |[u + v]| ≤ |[u]| + |[v]|. Hence

\frac{1}{(λ ε_{h})^{N−1}} ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(w_{ε_{h}})}(1 + |[w_{ε_{h}}]|) dH^{N−1}

≤ \frac{C}{(λ ε_{h})^{N−1}} ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(u_{x_{0},ν})}(1 + |[u_{x_{0},ν}]|) dH^{N−1}

+ \frac{C}{(λ ε_{h})^{N−1}} ∫_{(Q_{ν} (x_{0},λ ε_{h})\Q_{ν} (x_{0},sλε_{h}))∩S(u)}(1 + |[u]|) dH^{N−1}.

Both these terms can be treated as in (2.4.4). Finally, by (2.4.1), (2.4.3)_{1}, and (2.4.5),

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) ≤ lim inf_{λ→1^{−}} lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u_{x_{0},ν}; Q_{ν}(x_{0}, sλε_{h}))}{(λ ε_{h})^{N−1}}

≤ lim sup_{ε→0^{+}} \frac{m(u_{x_{0},ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} = g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})).

We conclude this section by proving that equality holds in (2.4.1).

Lemma 2.4.2.

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) ≥ g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})).

Proof. The proof of this lemma is very similar to the previous one, and thus we will indicate only the main changes. Take ε_{h} → 0^{+} such that

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) = lim sup_{ε→0^{+}} \frac{m(u_{x_{0},ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} = lim_{ε_{h}→0^{+}} \frac{m(u_{x_{0},ν}; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ,  (2.4.6)

fix 0 < s < 1, and let z_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s ε_{h}); ℝ^{d}) be such that z_{ε_{h}} = v_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, sε_{h}) and

F(z_{ε_{h}}; Q_{ν}(x_{0}, sε_{h})) ≤ m(v_{ε_{h}}; Q_{ν}(x_{0}, sε_{h})) + ε_{h}^{N}.

Extend z_{ε_{h}} to be v_{ε_{h}} outside Q_{ν}(x_{0}, s ε_{h}) and consider a layer L of width (1−s)ε_{h}/2 and a smooth cut-off function φ such that 0 < φ < 1 in L, φ ≡ 1 on Q_{ν}(x_{0}, sε_{h}), φ = 0 on ∂Q_{ν}(x_{0}, ε_{h}) and ||∇φ||_{∞} ≤ 3/(1 − s)ε_{h}. Define w_{ε_{h}} := φ z_{ε_{h}} + (1 − φ)u_{x_{0},ν}. Since w_{ε_{h}} = u_{x_{0},ν} on a neighborhood of ∂Q_{ν}(x_{0}, ε_{h}), we obtain, as in the second part of the proof of the previous lemma, and in view of Remark 2.2.4,

\frac{m(u_{x_{0},ν}(x); Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ≤ \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N−1}}

+ ε_{h} + \frac{C}{ε_{h}^{N−1}} ∫_{Q_{ν} (x_{0},ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx + \frac{C}{(1 − s)^{p}ε_{h}^{N−1+p}} ∫_{Q_{ν} (x_{0},ε_{h})} |v_{ε_{h}}(x) − u_{x_{0},ν}(x)|^{p}dx

+ \frac{C}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},sε_{h}))∩S(w_{ε_{h}})}(1 + |[w_{ε_{h}}]|) dH^{N−1}.

As before, and also by (2.4.6), we may conclude that

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) ≤ lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N−1}} .  (2.4.7)

Let s_{j} → 1^{−} be such that

lim inf_{s→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, sε_{h}))}{ε_{h}^{N−1}} = lim_{s_{j} →1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, s_{j}ε_{h}))}{ε_{h}^{N−1}} ,  (2.4.8)

and, by virtue of (2.4.3), choose a subsequence of {ε_{h}} (not relabeled) and λ ∈ (0, 1) such that, for every j ∈ ℕ,

\begin{matrix}μ & (∂Q_{ν}(x_{0}, s_{j}λε_{h})) = H^{N−1}(∂Q_{ν}(x_{0}, s_{j}λε_{h}) ∩ S(v̄_{ε_{h}})) = 0, \\ lim_{ε_{h}→0^{+}} \frac{1}{(s_{j}ε)^{N−1}} ∫_{∂Q_{ν} (x_{0},s_{j} λ ε_{h})∩\{u≠v_{ε_{h}} \}}(1 + |u − v_{ε_{h}}|)d H^{N−1} = 0. \end{matrix}  (2.4.9)

Given z_{ε_{h}} ∈ SBV_{p}(Q_{ν}(x_{0}, s_{j}ε_{h}); ℝ^{d}) such that z_{ε_{h}} = u on a neighborhood of ∂Q_{ν}(x_{0}, s_{j}λε_{h}) and

F(z_{ε_{h}}; Q_{ν}(x_{0}, s_{j}λε_{h})) ≤ m(u; Q_{ν}(x_{0}, s_{j}λε_{h})) + (s_{j}ε_{h})^{N},

define

w_{ε_{h}} := \begin{cases}z_{ε_{h}} & in Q_{ν}(x_{0}, s_{j}λε_{h}), \\ v_{ε_{h}} & in Q_{ν}(x_{0}, s_{j}ε_{h})\\overline{Q_{ν}(x_{0}, s_{j}λε_{h})}. \end{cases}

Since w_{ε_{h}} = v_{ε_{h}} on a neighborhood of ∂Q_{ν}(x_{0}, s_{j}ε_{h}), we have by Remark 2.2.4,

\frac{m(v_{ε_{h}}; Q_{ν}(x_{0}, s_{j}ε_{h}))}{(s_{j}ε_{h})^{N−1}} ≤ \frac{m(u; Q_{ν}(x_{0}, s_{j}λε_{h}))}{(s_{j}ε_{h})^{N−1}} + s_{j}ε_{h}

+ \frac{C}{(s_{j}ε_{h})^{N−1}} ∫_{Q_{ν} (x_{0},s_{j} ε_{h})}(1 + |∇v_{ε_{h}}|^{p}) dx + \frac{C}{(s_{j}ε_{h})^{N−1}} ∫_{∂Q_{ν} (x_{0},s_{j} λ ε_{h})∩\{u≠v_{ε_{h}} \}}(1 + |u − v_{ε_{h}}|)d H^{N−1}

+ \frac{C}{(s_{j}ε_{h})^{N−1}} ∫_{(Q_{ν} (x_{0},s_{j} ε_{h})\Q_{ν} (x_{0},s_{j} λ ε_{h}))∩S(v_{ε_{h}})}(1 + |[v_{ε_{h}}]|) dH^{N−1}.

As before, and by (2.4.7)–(2.4.9), we conclude that

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) ≤ lim inf_{s_{j} →1^{−}} lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m(u; Q_{ν}(x_{0}, s_{j}λε_{h}))}{(s_{j}ε_{h})^{N−1}} = \frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}).

Remark 2.4.3. We remark that Lemma 2.4.1 and Lemma 2.4.2 are strongly hinged on the construction of the sequence {v_{ε}} in Lemma 2.1.8, which, in turn, depends on the choice of sign for ν_{u}(x_{0}) (see 2.1.19). Had we taken the normal with opposite sign, we would have obtained (see 2.1.15)

lim_{ε→0^{+}} \frac{1}{ε^{N−1+p}} ∫_{Q_{ν} (x_{0},ε)} |v_{ε}(x) − u_{x_{0},u^{−}(x_{0}),u^{+}(x_{0}),−ν}(x)|^{p} dx = 0.

Lemmas 2.4.1 and 2.4.2 would now yield

\frac{dF(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) = g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), −ν_{u}(x_{0})).

We conclude, therefore, that g(x_{0}, λ, θ, ν) = g(x_{0}, λ, θ, −ν) for L^{N} a.e. x ∈ Ω and all λ, θ ∈ ℝ^{d}, ν ∈ S^{N−1}.

2.5 Proofs of Theorems 1.0.2 and 1.0.3.

Proof of Theorem 1.0.2. To prove Theorem 1.0.2 we replace m(u; A), as defined in (1.0.1), by m_{1}(u; A), which is defined in (1.0.7). Here we take μ := L^{N}⌊Ω. Lemma 2.2.1 then continues to hold, with the exception that we need to show that the function v^{δ} defined in Lemma 3.3 of [17] now belongs to W^{1,p}(Ω; ℝ^{d}). Fix δ > 0 and let {Q_{i}^{δ}} be an admissible sequence, in the sense of the definition of m_{1}^{δ}(u; A), such that

∑_{i=1}^{∞} m_{1}(u; Q_{i}^{δ}) ≤ m_{1}^{δ}(u; A) + δ.  (2.5.1)

By the definition of m_{1} we can choose v_{i}^{δ} ∈ W^{1,p}(Ω; ℝ^{d}) such that

v_{i}^{δ} = u on ∂Q_{i}^{δ},  F_{1}(v_{i}^{δ}; Q_{i}^{δ}) ≤ m_{1}(u; Q_{i}^{δ}) + δ L^{N}(Q_{i}^{δ}).  (2.5.2)

For each k ∈ ℕ set

v^{δ,k} := ∑_{i=1}^{k} v_{i}^{δ}χ_{Q_{i}^{δ}} + u χ_{Ω\∪_{j=1}^{k}Q_{j}^{δ}}.  (2.5.3)

Then v^{δ,k} ∈ W^{1,p}(Ω; ℝ^{d}), and by Poincaré’s inequality, and since v_{i}^{δ} = u on ∂Q_{i}^{δ}, we deduce that

\begin{aligned}||v^{δ,k} − u||_{W^{1,p}(Ω)}^{p} & ≤ ∑_{i=1}^{k} ||v_{i}^{δ} − u||_{W^{1,p}(Q_{i}^{δ})}^{p} ≤ C(N, δ) ∑_{i=1}^{k} ||∇v_{i}^{δ} − ∇u||_{L^{p}(Q_{i}^{δ})}^{p} \\ & ≤ C(N, δ)(m_{1}^{δ}(u; A) + δ + ||∇u||_{L^{p}(A)}^{p}), \end{aligned}

where we have used the coercitivity hypothesis (1.0.6) for F_{1}, (2.5.1) and (2.5.2). Hence, for fixed δ, the sequence {v^{δ,k}} is bounded in W^{1,p}(Ω; ℝ^{d}), and thus it converges weakly to some function v^{δ} ∈ W^{1,p}(Ω; ℝ^{d}). Clearly

v^{δ} := ∑_{i=1}^{∞} v_{i}^{δ}χ_{Q_{i}^{δ}} + u χ_{Ω\∪_{j=1}^{∞}Q_{j}^{δ}}.

We may now proceed as in Lemma 3.3 of [17].

Lemma 2.2.3 of [17] continues to hold in the present context, since the function w defined in (2.2.1) clearly belongs to W^{1,p}(Ω; ℝ^{d}), as u and v coincide on a neighborhood of ∂A_{δ} (contained in A_{δ}). In turn, also Lemma 2.2.1 continues to hold.

The proofs of Lemmas 2.3.1 and 2.3.2 become significantly simpler in the Sobolev case. Indeed, since H^{N−1}(S(u) ∩ Ω) = 0 then by Remark 2.1.2 the function ū_{ε} defined in Lemma 2.1.7 coincides with u_{ε}. Consequently the first part of Lemma 2.3.1, up to (2.3.4), and the second part of Lemma 2.3.2, after (2.3.7), are not needed in this case.

Proof of Theorem 1.0.3. This time we replace m(u; A) by m_{2}(u; A), which is defined in (1.0.8). Clearly the functions v^{δ,k} defined in (2.5.3) belong to BV (Ω; T), and so does their limit v^{δ} as k → ∞.

Lemmas 2.2.1 and 2.2.2 continue to hold, while Lemmas 2.4.1 and 2.4.2 should be replaced by Lemma 2.5.1 below.

Lemma 2.5.1.

\frac{dF_{2}(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) = g_{2}(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})),

where g_{2} is the function defined in Theorem 1.0.3.

Proof. Let x_{0} satisfy (2.1.18), and set ν := ν_{u}(x_{0}). By Lemma 2.2.2 for H^{N−1} a.e. x_{0} ∈ Ω ∩ S(u)

\frac{dF_{2}(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) = lim_{ε→0^{+}} \frac{F_{2}(u; Q_{ν}(x_{0}, ε))}{ε^{N−1}} = lim_{ε→0^{+}} \frac{m_{2}(u; Q_{ν}(x_{0}, ε))}{ε^{N−1}} < ∞.

Since T is finite,

d := min{|u_{1} − u_{2}| : u_{1}, u_{2} ∈ T, u_{1} ≠ u_{2}} > 0.

Hence, by (2.1.18),

\begin{aligned}0 & ≤ \frac{1}{ε^{N−1}} ∫_{0}^{1} ∫_{∂Q_{ν} (x_{0},λ ε)∩\{u≠u_{x_{0},ν} \}}(1 + |u − u_{x_{0},ν}|)d H^{N−1}d λ \\ & ≤ \frac{1/d + 1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u − u_{x_{0},ν}| dx → 0 as ε → 0^{+}. \end{aligned}

Thus for any sequence ε_{h} → 0^{+} we may find a subsequence (not relabeled) such that for L^{1} a.e λ ∈ (0, 1)

μ(∂Q^{ν}(x^{0}, λε^{h})) = 0, lim_{ε_{h}→0^{+}} \frac{1}{ε_{h}^{N−1}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u≠u_{x_{0},ν} \}}(1 + |u − u^{x_{0},ν}|)d H^{N−1} = 0.  (2.5.4)

Fix a sequence ε_{h} → 0^{+} and λ ∈ (0, 1) for which (2.5.4) holds. Given z_{ε_{h}} ∈ BV (Q_{ν}(x_{0}, ε_{h}); T) such that z_{ε_{h}} = u_{x_{0},ν} on a neighborhood of ∂Q_{ν}(x_{0}, λε_{h}) and

F_{2}(z_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) ≤ m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N},

define

w_{ε_{h}} := \begin{cases}z_{ε_{h}} & in Q_{ν}(x_{0}, λε_{h}), \\ u & in Q_{ν}(x_{0}, ε_{h})\\overline{Q_{ν}(x_{0}, λε_{h})}. \end{cases}

Since w_{ε_{h}} = u on ∂Q_{ν}(x_{0}, ε_{h}) we have by Remark 2.2.4

m_{2}(u; Q_{ν}(x_{0}, ε_{h})) ≤ F_{2}(w_{ε_{h}}; Q_{ν}(x_{0}, ε_{h})) ≤ F_{2}(z_{ε_{h}}; Q_{ν}(x_{0}, λ ε_{h}))
+ C ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u≠u_{x_{0},ν} }}(1 + |u − u_{x_{0},ν}|)d H^{N−1} + C ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|)dH^{N−1}
≤ m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N}
+ C ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩{u≠u_{x_{0},ν} }}(1 + |u − u_{x_{0},ν}|)dH^{N−1} + C ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|) dH^{N−1}.

Dividing by ε_{h}^{N−1} yields

\frac{m_{2}(u; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ≤ \frac{m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} + ε_{h}

+ \frac{C}{ε_{h}^{N−1}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u≠u_{x_{0},ν} \}}(1 + |u − u_{x_{0},ν}|)dH^{N−1}

+ \frac{C}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u)}(1 + |[u]|)dH^{N−1}.

We can now proceed as in the first part of Lemma 2.4.1 to conclude that

\frac{dF_{2}(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}) ≤ lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} ≤ g_{2}(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})).

To prove the converse inequality take ε_{h} → 0^{+} such that

g_{2}(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) = lim sup_{ε→0^{+}} \frac{m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} = lim_{ε_{h}→0^{+}} \frac{m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ,

Extract a subsequence of {ε_{h}} (not relabeled) and fix λ ∈ (0, 1) for which (2.5.4) holds. Choosing admissible test functions z_{ε_{h}} ∈ BV (Q_{ν}(x_{0}, ε_{h}); T) such that z_{ε_{h}} = u on a neighborhood of ∂Q_{ν}(x_{0}, λε_{h}) and

F_{2}(z_{ε_{h}}; Q_{ν}(x_{0}, λε_{h})) ≤ m_{2}(u; Q_{ν}(x_{0}, λε_{h})) + ε_{h}^{N},

define

w_{ε_{h}} := \begin{cases}z_{ε_{h}} & in Q_{ν}(x_{0}, λε_{h}), \\ u_{x_{0},ν} & in Q_{ν}(x_{0}, ε_{h})\\overline{Q_{ν}(x_{0}, λε_{h})}. \end{cases}

Since w_{ε_{h}} = u_{x_{0},ν} on ∂Q_{ν}(x_{0}, ε_{h}) we have by Remark 2.2.4

\frac{m_{2}(u_{x_{0},ν}; Q_{ν}(x_{0}, ε_{h}))}{ε_{h}^{N−1}} ≤ \frac{m_{2}(u; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} + ε_{h}

+ \frac{C}{ε_{h}^{N−1}} ∫_{∂Q_{ν} (x_{0},λ ε_{h})∩\{u≠u_{x_{0},ν} \}}(1 + |u − u_{x_{0},ν}|)d H^{N−1}

+ \frac{C}{ε_{h}^{N−1}} ∫_{(Q_{ν} (x_{0},ε_{h})\Q_{ν} (x_{0},λ ε_{h}))∩S(u_{x_{0},ν})}(1 + |[u_{x_{0},ν}]|) dH^{N−1}.

As before, we conclude that

g_{2}(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) ≤ lim inf_{λ→1^{−}} lim inf_{ε_{h}→0^{+}} \frac{m_{2}(u; Q_{ν}(x_{0}, λε_{h}))}{ε_{h}^{N−1}} = \frac{dF_{2}(u; ·)}{dH^{N−1}⌊S(u)} (x_{0}).

3 Part II: Applications

3.1 Preliminary Results.

In the next theorem we use the notation x = (x′, x_{N}) ∈ ℝ^{N} and ν = (ν′, ν_{N}) ∈ S^{N−1}, where x′, ν′ ∈ ℝ^{N−1} and x_{N}, ν_{N} ∈ ℝ.

Theorem 3.1.1 (Slicing Theorem). Let u ∈ SBV_{p}(ℝ^{N}; ℝ^{d}). Then for H^{1} a.e. t ∈ ℝ the function u^{t}(x′) := u(x′, t) ∈ SBV_{p}(ℝ^{N−1}; ℝ^{d}) with ∇_{x′}u^{t}(x′) = ∇_{x′}u(x′, t),

S(u^{t}) = {x′ ∈ ℝ^{N−1} : (x′, t) ∈ S(u)},

and if H^{N−2}(S(u^{t})) > 0 then for H^{N−2} a.e. x′ ∈ S(u^{t}) we have (u^{t}(x′))^{±} = u^{±}(x′, t) and

ν′_{u^{t}(x′)} = \frac{ν′_{u(x′,t)}}{|ν′_{u(x′,t)}|} .

Moreover, if g : ℝ^{N} → [0, +∞) is a Borel measurable function, then for every open set A ⊂ ℝ^{N} we have

∫_{A} g(x) dμ(x) ≥ ∫_{ℝ} (∫_{A_{t}} g(x′, t)dμ_{t}(x′)) dt,  (3.1.1)

where

A_{t} := {x′ ∈ ℝ^{N−1} : (x′, t) ∈ A},

μ := C_{1} |Du| + C_{2}H^{N−1}⌊S(u), μ_{t} := C_{1} |D_{x′}u_{t}| + C_{2}H^{N−2}⌊S(u_{t}), with C_{1}, C_{2} ≥ 0. In particular

∫_{S(u)∩A} g(x) dH^{N−1} ≥ ∫_{ℝ} (∫_{S(u^{t})∩A_{t}} g(x′, t)dH^{N−2}(x′)) dt.  (3.1.2)

Proof. A detailed proof of the first part of the theorem in the BD (Bounded Deformation) setting may be found in Proposition 3.4 and in the proof of Theorem 4.5 of [10]. The BV case is analogous and rests on Theorem 3.3 of [44] (see also page 204 of [12]). To prove the second part note that, following the proof of Theorem 3.3 of [44], it may be shown that

∫_{A} |Du| ≥ ∫_{A} |D_{x′}u| = ∫_{ℝ} (∫_{A_{t}} |D_{x′}u_{t}|) dt.  (3.1.3)

This inequality is straightforward if u ∈ C^{1} (ℝ^{N}; ℝ^{d}), and the general case where u ∈ BV (ℝ^{N}; ℝ^{d}) follows by a regularization argument.

By rewriting (3.1.3) as

∫_{ℝ^{N}} χ_{A}(x) d|Du|(x) ≥ ∫_{ℝ} (∫_{{x_{n}=t}} χ_{A}(x)d|D_{x′}u_{t}|(x′)) dt,

one may extend the validity of (3.1.3) to characteristic functions of Borel sets B ⊂ ℝ^{N}, to simple functions and then to any non negative Borel measurable function h. Hence we have

∫_{A} h(x) d|Du|(x) ≥ ∫_{ℝ} (∫_{A_{t}} h(x′, t)d|D_{x′}u_{t}|(x′)) dt.  (3.1.4)

Let now g : ℝ^{N} → [0, +∞) be a Borel measurable function, and set

h(x) := \frac{g(x)}{|[u(x)]|} χ_{S(u)∩A}(x).

Applying (3.1.4) to the function g, we can see easily that the left-hand side of the inequality reduces to

∫_{S(u)∩A} g(x) dH^{N−1},

while the right-hand side reads

∫_{ℝ} (∫_{A_{t}} \frac{g(x′, t)}{|[u(x′, t)]|} χ_{S(u)∩A}(x′, t)d|D_{x′}u_{t}|(x′)) dt.

Recalling that

χ_{S(u)∩A}(x′, t) = χ_{S(u^{t})∩A_{t}}(x′),
d|D_{x′}u_{t}| = |∇_{x′}u^{t}(x′)| L^{N−1} + |[u^{t}(x′)]|H^{N−2}⌊S(u^{t}),

and that

|[u(x′, t)]| = |[u^{t}(x′)]| for H^{N−2} a.e. x′ ∈ S(u^{t}),

we conclude that

∫_{ℝ} (∫_{A_{t}} \frac{g(x′, t)}{|[u(x′, t)]|} χ_{S(u)∩A}(x′, t)d|D_{x′}u_{t}|(x′)) dt

= ∫_{ℝ} (∫_{S(u^{t})∩A_{t}} \frac{g(x′, t)}{|[u^{t}(x′)]|} (|∇_{x′}u^{t}(x′)| dx′ + |[u^{t}(x′)]| dH^{N−2})) dt

= ∫_{ℝ} (∫_{S(u^{t})∩A_{t}} g(x′, t)dH^{N−2}(x′)) dt.

This establishes (3.1.2), which, together with (3.1.4) now yields (3.1.1).

We recall that a locally bounded, Borel measurable function f : ℝ^{d×N} → ℝ is said to be quasiconvex if

∫_{(0,1)^{N}} f (ξ + ∇φ(y)) dy ≥ f(ξ)

for all ξ ∈ ℝ^{d×N}, φ ∈ W_{0}^{1,∞}((0, 1)^{N}; ℝ^{d}). The quasiconvex envelope Qf of a function f : ℝ^{d×N} → ℝ is defined as

Qf := sup{G : G is quasiconvex, G ≤ f}.

If f is Borel measurable and 0 ≤ f(ξ) ≤ C(1 + |ξ|^{p}) for some p ≥ 1 then it can be shown that

Qf(ξ) = inf {∫_{(0,1)^{N}} f (ξ + ∇φ(y)) dy : φ ∈ W_{0}^{1,∞}((0, 1)^{N}; ℝ^{d})}.

For a detailed study of quasiconvexity we refer the reader to [30].

We say that a locally bounded, Borel function g : ℝ^{2d} ×S^{N−1} → ℝ is BV-elliptic if for all (a, b, ν) ∈ ℝ^{2d} ×S^{d−1} and for any finite subset T of ℝ^{d} we have

∫_{S(w)∩Q_{ν}} g(w^{+}, w^{−}, ν_{w}) dH^{N−1} ≥ g(a, b, ν)

for all w ∈ BV (Q_{ν}; T) such that w = u_{a,b,ν} on ∂Q_{ν}, where u_{a,b,ν} := u_{0,a,b,ν}, i.e.

u_{a,b,ν}(x) := \begin{cases}a & if x · ν > 0, \\ b & if x · ν ≤ 0. \end{cases}

Given a function g : ℝ^{2d} × S^{N−1} → ℝ we define Rg as the greatest BV -elliptic function less than g. If g is continuous and satisfies a growth condition of the type (1.0.9) then it can be shown that (see [22], [29])

Rg(a, b, ν) = inf{∫_{S(w)∩Q_{ν}} g(w^{+}, w^{−}, ν_{w}) dH^{N−1} : w ∈ SBV_{0}(Q_{ν}; ℝ^{d}) ∩ L^{∞}(Q_{ν}; ℝ^{d}), w = u_{a,b,ν} on ∂Q_{ν}},

where, we recall, the space SBV_{0}(Ω; ℝ^{d}) is defined by

SBV_{0}(Ω; ℝ^{d}) := {w ∈ SBV (Ω; ℝ^{d}) : ∇w = 0 a.e. in Ω and H^{N−1}(S(w) ∩ Ω) < ∞}.

We say that f : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞] is a normal integrand if for L^{N} a.e. x ∈ Ω the function f(x, ·, ·) is lower semicontinuous and there exists a Borel integrand f̃ : Ω×ℝ^{N} ×ℝ^{d×N} → [0, +∞] such that f(x, ·, ·) = f̃(x, ·, ·) for L^{N} a.e. x ∈ Ω.

The following lower semicontinuity result was proved by Ambrosio in [6] and [7] in the case where the integrand f_{0} is Carathéodory and g_{0} continuous, and by Kristensen [42] when f_{0} is a normal integrand. For the convenience of the reader, we present a slightly modified version of Ambrosio’s proof in the Appendix.

Theorem 3.1.2. Assume that f_{0} : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞) is a normal integrand, quasiconvex in ξ, and that g_{0} : Ω × (ℝ^{d})^{2} × S^{N−1} → [0, +∞) is a BV -elliptic function, lower semicontinuous in x and continuous in the remaining variables. Suppose also that the following growth conditions are satisfied:

0 ≤ f_{0}(x, u, ξ) ≤ C(1 + |ξ|^{p}) and 0 ≤ g_{0}(x, λ, θ, ν) ≤ C (1 + |λ − θ|)  (3.1.5)

for L^{N} a.e. x ∈ Ω, all u, λ, θ ∈ ℝ^{d}, ξ ∈ ℝ^{d×N}, ν ∈ S^{N−1}, and for some C > 0. Then for every u ∈ SBV_{p}(Ω; ℝ^{d}) and any sequence {u_{n}} ⊂ SBV_{p}(Ω; ℝ^{d}) converging to u in L^{1}(Ω; ℝ^{d}) and such that

sup_{n} (∫_{Ω} |∇u_{n}|^{p} dx + ∫_{S(u_{n})∩Ω} (1 + |[u_{n}]|) dH^{N−1}) < ∞,  (3.1.6)

we have

∫_{Ω} f_{0}(x, u, ∇u) dx + ∫_{S(u)∩Ω} g_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1}

≤ lim inf_{n→∞} (∫_{Ω} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩Ω} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1}) .

Consider a functional F : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞]. For any u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) define

J (u; A) := inf_{{u_{n}}}{lim inf_{n→∞} F (u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{1}(A; ℝ^{d})},

J_{p}(u; A) := inf_{{u_{n}}}{lim inf_{n→∞} F (u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{p}(A; ℝ^{d})}.

(3.1.7)

Lemma 3.1.3. Let F : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞] be a functional satisfying conditions (H_{1}), (H_{2}) and (H_{4}). Then for any u ∈ SBV_{p}(Ω; ℝ^{d}) ∩ L^{p}(Ω; ℝ^{d})

J_{p}(u; A) = inf_{{u_{n}}}{lim inf_{n→∞} F (u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{p}(A; ℝ^{d}), u_{n} = u on a neighborhood of ∂A}.

Proof. Fix u ∈ SBV_{p}(Ω; ℝ^{d}) and write

J_{p}^{∗}(u; A) := inf_{{u_{n}}}{lim inf_{n→∞} F (u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{p}(A; ℝ^{d}), u_{n} = u on a neighborhood of ∂A}.

It is clear that J_{p}(u; A) ≤ J_{p}^{∗}(u; A). We now prove the converse inequality. For every v ∈ SBV_{p}(Ω; ℝ^{d}) define

G_{p}(v; A) := ∫_{A} (1 + |∇v|^{p}) dx + ∫_{A∩S(v)}(1 + |[v]|) dH^{N−1}.

Given ε > 0 and consider a sequence {u_{n}} in SBV_{p}(Ω; ℝ^{d}) such that u_{n} → u in L^{p}(A; ℝ^{d}) and

lim inf_{n→+∞} F (u_{n}; A) ≤ J_{p}(u; A) + ε.

Due to the coercivity hypothesis (H_{4}), we may extract a subsequence {u_{n_{k}}} of {u_{n}} such that

lim_{k→+∞} F (u_{n_{k}}; A) = lim inf_{n→+∞} F (u_{n}; A)  (3.1.8)

and the sequence of measures ν_{k} := G_{p}(u; ·) + G_{p}(u_{n_{k}}; ·) converges ⋆-weakly to some Radon measure ν. Denoting, for every t > 0, A_{t} := {x ∈ A| dist(x, ∂A) > t}, we fix some η > 0 and for every 0 < δ < η we define the subsets L_{δ} := A_{η−2δ} \ \overline{A_{η+δ}}. Consider a smooth cut-off function φ_{δ} ∈ C_{0}^{∞}(A_{η−δ}; [0, 1]) such that φ_{δ} = 1 on A_{η}. As the thickness of the strip L_{δ} is of order δ, we have an upper bound of the form ∥∇φ_{δ}∥_{L^{∞}(A_{η−δ})} ≤ C/δ. Define

w_{n} := u_{n}φ_{δ} + u(1 − φ_{δ}).

Clearly this sequence converges to u in L^{p}(A; ℝ^{d}) and satisfies w_{n} = u on ∂A. On the other hand, by (H_{1}), the locality hypothesis (H_{2}), and the growth condition (H_{4}), we have the estimate

\begin{aligned}F (w_{n_{k}}; A) & ≤ F (w_{n_{k}}; A_{η}) + F (w_{n_{k}}; A \ \overline{A_{η−δ}}) + F (w_{n_{k}}; A_{η−2δ} \ \overline{A_{η+δ}}) \\ & ≤ F (u_{n_{k}}; A_{η}) + F (u; A \ \overline{A_{η−δ}}) + C (G_{p}(u_{n_{k}}; L_{δ}) + G_{p}(u; L_{δ}) + \frac{1}{δ^{p}} ∫_{L_{δ}} |v_{n} − u|^{p} dx) . \end{aligned}  (3.1.9)

Then by (3.1.8)

lim inf_{k→∞} F (w_{n_{k}}; A) ≤ lim_{k→∞} F (u_{n_{k}}; A) + Cν(A \ \overline{A_{η−δ}}) + Cν(\overline{L_{δ}})
≤ J_{p}(u; A) + ε + Cν(u; A \ \overline{A_{η−δ}}) + Cν(\overline{L_{δ}}).

Letting δ go to zero we obtain

J_{p}^{∗}(u; A) ≤ J_{p}(u; A) + ε + Cν(A \ A_{η}) + Cν(∂A_{η}).

Choose a subsequence {η_{k}} such that η_{k} → 0^{+} and ν(∂A_{η_{k}}) = 0. By letting first k → ∞ and then ε → 0^{+} we conclude that J_{p}^{∗}(u; A) ≤ J_{p}(u; A).

Remark 3.1.4. Given a sequence of functionals F_{n} : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞], for any u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) define Γ^{−} and Γ_{p}^{−} as the Γ-lower limits of the sequence {F_{n}} with respect to the L^{1}(A; ℝ^{d}) and L^{p}(A; ℝ^{d}) topology, respectively, that is

Γ^{−}(u; A) := inf_{{u_{n}}}{lim inf_{n→∞} F_{n}(u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{1}(A; ℝ^{d})},

Γ_{p}^{−}(u; A) := inf_{{u_{n}}}{lim inf_{n→∞} F_{n}(u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), u_{n} → u in L^{p}(A; ℝ^{d})}.

(3.1.10)

Assume that the sequence {F_{n}} satisfies conditions (H_{1}), (H_{2}) and (H_{4}) uniformly in n. Then Lemma 3.1.3 still holds for H_{p}.

The next lemma is essentially due to Braides, Defranceschi and Vitali [22] and is based on De Giorgi’s Slicing Lemma (see also [15] and [28]). In what follows we use the notation.

∂_{l}u := \frac{∂u}{∂x_{l}} , l = 1, …, N.

Lemma 3.1.5. Let F : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞] be a functional satisfying condition (H_{1}). Assume that there exist C, C_{1}, … C_{N}, D_{1}, … D_{N} > 0 such that

0 ≤ F (u; A) ≤ C(|A| + ∑_{l=1}^{N} C_{l} ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1})

for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ ×A(Ω), and for every u, v ∈ SBV_{p}(Ω; ℝ^{d})

F (u; B) = F (v; B) whenever u = v H^{N−1} a.e. on a Borel set B ⊂ Ω.  (3.1.11)

Given j integers {λ_{1}, …, λ_{j}}, with 0 < λ_{1} < … < λ_{j}, for any u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) we may find an integer i ∈ {1, …, j − 1} such that

\begin{aligned}F (τ_{i} & (u); A) ≤ F (u; A) \\ & + \frac{C}{j − 1} (|A| + ∑_{l=1}^{N} C_{l} ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) + \frac{C}{e^{λ_{1}}} ∫_{A} |u| dx, \end{aligned}

where τ_{i} : ℝ^{d} → ℝ^{d} is a smooth truncation function such that

τ_{i}(z) := \{ \begin{matrix}z & if |z| ≤ e^{λ_{i}}, \\ 0 & if |z| > e^{λ_{i+1}}. \end{matrix}

and |∇τ_{i}(z)| ≤ C for all z ∈ ℝ^{d}.

Proof. Given v ∈ SBV_{p}(Ω; ℝ^{d}) define

G(v; A) := |A| + ∑_{l=1}^{N} C_{l} ∫_{A} |∂_{l}v|^{p} dx + ∑_{l=1}^{N} D_{l} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}.  (3.1.12)

For each integer i ∈ {1, …, j − 1} set w_{i}(x) := τ_{i}(u(x)); then the following formula holds (see e.g. [11])

Dw_{i} = ∇τ_{i}(u) ∇u L^{N} + (τ_{i}(u^{+}) − τ_{i}(u^{−})) ⊗ H^{N−1}⌊S(u).

Since |∇τ_{i}(z)| ≤ C, for every l = 1, …, N we have that |∂_{l}w_{i}(x)| ≤ C|∂_{l}u| L^{N} a.e. on Ω and |τ_{i}(u^{+}) − τ_{i}(u^{−})| ≤ C|u^{+} − u^{−}|. Using the facts that τ_{i}(u(x)) = u(x) H^{N−1} a.e. in the Borel set {x ∈ A : |u| ≤ e^{λ_{i}}} and that in the Borel set {x ∈ A : |u| ≥ e^{λ_{i+1}}} we have ∇τ_{i}(u(x)) = 0 L^{N} a.e. and τ_{i}(u(x)) = 0 H^{N−1} a.e., we get

F (w_{i}; A) = F (w_{i}; {x ∈ A : |u| ≤ e^{λ_{i}}}) + F (w_{i}; {x ∈ A : e^{λ_{i}} < |u| < e^{λ_{i+1}}})
+ F (w_{i}; {x ∈ A : |u| ≥ e^{λ_{i+1}}})
≤ F (u; A) + CG(u; {x ∈ A : e^{λ_{i}} < |u| < e^{λ_{i+1}}}) + C |{x ∈ A : |u| ≥ e^{λ_{i+1}}}| .

Since

|\{x ∈ A : |u| ≥ e^{λ_{i+1}}\}| ≤ \frac{1}{e^{λ_{i+1}}} ∫_{A} |u| dx

we have

\frac{1}{j − 1} ∑_{i=1}^{j−1} F (w_{i}; A) ≤ F (u; A) + \frac{C}{j − 1} G(u; A) + \frac{C}{e^{λ_{j}}} ∫_{A} |u| dx.

It now suffices to choose i_{j} ∈ {1, …, j} such that

F (w_{i_{j}}; A) ≤ \frac{1}{j − 1} ∑_{i=1}^{j−1} F (w_{i}; A).

Remark 3.1.6. The previous proof continues to work if instead of positive integers we consider a set of positive real numbers {λ_{1}, …, λ_{j}}, with 0 < λ_{1} < … < λ_{j} and such that λ_{i+1} − λ_{i} ≥ α > 0 for all i ∈ {1, …, j − 1}. In this case ||∇τ_{i}(z)||_{∞} ≤ C(α).

Lemma 3.1.7. (cf. [15]). Let F satisfy conditions (H_{1}), (3.1.11). Further assume that

\begin{aligned}\frac{1}{C} (∑_{l=1}^{N} C_{l} & ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) \\ & ≤ F (u; A) ≤ C(|A| + ∑_{l=1}^{N} C_{l} ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) \end{aligned}  (3.1.13)

for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ ×A(Ω), and for some positive constants C, C_{1}, … C_{N}, D_{1}, … D_{N} > 0. Then for any u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω) we have

J (u; A) = lim_{j→∞} J (τ_{j}(u); A),

where

τ_{j}(z) := \{ \begin{matrix}z & if |z| ≤ e^{j}, \\ 0 & if |z| > e^{j+1}. \end{matrix}

Moreover, for any u ∈ SBV_{p}(Ω; ℝ^{d}) ∩ L^{∞}(Ω; ℝ^{d}) we have

J (u; A) = J_{p}(u; A) = inf_{{v_{n}}}{lim inf_{n→∞} F (v_{n}; A) : v_{n} ∈ SBV_{p}(A; ℝ^{d}), sup_{n} ∥v_{n}∥_{L^{∞}(A;ℝ^{d})} < ∞, v_{n} → u in L^{p}(A; ℝ^{d})}.

Proof. Fix an integer j ∈ ℕ and let {u_{m}} ⊂ SBV_{p}(A; ℝ^{d}) be a sequence converging to u in L^{1}(A; ℝ^{d}) and such that

lim inf_{m→+∞} F (u_{m}; A) ≤ J (u; A) + \frac{1}{j} .

Upon extracting a subsequence {u_{m_{n}}} of {u_{m}}, if necessary, we may assume that

lim_{n→+∞} F (u_{m_{n}}; A) = lim inf_{m→+∞} F (u_{m}; A) < ∞.  (3.1.14)

For simplicity in the notation we write u_{n} instead of u_{m_{n}}. By (3.1.13) and the fact that u_{n} → u in L^{1}(A; ℝ^{d}), we may find a constant M > 0 (independent of j) such that

sup_{n} (∫_{A} |u_{n}| dx + G(u_{n}; A)) ≤ M,

where G has been introduced in (3.1.12). Let {λ_{n}} be an increasing sequence of positive integers such that λ_{n} → ∞. Applying Lemma 3.1.5 with the integers {λ_{j+1}, …, λ_{2j}} to each u_{n}, we may find i_{n} ∈ {j + 1, …, 2j − 1} such that

F (τ_{i_{n}}(u_{n}); A) ≤ F (u_{n}; A) + \frac{C}{j − 1} G(u_{n}; A) + \frac{C}{e^{λ_{j+1}}} ∫_{A} |u_{n}| dx ≤ F (u_{n}; A) + \frac{CM}{j − 1} + \frac{CM}{e^{λ_{j+1}}} ,

where

τ_{i_{n}}(z) := \{ \begin{matrix}z & if |z| ≤ e^{λ_{in}} , \\ 0 & if |z| > e^{λ_{in+1}}. \end{matrix}

Since i_{n} ∈ {j + 1, …, 2j − 1} for all n ∈ ℕ, there must exist an integer l_{j} ∈ {j + 1, …, 2j − 1} such that i_{n} = l_{j} for infinitely indices n ∈ ℕ. Select, therefore, an increasing sequence {n_{k}} of integers such that i_{n_{k}} = l_{j} for all k. Then by (3.1.14)

lim inf_{n→∞} F (τ_{l_{j}} (u_{n}); A) ≤ lim inf_{k→∞} F (τ_{l_{j}} (u_{n_{k}}); A)

≤ lim_{k→∞} F (u_{n_{k}}; A) + \frac{CM}{j − 1} + \frac{CM}{e^{λ_{j+1}}}

≤ J (u; A) + \frac{1}{j} + \frac{CM}{j − 1} + \frac{CM}{e^{λ_{j+1}}} .

(3.1.15)

Clearly {τ_{l_{j}} (u_{n})} converges in L^{1}(A; ℝ^{d}) to τ_{l_{j}} (u), and a standard diagonalization argument shows that J (·; A) is L^{1}-lower semicontinuous, and so, by (3.1.15),

J (τ_{l_{j}} (u); A) ≤ lim inf_{n→∞} J (τ_{l_{j}} (u_{n}); A) ≤ lim inf_{n→∞} F (τ_{l_{j}} (u_{n}); A) ≤ J (u; A) + \frac{1}{j} + \frac{CM}{j − 1} + \frac{CM}{e^{λ_{j+1}}} .

Since {τ_{l_{j}} (u)} converges to u in L^{1}(A; ℝ^{d}) as j → ∞, we get

J (u; A) ≤ lim inf_{j→∞} J (τ_{l_{j}} (u); A) ≤ lim sup_{j→∞} J (τ_{l_{j}} (u); A) ≤ J (u; A).

Since {λ_{n}} was arbitrary, this proves the first part of the Lemma.

To prove the second part of the lemma, assume that u ∈ SBV_{p}(Ω; ℝ^{d}) ∩ L^{∞}(Ω; ℝ^{d}), consider an increasing sequence of positive integers {λ_{n}} such that λ_{n} → ∞ and fix j ∈ ℕ such that e^{λ_{j}} > ∥u∥_{L^{∞}(A;ℝ^{d})} + 1. Note that τ_{i_{n}}(u(x)) ≡ u(x) for all x ∈ A and that, by construnction, ∥τ_{i_{n}}(u_{n})∥_{L^{∞}(A;ℝ^{d})} ≤ e^{λ_{2j}}. Using the fact that |∇τ_{i_{n}}(z)| ≤ C for all z ∈ ℝ^{d}, we get

\begin{aligned}|τ_{i_{n}}(u_{n}) − u|^{p} = |τ_{i_{n}}(u_{n}) − τ_{i_{n}}(u)|^{p} & ≤ (1 + |τ_{i_{n}}(u_{n})|^{p−1} + |τ_{i_{n}}(u)|^{p−1}) |τ_{i_{n}}(u_{n}) − τ_{i_{n}}(u)| \\ & ≤ C (1 + 2e^{(p−1)λ_{2j}} ) |u_{n} − u|, \end{aligned}

and so ∥τ_{i_{n}}(u_{n}) − u∥_{L^{p}(A;ℝ^{d})} → 0. Thus, by (3.1.15),

inf_{{v_{n}}}{lim inf_{n→∞} F (v_{n}; A) : v_{n} ∈ SBV_{p}(A; ℝ^{d}), sup_{n} ∥v_{n}∥_{L^{∞}(A;ℝ^{d})} < ∞, v_{n} → u in L^{p}(A; ℝ^{d})}

≤ J (u; A) + \frac{1}{j} + \frac{CM}{j − 1} + \frac{CM}{e^{λ_{j+1}}} .

By letting j → ∞ we obtain

J_{∞}(u; A) := inf_{{v_{n}}}{lim inf_{n→∞} F (v_{n}; A) : v_{n} ∈ SBV_{p}(A; ℝ^{d}), sup_{n} ∥v_{n}∥_{L^{∞}(A;ℝ^{d})} < ∞, v_{n} → u in L^{p}(A; ℝ^{d})} ≤ J (u; A).

The converse inequality is trivial, since

J (u; A) ≤ J_{p}(u; A) ≤ J_{∞}(u; A).

Remark 3.1.8. Since (H_{4}) and (3.1.13) are clearly equivalent, it is easy to see from Lemmas 3.1.3 and 3.1.7 that for any u ∈ SBV_{p}(Ω; ℝ^{d}) ∩ L^{∞}(Ω; ℝ^{d}) we have

J (u; A) = J_{p}(u; A) = inf_{{u_{n}}}{lim inf_{n→∞} F (u_{n}; A) : u_{n} ∈ SBV_{p}(A; ℝ^{d}), sup_{n} ∥u_{n}∥_{L^{∞}(A;ℝ^{d})} < ∞,
u_{n} → u in L^{p}(A; ℝ^{d}), u_{n} = u on a neighborhood of ∂A}.

Remark 3.1.9. Given a sequence of functionals F_{n} : SBV_{p}(Ω; ℝ^{d}) × A(Ω) → [0, ∞] satisfying conditions (H_{1}), (H_{2}) and

\begin{aligned}\frac{1}{C} (∑_{l=1}^{N} C_{l,n} & ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l,n} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) \\ & ≤ F_{n}(u; A) ≤ C(|A| + ∑_{l=1}^{N} C_{l,n} ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l,n} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) \end{aligned}

for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ ×A(Ω), and for some constants C, C_{l,n}, C_{l,n} > 0, l = 1, …, N, n ∈ ℕ, then Lemma 3.1.7 still holds for Γ^{−} and Γ_{p}^{−} (see (3.1.10)) for those u for which

sup_{n} (∑_{l=1}^{N} C_{l,n} ∫_{A} |∂_{l}u|^{p} dx + ∑_{l=1}^{N} D_{l,n} ∫_{S(u)∩A}(1 + |[u]|)|ν_{l}(u)| dH^{N−1}) < ∞.

3.2 Relaxed Energy: Proof of Theorem 1.0.4

Proof of Theorem 1.0.4. Let {u_{n}} ⊂ SBV_{p}(Ω; ℝ^{d}) be any sequence converging to u ∈ SBV_{p}(Ω; ℝ^{d}) in L^{1}(Ω; ℝ^{d}) and such that

sup_{n} (∫_{A} Qf_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩A} Rg_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1}) < ∞.

By (1.0.9) we may apply Ambrosio’s lower semicontinuity results (see Theorem 3.1.2) to obtain

\begin{matrix}∫_{A} Qf_{0}(x, u, ∇u) dx & + ∫_{S(u)∩A} Rg_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1} \\ ≤ & lim inf_{n→∞} (∫_{A} Qf_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩A} Rg_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1}) \end{matrix}

and taking the infimum over all such sequences {u_{n}} we get

∫_{A} Qf_{0}(x, u, ∇u) dx + ∫_{S(u)∩A} Rg_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1} ≤ H(u; A).

Conversely, define, for every u ∈ SBV_{p}(Ω; ℝ^{d}) and A ∈ A(Ω),

F (u; A) := ∫_{A} f_{0}(x, u, ∇u) dx + ∫_{S(u)∩A} g_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1}.

We claim that the functional H defined by (3.1.7) satisfies conditions (H_{1})–(H_{4}). Indeed hypotheses (H_{2}) and (H_{3}) are immediate consequences of the definition of H. The upper bound of H in (H_{4}) is obvious, since H(u; A) ≤ F (u; A). The lower bound is a consequence of the lower semicontinuity of

u ↦ ∫_{A} |∇u|^{p} dx + ∫_{A∩S(u)}(1 + |[u]|) dH^{N−1}

with respect to the L^{1}(A; ℝ^{d}) topology, as it is guaranteed by Theorem 3.1.2.

To prove (H_{1}), and as it is usual (see e.g. [13], [39]), it suffices to prove that

H(u; A) ≤ H(u; B) + H(u; A \ \overline{C})

for all A, B, C ∈ A(Ω), with C ⊂⊂ B ⊂⊂ A. Assume that u ∈ SBV_{p}(Ω; ℝ^{d}) ∩ L^{∞}(Ω; ℝ^{d}), and fix η > 0 and B_{0} ∈ A_{∞}(Ω) such that C ⊂⊂ B_{0} ⊂⊂ B and μ(∂B_{0}) = 0, where

μ := L^{N}⌊Ω + (1 + |[u]|)H^{N−1}⌊(S(u) ∩ Ω).

By Remark 3.1.8 there exist two sequences \{v_{k}\}, \{u_{k}\} ⊂ SBV_{p}(Ω; ℝ^{d}) such that v_{k} → u in L^{p}(A\\overline{B_{0}}; ℝ^{d}), u_{k} → u in L^{p}(B_{0}; ℝ^{d}), v_{k} = u_{k} = u on ∂B_{0} and

lim_{k→∞} F (v_{k}; A \ \overline{B_{0}}) ≤ H(u; A \ \overline{B_{0}}) + η,  lim_{k→∞} F (u_{k}; B_{0}) ≤ H(u; B_{0}) + η.  (3.2.1)

Define w_{k} = u_{k} in B_{0} and w_{k} = v_{k} in A \ \overline{B_{0}}. Then \{w_{k}\} ⊂ SBV_{p}(Ω; ℝ^{d}) and w_{k} → u in L^{p}(A; ℝ^{d}). Moreover, by the growth conditions (1.0.9) and the fact that μ(∂B_{0}) = 0

F (w_{k}; A) = F (w_{k}; B_{0}) + F (w_{k}; A \ \overline{B_{0}})
= F (u_{k}; B_{0}) + F (v_{k}; A \ \overline{B_{0}}).

Thus, passing to the limit in k we have

\begin{aligned}H(u; A) & ≤ H(u; B_{0}) + H(u; A \ \overline{B_{0}}) + 2η \\ & ≤ H(u; B) + H(u; A \ \overline{C}) + 2η, \end{aligned}

where we have used (3.2.1) and the fact that H(u; ·) is an increasing set function. It now suffices to let η → 0^{+}. In the general case, by Lemma 3.1.7

H(u; B) + H(u; A \ \overline{C}) = lim_{j→∞} (H(τ_{j}(u); B) + H(τ_{j}(u); A \ \overline{C})) .

Since τ_{j}(u) is bounded by the first part we have

H(τ_{j}(u); A) ≤ H(τ_{j}(u); B) + H(τ_{j}(u); A \ \overline{C}),

and letting j → ∞ we get, again by Lemma 3.1.7,

H(u; A) = lim_{j→∞} H(τ_{j}(u); A) ≤ lim_{j→∞} (H(τ_{j}(u); B) + H(τ_{j}(u); A \ \overline{C})) = H(u; B) + H(u; A \ \overline{C}).

By Theorem 1.0.1 we now have

H(u; A) = ∫_{A} f(x, u, ∇u) dx + ∫_{S(u)∩A} g(x, u^{+}, u^{−}, ν_{u}) dH^{N−1},

where f and g are given by (1.0.2) and (1.0.3).

\underline{Step 1}: We claim that

f(x_{0}, u_{0}, ξ_{0}) ≤ Qf_{0}(x_{0}, u_{0}, ξ_{0}).

for L^{N} a.e. x_{0} ∈ Ω and all (u_{0}, ξ_{0}) ∈ ℝ^{d} × ℝ^{d×N}. Since f_{0} is a Carathéodory function, by the Scorza-Dragoni Theorem for each j ∈ ℕ there exists a compact set K_{j} ⊂ Ω, with |Ω \ K_{j}| ≤ 1/j, such that

f_{0} : K_{j} × ℝ^{N} × ℝ^{d×N} → [0, +∞)

is continuous. Let K_{j}^{∗} be the set of Lebesgue points of χ_{K_{j}}, and set

ω := ⋃_{j}^{∞}(K_{j} ∩ K_{j}^{∗}).  (3.2.2)

Then

|Ω \ ω| ≤ |Ω \ K_{j}| ≤ \frac{1}{j} → 0 as j → ∞.

Fix (x_{0}, u_{0}, ξ_{0}) ∈ ω × ℝ^{d} × ℝ^{d×N} and η > 0, and let w ∈ C_{0}^{∞}(Q; ℝ^{d}) be such that

∫_{Q} f_{0}(x_{0}, u_{0}, ξ_{0} + ∇w(y)) dy ≤ Qf_{0}(x_{0}, u_{0}, ξ_{0}) + η.  (3.2.3)

For any fixed ε > 0 set v_{ε}(x) := ε w((x − x_{0})/ε). Then v_{ε} ∈ W_{0}^{1,p}(Q(x_{0}, ε); ℝ^{d}), and by (1.0.2) and the definitions of m and H

f(x_{0}, u_{0}, ξ_{0}) := lim sup_{ε→0^{+}} \frac{m(u_{0} + ξ_{0}(· − x_{0}); Q(x_{0}, ε))}{ε^{N}} ≤ lim sup_{ε→0^{+}} \frac{H(u_{0} + ξ_{0}(· − x_{0}) + v_{ε}; Q(x_{0}, ε))}{ε^{N}}
≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f_{0}(x, u_{0} + ξ_{0}(x − x_{0}) + v_{ε}(x), ξ_{0} + ∇v_{ε}(x)) dx.

(3.2.4)

Fix L > max{|u_{0}|, |ξ_{0}| + ||∇w||_{L^{∞}}} + 1, and let j be such that x_{0} ∈ K_{j} ∩ K_{j}^{∗}, where we are using the notation introduced in (3.2.2). Since

f_{0} : K_{j} × \overline{B_{d}(0, L)} × \overline{B_{d×N}(0, L)} → [0, ∞)

is uniformly continuous, there exists 0 < ρ < 1 such that

|f_{0}(x, u, ξ) − f_{0}(x_{1}, u_{1}, ξ_{1})| ≤ η  (3.2.5)

for all (x, u, ξ), (x_{1}, u_{1}, ξ_{1}) ∈ K_{j} × \overline{B_{d}(0, L)} × \overline{B_{d×N}(0, L)}, with |x − x_{1}| ≤ ρ, |u − u_{1}| ≤ ρ and |ξ − ξ_{1}| ≤ ρ. For ε > 0 sufficiently small

(x, u_{0} + ξ_{0}(x − x_{0}) + v_{ε}(x)) ∈ B_{N}(x_{0}, ρ) × B_{d}(u_{0}, ρ)

for all x ∈ Q(x_{0}, ε), and thus by (3.2.4), (1.0.9), (3.2.5), (3.2.3),

f(x_{0}, u_{0}, ξ_{0}) ≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)∩K_{j}} f_{0}(x, u_{0} + ξ_{0}(x − x_{0}) + v_{ε}(x), ξ_{0} + ∇v_{ε}(x)) dx

+ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)\K_{j}} C(1 + |ξ_{0}|^{q} + ||∇w||_{L^{∞}}^{q}) dx

≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)∩K_{j}} f_{0}(x_{0}, u_{0}, ξ_{0} + ∇v_{ε}(x)) dx + η
≤ ∫_{Q} f_{0}(x_{0}, u_{0}, ξ_{0} + ∇w(y)) dy + η ≤ Qf(x_{0}, u(x_{0}), v(x_{0})) + 2η,

where we have used the fact that

\frac{|Q(x_{0}, ε) \ K_{j}|}{ε^{N}} → 0 as ε → 0^{+},

since x_{0} is a Lebesgue point of χ_{K_{j}}. It now suffices to let η → 0^{+}.

\underline{Step 2}: To conclude the proof it remains to show that

g(x_{0}, a, b, ν) ≤ Rg_{0}(x_{0}, a, b, ν)

for all (x_{0}, a, b, ν) ∈ Ω × ℝ^{2d} × S^{d−1}.

Fix (x_{0}, a, b, ν) ∈ Ω × ℝ^{2d} × S^{d−1} and η > 0, and let w ∈ SBV_{p}(Q_{ν}; ℝ^{d}) ∩ L^{∞}(Q_{ν}; ℝ^{d}) be such that ∇w = 0 L^{N} a.e in Q_{ν}, w = u_{x_{0},a,b,ν} in a neighborhood of ∂Q_{ν} and

∫_{S(w)∩Q_{ν}} g(x_{0}, w^{+}(y), w^{−}(y), ν_{w}(y)) dH^{N−1}(y) ≤ Rg_{0}(x_{0}, a, b, ν) + η,  (3.2.6)

where, we recall,

u_{x_{0},a,b,ν}(x) := \begin{cases}a & if (x − x_{0}) · ν > 0, \\ b & if (x − x_{0}) · ν ≤ 0. \end{cases}

For any fixed ε > 0 set v_{ε}(x) := w((x − x_{0})/ε). Then by (1.0.3) and the definition of m

g(x_{0}, a, b, ν) := lim sup_{ε→0^{+}} \frac{m(u_{x_{0},a,b,ν}; Q_{ν}(x_{0}, ε))}{ε^{N−1}} ≤ lim sup_{ε→0^{+}} \frac{H(v_{ε}; Q_{ν}(x_{0}, ε))}{ε^{N−1}}
≤ lim sup_{ε→0^{+}} \frac{1}{ε^{N−1}} ∫_{Q_{ν} (x_{0},ε)∩S(v_{ε})} g(x, v_{ε}^{+}(x), v_{ε}^{−}(x), ν_{v_{ε}}(x)) dH^{N−1}(x).

(3.2.7)

Fix L > ||w||_{L^{∞}} + 1. Since

g_{0} : \overline{B_{N}(x_{0}, 1)} × \overline{B_{d}(0, L)} × \overline{B_{d}(0, L)} × S^{d−1} → [0, ∞)

is uniformly continuous, there exists 0 < ρ < 1 such that

|g_{0}(x, u, v, ν) − g_{0}(x_{1}, u_{1}, v_{1}, ν_{1})| ≤ η  (3.2.8)

for all (x, u, v, ν), (x_{1}, u_{1}, v_{1}, ν_{1}) ∈ \overline{B_{N}(x_{0}, 1)} × \overline{B_{d}(0, L)} × \overline{B_{d}(0, L)} × S^{d−1}, with |x − x_{1}| ≤ ρ, |u − u_{1}| ≤ ρ, |v − v_{1}| ≤ ρ and |ν − ν_{1}| ≤ ρ. Thus by (3.2.7), (3.2.8), (3.2.6),

\begin{matrix}g(x_{0}, a, b, ν) ≤ & lim sup_{ε→0^{+}} \frac{1}{ε^{N−1}} ∫_{Q_{ν} (x_{0},ε)∩S(v_{ε})} g(x_{0}, v_{ε}^{+}(x), v_{ε}^{−}(x), ν_{v_{ε}}(x)) dH^{N−1}(x) + η \\ = & ∫_{S(w)∩Q_{ν}} g(x_{0}, w^{+}(y), w^{−}(y), ν_{w}(y)) dH^{N−1}(y) + η ≤ Rg_{0}(x_{0}, a, b, ν) + 2η. \end{matrix}

It now suffices to let η → 0^{+}.

3.3 Brittle Thin Films.

In this section we deduce, via Γ-convergence, the 2-dimensional model for brittle thin films from the 3-dimensional model allowing fracture. Using the global method introduced in Section 2.2, we recover easily the result recently obtained by Braides and Fonseca in [23], now holding without having to assume some technical Lipschitz regularity hypotheses imposed by those authors.

Considering a sequence of positive real numbers ε → 0^{+} corresponding to the thickness of a 3-dimensional cilinder of fixed 2-dimensional cross section ω ⊂ ℝ^{2}, we analyze the asymptotic behavior of the 3-dimensional energy

v ∈ SBV_{p}(Ω_{ε}; ℝ^{3}) ↦ ∫_{Ω_{ε}} f_{0}(∇v) dx + ∫_{S(u)∩Ω_{ε}} g_{0}([v], ν(v)) dH^{2},  (3.3.1)

where Ω_{ε} := ω × (−ε/2, ε/2) and

f_{0} : ℝ^{3×3} → [0, ∞), g_{0} : ℝ^{3} × S^{2} → [0, ∞),

are the bulk and the surface energy density, respectively. In order to set the problem in a fixed domain, we use the rescaling (see [23])

u(x_{α}, x_{3}) := v(x_{α}, ε x_{3}), x_{α} := (x_{1}, x_{2}),

so that

\begin{matrix}\frac{1}{ε} & (∫_{Ω_{ε}} f_{0}(∇v) dx + ∫_{S(u)∩Ω_{ε}} g_{0}([v], ν(v)) dH^{2}) \\ = ∫_{Ω} f_{0} (∇_{α}u, \frac{1}{ε} ∇_{3}u) dx + ∫_{S(u)∩Ω} g_{0} ([u], \frac{(ν_{α}(u), \frac{1}{ε} ν_{3}(u))}{|(ν_{α}(u), \frac{1}{ε} ν_{3}(u))|} ) |(ν_{α}(u), \frac{1}{ε} ν_{3}(u))| dH^{2}, \end{matrix}

where Ω := ω × (−1/2, 1/2). Here, and in what follows, (ξ_{α}, ξ_{3}) denotes the 3 × 3 matrix ξ with columns ξ_{1}, ξ_{2} and ξ_{3}. To simplify the notation we extend g_{0} by homogeneity of degree one to ℝ^{3} × ℝ^{3} by setting

g_{0}(z, ν) := \begin{cases}|ν|g_{0} (z, \frac{ν}{|ν|} ) & if ν ≠ 0, \\ 0 & if ν = 0, \end{cases}

and thus we are led to the problem of Γ-convergence, as ε goes to zero, of the sequence of functionals

F_{ε} : SBV_{p}(Ω; ℝ^{3}) × A(ω) → [0, ∞]

defined by

F_{ε}(u; A) := ∫_{A×(− \frac{1}{2} , \frac{1}{2} )} f_{0} (∇_{α}u, \frac{1}{ε} ∇_{3}u) dx + ∫_{S(u)∩(A×(− \frac{1}{2} , \frac{1}{2} ))} g_{0} ([u], ν_{α}(u), \frac{1}{ε} ν_{3}(u)) dH^{2}.  (3.3.2)

Given a family of functions {u_{ε}} ⊂ SBV_{p}(Ω; ℝ^{3}) such that (cf. [23] Prop. 2.3)

sup_{ε} F_{ε}(u_{ε}; ω) < ∞ and u_{ε} → u in L^{1}(Ω; ℝ^{3}),

by the growth conditions (3.3.6) we obtain

sup_{ε} (∫_{Ω} |(∇_{α}u_{ε}, \frac{1}{ε} ∇_{3}u_{ε})|^{p} dx + ∫_{S(u_{ε})∩Ω} (1 + |[u_{ε}]|) |(ν_{α}(u_{ε}), \frac{1}{ε} ν_{3}(u_{ε}))| dH^{2}) ≤ M < ∞

for some M > 0. Thus for 0 < ε < 1 we have

sup_{ε} (∫_{Ω} |∇u_{ε}|^{p} dx + ∫_{S(u_{ε})∩Ω} (1 + |[u_{ε}]|) dH^{2}) ≤ M < ∞,

and we may apply Theorem 3.1.2 to the functional

v ↦ ∫_{Ω} |∇_{3}v|^{p} dx + ∫_{S(v)∩Ω} |ν_{3}(v)| dH^{2}

to obtain

∫_{Ω} |∇_{3}u|^{p} dx + ∫_{S(u)∩Ω} |ν_{3}(u)| dH^{2} ≤ lim inf_{ε→0^{+}} (∫_{Ω} |∇_{3}u_{ε}|^{p} dx + ∫_{S(u_{ε})∩Ω} |ν_{3}(u_{ε})| dH^{2}) ≤ lim inf_{ε→0^{+}} εF_{ε}(u_{ε}; ω) = 0,

which implies that u ∈ V, where

V := {u ∈ SBV_{p}(Ω; ℝ^{3}) : ∇_{3}u = 0 L^{3} a.e. and ν_{3}(u) = 0 H^{2} a.e. in Ω}.

Since functions in V are independent of x_{3}, in what follows V is identified with SBV_{p}(ω; ℝ^{3}).

Set

f̄_{0}(ξ_{α}) := inf {f_{0}(ξ) : ξ := (ξ_{α}, ξ_{3}), ξ_{3} ∈ ℝ}  (3.3.3)

where ξ_{α} ∈ ℝ^{3×2}, and

ḡ_{0}(z, μ) := inf {g_{0}(z, μ, t) : t ∈ ℝ},  (3.3.4)

where z ∈ ℝ^{3} and μ ∈ ℝ^{2} \ {0}. Note that ḡ_{0}(z, ·) is homogeneous of degree one, and thus we may define ḡ_{0}(z, 0) := 0.

Given u ∈ SBV_{p}(Ω; ℝ^{3}), by Theorem 3.1.1 for L^{1} a.e. t ∈ (−1/2, 1/2) the function u^{t}(x_{α}) := u(x_{α}, t) ∈ SBV_{p}(ω; ℝ^{3}) and

\begin{aligned}F_{ε}(u; A) & = ∫_{A×(− \frac{1}{2} , \frac{1}{2} )} f_{0} (∇_{α}u, \frac{1}{ε} ∇_{3}u) dx + ∫_{S(u)∩(A×(− \frac{1}{2} , \frac{1}{2} ))} g_{0} ([u], ν_{α}(u), \frac{1}{ε} ν_{3}(u)) dH^{2} \\ & ≥ ∫_{A×(− \frac{1}{2} , \frac{1}{2} )} f̄_{0} (∇_{α}u) dx + ∫_{S(u)∩(A×(− \frac{1}{2} , \frac{1}{2} ))} ḡ_{0} ([u], ν_{α}(u)) dH^{2} \\ & ≥ ∫_{−1/2}^{1/2} (∫_{A} f̄_{0} (∇_{α}u^{t}) dx_{α} + ∫_{S(u^{t})∩A} ḡ_{0} ([u^{t}], ν̄(u^{t})) |ν_{α}(u)| dH^{1}) dt \\ & ≥ ∫_{−1/2}^{1/2} (∫_{A} Q_{2} f̄_{0} (∇_{α}u^{t}) dx_{α} + ∫_{S(u^{t})∩A} R_{2} ḡ_{0} ([u^{t}], ν̄(u^{t})) |ν_{α}(u)| dH^{1}) dt \\ & = ∫_{−1/2}^{1/2} (∫_{A} Q_{2} f̄_{0} (∇_{α}u^{t}) dx_{α} + ∫_{S(u^{t})∩A} R_{2} ḡ_{0} ([u^{t}], ν̄(u^{t})) \sqrt{1 − |ν_{3}(u)|^{2}} dH^{1}) dt, \end{aligned}  (3.3.5)

where we have used Fubini’s Theorem and the fact that ḡ_{0}(z, ·) is homogeneous of degree one. Here Q_{2} f̄_{0} and R_{2} ḡ_{0} are, respectively, the quasiconvex envelope of f̄_{0} and the BV -elliptic envelope of ḡ_{0}(z, μ) corresponding to the ambient space SBV_{p}(ω; ℝ^{3}), precisely for ξ_{α} ∈ ℝ^{3×2}

Q_{2}f̄_{0}(ξ_{α}) := inf {∫_{(0,1)^{2}} f̄_{0} (ξ_{α} + ∇_{α}φ(y_{α})) dy_{α} : φ ∈ W_{0}^{1,∞}((0, 1)^{2}; ℝ^{3})},

while for z ∈ ℝ^{3} and μ ∈ S^{1}

R_{2} ḡ_{0}(z, μ) := inf{∫_{S(v)∩Q′_{μ}} ḡ_{0}([v], ν(v)) dH^{1} : v ∈ SBV_{0}(Q′_{μ}; ℝ^{3}) ∩ L^{∞}(Q′_{μ}; ℝ^{3}), v = u_{z,μ} on ∂Q′_{μ}},

where Q′_{μ} is the unit square in ℝ^{2} centered at zero with two of its faces orthogonal to μ, and

u_{z,μ}(x_{α}) := \begin{cases}z & if x_{α} · μ > 0, \\ 0 & if x_{α} · μ ≤ 0. \end{cases}

These considerations now lead us to the following result.

Theorem 3.3.1. Assume that

f_{0} : ℝ^{3×3} → [0, ∞), g_{0} : ℝ^{3} × ℝ^{3} → [0, ∞)

are continuous functions, g_{0}(z, ·) is even and positively homogeneous of degree one, and

\frac{1}{C} |ξ|^{p} ≤ f_{0}(ξ) ≤ C(1 + |ξ|^{p}),  \frac{1}{C} (1 + |z|) ≤ g_{0}(z, ν) ≤ C (1 + |z|),  g_{0}(z, ν) = g_{0}(−z, −ν), (3.3.6)

for all ξ ∈ ℝ^{3×3}, z ∈ ℝ^{3}, ν ∈ S^{2}, and for some C > 0. Then for every u ∈ V and A ∈ A(ω) the functionals

v ∈ SBV_{p}(Ω; ℝ^{3}) ↦ F_{ε}(v; A)

Γ-converge with respect to the L^{1}(Ω; ℝ^{3}) convergence as ε → 0^{+} to the functional

F_{0}(u; A) := ∫_{A} Q_{2} f̄_{0}(∇_{α}u) dx_{α} + ∫_{S(u)∩A} R_{2} ḡ_{0}([u], ν_{α}(u)) dH^{1}.

Proof. In order to prove that the sequence {F_{ε}(u; A)} Γ-converges to F_{0}(u; A) for all u ∈ V and A ∈ A(ω), it is enough to show that for any given sequence {ε_{n}}, with ε_{n} → 0^{+}, there exists a subsequence {ε_{n_{k}}} of {ε_{n}} such that the Γ-lower limit defined by

F^{−}(u; A) := inf {lim inf_{k→∞} F_{ε_{n_{k}}} (u_{k}; A) : {u_{k}} ⊂ SBV_{p}(Ω; ℝ^{3}), u_{k} → u in L^{1}(Ω; ℝ^{3})}  (3.3.7)

coincides with F_{0}(u; A) for all u ∈ V and A ∈ A(ω).

Fix a sequence {ε_{n}}, with ε_{n} → 0^{+}, and let R(ω) be the countable subfamily of A(ω) obtained by taking all finite unions of open squares in ω with faces parallel to the axes, centered at x_{α} ∈ ω ∩ ℚ^{2} and with rational edge length. Since L^{1}(Ω; ℝ^{3}) is a separable metric space, using Kuratowski’s Compactness Theorem and a diagonal argument (see Prop. 7.9 in [21]), we may find a subsequence {ε_{n_{k}}} of {ε_{n}} such that

Γ − lim_{k→∞} F_{ε_{n_{k}}} (u; A′) exists for all u ∈ V and for all A′ ∈ R(ω).  (3.3.8)

In order to apply Theorem 1.0.1 we need to show that the functional defined in (3.3.7) satisfies (H_{1})-(H_{4}) for all A ∈ A(ω) and u ∈ V. For simplicity of notation we write ε_{k} in place of ε_{n_{k}}.

Hypotheses (H_{2}) and (H_{3}) are immediate consequences of the definition, and in view of (3.3.6) the upper bound in (H_{4}) is obtained by taking u_{k} ≡ u ∈ V. In order to establish the lower bound in (H_{4}), let {u_{j}} ⊂ SBV_{p}(Ω; ℝ^{3}) be such that u_{j} → u in L^{1}(Ω; ℝ^{3}) and F^{−}(u; A) = lim_{j→∞} F_{ε_{k_{j}}} (u_{j}; A) for a suitable subsequence {ε_{k_{j}} } of {ε_{k}}. By (3.3.6) and using the lower semicontinuity result of Theorem 3.1.2 we have

F^{−}(u; A) = lim_{j→∞} F_{ε_{k_{j}}} (u_{j}; A) ≥ \frac{1}{C} lim inf_{j→∞} (∫_{A×(− \frac{1}{2} , \frac{1}{2} )} |∇u_{j}|^{p} dx + ∫_{S(u_{j})∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[u_{j}]|) dH^{2})

≥ \frac{1}{C} (∫_{A×(− \frac{1}{2} , \frac{1}{2} )} |∇u|^{p} dx + ∫_{S(u)∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[u]|) dH^{2})

= \frac{1}{C} (∫_{A} |∇_{α}u|^{p} dx_{α} + ∫_{S(u)∩A} (1 + |[u]|) dH^{1}) .

To prove that F^{−}(u; ·) is the trace of a Borel measure, we first observe that in view of Remark 3.1.9, and as in the proof of Theorem 1.0.4, it is enough to show that

F^{−}(u; A) ≤ F^{−}(u; B) + F^{−}(u; A \ \overline{C})

for all A, B, C ∈ A(ω), with C ⊂⊂ B ⊂⊂ A, and for all u ∈ V ∩ L^{∞}(Ω; ℝ^{3}), and with convergence in L^{1}(Ω; ℝ^{3}) replaced by convergence in L^{p}(Ω; ℝ^{3}) in the definition of F^{−}(u; A). Fix η > 0 and find {v_{k}} ⊂ SBV_{p}(Ω; ℝ^{3}) such that v_{k} → u in L^{p}((A \ \overline{C}) × (−1/2, 1/2); ℝ^{3}) and

lim inf_{k→∞} F_{ε_{k}}(v_{k}; A \ \overline{C}) ≤ F^{−}(u; A \ \overline{C}) + η.

Extract a subsequence {k_{j}} for which

lim_{j→+∞} F_{ε_{k_{j}}} (v_{k_{j}}; A \ \overline{C}) ≤ F^{−}(u; A \ \overline{C}) + η.  (3.3.9)

Let B_{0} ∈ R(ω) be such that C ⊂⊂ B_{0} ⊂⊂ B. By (3.3.8) F^{−}(u; B_{0}) = Γ − lim_{j→∞} F_{ε_{k_{j}}} (u; B_{0}) and thus, by Remark 3.1.8, there exists a sequence {u_{j}} ⊂ SBV_{p}(Ω; ℝ^{3}) such that u_{j} → u in L^{p}(B_{0} × (−1/2, 1/2); ℝ^{3}) and

lim_{j→∞} F_{ε_{k_{j}}} (u_{j}; B_{0}) = F^{−}(u; B_{0}).  (3.3.10)

For every t > 0 let B_{t} := {x_{α} ∈ B_{0}| dist(x_{α}, ∂B_{0}) > t}, and

G_{ε}(v; A) := ∫_{A×(− \frac{1}{2} , \frac{1}{2} )} |(∇_{α}v, \frac{1}{ε} ∇_{3}v)|^{p} dx + ∫_{S(v)∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[v]|) |(ν_{α}(v), \frac{1}{ε} ν_{3}(v))|^{2} dH^{2},

v ∈ SBV_{p}(Ω; ℝ^{3}). Due to the coercivity hypothesis (3.3.6), we may extract a bounded subsequence from the sequences of measures ν_{j} := G_{ε_{k_{j}}} (v_{k_{j}}; ·) + G_{ε_{k_{j}}} (u_{j}; ·) restricted to B_{0} \ \overline{C} converging ⋆-weakly to some Radon measure ν defined on B_{0} \ \overline{C}. For every 0 < δ < η we define the subsets L_{δ} := B_{η−2δ} \ \overline{B_{η+δ}}. Consider a smooth cut-off function φ_{δ} ∈ C_{0}^{∞}(B_{η−δ}; [0, 1]) such that φ_{δ}(x_{α}) = 1 on B_{η}. As the thickness of the strip L_{δ} is of order δ, we have an upper bound of the form ∥∇φ_{δ}∥_{L^{∞}(B_{η−δ})} ≤ C/δ. Define

w_{j}(x) := u_{j}(x)φ_{δ}(x_{α}) + v_{k_{j}} (x)(1 − φ_{δ}(x_{α})).

Clearly this sequence converges to u in L^{p}(A × (−1/2, 1/2); ℝ^{3}) as j → ∞, and satisfies

∇_{α}w_{j} = φ_{δ}∇_{α}u_{j} + (1 − φ_{δ})∇_{α}v_{k_{j}} + ∇_{α}φ_{δ} ⊗ (u_{j} − v_{k_{j}}), ∇_{3}w_{j} = φ_{δ}∇_{3}u_{j} + (1 − φ_{δ})∇_{3}v_{k_{j}} .

By the growth condition (3.3.6), we have the estimate

F_{ε_{k_{j}}} (w_{j}; A) ≤F_{ε_{k_{j}}} (u_{j}; B_{η}) + F_{ε_{k_{j}}} (v_{k_{j}}; A \ \overline{B_{η−δ}})

+ C (G_{ε_{k_{j}}} (u_{j}; L_{δ}) + G_{ε_{k_{j}}} (u_{j}; L_{δ}) + \frac{1}{δ^{p}} ∫_{L_{δ}×(− \frac{1}{2} , \frac{1}{2} )} |u_{j} − v_{k_{j}} |^{p} dx)

≤F_{ε_{k_{j}}} (u_{j}; B_{0}) + F_{ε_{k_{j}}} (v_{k_{j}}; A \ \overline{C})

+ C (G_{ε_{k_{j}}} (u_{j}; L_{δ}) + G_{ε_{k_{j}}} (u_{j}; L_{δ}) + \frac{1}{δ^{p}} ∫_{L_{δ}×(− \frac{1}{2} , \frac{1}{2} )} |u_{j} − v_{k_{j}} |^{p} dx) .

Thus, passing to the limit in k and using (3.3.9) and (3.3.10), we have

\begin{matrix}F^{−}(u; A) ≤ F^{−}(u; B_{0}) + F^{−}(u; A \ \overline{C}) + η + Cν(\overline{L_{δ}}) \\ ≤ F^{−}(u; B) + F^{−}(u; A \ \overline{C}) + η + Cν(\overline{L_{δ}}), \end{matrix}

where we have used the fact that F^{−}(u; ·) is clearly an increasing set function. We let δ go to zero to obtain

F^{−}(u; A) ≤ F^{−}(u; B) + F^{−}(u; A \ \overline{C}) + η + Cν(∂B_{η}).

It suffices to choose a subsequence {η_{k}} such that η_{k} → 0^{+} and ν(∂B_{η_{k}}) = 0, to conclude that for u ∈ V∩L^{∞}(Ω; ℝ^{3}) we have

F^{−}(u; A) ≤ F^{−}(u; B) + F^{−}(u; A \ \overline{C}).

We are now in position to apply Theorem 1.0.1 to the functional F^{−} : V × A(ω) → [0, ∞) to obtain the existence of two Borel functions

f : ω × ℝ^{3} × ℝ^{3×2} → [0, ∞) and g : ω × ℝ^{3} × ℝ^{3} × S^{1} → [0, ∞)

such that

F^{−}(u; A) := ∫_{A} f (x_{α}, u, ∇_{α}u) dx_{α} + ∫_{S(u)∩A} g (x_{α}, u^{+}, u^{−}, ν_{α}(u)) dH^{1}

for all u ∈ V and A ∈ A(ω). Since F^{−}(u + c; A) = F^{−}(u; A) for all c ∈ ℝ^{3}, and f_{0} and g_{0} are independent of x, it follows immediately from (1.0.2) and (1.0.3) that f = f (ξ_{α}) and g = g (λ − θ, ν_{α}). We prove that f = Q_{2}f̄_{0} and g = R_{2}ḡ_{0}, where f̄_{0} and ḡ_{0} are given by (3.3.3) and (3.3.4), respectively.

Given ξ_{α} ∈ ℝ^{3×2} let ξ_{3} ∈ ℝ^{3} be such that f̄_{0}(ξ_{α}) = min {f_{0}(ξ) : ξ := (ξ_{α}, ξ_{3}), ξ_{3} ∈ ℝ}. Note that the minimum exists by (3.3.6) and the continuity of f_{0}. Consider the sequence u_{k}(x) := ξ_{α} x_{α} + ε_{k} ξ_{3}x_{3}, and fix A ∈ A(ω). Clearly u_{k} → u in L^{1} (A × (− \frac{1}{2} , \frac{1}{2} ); ℝ^{3}), where u(x) = u(x_{α}) = ξ_{α} x_{α}, and so

F^{−}(u; A) = f(ξ_{α})L^{2}(A) ≤ lim inf_{k→∞} F_{1/k}(u_{k}; A) = f_{0}(ξ_{α}, ξ_{3})L^{2}(A) = f̄_{0}(ξ_{α})L^{2}(A),

that is

f(ξ_{α}) ≤ f̄_{0}(ξ_{α}).  (3.3.11)

Similarly, for μ ∈ S^{1}, z ∈ ℝ^{3}, let t ∈ ℝ be such that ḡ_{0}(z, μ) = g_{0}(z, μ, t). Consider the sequence

v_{k}(x) := \begin{cases}z if x_{α} · μ + ε_{k} x_{3}t > 0, \\ 0 if x_{α} · μ + ε_{k} x_{3}t ≤ 0. \end{cases}

Then v_{k} → v in L^{1} (A × (− \frac{1}{2} , \frac{1}{2} ); ℝ^{3}), where A ∈ A(ω), and

v = v(x_{α}) := \begin{cases}z if x_{α} · ν > 0, \\ 0 if x_{α} · ν ≤ 0. \end{cases}

We have

F^{−}(v; A) = f(0)L^{2}(A) + g(z, μ)H^{1} ({x_{α} ∈ A : x_{α} · μ = 0})

≤ lim inf_{k→∞} F_{ε_{k}}(v_{k}; A)

= f_{0}(0)L^{2}(A) + lim inf_{k→∞} g_{0} (z, \frac{μ}{\sqrt{1 + ε_{k}^{2} t^{2}}} , \frac{t}{\sqrt{1 + ε_{k}^{2} t^{2}}} ) H^{2} (\{x ∈ A × (− \frac{1}{2} , \frac{1}{2} ) : x_{α} · μ + ε_{k} x_{3}t = 0\})

= f_{0}(0)L^{2}(A) + g_{0}(z, μ, t) H^{1} ({x_{α} ∈ A : x_{α} · μ = 0}) .

We conclude that

f(0)L^{2}(A) + g(z, μ)H^{1} (A ∩ { x_{α} · μ = 0}) ≤ f_{0}(0)L^{2}(A) + ḡ_{0}(z, μ) H^{1} (A ∩ {x_{α} · μ = 0}) .  (3.3.12)

Taking A = A_{η} := {x_{α} ∈ ω : |x_{α} · μ| < η}, η > 0, letting η → 0^{+} in (3.3.12), yields

g(z, μ) ≤ ḡ_{0}(z, μ).  (3.3.13)

By (3.3.11) and (3.3.13) we now have

F^{−}(u; A) ≤ ∫_{A} f̄_{0}(∇_{α}u) dx_{α} + ∫_{S(u)∩A} ḡ_{0} ([u]; ν_{α}(u)) dH^{1},

for every u ∈ SBV_{p}(ω; ℝ^{3}) and all A ∈ A(ω). Since f̄_{0} and ḡ_{0} satisfy the hypotheses of Theorem 1.0.4, and in view of the L^{1}(ω)-lower semicontinuity of F^{−}(·; A), we conclude that

f ≤ Q_{2}f̄_{0} and g ≤ R_{2}ḡ_{0}.  (3.3.14)

Conversely, let A ∈ A(ω), u ∈ V, and consider any sequence {u_{k}} ⊂ SBV_{p}(Ω; ℝ^{3}) such that

sup_{k} F_{ε_{k}}(u_{k}; A) < ∞ and u_{k} → u in L^{1} (A × (− \frac{1}{2} , \frac{1}{2} ); ℝ^{3}).

In light of the coercivity hypotheses (3.3.6) we obtain

sup_{k} (∫_{A×(− \frac{1}{2} , \frac{1}{2} )} |(∇_{α}u_{k}, \frac{1}{ε} ∇_{3}u_{k})|^{p} dx + ∫_{S(u_{k})∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[u_{k}]|) |(ν_{α}(u_{k}), \frac{1}{ε_{k}} ν_{3}(u_{k}))| dH^{2}) < ∞.

Thus for 0 < ε_{k} < 1 condition (3.1.6) is satisfied, and

lim_{k→∞} ∫_{S(u_{k})∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[u^{k}]|) |ν^{3}(u^{k})| dH^{2} = 0.  (3.3.15)

By (3.3.5) and Theorem 3.1.1

F_{ε_{k}}(u_{k}; A) ≥ ∫_{−1/2}^{1/2} (∫_{A} Q_{2} f̄_{0} (∇_{α}u_{k}^{t}) dx_{α} + ∫_{S(u_{k}^{t})∩A} R_{2} ḡ_{0} ([u_{k}^{t}], ν̄(u_{k}^{t})) \sqrt{1 − |ν_{3}(u_{k})|^{2}} dH^{1}) dt

≥ ∫_{−1/2}^{1/2} (∫_{A} Q_{2} f̄_{0} (∇_{α}u_{k}^{t}) dx_{α} + ∫_{S(u_{k}^{t})∩A} R_{2} ḡ_{0} ([u_{k}^{t}], ν̄(u_{k}^{t})) dH^{1}) dt

− C ∫_{S(u_{k})∩(A×(− \frac{1}{2} , \frac{1}{2} ))} (1 + |[u_{k}]|) |ν_{3}(u_{k})| dH^{2},

where we have used (3.3.6) and the fact that

1 − |ν_{3}(u_{k})| ≤ 1 − \sqrt{1 − |ν_{3}(u_{k})|^{2}}.

Letting k → ∞ yields, by Fatou’s Lemma and (3.3.15)

lim inf_{k→∞} F_{ε_{k}}(u_{k}; A) ≥ ∫_{−1/2}^{1/2} lim inf_{k→∞} (∫_{A} Q_{2} f̄_{0}(∇_{α}u_{k}^{t}) dx_{α} + ∫_{S(u_{k}^{t})∩A} R_{2} ḡ_{0}([u_{k}^{t}], ν̄(u_{k}^{t})) dH^{1}) dt

≥ ∫_{−1/2}^{1/2} (∫_{A} Q_{2} f̄_{0} (∇_{α}u^{t}) dx_{α} + ∫_{S(u^{t})∩A} R_{2} ḡ_{0} ([u^{t}], ν̄(u^{t})) dH^{1}) dt

= ∫_{A} Q_{2} f̄_{0}(∇_{α}u) dx_{α} + ∫_{S(u)∩A} R_{2} ḡ_{0} ([u]; ν_{α}(u)) dH^{1},

where we have used Theorem 3.1.2. Given the arbitrariness of the sequence {u_{k}}, u, and A ∈ A(ω), we deduce that

f(ξ_{α}) ≥ Q_{2} f̄_{0}(ξ_{α}) and g(z, μ) ≥ R_{2} ḡ_{0}(z, μ).

This, together with (3.3.14), establishes the identity

f(ξ_{α}) = Q_{2} f̄_{0}(ξ_{α}) and g(z, μ) = R_{2} ḡ_{0}(z, μ).

A Appendix

In this Appendix, and for the convenience of the reader, we provide a proof of Theorem 3.1.2 which departs slightly from the original proof of Ambrosio. Lemma A.0.2. Let f : Ω × ℝ^{N} × ℝ^{d×N} → [0, +∞) be a normal integrand. Then for each ε_{0} > 0 there exists a compact set K ⊂ Ω such that |Ω \ K| ≤ ε_{0} and f : K × ℝ^{N} × ℝ^{d×N} → [0, +∞) is lower semicontinuous. Moreover, if (x_{0}, u_{0}) ∈ K × ℝ^{d} and if f(x_{0}, u_{0}, ·) is continuous, then for each ε > 0 and L > 0 there exists δ = δ(x_{0}, u_{0}, ε, K, L) ∈ (0, 1) such that

f(x_{0}, u_{0}, ξ) ≤ f(x, u, ξ) + ε

for all (x, u) ∈ K × ℝ^{d}, with |x − x_{0}| + |u − u_{0}| ≤ δ and all |ξ| ≤ L.

Proof. The first part of the lemma follows from Theorem 1.1 in [35]. To prove the second claim assume, for contradiction, that there exist (x_{0}, u_{0}) ∈ K × ℝ^{d}, L̄ > 0, ε̄ > 0 and a sequence

\{(x_{n}, u_{n}, ξ_{n})\} ⊂ K × \overline{B_{d}(u_{0}, 1)} × \overline{B_{d×N}(0, L̄)},

such that

ε̄+ f(x_{n}, u_{n}, ξ_{n}) < f(x_{0}, u_{0}, ξ_{n})  (A.0.16)

and (x_{n}, u_{n}, ξ_{n}) → (x_{0}, u_{0}, ξ_{0}) as n → ∞, for some ξ_{0} ∈ \overline{B_{d×N}(0, L̄)}. Since f(x_{0}, u_{0}, ·) is continuous and f restricted to K × ℝ^{d} × ℝ^{d×N} is lower semicontinuous, for any ε < \frac{1}{2} ε̄ there exists δ > 0 such that

|f(x_{0}, u_{0}, ξ) − f(x_{0}, u_{0}, ξ_{0})| ≤ ε, f(x_{0}, u_{0}, ξ_{0}) ≤ f(x, u, ξ) + ε

for all (x, u, ξ) ∈ K ×\overline{B_{d}(u_{0}, 1)}×\overline{B_{d×N}(0, L̄)} with |x−x_{0}|+|u−u_{0}| ≤ δ and |ξ −ξ_{0}| ≤ δ. Thus for all n sufficiently large, and in view of (A.0.16), we have

ε̄+ f(x_{n}, u_{n}, ξ_{n}) < f(x_{0}, u_{0}, ξ_{n}) ≤ f(x_{0}, u_{0}, ξ_{0}) + ε ≤ f(x_{n}, u_{n}, ξ_{n}) + 2ε

which is a contradiction.

Remark A.0.3. With a similar proof we can show that if g : Ω×(ℝ^{d})^{2}×S^{N−1} → [0, +∞) is lower semicontinuous in x and continuous in the remaining variables, then for every x_{0} ∈ Ω, ε > 0, and L > 0, there exists δ = δ(x_{0}, ε, L) ∈ (0, 1) such that

g(x_{0}, λ, θ, ν) ≤ g(x, λ, θ, ν) + ε

for all x ∈ Ω, with |x − x_{0}| ≤ δ and all (λ, θ, ν) ∈ (ℝ^{d})^{2} × S^{N−1} with |λ|, |θ| ≤ L.

Given a non negative, finite Radon measure μ on ℝ^{N}, the maximal function M(μ) of μ is defined by

M(μ)(x) := sup_{r>0} \frac{μ(B(x, r))}{|B(x, r)|} .

It can be shown that (see [7]) if u ∈ SBV_{p}(ℝ^{N}; ℝ^{d}) then for any λ > 0 and for any Borel set E ⊂ ℝ^{N} we have that

\begin{aligned}|\{x ∈ E : M(|Du|) > 2λ\}| & ≤ \frac{1}{λ^{p}} ∫_{E∩\{M(|∇u|)>λ\}} |M(|∇u|)|^{p} dx + C(N, d) \frac{1}{λ} |D^{s}u(ℝ^{N})|, \\ ∫_{ℝ^{N}} |M(|∇u|)|^{p} dx & ≤ C(N, d, p) ∫_{ℝ^{N}} |∇u|^{p} dx. \end{aligned}  (A.0.17)

For a proof of the following theorem see [12].

Theorem A.0.4. Let u ∈ BV (ℝ^{N}; ℝ^{d}). Given λ > 1 there exists a Lipschitz function v such that u = v L^{N} a.e. on {x ∈ ℝ^{N} : M(|Du|) ≤ λ} and the Lipschitz constant for v is bounded by Cλ where C depends only on N and d.

The following lemma was proved by Fonseca and Müller [38] using Young measures.

Lemma A.0.5. Let 1 < p < +∞, let {u_{n}} be a bounded sequence in L^{p}(Ω; ℝ^{d}). For λ > 0 consider the truncation τ_{λ} : ℝ^{d} → ℝ^{d} given by

τ_{λ}(z) := \{ \begin{matrix}z & if |z| ≤ λ, \\ λ \frac{z}{|z|} & if |z| > λ. \end{matrix}

Then there exists a subsequence of {u_{n}} (not relabeled) and an increasing sequence λ_{n} → +∞ such that the truncated sequence {τ_{λ_{n}} ∘ u_{n}} is p-equi-integrable and

||τ_{λ_{n}} ∘ u_{n} − u_{n}||_{L^{q}(Ω)} → 0 for all 1 ≤ q < p.

Remark A.0.6. If ε_{n} > 0 is any decreasing sequence approaching zero, without loss of generality we may assume that

lim_{n→∞} λ_{n} ε_{n} = 0.

The following result was obtained by Kristensen in [42]. We present here a different proof based on the approach of Fonseca, Müller and Pedregal [41]. In what follows we denote by T_{λ} and τ_{λ} truncations in ℝ^{d} and ℝ, respectively.

Theorem A.0.7. Let Ω ⊂ ℝ^{N} be an open, bounded subset with Lipschitz boundary, and let {u_{n}} ⊂ SBV_{p}(Ω; ℝ^{d}) be a sequence converging in L^{1}(Ω; ℝ^{N}) to some function u_{0} and such that

sup_{n} (∫_{Ω} |∇u_{n}|^{p}dx + ∫_{S(u_{n})∩Ω} |[u_{n}]| dH^{N−1}) < ∞

and

H^{N−1}(S(u_{n}) ∩ Ω) → 0 as n → ∞.

Then there exists a subsequence {u_{n_{k}}} and a sequence {v_{k}} ⊂ W^{1,∞}(ℝ^{N}; ℝ^{d}) such that {v_{k}} converges to u_{0} weakly in W^{1,p}(Ω; ℝ^{N}), v_{k} = u_{0} on ∂Ω,

|{x ∈ Ω : v_{k}(x) ≠ u_{n_{k}}(x)}| → 0 as k → ∞

and {|∇v_{k}|^{p}} is equi-integrable.

Proof. Since Ω is Lipschitz we can extend the functions u_{n} outside Ω (see e.g. [12]) in such a way that the new functions U_{n} ∈ SBV_{p}(ℝ^{N}; ℝ^{d}) and

\begin{matrix}∫_{ℝ^{N}} (|U_{n}| + |∇U_{n}|^{p}) dx & + ∫_{S(U_{n})} |[U_{n}]| dH^{N−1} ≤ C(N, d) (∫_{Ω} (|u_{n}| + |∇u_{n}|^{p}) dx + ∫_{S(u_{n})∩Ω} |[u_{n}]| dH^{N−1}) , \\ H^{N−1}(S(U_{n})) → 0 as n → ∞. \end{matrix} (A.0.18)

By (A.0.17) we have that

sup_{n} ∫_{ℝ^{N}} |M(|∇U_{n}|)|^{p}dx < ∞,  (A.0.19)

and thus by Lemma A.0.5 and Remark A.0.6 there exists an increasing sequence λ_{n} → +∞ such that

lim_{n→∞} λ_{n}^{p}H^{N−1}(S(U_{n})) = 0,  (A.0.20)

and the truncated sequence {τ_{λ_{n}} ∘ M(|∇U_{n}|)} is p-equi-integrable. Define V_{n}(x) := T_{λ_{n}}(U_{n}). Then (see [11]) S(V_{n}) ⊂ S(U_{n}),

|∇V_{n}(x)| ≤ |∇U_{n}(x)| L^{N} a.e.,  (A.0.21)

and, in turn M(|∇V_{n}|) ≤ M(|∇U_{n}|), so that

{τ_{λ_{n}} ∘ M(|∇V_{n}|)} is p-equi-integrable.  (A.0.22)

Set

E_{n} := {x ∈ ℝ^{N} : M(|DV_{n}|) ≤ 2λ_{n}}.

By Theorem A.0.4 there exists a Lipschitz function v_{n} such that v_{n} = V_{n} L^{N} a.e. on E_{n} and |∇v_{n}| ≤ Cλ_{n}, where C depends only on N and d. Hence L^{N} a.e. on E_{n}

|∇v_{n}| = |∇V_{n}| ≤ |M(|∇V_{n}|)| ≤ \begin{cases}|τ_{λ_{n}} ∘ M(|∇V_{n}|)| & if M(|∇V_{n}|) < λ_{n}, \\ 2|τ_{λ_{n}} ∘ M(|∇V_{n}|)| & if λ_{n} ≤ M(|∇V_{n}|) < 2λ_{n}, \end{cases}  (A.0.23)

while if M(|∇V_{n}|) ≥ 2λ_{n} then

|∇v_{n}| ≤ Cλ_{n} = 2C|τ_{λ_{n}} ∘ M(|∇V_{n}|)|.  (A.0.24)

In view of (A.0.22), (A.0.23) and (A.0.24) to prove the equi-integrability of {|∇v_{n}|^{p}} it remains to study the sequence {|∇v_{n}|^{p}L^{N}⌊F_{n}} where

F_{n} := {x ∈ Ω \ E_{n} : M(|∇V_{n}|) < 2λ_{n}} .

By (A.0.17) and (A.0.20)

∫_{F_{n}} |∇v_{n}|^{p}dx ≤ Cλ_{n}^{p}|F_{n}| + Cλ_{n}^{p−1} ∫_{S(U_{n})} |[V_{n}]| dH^{N−1} ≤ Cλ_{n}^{p}H^{N−1}(S(U_{n})) → 0

as n → ∞. Moreover by (A.0.17)

|{x ∈ ℝ^{N} : v_{n} ≠ U_{n}}| ≤ |{x ∈ ℝ^{N} : |U_{n}| > λ_{n}}| + |{x ∈ ℝ^{N} : M(|DV_{n}|) > 2λ_{n}}|

≤ |\{x ∈ ℝ^{N} : |U_{n}| > λ_{n}\}| + \frac{1}{λ_{n}^{p}} ∫_{\{M(|∇V_{n}|)>λ_{n}\}} |M(|∇V_{n}|)|^{p} dx + \frac{C}{λ_{n}} ∫_{S(U_{n})} |[V_{n}]| dH^{N−1}

≤ \frac{1}{λ_{n}} ∫_{ℝ^{N}} |U_{n}| dx + \frac{1}{λ_{n}^{p}} ∫_{\{M(|∇V_{n}|)>λ_{n}\}} |M(|∇V_{n}|)|^{p} dx + CH^{N−1}(S(U_{n})) → 0

as k → ∞, where we have used the fact that the sequence {U_{n}} is uniformly bounded in L^{1}(ℝ^{N}; ℝ^{d}) by construction, (A.0.19) and (A.0.21), as well as (A.0.18).

To prove that {v_{n}} still converges to u_{0} in L^{1}(Ω; ℝ^{N}), we fix n and a ball B(0, R) such that Ω ⊂ B(0, R). Define z_{n} := v_{n} − V_{n} and consider a sequence {z_{n,j}} ⊂ BV (B(0, 4R); ℝ^{d}) ∩ C^{∞}(B(0, 4R); ℝ^{d}) such that

z_{n,j} → z_{n} in L^{1}(B(0, 4R); ℝ^{d}), ||Dz_{n,j}||(B(0, 4R)) → ||Dz_{n}||(B(0, 4R))

as j → ∞. Without loss of generality we may assume that {z_{n,j}} converges to z_{n} pointwise L^{N} a.e. on B(0, 4R). Since z_{n}(x) = 0 L^{N} a.e. on E_{n} ∩ B(0, 4R), we may find a point x_{0} ∈ E_{n} ∩ Ω such that z_{n}(x_{0}) = 0 and z_{n,j}(x_{0}) converges to z_{n}(x_{0}). Then B(x_{0}, 2R) ⊂ B(0, 4R), and by Poincaré’s inequality on balls

∫_{B(x_{0},2R)} |z_{n,j}(x) − z_{n,j}(x_{0})| dx ≤ C(N, R) ∫_{B(x_{0},2R)} |∇z_{n,j}(x)| dx.

Hence

∫_{B(x_{0},2R)} |z_{n,j}(x)| ≤ |z_{n,j}(x_{0})| |B(x_{0}, 2R)| + C(N, R) ∫_{B(x_{0},2R)} |∇z_{n,j}(x)| dx,

and if we now let j → ∞ we get

∫_{B(x_{0},2R)} |z_{n}(x)| dx ≤ C(N, R)||Dz_{n}||(B(0, 4R)),

and, in turn,

∫_{Ω} |v_{n}(x)| dx ≤ ∫_{Ω} |U_{n}(x)| dx + C(N, R)||Dz_{n}||(B(0, 4R)).

Thus the sequence {v_{n}} is uniformly bounded in W^{1,p}(Ω; ℝ^{N}), and, up to a subsequence, it converges weakly in W^{1,p}(Ω; ℝ^{N}) to some function v_{0} ∈ W^{1,p}(Ω; ℝ^{N}). On the other hand, since

|{x ∈ Ω : v_{n}(x) ≠ U_{n}(x)}| → 0

we conclude that v_{0} = u_{0} L^{N} a.e. on Ω.

Finally, in order to match the boundary conditions for k ∈ ℕ let

Ω_{k} := \{x ∈ Ω : dist(x, ∂Ω) < \frac{1}{k} \} ,

and consider smooth cut-off functions θ_{k} ∈ C_{c}^{∞}(Ω; [0, 1]) such that θ_{k} ≡ 1 in Ω \ Ω_{k}. Set v_{k,n} := θ_{k}v_{n} + (1 − θ_{k})u_{0}. Then v_{k,n} = v_{n} on Ω \ Ω_{k} and

∇v_{k,n} = (v_{n} − u_{0}) ⊗ ∇θ_{k} + θ_{k}(∇v_{n} − ∇u_{0}),

thus

lim_{k→∞} lim_{n→∞} |{x ∈ Ω : v_{k,n}(x) ≠ u_{n}(x), ∇v_{k,n}(x) ≠ ∇u_{n}(x)}| ≤ lim_{k→∞} |Ω_{k}| = 0,

lim_{k→∞} lim_{n→∞} ||v_{k,n} − u_{0}||_{L^{p}(Ω;ℝ^{d})} ≤ lim_{n→∞} ||v_{n} − u_{0}||_{L^{p}(Ω;ℝ^{d})} = 0,

lim_{k→∞} lim_{n→∞} ||∇v_{k,n} − (∇v_{n} − ∇u_{0})||_{L^{p}(Ω;ℝ^{d×N})} ≤ lim_{k→∞} lim_{n→∞} ||∇θ_{k}||_{L^{∞}(Ω;ℝ^{N}}||v_{n} − u_{0}||_{L^{p}(Ω;ℝ^{d})}

+ lim_{k→∞} lim_{n→∞} ||∇v_{n} − ∇u_{0}||_{L^{p}(Ω_{k};ℝ^{d})} = 0,

where we have used the equi-integrability of {|∇v_{n}|^{p}}. Thus we may extract a subsequence with all the required properties.

Next we present a result due to Braides, Defranceschi and Vitali [22] (see also [6]).

Lemma A.0.8. Let {u_{n}} ⊂ SBV_{p}(Q_{ν}; ℝ^{d}) ∩ L^{∞}(Q_{ν}; ℝ^{d}) be a sequence satisfying

∥u_{n}∥_{L^{∞}(Q_{ν};ℝ^{d})} + H^{N−1}(Q_{ν} ∩ S(u_{n})) ≤ C, lim_{n→∞} ∫_{Q_{ν}} |∇u_{n}|^{p} dx = 0,

and u_{n}(y) = u_{a,b,ν}(y) on ∂Q_{ν}. Then there exists a sequence {v_{n}} ⊂ SBV_{0}(Q_{ν}; ℝ^{d}) such that

lim_{n→∞} ∥v_{n} − u_{n}∥_{L^{∞}(Q_{ν};ℝ^{d})} = 0,  lim_{n→∞} H^{N−1}(S(v_{n}) \ S(u_{n})) = 0

and v_{n}(y) = u_{a,b,ν}(y) on ∂Q_{ν}.

Proof. Let k_{n} be the integer part of

(∫_{Q_{ν}} |∇u_{n}| dx)^{−1/2} ,

so that k_{n} → ∞ and k_{n} ∫_{Q_{ν}} |∇u_{n}| dx → 0 as n → ∞. Take n large enough so that 2/k_{n} < |b^{(i)} − a^{(i)}|, whenever b^{(i)} ≠ a^{(i)}, i = 1, …, d, where a^{(i)} and b^{(i)} are the components of the vectors a and b, respectively. By the Coarea Formula for each i = 1, …, d we have

\begin{aligned}∫_{Q_{ν}} |∇u_{n}^{(i)}| dx & = ∫_{Q_{ν} \S(u_{n})} |∇u_{n}^{(i)}| dx = ∫_{Q_{ν} \S(u_{n})} |Du_{n}^{(i)}| \\ & = ∑_{j=0}^{k_{n}−1} ∫_{α_{j}}^{α_{j+1}} H^{N−1} ((Q_{ν} \ S(u_{n})) ∩ ∂^{∗}\{x ∈ Q_{ν} : u_{n}^{(i)} > t\}) dt, \end{aligned}

where α_{0} := −C, α_{1} := −C + \frac{1}{k_{n}}, …, α_{k_{n}} := C. It is clear that for each j ∈ \{0, …, k_{n} − 1\} we may find t_{j}^{(i)} ∈ (α_{j}, α_{j+1}) such that

H^{N−1} (Q_{ν} \ S(u_{n}) ∩ ∂^{∗}{x ∈ Q_{ν} : u_{n}^{(i)} > t_{j}^{(i)}}) < 2 ∫_{Q_{ν}} |∇u_{n}^{(i)}| dx.

Let t_{−1}^{(i)} := −C, t_{k_{n}}^{(i)} := C and for j ∈ {−1, …, k_{n} − 1} set

E_{j}^{(i)} := \{x ∈ Q_{ν} : t_{j}^{(i)} < u_{n}^{(i)} ≤ t_{j+1}^{(i)}\} , v_{n}^{(i)}(x) := \begin{cases}a^{(i)} & if x ∈ E_{j}^{(i)} and t_{j}^{(i)} < a^{(i)} ≤ t_{j+1}^{(i)} \\ b^{(i)} & if x ∈ E_{j}^{(i)} and t_{j}^{(i)} < b^{(i)} ≤ t_{j+1}^{(i)} \\ t_{j}^{(i)} & otherwise in E_{j}^{(i)}, \end{cases}

where we have used the fact that 2/k_{n} < |b^{(i)} − a^{(i)}| whenever b^{(i)} ≠ a^{(i)}. Clearly the sets E_{j}^{(i)} = {x ∈ Q_{ν} : u_{n}^{(i)} > t_{j}^{(i)}} \ {x ∈ Q_{ν} : u_{n}^{(i)} > t_{j+1}^{(i)}} are of finite perimeter, with Q_{ν} = ∪_{j=−1}^{k_{n}−1}E_{j}^{(i)}. Moreover v_{n}(y) = u_{a,b,ν}(y) on ∂Q_{ν}, ∥v_{n} − u_{n}∥_{∞} ≤ 1/k_{n},

S(v_{n}) ⊂ ⋃_{i=1}^{d} ⋃_{j=−1}^{k_{n}−1} ∂^{∗}E_{j}^{(i)} ⊂ ⋃_{i=1}^{d} ⋃_{j=−1}^{k_{n}−1} ∂^{∗}{x ∈ Q_{ν} : u_{n}^{(i)} > t_{j}^{(i)}} ∪ ∂^{∗}{x ∈ Q_{ν} : u_{n}^{(i)} > t_{j+1}^{(i)}},

so that

\begin{matrix}H^{N−1}(S(v_{n}) \ S(u_{n})) ≤ ∑_{i=1}^{d} ∑_{j=−1}^{k_{n}−1}[H^{N−1} ((Q_{ν} \ S(u_{n})) ∩ ∂^{∗}\{x ∈ Q_{ν} : u_{n}^{(i)} > t_{j}^{(i)}\}) \\ +H^{N−1} ((Q_{ν} \ S(u_{n})) ∩ ∂^{∗}\{x ∈ Q_{ν} : u_{n}^{(i)} > t_{j+1}^{(i)}\})] ≤ C(k_{n} + 1) ∫_{Q_{ν}} |∇u_{n}| dx → 0. \end{matrix}

Proof of Theorem 3.1.2. Step 1. We first prove Theorem 3.1.2 under the additional hypotheses

f_{0}(x, u, ξ) ≥ C_{1}|ξ|^{p} and g_{0}(x, λ, θ, ν) ≥ C_{1} (1 + |λ − θ|)  (A.0.25)

for L^{N} a.e. x ∈ Ω, all u, λ, θ ∈ ℝ^{d}, ξ ∈ ℝ^{d×N}, ν ∈ S^{N−1}, and for some C_{1} > 0.

Without loss of generality we may assume that

lim inf_{n→∞} (∫_{Ω} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩Ω} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1})

= lim_{n→∞} ∫_{Ω} f_{0}(x, u_{n}, ∇u_{n}) dx + lim_{n→∞} ∫_{S(u_{n})∩Ω} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1} < ∞.

Passing to a subsequence, if necessary, there exists a nonnegative Radon measure μ such that

f_{0}(x, u_{n}(x), ∇u_{n}(x)) L^{N}⌊ Ω + g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) H^{N−1}⌊ S(u_{n}) ∩ Ω ⇀^{⋆} μ

as n → ∞, weakly ⋆ in the sense of measures. We claim that

\frac{dμ}{dL^{N}} (x_{0}) ≥ f(x_{0}, u_{n}(x_{0}), ∇u_{n}(x_{0}))  for L^{N} a.e. x_{0} ∈ Ω,  (A.0.26)
\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ g_{0}(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0}))  for H^{N−1} a.e. x_{0} ∈ Ω ∩ S(u).  (A.0.27)

If (A.0.26) and (A.0.27) hold, then the conclusion of the theorem follows immediately. Indeed, let φ ∈ C_{c}(Ω; ℝ), 0 ≤ φ ≤ 1. We have

lim inf_{n→∞} (∫_{Ω} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩Ω} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1}) ≥ lim inf_{n→∞}(∫_{Ω} φ(x) f_{0}(x, u_{n}, ∇u_{n}) dx

+ ∫_{S(u_{n})∩Ω} φ(x) g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1}) = ∫_{Ω} φ dμ ≥ ∫_{Ω} φ \frac{dμ}{dL^{N}} dx + ∫_{S(u)∩Ω} φ \frac{dμ}{dH^{N−1}⌊S(u)} dH^{N−1}

≥ ∫_{Ω} φ(x) f_{0}(x, u, ∇u) dx + ∫_{S(u)∩Ω} φ(x) g_{0}(x, u^{+}, u^{−}, ν_{u}) dH^{N−1}.

By letting φ → 1, and using Monotone Convergence Theorem, we obtain the desired result.

To show (A.0.26) for each j ∈ ℕ we apply Lemma A.0.2 to obtain a compact set K_{j} ⊂ Ω, with |Ω \ K_{j}| ≤ 1/j, such that

f_{0} : K_{j} × ℝ^{N} × ℝ^{d×N} → [0, +∞)

is lower semicontinuous. Let K_{j}^{∗} be the set of Lebesgue points of χ_{K_{j}}, and set

ω := ⋃_{j}^{∞}(K_{j} ∩ K_{j}^{∗}).  (A.0.28)

Then

|Ω \ ω| ≤ |Ω \ K_{j}| ≤ \frac{1}{j} → 0 as j → ∞.

Fix x_{0} ∈ ω such that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q(x_{0}, ε))}{ε^{N}} < ∞, lim_{ε→0^{+}} \frac{1}{ε^{N+1}} ∫_{Q(x_{0},ε)} |u(x) − u(x_{0}) − ∇u(x_{0})(x − x_{0})|dx = 0. (A.0.29)

Choosing ε_{k} ↘ 0 such that μ(∂Q(x_{0}, ε_{k})) = 0, we have

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} \frac{μ(Q(x_{0}, ε_{k}))}{ε_{k}^{N}}

= lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{N}} (∫_{Q(x_{0},ε_{k})} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩Q(x_{0},ε_{k})} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1})

= lim_{k→∞} lim_{n→∞}(∫_{Q} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{n,k}(y), ∇w_{n,k}(y)) dy

+ \frac{1}{ε_{k}} ∫_{S(w_{n,k})∩Q} g_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{n,k}^{+}(y), u(x_{0}) + ε_{k}w_{n,k}^{−}(y), ν_{w_{n,k}}) dH^{N−1}),

(A.0.30)

where

w_{n,k}(y) := \frac{u_{n}(x_{0} + ε_{k}y) − u(x_{0})}{ε_{k}} .

Clearly w_{n,k} ∈ SBV_{p}(Q; ℝ^{d}), and by (A.0.29)_{2}, lim_{k→∞} lim_{n→∞} ||w_{n,k} − w_{0}||_{L^{1}(Q;ℝ^{d})} = 0, where w_{0}(y) := ∇u(x_{0})y.

Moreover, by (A.0.29)_{1}, (A.0.30) and the coercitivity hypotheses (A.0.25 on f_{0} and g_{0}, we have that

sup_{k} ∫_{Q} |∇w_{n,k}|^{p} dy < ∞,  lim_{k→∞} lim_{n→∞} ∫_{S(w_{n,k})∩Q}(1/ε_{k} + |[w_{n,k}]|) dH^{N−1} < ∞.

By a standard diagonalization argument, we may extract a subsequence w_{k} := w_{n_{k},k} which converges to w_{0} in L^{1}(Q; ℝ^{d}) and such that

lim_{k→∞} H^{N−1}(S(w_{k}) ∩ Q) = 0,  sup_{k} (∫_{Q} |∇w_{k}|^{p} dy + ∫_{S(w_{k})∩Q} |[w_{k}]| dH^{N−1}) < ∞,

and

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim_{k→∞} ∫_{Q} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}w_{k}(y), ∇w_{k}(y)) dy.

By Theorem A.0.7 we may find a subsequence (not relabeled) of {w_{k}} and a sequence {v_{k}} ⊂ W^{1,∞}(Q; ℝ^{d}) weakly converging to w_{0} in W^{1,p}(Q; ℝ^{d}) such that v_{k} = w_{0} on ∂Q,

|{y ∈ Q : v_{k}(y) ≠ w_{k}(y), ∇v_{k}(y) ≠ ∇w_{k}(y)}| → 0 as k → ∞,  (A.0.31)

and {|∇v_{k}|^{p}} is equi-integrable. Hence

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{k→∞} ∫_{\{v_{k}=w_{k}\}} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy

= lim_{k→∞} ∫_{Q} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy,

where we have also used the growth condition on f_{0}, the p-equi-integrability of {v_{k}} and (A.0.31). As x_{0} ∈ ω there exists an integer j_{0} such that x_{0} ∈ K_{j_{0}} ∩ K_{j_{0}}^{∗}. By Lemma A.0.2 for any fixed integer j ∈ ℕ there exists 0 < ρ_{j} < 1 such that

f_{0}(x_{0}, u(x_{0}), ξ) ≤ f_{0}(x, u, ξ) + \frac{1}{j}  (A.0.32)

for all (x, u, ξ) ∈ K_{j_{0}} × \overline{B_{d}(u(x_{0}), 1)} × \overline{B_{d×N}(0, j)}, with |x − x_{0}| + |u − u(x_{0})| ≤ ρ_{j}. Set

E_{k,j} := {y ∈ Q : |ε_{k}v_{k}(y)| ≤ ρ_{j}, |∇v_{k}(y)| ≤ j}.

Since {v_{k}} is bounded in W^{1,1}(Q; ℝ^{d}), we have that

lim_{j→∞} lim_{k→∞} |Q \ E_{k,j}| = 0.  (A.0.33)

Thus

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim_{k→∞} ∫_{Q} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy

≥ lim_{j→∞} lim_{k→∞} ∫_{E_{k,j}} f_{0}(x_{0} + ε_{k}y, u(x_{0}) + ε_{k}v_{k}(y), ∇v_{k}(y)) dy,

= lim_{j→∞} lim_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{x_{0}+ε_{k}E_{k,j}} f_{0}(x, u(x_{0}) + ε_{k}v_{k}((x − x_{0})/ε_{k}), ∇v_{k}((x − x_{0})/ε_{k})) dx.

By (A.0.32)

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{K_{j_{0}} ∩(x_{0}+ε_{k}E_{k,j})} f_{0}(x, u(x_{0}) + ε_{k}v_{k}((x − x_{0})/ε_{k}), ∇v_{k}((x − x_{0})/ε_{k})) dx

≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{K_{j_{0}} ∩(x_{0}+ε_{k}E_{k,j})} f_{0}(x_{0}, u(x_{0}), ∇v_{k}((x − x_{0})/ε_{k})) dx.

(A.0.34)

Using the fact that |∇v_{k}| ≤ j in E_{k,j} and the growth condition on f_{0}, we have that for each fixed j

\frac{1}{ε_{k}^{N}} ∫_{(x_{0}+ε_{k}E_{k,j})\K_{j_{0}}} f_{0}(x_{0}, u(x_{0}), ∇v_{k}((x − x_{0})/ε_{k})) dx ≤ C(1 + j^{p}) \frac{|Q(x_{0}, ε_{k}) \ K_{j_{0}}|}{ε_{k}^{N}} → 0

as k → ∞, because x_{0} is a Lebesgue point of χ_{K_{j_{0}}} . Consequently, from (A.0.34) we get

\frac{dμ}{dL^{N}} (x_{0}) ≥ lim inf_{j→∞} lim inf_{k→∞} \frac{1}{ε_{k}^{N}} ∫_{x_{0}+ε_{k}E_{k,j}} f_{0}(x_{0}, u(x_{0}), ∇v_{k}((x − x_{0})/ε_{k})) dx

= lim_{j→∞} lim_{k→∞} ∫_{E_{k,j}} f_{0}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy = lim_{k→∞} ∫_{Q} f_{0}(x_{0}, u(x_{0}), ∇v_{k}(y)) dy,

where we have used, once again, the growth condition on f_{0}, the equi-integrability of {|∇v_{k}|^{p}} and (A.0.33). Since v_{k} = w_{0} on ∂Q it now follows from the quasiconvexity of f_{0} that

\frac{dμ}{dL^{N}} (x_{0}) ≥ f_{0}(x_{0}, u(x_{0}), ∇u(x_{0})),

thus establishing (A.0.26).

In order to prove (A.0.27) fix x_{0} ∈ Ω such that

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) = lim_{ε→0^{+}} \frac{μ(Q_{ν}(x_{0}, ε))}{ε^{N−1}} < ∞, lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν} (x_{0},ε)} |u(x) − u_{x_{0},ν}(x)| dx = 0.  (A.0.35)

Choosing ε_{k} ↘ 0 such that μ(∂Q_{ν}(x_{0}, ε_{k})) = 0, we have

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) = lim_{k→∞} \frac{μ(Q_{ν}(x_{0}, ε_{k}))}{ε_{k}^{N−1}}

= lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{N−1}} (∫_{Q_{ν} (x_{0},ε_{k})} f_{0}(x, u_{n}, ∇u_{n}) dx + ∫_{S(u_{n})∩Q_{ν} (x_{0},ε_{k})} g_{0}(x, u_{n}^{+}, u_{n}^{−}, ν_{u_{n}}) dH^{N−1})

= lim_{k→∞} lim_{n→∞}(∫_{Q_{ν}} ε_{k} f_{0} (x_{0} + ε_{k}y, w_{n,k}(y), \frac{1}{ε_{k}} ∇w_{n,k}(y)) dy
+ ∫_{S(w_{n,k})∩Q_{ν}} g_{0}(x_{0} + ε_{k}y, w_{n,k}^{+}(y), w_{n,k}^{−}(y), ν_{w_{n,k}}) dH^{N−1}),

(A.0.36)

where

w_{n,k}(y) := u_{n}(x_{0} + ε_{k}y).

Clearly w_{n,k} ∈ SBV_{p}(Q_{ν}; ℝ^{d}), and by (A.0.35)_{2}, lim_{k→∞} lim_{n→∞} ||w_{n,k} − u_{x_{0},ν}||_{L^{1}(Q_{ν};ℝ^{d})} = 0. Moreover by (A.0.35)_{1},

(A.0.36) and the coercitivity hypotheses on f_{0} and g_{0} we have that

lim_{k→∞} lim_{n→∞} \frac{1}{ε_{k}^{p−1}} ∫_{Q_{ν}} |∇w_{n,k}|^{p} dy < ∞,  sup_{k} ∫_{S(w_{n,k})∩Q_{ν}} (1 + |[w_{n,k}]|) dH^{N−1} < ∞.

By a standard diagonalization argument, we may extract a subsequence w_{k} := w_{n_{k},k} which converges to u_{x_{0},ν} in L^{1}(Q_{ν}; ℝ^{d}) and such that

lim_{k→∞} ∫_{Q_{ν}} |∇w_{k}|^{p} dy = 0,  sup_{k} ∫_{S(w_{k})∩Q_{ν}} (1 + |[w_{k}]|) dH^{N−1} < ∞  (A.0.37)

and

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ lim_{k→∞} ∫_{S(w_{k})∩Q_{ν}} g_{0}(x_{0} + ε_{k}y, w_{k}^{+}(y), w_{k}^{−}(y), ν_{w_{k}}) dH^{N−1}.

Using Remarks 3.1.4 and 3.1.9 (with C_{l,k} = D_{l,k} ≡ 1), we may assume that sup_{k} ∥w_{k}∥_{L^{∞}(Q_{ν};ℝ^{d})} ≤ M, for some M, and that w_{k} = u_{x_{0},ν} on ∂Q_{ν}. Fix η > 0. By Remark A.0.3 there exists δ = δ(x_{0}, η, M) ∈ (0, 1) such that

g(x_{0}, λ, θ, ν) ≤ g(x, λ, θ, ν) + η

for all x ∈ Ω, with |x − x_{0}| ≤ δ and all (λ, θ, ν) ∈ (ℝ^{d})^{2} × S^{N−1} with |λ|, |θ| ≤ M. Hence

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ lim sup_{k→∞} ∫_{S(w_{k})∩Q_{ν}} g_{0}(x_{0}, w_{k}^{+}(y), w_{k}^{−}(y), ν_{w_{k}}) dH^{N−1} − η sup_{j} H^{N−1}(S(w_{j}) ∩ Q_{ν}).

By letting η → 0^{+}, and in view of A.0.37, we conclude that

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ lim sup_{k→∞} ∫_{S(w_{k})∩Q_{ν}} g_{0}(x_{0}, w_{k}^{+}(y), w_{k}^{−}(y), ν_{w_{k}}) dH^{N−1}.

By Lemma A.0.8, there exists a sequence {v_{k}} ⊂ SBV_{0}(Q_{ν}; ℝ^{d}) such that

lim_{n→∞} ∥v_{k} − w_{k}∥_{L^{∞}(Q_{ν};ℝ^{d})} = 0,  lim_{n→∞} H^{N−1}(S(v_{k}) \ S(w_{k})) = 0

and v_{k} = u_{x_{0},ν} on ∂Q_{ν}. Then

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ lim sup_{k→∞} ∫_{Q_{ν} ∩S(w_{k})∩S(v_{k})} g_{0}(x_{0}, w_{k}^{+}, w_{k}^{−}, ν_{w_{k}}) dH^{N−1}

≥ lim sup_{k→∞} ∫_{Q_{ν} ∩S(v_{k})} g_{0}(x_{0}, v_{k}^{+}, v_{k}^{−}, ν_{v_{k}}) dH^{N−1} − I_{1} − I_{2},

(A.0.38)

where

I_{1} := lim sup_{k→∞} ∫_{Q_{ν} ∩(S(v_{k})\S(w_{k}))} g_{0}(x_{0}, v_{k}^{+}, v_{k}^{−}, ν_{v_{k}}) dH^{N−1},
I_{2} := lim sup_{k→∞} ∫_{Q_{ν} ∩S(w_{k})∩S(v_{k})} |g_{0}(x_{0}, w_{k}^{+}, w_{k}^{−}, ν_{w_{k}}) − g_{0}(x_{0}, v_{k}^{+}, v_{k}^{−}, ν_{v_{k}})| dH^{N−1}.

Since H^{N−1}(S(v_{k}) \ S(w_{k})) → 0 and, by the growth condition (3.1.5), {g_{0}(x_{0}, v_{k}^{+}, v_{k}^{−}, ν_{v_{k}})} is bounded, we deduce that I_{1} = O (1/k). Also, as g_{0}(x_{0}, ·, ·, ·) is uniformly continuous in B_{d}(0, M) × B_{d}(0, M) × S^{N−1}, ν_{v_{k}} = ν_{w_{k}} H^{N−1}a.e. in S(w_{k}) ∩ S(v_{k}) and ∥v_{k} − u_{k}∥_{L^{∞}(Q_{ν};ℝ^{d})} → 0, we conclude that I_{2} = O (1/k). Passing to the limit as k → +∞ in (A.0.38), we get

\frac{dμ}{dH^{N−1}⌊S(u)} (x_{0}) ≥ lim sup_{k→∞} ∫_{S(v_{k})∩Q_{ν}} g_{0}(x_{0}, v_{k}^{+}, v_{k}^{−}, ν_{v_{k}}) dH^{N−1} ≥ g_{0}(x_{0}, u^{+}(x_{0})), u^{−}(x_{0}), ν_{u}(x_{0}))

due to the BV -ellipticity of g_{0}.

Step 2. The proof of Theorem 3.1.2 in the general case is now standard. We fix ε > 0 and consider the new integrands

f_{ε}(x, u, ξ) := f_{0}(x, u, ξ) + ε |ξ|^{p} and g_{ε}(x, λ, θ, ν) := g_{0}(x, λ, θ, ν) + ε (1 + |λ − θ|).

We may now apply the Step 1 and then by (3.1.6) we may let ε → 0^{+}. We omit the details.

Acknowledgments. The research of G. Bouchitté was partially supported by ICCTI-MENESR 166-BZ, the research of I. Fonseca by the National Science Foundation under Grant No. DMS–9731957, the research of G. Leoni by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria” and by GNAFA, the research of L. Mascarenhas by ICCTI-PRAXIS XXI, FEDER-PRAXIS/2/2.1/MAT/125/94, PRAXIS-FEDER/3/3.1/CTM/10/94 and ICCTI-MENESR 166-82. The authors acknowledge the hospitality of the Universities of Toulon (ANLA), Carnegie Mellon and its Center for Nonlinear (CNA, Grant No. DMS–9803791) and Analysis and Lisbon (CMAF) where the present work was undertaken.

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