A RELAXATION RESULT IN THE FRAMEWORK OF STRUCTURED DEFORMATIONS IN THE BV-SETTING

MARGARIDA BAÍA, JOSÉ MATIAS & PEDRO MIGUEL SANTOS

Abstract. In this work we obtain an integral representation of an energy for structured deformations of continua in the space of functions of bounded variation, as a first step to the study of asymptotic models for thin defective crystalline structures, where phenomena as slips, vacancies and dislocations prevent the effectiveness of classical theories.

Keywords: Structured Deformations, Relaxation, Functions of Bounded Variation.

2000 Mathematics Subject Classification: 74E15, 49J45, 28A33, 46T30.

1. Introduction

The theory of first order structured deformations introduced by Del Piero & Owen [21] sets a basis to address a large variety of problems in continuum mechanics where geometrical changes can be associated to smooth-classical deformations, piecewise deformations and more complex deformations for which an analysis at macroscopic and microscopic level is required. Namely, this theory can be applied to problems related to plasticity, crystals with defects, liquid crystals and material composites.

From a variational point of view, the problem of assigning a free energy to a body that undergoes a structured deformation was first studied by Choksi and Fonseca [14] in the context of functions of special bounded variation, shortly denoted by SBV (we refer to Section 2 for the main notations and concepts used throughout this work). This energy was defined as the more effective way to build up the deformation using sequences of approximating simple deformations in SBV. More precisely, the authors defined a structured deformation in an open and bounded set Ω ⊂ ℝ^{N}, N ≥ 1, as a pair (g, G) where the macroscopic deformation g is an element of SBV (Ω; ℝ^{d}) and G (deformation without disarrangements) is an integrable tensor field in Ω. Using a Lusin-type result due to Alberti [2] (Theorem 2.5 below) it was proven in [14] that given such a pair there exist deformations u_{n} in SBV (Ω; ℝ^{d}) with

u_{n} ⟶^{L^{1}} g and ∇u_{n} ⇀^{M(Ω)} G.  (1.1)

As a result the energy associated to (g, G), I(g, G), was defined as the relaxation with respect to the topology given in (1.1) of the functional

E(u) = ∫_{Ω} W(∇u) dx + ∫_{S_{u}} Ψ([u], ν_{u}) dH^{N−1}, u ∈ SBV (Ω; ℝ^{d}),

for appropriate bulk and interfacial densities W and Ψ, that is

I(g, G) := inf_{{u_{n}}⊂SBV (Ω;ℝ^{d})} {lim inf_{n→∞} E(u_{n}), u_{n} ⟶^{L^{1}} g, ∇u_{n} ⇀^{M(Ω)} G} .  (1.2)

The main objective in [14] was then to characterize I(g, G) through an integral representation formula. The principal difference of this characterization with respect to previous integral representation results for similar relaxed energies, where relaxation is taken with respect to the L^{1} (BV weak) topology (cf. Ambrosio & Dal Maso [4], Barroso, Bouchitté, Buttazo & Fonseca [7], Braides & Coscia [9], Bouchitté, Fonseca & Mascarenhas [11], Bouchitté, Fonseca, Leoni & Mascarenhas [12], Fonseca & Müller [26, 27]), is the fact that gradients of approximating sequences {u_{n}} in (1.1) are constrained to converge to the given function G (not necessarily ∇g). In this case, if ∇u_{n} → G in L^{1}, the difference G − Dg is achieved by the limit of singular measures since Du_{n} → Dg in the sense of distributions.

Moreover, the Hausdorff measure of the jump set of u_{n} tends necessarily to infinity, otherwise Theorem 2.1 in Ambrosio [1] asserts that G = ∇g almost everywhere.

In the context of defective crystals, as already mentioned in [14], (1.2) can be interpreted as a way to realize the deformed crystal by piecing together elastic crystals at a finer and finer scale. We refer to Choksi, Del Piero, Fonseca & Owen [13] where the framework introduced in [14] has been used to predict in simple models the origins and main characteristics of phenomena as fracture, yielding and hysterisis, with applications to single deffective crystals (see Section 4.3 in [13]).

Our objective in this work is to generalize the relaxation result derived in [14] to the full BVsetting for a class of second order energies suitable to the study of equilibrium configurations of thin defective crystalline structures as addressed by Matias and Santos [33]. As noticed in [33] the energy considered in [14] is not appropriate to this study since some control in the second order derivatives of the deformation is needed to avoid geometrical obstacles in the thin film limit.

To present our main result, Theorem 4.1, whose complete statement is given in Section 4, we denote by BV^{2}(Ω; ℝ^{d}) (SBV^{2}(Ω; ℝ^{d})), d ≥ 1, the space of functions u ∈ BV (Ω; ℝ^{d}) such that ∇u ∈ BV (Ω; ℝ^{d×N}) (respectively for SBV^{2}(Ω; ℝ^{d})) and we start by introducing the space of generalized structured deformations

GSD(Ω; ℝ^{d}) := BV^{2}(Ω; ℝ^{d}) × BV (Ω; ℝ^{d×N}).

For any (g, G) ∈ GSD(Ω; ℝ^{d}) we consider the relaxed energy

I(g, G) = inf_{{u_{n}}⊂SBV^{2}(Ω;ℝ^{d})} {lim inf_{n→∞} E(u_{n}), u_{n} ⟶^{L^{1}} g, ∇u_{n} ⟶^{L^{1}} G}  (1.3)

where

E(u) = ∫_{Ω} W (∇u, ∇^{2}u) dx + ∫_{S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1} + ∫_{S_{∇u}} Ψ_{2}([∇u], ν_{∇u}) dH^{N−1}

for u ∈ SBV^{2}(Ω; ℝ^{d}), and the functions W, Ψ_{1}, Ψ_{2} satisfy the hypotheses (H_{1}) − (H_{7}) introduced in Section 3. Under hypotheses (H_{1}) − (H_{7}), Theorem 4.1 asserts that there exist bulk and interfacial densities W_{1}, W_{2}, γ_{1}, γ_{2} (see (3.2)-(3.5)) such that for all (g, G) ∈ GSD(Ω; ℝ^{d})

I(g, G) = ∫_{Ω} (W_{1}(G − ∇g) + W_{2}(G, ∇G)) dx + ∫_{S_{g}} γ_{1}([g], ν_{g})dH^{N−1}

+ ∫_{S_{G}} γ_{2}(G^{+}, G^{−}, ν_{G}) dH^{N−1} + ∫_{Ω} W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}) d|D^{c}g|

+ ∫_{Ω} W_{2}^{∞} (G, \frac{dD^{c}G}{d|D^{c}G|}) d|D^{c}G|  (1.4)

where W_{2}^{∞}, as usual, denotes the recession function of W_{2} in the second variable, that is,

W_{2}^{∞}(A, B) = lim sup_{t→∞} \frac{W (A, tB)}{t}, A ∈ ℝ^{d×N}, B ∈ ℝ^{d×N×N}.

To show (1.4) we start by deriving a similar relaxation result in the SBV -setting, Theorem 3.2. Althought this characterization could have been derived directly for the whole energy I(g, G) using localization and blow up methods, Alberti’s theorem (Theorem 2.5 below) allows us to divide this energy into two first order relaxed energies I_{1}(g, G) and I_{2}(G), rendering our arguments more concise. The effect of the structured deformation is captured in the first energy I_{1}(g, G) through the limit energy density W_{1} that depends on G − ∇g (deformation due to disarrangements at the microscopic level). The full BV characterization (1.4) follows then from Theorem 3.2 together with a sequential characterization of the energy I_{1}(g, G) (cf. Lemma 4.3 below), Reshetnyak’s Theorem (cf. Theorem 2.2) and Alberti’s Rank One Theorem (cf. [3]).

We finish this introduction by referring to Carriero, Leaci and Tomarelli (see [15], [16] and the references therein) for other second order variational problems arising from some models in image segmentation and in continuum mechanics.

The overall plan of this work in the ensuing sections will be as follows: Section 2 collects the main notations and results used throughout. Section 3 is devoted to the proof of the SBV counterpart of our main result, Theorem 3.2. The proof of Theorem 4.1 is obtained in Section 4.

2. Preliminaries

The purpose of this part is to fix some notation and to give a brief overview of the concepts and main results that are used in subsequent sections. All these results are stated without proofs as they can be readily found in the references given below.

2.1. Notation. Throughout the text Ω ⊂ ℝ^{N}, N ≥ 1, will denote an open bounded set and we will use the following notations:

- A(Ω) is the family of all open subsets of Ω,

- M(Ω) is the set of finite Radon measures on Ω,

- ||μ|| stands for the total variation of a measure μ ∈ M(Ω),

- S^{N−1} stands for the unit sphere in ℝ^{N},

- e_{i} denotes the i^{th} element of the canonical basis of ℝ^{N} for i = 1, … N,

- Q denotes the unit cube centered at the origin with one side orthogonal to e_{N},

- Q(x, δ) denotes a cube centered at x ∈ Ω with side length δ and with one side orthogonal to e_{N},

- Q_{ν}(x, δ) is the cube centered at x ∈ Ω with side length δ and with one side orthogonal to ν ∈ S^{N−1},

- Q_{ν} := Q_{ν}(0, 1),

- ℝ^{d×N×N} is the set of real tensors of order d × N × N, d ≥ 1,

- C represents a generic constant,

- lim_{n,m} := lim_{n} lim_{m} while lim_{m,n} := lim_{m} lim_{n} .

2.2. Measure Theory. We start by recalling a generalization of the Besicovitch Differentiation Theorem due to Ambrosio and Dal Maso [4].

Theorem 2.1. If λ and μ are Radon measures in Ω, μ ≥ 0, then there exists a Borel set E ⊂ Ω such that μ(E) = 0, and for every x ∈ supp μ\E

\frac{dλ}{dμ}(x) := lim_{ϵ→0} \frac{λ(x + ϵC)}{μ(x + ϵC)}

exists and is finite whenever C is a bounded, convex, open set containing the origin.

We also recall Reshetnyak’s Theorem on weak convergence of vector measures (see Reshetnyak [35]; see also Ambrosio, Fusco and Pallara [5]).

Theorem 2.2. Let μ, μ_{n} be ℝ^{d}−valued finite Radon measures in Ω such that μ_{n} ⇀^{∗} μ in Ω and such that ||μ_{n}||(Ω) → ||μ||(Ω). Then

lim_{n→∞} ∫_{Ω} f (x, \frac{μ_{n}}{||μ_{n}||}(x)) d||μ_{n}||(x) = ∫_{Ω} f (x, \frac{μ}{||μ||}(x)) d||μ||(x)

for every continuous and bounded function f : Ω × S^{d−1} → ℝ.

2.3. BV-functions. In this part we briefly summarize some facts on functions of functions of bounded variation to be used afterwards. We refer to Ambrosio, Fusco and Pallara [5], Evans and Gariepy [23], Federer [24], Giusti [29] and Ziemer [36] for a detailed description of this subject.

A function u ∈ L^{1}(Ω; ℝ^{d}) is said to be of bounded variation, and we write u ∈ BV (Ω; ℝ^{d}), if all its first distributional derivatives D_{j}u_{i} ∈ M(Ω) for i = 1, ..., d and j = 1, ..., N. The matrix-valued measure whose entries are D_{j}u_{i} is denoted by Du. The space BV (Ω; ℝ^{d}) is a Banach space when endowed with the norm

∥u∥_{BV} = ∥u∥_{L^{1}} + ||Du||(Ω).

By the Lebesgue Decomposition Theorem Du can be split into the sum of two mutually singular measures D^{a}u and D^{s}u (the absolutely continous part and singular part, respectively, of Du with respect to the Lebesgue measure L^{N}). By ∇u we denote the Radon-Nikodým derivative of D^{a}u with respect to L^{N}, so that we can write

Du = ∇uL^{N}⌊Ω + D^{s}u.

Let Ω_{u} be the set of points where the approximate limit of u exists, i.e., x ∈ Ω such that there exist z ∈ ℝ^{N} with

lim_{ε→0} − ∫_{Q(x,ε)} |u(y) − z| dy = 0.

If x ∈ Ω_{u} and z = u(x) we say that u is approximately continuous at x (or that x is a Lebesgue point of u). The function u is approximately continuous L^{N}-a.e. x ∈ Ω_{u} and

L^{N}(Ω \ Ω_{u}) = 0.  (2.1)

Let S_{u} be the jump set of this function, i.e., the set of points x ∈ Ω \ Ω_{u} for which there exists a, b ∈ ℝ^{N} and a unit vector ν ∈ S^{N−1}, normal to S_{u} at x, such that a ≠ b and

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{\{y∈Q_{ν}(x,ε) : (y−x)·ν>0\}} |u(y) − a| dy = 0  (2.2)

and

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{\{y∈Q_{ν}(x,ε) : (y−x)·ν<0\}} |u(y) − b| dy = 0.  (2.3)

The triple (a, b, ν) uniquely determined by (2.2) and (2.3) up to permutation of (a, b), and a change of sign of ν and is denoted by (u^{+}(x), u^{−}(x), ν_{u}(x)).

If u ∈ BV (Ω) it is well known that S_{u} is countably N − 1 rectifiable, i.e.

S_{u} = ⋃_{n=1}^{∞} K_{n} ∪ E,

where H^{N−1}(E) = 0 and K_{n} are compact subsets of C^{1}-hypersurfaces. Furthermore, H^{N−1}((Ω\Ω_{u})\ S_{u}) = 0 and the following decomposition holds

Du = ∇uL^{N}⌊Ω + [u] ⊗ ν_{u}H^{N−1}⌊S_{u} + D^{c}u,

where [u] := u^{+} − u^{−} and D^{c}u is the Cantor part of the measure Du, i.e., D^{c}u = D^{s}u⌊(Ω_{u}).

If Ω is an open and bounded set with Lipschitz boundary then the outer unit normal to ∂Ω (denoted by ν) exists H^{N−1} a.e. and the trace for functions in BV (Ω; ℝ^{d}) is defined.

We next recall some useful results on BV functions used in the sequel.

Theorem 2.3. (Approximate Differentiability) If u ∈ BV (Ω; ℝ^{d}), then for L^{N}-a.e. x ∈ Ω

lim_{ϵ→0} \frac{1}{ϵ^{N}} \{∫_{Q(x,ϵ)} |u(y) − u(x) − ∇u(x).(y − x)|^{\frac{N}{N−1}} dy\}^{\frac{N−1}{N}} = 0.

Lemma 2.4. Let u ∈ BV (Ω; ℝ^{d}). There exist piecewise constant functions u_{n} such that u_{n} → u in L^{1}(Ω; ℝ^{d}) and

||Du||(Ω) = lim_{n→∞} ||Du_{n}||(Ω) = lim_{n→∞} ∫_{S_{un}} |[u_{n}](x)| dH^{N−1}(x).

The space of special functions of bounded variation introduced by De Giorgi and Ambrosio [17] in problems arising from pattern recognition and the mathematical theory of liquid crystals, SBV (Ω; ℝ^{d}), is the space of functions u ∈ BV (Ω; ℝ^{d}) such that D^{c}u = 0, i.e. for which

Du = ∇uL^{N} + [u] ⊗ ν_{u}H^{N−1}⌊S_{u}.

The next result is a Lusin type theorem for gradients due to Alberti [2] and it is essential for our arguments.

Theorem 2.5. Given f ∈ L^{1}(Ω; ℝ^{d×N}) there exists u ∈ SBV (Ω; ℝ^{d}) and a Borel function g : Ω → ℝ^{d×N} such that

Du = fL^{N} + gH^{N−1}⌊S_{u},
∫_{S_{u}} |g| dH^{N−1} ≤ C||f||_{L^{1}(Ω;ℝ^{d×N})}.

Remark 2.6. From the proof of Theorem 2.5 it follows also that

||u||_{L^{1}(Ω)} ≤ C||f||_{L^{1}(Ω;ℝ^{d×N})}.

The following technical result is a simplified version of Lemma 4.3 in Matias [32].

Lemma 2.7. Let Ω ⊂ ℝ^{N} be open and bounded and let A ∈ ℝ^{d×N}. Then there exists u ∈ SBV (Ω; ℝ^{d}) such that u_{⌊∂Q} = 0 and ∇u = A a.e in Ω. In addition

|D^{s}u|(Ω) ≤ C(N)|A| |Ω|.

Following Carriero, Leaci and Tomarelli (see [15] and [16]) we define, as already presented in the introduction,

SBV^{2}(Ω; ℝ^{d}) = {v ∈ SBV (Ω; ℝ^{d}), ∇v ∈ SBV (Ω; ℝ^{d×N})}.

If u ∈ SBV^{2}(Ω; ℝ^{d}) we use the notation ∇^{2}u = ∇(∇u), that is, ∇^{2}u is the absolutely continuous part of D(∇u) with respect to Lebesgue measure. We will also denote by

BV^{2}(Ω; ℝ^{d}) = {v ∈ BV (Ω; ℝ^{d}), ∇v ∈ BV (Ω; ℝ^{d×N})}.

3. Integral representation in SBV

The aim of this section is to derive an integral representation of the relaxed functional energy defined in (1.3)

I(g, G) = inf_{{u_{n}}⊂SBV^{2}(Ω;ℝ^{d})} {lim inf_{n→∞} E(u_{n}), u_{n} ⟶^{L^{1}} g, ∇u_{n} ⟶^{L^{1}} G} ,

E(u) = ∫_{Ω} W (∇u, ∇^{2}u) dx+∫_{S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1}+∫_{S_{∇u}} Ψ_{2}([∇u], ν_{∇u}) dH^{N−1}, u ∈ SBV^{2}(Ω; ℝ^{d}),

for (g, G) ∈ SD(Ω; ℝ^{d}) where

SD(Ω; ℝ^{d}) := SBV^{2}(Ω; ℝ^{d}) × SBV (Ω; ℝ^{d×N}).

In a similar way than Choksi and Fonseca [14] we assume the density functions W, Ψ_{1}, Ψ_{2} to satisfy the following conditions (H_{1}) : there exists C > 0 such that

\frac{1}{C} |B| − C ≤ W (A, B) ≤ C(1 + |B|)

for all A ∈ ℝ^{d×N} and B ∈ ℝ^{d×N×N};

(H_{2}) : there exists C > 0 such that

|W (A_{1}, B_{1}) − W (A_{2}, B_{2})| ≤ C(|A_{1} − A_{2}| + |B_{1} − B_{2}|)
for all A_{i} ∈ ℝ^{d×N} and B_{i} ∈ ℝ^{d×N×N}, i = 1, 2;

(H_{3}) : there exists 0 < α < 1 and L > 0 such that

|W^{∞}(A, B) − \frac{W (A, tB)}{t}| ≤ \frac{C}{t^{α}}

for all t > L, A ∈ ℝ^{d×N}, B ∈ ℝ^{d×N×N} with |B| = 1;

(H_{4}) : there exist c_{1} > 0 and C_{1} > 0 such that

c_{1}|λ| ≤ Ψ_{1}(λ, ν) ≤ C_{1}|λ|

for all λ ∈ ℝ^{d} and ν ∈ S^{N−1}; (H_{5}): there exist c_{2} > 0 and C_{2} > 0 such that

c_{2}|Λ| ≤ Ψ_{2}(Λ, ν) ≤ C_{2}|Λ|

for all ν ∈ S^{N−1} and Λ ∈ ℝ^{d×N}; (H_{6}): (homogeneity of degree one)

Ψ_{1}(tλ, ν) = tΨ_{1}(λ, ν), Ψ_{2}(tΛ, ν) = tΨ_{2}(Λ, ν)

for all ν ∈ S^{N−1}, λ ∈ ℝ^{d}, Λ ∈ ℝ^{d×N} and t > 0; (H_{7}): (sub-additivity)

Ψ_{1}(λ_{1} + λ_{2}, ν) ≤ Ψ_{1}(λ_{1}, ν) + Ψ_{1}(λ_{2}, ν),
Ψ_{2}(Λ_{1} + Λ_{2}, ν) ≤ Ψ_{2}(Λ_{1}, ν) + Ψ_{2}(Λ_{2}, ν)

for all ν ∈ S^{N−1}, λ_{i} ∈ ℝ^{d}, Λ_{i} ∈ ℝ^{d×N}, i = 1, 2.

We observe that, as already mentioned in [14], the coercivity hypotheses above and the homogeneity condition (H_{6}) are of technical order and can be relaxed (see Remark 3.3 in [14]). In addition, the linear growth and subadditivity requirements on the interfacial densities could be weakened by assuming a Lipschitz continuity property instead.

Remark 3.1.

(1) In what follows we extend Ψ_{i}, i = 1, 2, as homogeneous functions of degree one in the second variable to all of ℝ^{N}.

(2) We note that the class of functions over which the infimum in the definition of I(g, G) is taken is nonempty and, in addition, under hypotheses (H_{1}), (H_{4}) and (H_{5}) the energy I(g, G) < ∞. More precisely, there exists C > 0 such that

I(g, G) ≤ C [∫_{Ω}(1 + |∇g| + |G| + |∇G|) dx + ||D^{s}g||(Ω) + ||D^{s}G||(Ω)]  (3.1)

for all (g, G) ∈ SD(Ω; ℝ^{d}).

In fact, given (g, G) ∈ SD(Ω; ℝ^{d}) by Theorem 2.5 there exists h ∈ SBV (Ω; ℝ^{N}) such that

∇h = G L^{N} − a.e. in Ω

and

||D^{s}h||(Ω) ≤ C_{1}||G||_{L^{1}}.

Moreover by Lemma 2.4 there exist {ū_{n}} piecewise constant such that

ū_{n} ⟶_{n→∞}^{L^{1}} g − h, ||D^{s}ū_{n}||(Ω) ⟶_{n→∞} ||Dg − Dh||(Ω).

Define now u_{n} ∈ SBV (Ω; ℝ^{N}) by

u_{n} := ū_{n} + h.

Clearly ∇u_{n}(x) = G(x) for L^{N} − almost every x ∈ Ω and u_{n} → g in L^{1}. Thus by hypotheses

(H_{1}), (H_{4}) and (H_{5})
I(g, G) ≤ lim inf_{n→∞} {∫_{Ω} W (∇u_{n}, ∇^{2}u_{n}) dx + ∫_{S_{un}} Ψ_{1}([u_{n}](x), ν(u_{n}(x))) dH^{N−1}

+ ∫_{S_{∇un}} Ψ_{2}([∇u_{n}](x), ν(∇u_{n}(x))) dH^{N−1}}

≤ C lim inf_{n→∞} {∫_{Ω} W (G, ∇G) dx + ||D^{s}u_{n}||(Ω) + ||D^{s}∇u_{n}||(Ω)}
≤ C [∫_{Ω}(1 + |∇g| + |G| + |∇G|) dx + ||D^{s}g||(Ω) + ||D^{s}G||(Ω)]

The main result of this section states the following.

Theorem 3.2. Under hypotheses (H_{1}) − (H_{7}) we have that for all (g, G) ∈ SD(Ω; ℝ^{d})

I(g, G) = ∫_{Ω} (W_{1}(G − ∇g) + W_{2}(G, ∇G)) dx + ∫_{S_{g}} γ_{1}([g], ν_{g})dH^{N−1}
+ ∫_{S_{G}} γ_{2}(G^{+}, G^{−}, ν_{G}) dH^{N−1}

where given A, B, Λ, Γ ∈ ℝ^{d×N}, C ∈ ℝ^{d×N×N}, λ ∈ ℝ^{d} and ν ∈ S^{N−1}

W_{1}(A) = inf_{u∈SBV^{2}(Q;ℝ^{d})} {∫_{S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1}, u|_{∂Q} = 0, ∇u = A a.e. in Q}  (3.2)

W_{2}(B, C) =  inf_{v∈SBV (Q;ℝ^{d×N})} {∫_{Q} W (B, ∇v(x)) dx + ∫_{S_{v}} Ψ_{2}([v], ν(v)) dH^{N−1}, v|_{∂Q}(x) = C x} (3.3)

γ_{1}(λ, ν) = inf_{u∈SBV^{2}(Q_{ν};ℝ^{d})} {∫_{S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1}, u|_{∂Q_{ν}} = γ_{(λ,ν)}, ∇u = 0 a.e. in Q_{ν}}  (3.4)

γ_{(λ,ν)}(x) := \{ \begin{matrix}λ & if x · ν > 0 \\ 0 & if x · ν < 0 \end{matrix}

γ_{2}(Λ, Γ, ν) =  inf_{v∈SBV (Q_{ν};ℝ^{d×N})}{∫_{Q_{ν}} W^{∞}(v, ∇v) dx + ∫_{S_{v}} Ψ_{2}([v], ν(v)) dH^{N−1}, v|_{∂Q_{ν}} = γ_{(Λ,Γ,ν)}} (3.5)

γ_{(Λ,Γ,ν)}(x) := \{ \begin{matrix}Λ & if x · ν > 0 \\ Γ & if x · ν < 0. \end{matrix}

Remark 3.3. We observe that as a consequence of hypothesis (H_{6}) the function W_{1} is homogeneous of degree one. Moreover, it is easy to check that the recession function of W_{2} in the second variable is given by

W_{2}^{∞}(B, C) = inf_{v∈SBV (Q;ℝ^{d×N})} {∫_{Q} W^{∞}(B, ∇v(x)) dx + ∫_{S_{v}} Ψ_{2}([v], ν(v)) dH^{N−1}, v|_{∂Q}(x) = C x}

for all (B, C) ∈ ℝ^{d×N} × ℝ^{d×N×N}.

The proof of Theorem 3.2 is an inmediate consequence of Lemma 3.4, Proposition 3.5 and Theorem 3.6 below.

Lemma 3.4. Under hypotheses (H_{4}) and (H_{7}), for all (g, G) ∈ SD(Ω; ℝ^{d})

I(g, G) = I_{1}(g, G) + I_{2}(G)  (3.6)

where

I_{1}(g, G) = inf_{{u_{n}}⊂SBV^{2}(Ω;ℝ^{d})} {lim inf_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}, u_{n} ⟶^{L^{1}} g, ∇u_{n} ⟶^{L^{1}} G}

and

I_{2}(G) = inf_{{v_{n}}⊂SBV (Ω;ℝ^{d×N})} {lim inf_{n→∞} ∫_{Ω} W (v_{n}, ∇v_{n}) dx + ∫_{S_{vn}} Ψ_{2}([v_{n}], ν(v_{n})) dH^{N−1}, v_{n} ⟶^{L^{1}} G} .

Proof. In fact, it is immediate to see that

I_{1}(g, G) + I_{2}(G) ≤ I(g, G).

On the other hand, let u_{n} ∈ SBV^{2}(Ω; ℝ^{d}) with u_{n} → g in L^{1} and ∇u_{n} → G in L^{1} be such that

I_{1}(g, G) = lim_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}

and let v_{n} ∈ SBV (Ω; ℝ^{d×N}) with v_{n} → G in L^{1} be such that

I_{2}(G) = lim_{n→∞} [∫_{Ω} W (v_{n}, ∇v_{n}) dx + ∫_{S_{vn}} Ψ_{2}([v_{n}], ν(v_{n})) dH^{N−1}] .

By Theorem 2.5 let h_{n} ∈ SBV (Ω; ℝ^{d}) be such that ∇h_{n} = v_{n} − ∇u_{n} and by Lemma 2.4 let h̃_{n} be a piecewise constant function with ||h̃_{n} − h_{n}||_{L^{1}} < \frac{1}{n} and | ||Dh̃_{n}||(Ω) − ||Dh_{n}||(Ω)| < \frac{1}{n}. Then the sequence

w_{n} = u_{n} + h_{n} − h̃_{n}

is admissible for I(g, G) and

I(g, G) ≤ lim inf_{n→∞} [∫_{Ω} W (∇w_{n}, ∇^{2}w_{n}) dx + ∫_{S_{wn}} Ψ_{1}([w_{n}], ν(w_{n})) dH^{N−1}

+ ∫_{S_{∇wn}} Ψ_{2}([∇w_{n}], ν(∇w_{n})) dH^{N−1}]

≤ lim_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}

+ lim sup_{n→∞} ∫_{S_{hn}} Ψ_{1}([h_{n}], ν(h_{n})) dH^{N−1} + lim sup_{n→∞} ∫_{Sh̃n} Ψ_{1}([h̃_{n}], ν(h̃_{n})) dH^{N−1}

+ lim_{n→∞} [∫_{Ω} W (v_{n}, ∇v_{n}) dx + ∫_{S_{vn}} Ψ_{2}([v_{n}], ν(v_{n})) dH^{N−1}]

≤ I_{1}(g, G) + I_{2}(G) + C ∫_{Ω} |v_{n} − ∇u_{n}| dx

by conditions (H_{4}), (H_{7}), Theorem 2.5 and Lemma 2.4. Inequality I_{1}(g, G) + I_{2}(G) ≥ I(g, G) follows by letting n → ∞ since v_{n} → G and ∇u_{n} → G in L^{1}.

□

Proposition 3.5. Under hypotheses (H_{1}) − (H_{3}) and (H_{5}), for all G ∈ SBV (Ω; ℝ^{d×N})

I_{2}(G) = ∫_{Ω} W_{2}(G, ∇G) dx + ∫_{S_{G}} γ_{2}(G^{+}, G^{−}, ν_{G}) dH^{N−1}.

Proof. It is a consequence of Theorem 4.2.2 in Bouchitté, Fonseca and Mascarenhas [11].

□

Theorem 3.6. Under hypotheses (H_{4}), (H_{6}) and (H_{7}), for all (g, G) ∈ SD(Ω; ℝ^{d})

I_{1}(g, G) = ∫_{Ω} W_{1}(G − ∇g) dx + ∫_{S_{g}} γ_{1}([g], ν_{g})dH^{N−1}.  (3.7)

The proof of Theorem 3.6 will be divided in three parts (Subsections 3.1-3.3 below). Precisely, in Subsection 3.1 we will introduce a local version of I_{1}(g, G) defined on A(Ω) and show that I_{1}(g, G, ·) is the restriction to A(Ω) of a Radon measure absolutely continuous with respect to L^{N} + H^{N−1}⌊S_{g}. In Subsections 3.2 and 3.3 we will prove that for all A ∈ A(Ω)

I_{1}(g, G, A) = ∫_{A} W_{1} (G − ∇g) dx + ∫_{A∩S(g)} γ_{1} ([g], ν_{g}) dH^{N−1}

from where, taking A = Ω, equality (3.7) follows ending the proof of Theorem 3.6.

3.1. Localization. We start by localizing I_{1}(g, G), i.e., we define for A ∈ A(Ω)

I_{1}(g, G, A) := inf_{{u_{n}}⊂SBV^{2}(A;ℝ^{d})} {lim inf_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}, u_{n} ⟶^{L^{1}(A;ℝ^{d})} g, ∇u_{n} ⟶^{L^{1}(A;ℝ^{d×N})} G} .

Our objective in this subsection is to show that I_{1}(g, G, ·)⌊A(Ω) is a Radon measure.

Remark 3.7. Following the argument in Remark 3.1 it is easy to see that there exists C > 0 such that for all A ∈ A(Ω)

I_{1}(g, G, A) ≤ C [∫_{A}(|∇g| + |G|) dx + ||D^{s}g||(A)] .  (3.8)

The following lemma shows that I_{1}(g, G, ·) is nested-subadditive.

Lemma 3.8. Let A, B, C ∈ A(Ω) with A ⊂⊂ B ⊂ C. Then

I_{1}(g, G, C) ≤ I_{1}(g, G, B) + I_{1}(g, G, C\Ā).  (3.9)

Proof. Let u_{n} ∈ SBV^{2}(B; ℝ^{d}) and v_{n} ∈ SBV^{2}(C\Ā; ℝ^{d}) be two sequences such that u_{n} → g in L^{1}(B; ℝ^{d}), ∇u_{n} → G in L^{1}(B; ℝ^{d×N}), v_{n} → g in L^{1}(C\Ā; ℝ^{d}), ∇v_{n} → G in L^{1}(C\Ā; ℝ^{d×N}), and that in addition

I_{1}(g, G, B) = lim_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}

and

I_{1}(g, G, C\Ā) = lim_{n→∞} ∫_{S_{vn}} Ψ_{1}([v_{n}], ν(v_{n})) dH^{N−1}.

Note that

u_{n} − v_{n} → 0 in L^{1}(B ∩ (C\Ā); ℝ^{d})  (3.10)

and

∇u_{n} − ∇v_{n} → 0 in L^{1}(B ∩ (C\Ā); ℝ^{d×N}).

For δ > 0 define

A_{δ} := {x ∈ B, dist(x, A) < δ}.

Let d(x) := dist(x, A), x ∈ C. Since the distance function to a fixed set is Lipschitz continuous (see Exercise 1.1 in Ziemer [36]), we can apply the change of variables formula (see Theorem 2, Section 3.4.3, in Evans and Gariepy [23]), to get that

∫_{A_{δ}\Ā} |u_{n} − v_{n}|Jd(x) dx = ∫_{0}^{δ} [∫_{d^{−1}(y)} |u_{n} − v_{n}|dH^{N−1}(x)] dy

and, as Jd(·) is bounded and (3.10) holds, then for almost every ρ ∈ [0, δ] it follows that

lim_{n→∞} ∫_{d^{−1}(ρ)} |u_{n} − v_{n}| dH^{N−1} = lim_{n→∞} ∫_{∂A_{ρ}} |u_{n} − v_{n}| dH^{N−1} = 0.  (3.11)

Fix ρ_{0} such that (3.11) holds. We observe that A_{ρ_{0}} is a set with locally Lipschitz boundary since it is a level set of a Lipschitz function (see e.g. Evans and Gariepy [30]). Hence we can consider u_{n}, v_{n}, ∇u_{n}, ∇v_{n} on ∂A_{ρ_{0}} in the sense of traces and define

w_{n} = \{ \begin{matrix}u_{n} & in Ā_{ρ_{0}} \\ v_{n} & in C\Ā_{ρ_{0}}. \end{matrix}

Then

I_{1}(g, G, C) ≤ lim inf_{n→∞} ∫_{S_{wn}} Ψ_{1}([w_{n}], ν(w_{n})) dH^{N−1}

and using (H4) and (3.10) we get (3.9).

□

Theorem 3.9. Assume that hypothesis (H_{4}) holds. Then I_{1}(g, G, ·)⌊A(Ω) is a Radon measure absolutely continuous with respect to L^{N} + H^{N−1}⌊S_{g}.

Proof. Let u_{n} ∈ SBV^{2}(Ω; ℝ^{d}) be such that u_{n} → g in L^{1}(Ω; ℝ^{d}), ∇u_{n} → G in L^{1}(Ω; ℝ^{d×N}) and

I_{1}(g, G, Ω) = lim_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1},

and define for all Borel set B ⊂ ℝ^{N}

μ_{n}(B) := ∫_{S_{un} ∩B} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}.

By (H4) the sequence of non-negative Radon measures {μ_{n}} is uniformly bounded in M(ℝ^{N}) and thus, upon passing if necessary to a subsequence, we conclude that

μ_{n} ⇀^{∗} μ in M(ℝ^{N}).

Let us show that for all V ∈ A(Ω)

μ(V) = I_{1}(g, G, V).  (3.12)

Given V ∈ A(Ω), let ϵ > 0 and take W ⊂⊂ V such that μ(V \W) < ϵ. It follows that

μ(V) ≤ μ(W) + ϵ
= μ(Ω) − μ(Ω\W) + ϵ
≤ I_{1}(g, G, Ω) − I_{1}(g, G, Ω\W̄) + ϵ
≤ I_{1}(g, G, V) + ϵ

where we have used the equality μ(Ω) = μ(\overline{Ω}) and Lemma 3.8. Thus, letting ϵ → 0, we get

μ(V) ≤ I_{1}(g, G, V).  (3.13)

Let us see now the reverse inequality. Define for A ∈ A(Ω)

λ(A) := ∫_{A}(|∇g| + |G|) dx + ||D^{s}g||(A).  (3.14)

Let K ⊂⊂ V be a compact set such that λ(V \K) < ϵ, and choose an open set W such that K ⊂⊂ W ⊂⊂ V. Using again Lemma 3.8, (3.14), and (3.8)

I_{1}(g, G, V) ≤ I_{1}(g, G, W) + I_{1}(g, G, V \K)
≤ μ(W̄) + Cλ(V \K)
≤ μ(V) + Cϵ,

which, together with (3.13), yields to (3.12) by letting ϵ → 0. □

3.2. Lower Bound. The objective of this part is to show that for all A ∈ A(Ω)

I_{1}(g, G, A) ≥ ∫_{A} W_{1} (G(x) − ∇g(x)) dx + ∫_{A∩S(g)} γ_{1} ([g], ν_{g}) dH^{N−1}.  (3.15)

To prove inequality (3.15) let u_{n} ∈ SBV^{2}(Ω; ℝ^{N}) be such that u_{n} ⟶_{n→∞}^{L^{1}} g, ∇u_{n} ⟶_{n→∞}^{L^{1}} G and

lim_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1} < ∞.

Define μ_{n} by

μ_{n}(B) = ∫_{B∩S(u_{n})} Ψ_{1}([u_{n}], ν(u_{n}))dH^{N−1},

for all Borel sets B ⊂ Ω. Since by the hypotheses on Ψ_{1} the sequence of Radon measures {μ_{n}} is bounded then there exists (up to a subsequence) μ ∈ M(Ω) with μ_{n} ⇀^{∗} μ in M(Ω). We now show that

\frac{dμ}{dL^{N}} (x_{0}) ≥ W_{1}(G(x_{0}) − ∇g(x_{0}))  (3.16)

for L^{N}-almost every x_{0} ∈ Ω and that

\frac{dμ}{dH^{N−1}⌊S_{g}} (x_{0}) ≥ γ_{1}([g](x_{0}), ν_{g}(x_{0}))  (3.17)

for H^{N−1}⌊S_{g}-almost every x_{0} ∈ Ω.

Proof of (3.16): Let x_{0} ∈ Ω be a point of approximate differentiability of g and of approximately continuity of G (see Theorem 2.3 and equality (2.1)) and such that \frac{dμ}{dL^{N}} (x_{0}) exists. Let \{δ_{k}\} → 0 be such that μ(∂Q(x_{0}, δ_{k})) = 0. Then lim_{n→∞} μ_{n}(Q(x_{0}, δ_{k})) = μ(Q(x_{0}, δ_{k}) and

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k→∞} \frac{μ(Q(x_{0}, δ_{k}))}{L^{N}(Q(x_{0}, δ_{k}))}

= lim_{k, n→∞} \frac{1}{δ_{k}^{N}} ∫_{S_{un} ∩Q(x_{0},δ_{k})} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}

= lim_{k, n→∞} \frac{1}{δ_{k}} ∫_{Q∩\{y: x_{0}+δ_{k}y ∈S_{un} \}} Ψ_{1}([u_{n}](x_{0} + δ_{k}y), ν(u_{n})(x_{0} + δ_{k}y))dH^{N−1}(y).

Defining

v_{n,k}(y) := \frac{u_{n}(x_{0} + δ_{k}y) − g(x_{0})}{δ_{k}}, y ∈ Q

we have that

\frac{dμ}{dL^{N}} (x_{0}) = lim_{k,n→∞} ∫_{Q∩S_{v_{n,k}}} Ψ_{1}([v_{n,k}], ν(v_{n,k})) dH^{N−1}.  (3.18)

As x_{0} is a point of approximate differentiability of g then

v_{n,k} ⟶_{k, n→∞}^{L^{1}} ∇g(x_{0})(·)

since

lim_{n→∞} ∫_{Q} |v_{n,k}(y) − ∇g(x_{0}) y| dy = lim_{n→∞} ∫_{Q} |\frac{u_{n}(x_{0} + δ_{k} y) − g(x_{0})}{δ_{k}} − ∇g(x_{0}) y| dy

= ∫_{Q} |\frac{g(x_{0} + δ_{k} y) − g(x_{0})}{δ_{k}} − ∇g(x_{0}) y| dy

= \frac{1}{δ_{k}^{N}} ∫_{Q(x_{0},δ_{k})} |\frac{g(z) − g(x_{0}) − ∇g(x_{0})(z − x_{0})}{δ_{k}} | dy

by a change of variables. Similarly, as x_{0} is also an approximately continuity point of G,

∇v_{k, n} ⟶_{k,n→∞}^{L^{1}} G(x_{0}).  (3.19)

We now change the sequence {v_{n,k}} to comply in (3.18) with the definition of W_{1} (see (3.2)). We start by setting

w_{n,k}(y) = v_{n,k}(y) − ∇g(x_{0})y, y ∈ Q,

and following the argument used in (3.11) we choose r_{n,k} ∈ ]0, 1[ such that

r_{n,k} ⟶_{k,n→∞} 1

and

∫_{∂Q(0,r_{n,k})} |w_{n,k}|dH^{N−1} ⟶_{k,n→∞} 0.  (3.20)

By Theorem 2.5 let ρ_{n,k} be such that

∇ρ_{n,k}(y) = G(x_{0}) − ∇v_{n,k}(y), y ∈ Q,

and define

z_{n,k} := w_{n,k} + ρ_{n,k} in Q(0, r_{n,k}).

Notice that

∇z_{n,k} = G(x_{0}) − ∇g(x_{0}) in Q(0, r_{n,k}).

In addition, by (3.19) ∇ρ_{n,k} ⟶_{k,n→∞} 0, and then by Theorem 2.5

|D^{s}ρ_{n,k}|(Q(0, r_{n,k})) ⟶_{k,n→∞} 0.  (3.21)

Thus by the continuity of the trace with respect to the intermediate topology (see Proposition 3.88 in Ambrosio, Fusco and Pallara [5]) it follows that

∫_{∂Q(0,r_{n,k})} |ρ_{n,k}| dH^{N−1} ⟶_{k,n→∞} 0.  (3.22)

Applying Lemma 2.7 in Q\(Q(0, r_{n,k})) let {η_{n,k}} be a sequence of functions such that

∇η_{n,k}(y) = G(x_{0}) − ∇g(x_{0}) in Q\(Q(0, r_{n,k})),
η_{n,k} = 0 on ∂(Q\(Q(0, r_{n,k}))  (3.23)

and

|D^{s}η_{n,k}|(Q\(Q(0, r_{n,k})) ≤ C(N)|Q\(Q(0, r_{n,k})|.  (3.24)

Then the sequence

z̃_{n,k}(y) := \{ \begin{matrix}z_{n,k}(y), y ∈ Q(0, r_{n,k}) \\ η_{n,k}(y), y ∈ Q\(Q(0, r_{n,k})) \end{matrix}

is admissible fot W^{1}(G(x_{0}) − ∇g(x_{0})), and in addition by (H_{4}), (H_{7}) and (3.23) we have, for any n and k, that

∫_{Q∩Sz̃_{n,k}} Ψ_{1}([z̃_{n,k}], ν(z̃_{n,k})) dH^{N−1} ≤ ∫_{Q(0,r_{n,k})∩S_{z_{n,k}}} Ψ_{1}([z_{n,k}], ν(z_{n,k})) dH^{N−1}

+C [∫_{∂Q(0,r_{n,k})} |z_{n,k}| dH^{N−1} + ∫_{[Q\Q(0,r_{n,k})]∩S_{η_{n,k}}} |[η_{n,k}]| dH^{N−1}]

≤ ∫_{Q(0,r_{n,k})∩S_{v_{n,k}}} Ψ_{1}([v_{n,k}], ν(v_{n,k})) dH^{N−1}

+C [∫_{∂Q(0,r_{n,k})} |w_{n,k}| dH^{N−1} + ∫_{∂Q(0,r_{n,k})} |ρ_{n,k}| dH^{N−1}

+ ∫_{Q(0,r_{n,k})∩S_{ρ_{n,k}}} |[ρ_{n,k}]| dH^{N−1} + ∫_{[Q\Q(0,r_{n,k})]∩S_{η_{n,k}}} |[η_{n,k}]| dH^{N−1}]

Therefore by (3.20), (3.21),(3.22) and (3.24) we get that

lim inf_{k, n→∞} ∫_{Q∩Sz̃_{n,k}} Ψ_{1}([z̃_{n,k}], ν(z̃_{n,k})) dH^{N−1} ≤ lim_{k, n→∞} ∫_{Q∩S_{v_{n,k}}} Ψ_{1}([v_{n,k}], ν(v_{n,k})) dH^{N−1}

which together with (3.18) implies (3.16).

□

Proof of (3.17): Let x_{0} ∈ S_{g} be such that \frac{dμ}{dH^{N−1}⌊S_{g}} (x_{0}) exists,

lim_{δ→0} \frac{H^{N−1}(S_{g} ∩ Q_{ν}(x_{0}, δ))}{δ^{N−1}} = 1  (3.25)

and

lim_{δ→0} \frac{1}{δ^{N−1}} ∫_{Q_{ν}(x_{0},δ)} |G(x)| dx = 0  (3.26)

where ν ≡ ν_{g}(x_{0}). We note that for H^{N−1}-a.e x_{0} ∈ S_{g} all the conditions above hold. In fact see Ambrosio, Fusco and Pallara [5] for (3.25). Equality (3.26) holds by (3.25) and the fact that

\frac{d|G|L^{N}}{dH^{N−1}⌊S_{g}} = 0.

Let {δ_{k}} → 0 be such that μ(∂Q_{ν}(x_{0}, δ_{k})) = 0. Then lim_{n→∞} μ_{n}(Q_{ν}(x_{0}, δ_{k})) = μ(Q_{ν}(x_{0}, δ_{k}) and

\frac{dμ}{dH^{N−1}⌊S_{g}} (x_{0}) = lim_{k→∞} lim_{n→∞} \frac{1}{H^{N−1}(S_{g} ∩ Q_{ν}(x_{0}, δ_{k}))}μ_{n}(Q_{ν}(x_{0}, δ_{k}))

= lim_{k→∞} lim_{n→∞} \frac{1}{H^{N−1}(S_{g} ∩ Q_{ν}(x_{0}, δ_{k}))} ∫_{S_{un} ∩Q_{ν}(x_{0},δ)} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}

= lim_{k→∞} lim_{n→∞}\frac{δ_{k}^{N−1}}{H^{N−1}(S_{g} ∩ Q_{ν}(x_{0}, δ_{k}))}∫_{Q_{ν}∩\{y: x_{0}+δ_{k}y ∈S_{un} \}} Ψ_{1}([u_{n}](x_{0} + δ_{k} y), ν(u_{n})(x_{0} + δ_{k}y)) dH^{N−1}

= lim_{k→∞} lim_{n→∞} ∫_{Q_{ν}∩{y: x_{0}+δ_{k}y ∈S_{un} }} Ψ_{1}([u_{n}](x_{0} + δ_{k} y), ν(u_{n})(x_{0} + δ_{k}y)) dH^{N−1}

by (3.25). Defining

w_{n,k}^{1}(y) = u_{n}(x_{0} + δ_{k}y) − g^{−}(x_{0}), y ∈ Q_{ν},

it follows that

\frac{dμ}{dH^{N−1}⌊S_{g}} (x_{0}) = lim_{k→∞} lim_{n→∞} ∫_{Q_{ν}∩S_{w{1 \atop n,k}}} Ψ_{1}([w_{n,k}^{1}](y), ν(w_{n,k}^{1})(y)) dH^{N−1}.

By (2.2) and (2.3) we have

w_{n,k}^{1} ⟶_{n,k→∞}^{L^{1}} γ_{([g](x_{0}),ν)},

and in addition by (3.26)

∇w_{n,k}^{1} ⟶_{n,k→∞}^{L^{1}} 0

since, for all k, ∇u_{n}(x_{0} + δ_{k}·) ⟶_{n→∞} G(x_{0} + δ_{k}·).

Using Theorem 2.5 and following the arguments of Lemma 3.8 we note that it is possible to modify w_{n,k}^{1} so that ∇w_{n,k}^{1} = 0 and w_{n,k}^{1}|_{∂Q_{ν}} = γ_{([g](x_{0}),ν)}. Thus by definition of γ_{1} (see (3.4)) inequality (3.17) holds.

□

As a consequence of (3.16) and (3.17) we now derive (3.15).

Proof of (3.15): Denote by μ_{a} the absolutely continuous part of μ with respect to Lebesgue measure and by μ_{g}^{s} the absolutely continuous part of μ with respect to H^{N−1}⌊S_{g}. Since μ is a positive measure we conclude that

lim inf_{n→∞} μ_{n}(A) ≥ μ(A)

≥ ∫_{A} μ_{a}(x) dx + ∫_{A∩S_{g}} μ_{g}^{s}(x) dH^{N−1}(x)
≥ ∫_{A} W_{1} (G(x) − ∇g(x)) dx + ∫_{A∩S_{g}} γ_{1} ([g], ν_{g}) dH^{N−1}.

Taking the infimum over all sequences u_{n} ∈ SBV^{2}(Ω; ℝ^{d}), u_{n} ⟶_{n→∞}^{L^{1}} g, ∇u_{n} ⟶_{n→∞}^{L^{1}} G inequality (3.15) holds.

□

A RELAXATION RESULT IN THE FRAMEWORK OF STRUCTURED DEFORMATIONS 15

3.3. Upper bound. Our objective here is to show that for all A ∈ A(Ω)

I_{1}(g, G, A) ≤ ∫_{A} W_{1} (G(x) − ∇g(x)) dx + ∫_{A∩S(g)} γ_{1} ([g], ν_{g}) dH^{N−1}.

For this purpose it is enough to prove that

\frac{dI_{1}(g, G, ·)}{dL^{N}}(x_{0}) ≤ W_{1}(G(x_{0}) − ∇g(x_{0})), L^{N} -a.e x_{0} ∈ Ω  (3.27)

and

\frac{dI_{1}(g, G, ·)}{dH^{N−1}⌊S_{g}} (x_{0}) ≤ γ_{1}([g](x_{0}), ν(g)(x_{0})), H^{N−1} -a.e x_{0} ∈ S_{g}.  (3.28)

We start with two auxiliary results. Proposition 3.10. Let Ψ_{1} satisfy (H_{4}) and (H_{7}). Then there exist constants C_{1}, C_{2} > 0 such that

|γ_{1}(λ, ν) − γ_{1}(λ′, ν)| ≤ C_{1}|λ − λ′|, ∀λ, λ′ ∈ ℝ^{d},  (3.29)
|W_{1}(A) − W_{1}(B)| ≤ C_{2}|A − B|, ∀A, B ∈ ℝ^{d×N}.  (3.30)

Moreover γ_{1} is upper semicontinuous with respect to ν. Proof. We show (3.30) and we refer to the proof of Proposition 4.3 in [14] for the rest of the statement. We start by showing that

W_{1}(A) ≤ W_{1}(B) + C_{1}|B − A|, ∀A, B ∈ ℝ^{d×N}.

Fixed ϵ > 0 let u ∈ SBV (Q; ℝ^{d}) be such that u|_{∂Q} = 0, ∇u = A and ϵ + W_{1}(A) ≥ ∫_{S_{u}∩Q} Ψ_{1}([u], ν_{u}) dH^{N−1}.

Let now v ∈ SBV (Q; ℝ^{d}) be such that v|_{∂Q} = 0, ∇v = B − A and |D^{s}v|(Q) ≤ C|B − A| (cf. Lemma 2.7), and set w = u + v. Then by (H_{4}) and (H_{7})

W_{1}(B) ≤ ∫_{S_{w}∩Q} Ψ_{1}([w], ν(w)) dH^{N−1}
≤ {∫_{S_{u}∩Q} Ψ_{1}([u], ν_{u}) dH^{N−1} + ∫_{S_{v}∩Q} Ψ_{1}([v], ν(v)) dH^{N−1}}
≤ W_{1}(A) + ϵ + C_{1}|B − A|.

The reverse inequality is proved in a similar way.

□

The following proposition easily follows from a diagonalization argument. Proposition 3.11. Let (g_{n}, G_{n}) ∈ SD(Ω; ℝ^{d}) be such that

g_{n} ⟶_{n→∞}^{L^{1}} g and G_{n} ⟶_{n→∞}^{L^{1}} G.

Then

I_{1}(g, G) ≤ lim inf_{n→∞} I_{1}(g_{n}, G_{n}).

We now show that inequalities (3.27) and (3.28) hold.

Proof of (3.27): Let x_{0} be a point of approximate continuity for G and ∇g, that is, such that

\frac{1}{δ^{N}} \{∫_{Q(x_{0},δ)} |G(x) − G(x_{0})| + |∇g(x) − ∇g(x_{0})| dx\} ⟶_{δ→0} 0.  (3.31)

Let ϵ > 0 and consider u ∈ SBV^{2}(Ω; ℝ^{N}) such that

W_{1}(G(x_{0}) − ∇g(x_{0})) + ϵ ≥ ∫_{Q∩Su} Ψ_{1}([u], ν_{u}) dH^{N−1},  (3.32)

u|_{∂Q} = 0 and ∇u(x) = G(x_{0}) − ∇g(x_{0}) for a.e. x ∈ Q. Extend u by periodicity to all of ℝ^{N} and define for n ∈ ℕ and δ > 0

u_{n,δ}(x) = \frac{δ}{n} u (\frac{n(x − x_{0})}{δ}) .

Given δ > 0 apply Theorem 2.5 and let ρ_{δ} ∈ SBV^{2}(Q(x_{0}, δ); ℝ^{N}) be a function such that

∇ρ_{δ}(x) = G(x) − G(x_{0}) + ∇g(x_{0}) − ∇g(x)  (3.33)

L^{N}-a.e. x ∈ Q(x_{0}, δ) and satisfying that

||Dρ_{δ}||(Q(x_{0}, δ)) ≤ C(N) ∫_{Q(x_{0},δ)} |G(x) − G(x_{0})| + |∇g(x_{0}) − ∇g(x)| dx.

Note that by (3.31)

\frac{||Dρ_{δ}||(Q(x_{0}, δ))}{δ^{N}} ⟶_{δ→0} 0.  (3.34)

In addition, using Lemma 2.4 define a sequence of piecewise constant functions ρ_{n,δ} such that for all δ > 0

ρ_{n,δ} ⟶_{n→∞}^{L^{1}} −ρ_{δ} and ||Dρ_{n,δ}||(Q(x_{0}, δ)) ⟶_{n→∞} ||Dρ_{δ}||(Q(x_{0}, δ)).  (3.35)

Let now define

w_{n,δ}(x) := g(x) + u_{n,δ}(x) + ρ_{δ}(x) + ρ_{n,δ}(x), x ∈ Q(x_{0}, δ).

Clearly w_{n,δ} ∈ SBV^{2}(Q(x_{0}, δ); ℝ^{d}), w_{n,δ} ⟶_{n→∞}^{L^{1}} g and ∇w_{n,δ} ⟶_{n→∞}^{L^{1}} G.

As for each δ > 0 the sequence w_{n,δ} is admissible for I_{1} and

\frac{dI_{1}(g, G, .)}{dL^{N}}(x_{0}) = lim_{δ→0} \frac{I_{1}(g, G, Q(x_{0}, δ))}{δ^{N}}.

Then

\frac{dI_{1}(g, G, .)}{dL^{N}}(x_{0}) ≤ lim inf_{δ→0} lim inf_{n→∞} \{\frac{1}{δ^{N}} ∫_{S_{w_{n,δ}} ∩Q(x_{0},δ)} ψ_{1}([w_{n,δ}], ν(w_{n,δ})) dH^{N−1}\}

and by (H_{7})

\frac{dI_{1}(g, G, .)}{dL^{N}}(x_{0}) ≤ lim inf_{δ→0} lim inf_{n→∞} \{\frac{1}{δ^{N}} ∫_{S_{g}∩Q(x_{0},δ)} Ψ_{1}([g], ν_{g}) dH^{N−1}

+ \frac{1}{δ^{N}} ∫_{\{x_{0}+ \frac{δ}{n} S_{u}\}∩Q(x_{0},δ)} Ψ_{1} (\frac{δ}{n}[u] (\frac{n(x − x_{0})}{δ}) , ν_{u} (\frac{n(x − x_{0})}{δ})) dH^{N−1}

+ \frac{1}{δ^{N}} ∫_{S_{ρ_{δ}} ∩Q(x_{0},δ)} Ψ_{1}([ρ_{δ}], ν(ρ_{δ})) dH^{N−1}

+ \frac{1}{δ^{N}} ∫_{S_{ρ_{n,δ}} ∩Q(x_{0},δ)} Ψ_{1}([ρ_{n,δ}], ν(ρ_{n,δ})) dH^{N−1}\} .

We observe that by (H_{4})

\frac{1}{δ^{N}} ∫_{S_{g}∩Q(x_{0},δ)} Ψ_{1}([g], ν_{g}) dH^{N−1} ⟶_{δ→0} 0

since

\frac{d|D^{s}g|}{dL^{N}}(x_{0}) = 0.

Moreover

\frac{1}{δ^{N}} ∫_{S_{ρ_{δ}} ∩Q(x_{0},δ)} Ψ_{1}([ρ_{δ}], ν(ρ_{δ})) dH^{N−1} ⟶_{δ→0} 0

and

lim_{δ→0} lim_{n→∞} \frac{1}{δ^{N}} ∫_{S_{ρ_{n,δ}} ∩Q(x_{0},δ)} Ψ_{1}([ρ_{n,δ}], ν(ρ_{n,δ})) dH^{N−1} = 0

by (H_{4}), (3.34) and (3.35). Finally, changing variables we obtain that

\frac{1}{δ^{N}} ∫_{\{x_{0}+ \frac{δ}{n} S_{u}\}∩Q(x_{0},δ)} Ψ_{1} (\frac{δ}{n}[u] (\frac{n(x − x_{0})}{δ}) , ν_{u} (\frac{n(x − x_{0})}{δ})) dH^{N−1} = ∫_{Q∩S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1},

from where

\frac{dI_{1}(g, G, .)}{dL^{N}}(x_{0}) ≤ ∫_{Q∩S_{u}} Ψ_{1}([u], ν_{u}) dH^{N−1}.

As a consequence, letting ε → 0 in (3.32) inequality (3.27) follows.

Proof of (3.28): Following an argument of Ambrosio, Mortola, Tortorelli [6] we note that it suffices to prove (3.28) for g = λχ_{E} with λ ∈ ℝ and where χ_{E} is the characteristic function of a set of finite perimeter E.

Step 1. We will start by addressing the case where E is a polyhedron. Let x_{0} ∈ S_{g} be such that

lim_{δ→0} \frac{1}{δ^{N−1}} ∫_{Q_{ν}(x_{0},δ)} |G(x)| dx = 0  (3.36)

where ν ≡ ν_{g}(x_{0}). Given ϵ > 0 let now u ∈ SBV (Q_{ν}; ℝ^{d}) be such that ∇u = 0, u|_{∂Q_{ν}} = γ_{(λ,ν)} and

γ_{1}(λ, ν) + ϵ ≥ ∫_{Q_{ν}} Ψ_{1}([u], ν_{u}) dH^{N−1}  (3.37)

(see (3.4)). For δ > 0 small enough define

D_{ν}^{n}(x_{0}, δ) := Q_{ν}(x_{0}, δ) ∩ \{x : \frac{|(x − x_{0}) · ν|}{δ} < \frac{1}{2n}\} ,

Q_{ν}^{+}(x_{0}, δ) = Q_{ν}(x_{0}, δ) ∩ \{x : \frac{(x − x_{0}) · ν}{δ} > 0\}

and Q_{ν}^{−}(x_{0}, δ) in an analogous way. Set now

u_{n,δ}(x) = \{ \begin{matrix}λ & x ∈ Q_{ν}^{+}(x_{0}, δ)\D_{ν}^{n}(x_{0}, δ) \\ u (\frac{n(x−x_{0})}{δ}) & x ∈ D_{ν}^{n}(x_{0}, δ) \\ 0 & x ∈ Q_{ν}^{−}(x_{0}, δ)\D_{ν}^{n}(x_{0}, δ) \end{matrix}

where u has been extended by periodicity to all of ℝ^{N}. Note that for x ∈ D_{ν}^{n}(x_{0}, δ) we have that

\frac{n}{δ}|(x − x^{0}) · ν| < \frac{1}{2}. Clearly u_{n,δ} ⟶_{n→∞,δ→0}^{L^{1}} γ̃_{(λ,ν)} where, for x ∈ Q_{ν}(x_{0}, δ),

γ̃_{(λ,ν)}(x) := \{ \begin{matrix}λ & if x · ν > 0 \\ 0 & if x · ν < 0. \end{matrix}

18 M. BAÍA, J. MATIAS & P. M. SANTOS

By Theorem 2.5 there exists ζ_{δ} ∈ SBV (Q_{ν}(x_{0}, δ); ℝ^{d}) such that ∇ζ_{δ} = G and

|D^{s}ζ_{δ}|(Q_{ν}(x_{0}, δ)) ≤ C||G||_{L^{1}(Q_{ν}(x_{0},δ);ℝ^{d×N})}.  (3.38)

Moreover by Lemma 2.4 there exists a sequence ζ_{n,δ} of piecewise constant functions defined on

Q_{ν}(x_{0}, δ) such that ζ_{n,δ} ⟶_{n→∞}^{L^{1}} ζ_{δ} and

|Dζ_{n,δ}|(Q_{ν}(x_{0}, δ)) ⟶_{n→∞} |Dζ_{δ}|(Q_{ν}(x_{0}, δ)).  (3.39)

Set

w_{n,δ} = u_{n,δ} + ζ_{δ} − ζ_{n,δ}.

Clearly w_{n,δ} is admissible for I_{1}(g, G, Q_{ν}(x_{0}, δ)). Therefore

\frac{dI_{1}(g, G, .)}{dH^{N−1}⌊S_{g}} (x_{0}) = lim_{δ→0} \frac{1}{δ^{N−1}} I_{1}(g, G, Q_{ν}(x_{0}, δ))
≤ lim_{δ→0} lim inf_{n→∞} \frac{1}{δ^{N−1}} \{∫_{Q_{ν}(x_{0},δ)∩S_{w_{n,δ}}} Ψ_{1}([w_{n,δ}](x), ν(w_{n,δ})(x)) dH^{N−1}\}
= lim_{δ→0} lim inf_{n→∞} \frac{1}{δ^{N−1}} \{∫_{Q_{ν}(x_{0},δ)∩S_{u_{n,δ}}} Ψ_{1}([u_{n,δ}](x), ν(u_{n,δ})(x)) dH^{N−1}
+ ∫_{Q_{ν}(x_{0},δ)∩S_{ζ_{δ}}} Ψ_{1}([ζ_{δ}](x), ν(ζ_{δ})(x)) dH^{N−1}
+ ∫_{Q_{ν}(x_{0},δ)∩S_{ζ_{n,δ}}} Ψ_{1}([ζ_{n,δ}](x), ν(ζ_{n,δ})(x)) dH^{N−1}}
=: J_{1} + J_{2} + J_{3}

The terms J_{2} and J_{3} go to zero due to (H_{4}), (3.38), (3.39) and (3.36). Moreover

J_{1} = lim_{δ→0} lim inf_{n→∞} \frac{1}{δ^{N−1}} ∫_{Q_{ν}(x_{0},δ)∩S_{u_{n,δ}}} Ψ_{1}([u_{n,δ}](x), ν(u_{n,δ})(x)) dH^{N−1}

= lim_{δ→0} lim inf_{n→∞} \frac{1}{δ^{N−1}} ∫_{D_{ν}^{n}(x_{0},δ)∩\{x: \frac{n(x−x_{0})}{δ} ∈S_{u}\}} Ψ_{1} ([u] (\frac{n(x − x_{0})}{δ}) , ν_{u} (\frac{n(x − x_{0})}{δ})) dH^{N−1}

= lim_{δ→0} lim inf_{n→∞} \frac{1}{n^{N−1}} ∫_{nQ_{ν}∩\{y: |y·ν|< \frac{1}{2}\}∩S_{u}} Ψ_{1}([u](y), ν_{u}(y)) dH^{N−1}(y)

= ∫_{Q_{ν}∩S_{u}} Ψ_{1}([u](y), ν_{u}(y)) dH^{N−1}(y). Thus

\frac{dI_{1}(g, G, .)}{dH^{N−1}⌊S_{g}} (x_{0}) ≤ ∫_{Q_{ν}∩S_{u}} Ψ_{1}([u](y), ν_{u}(y)) dH^{N−1}(y)

and consequently (3.28) follows by letting ϵ → 0 in (3.37).

Step 2. Let now E be a general set of finite perimeter and let g = λχ_{E}, λ ∈ ℝ. Consider E_{n} a sequence of polyhedra such that

Per(E_{n}) ⟶_{n→∞} Per(E),  (3.40)

L^{N}(E_{n}ΔE) ⟶_{n→∞} 0,

and

χ_{E_{n}} ⟶_{n→∞}^{L^{1}} χ_{E}  (3.41)

(see De Giorgi [18]). By Proposition 3.10 and Proposition 3.6 in Barroso, Bouchitté, Buttazzo and Fonseca [7] we obtain a sequence of functions γ_{1}^{m} : ℝ^{N} → [0, ∞) which are continuous, homogeneous of degree one and satisfy

γ_{1}(λ, y) ≤ γ_{1}^{m}(y) ≤ C|y|, ∀y ∈ ℝ^{N},  (3.42)

γ_{1}(λ, y) = inf_{m} γ_{1}^{m}(y),  (3.43)

where γ_{1}(λ, .) has been extended as an homogeneous function of degree one to all of ℝ^{N}. Let

g_{n} = λχ_{E_{n}}.  (3.44)

By (3.41) it is clear that g_{n} ⟶_{n→∞}^{L^{1}} g. Given A ∈ A(Ω), from the previous case and Proposition 3.11 we have that

I_{1}(g, G, A) ≤ lim inf_{n→∞} I_{1}(g_{n}, G, A)
≤ lim inf_{n→∞} {∫_{A} W_{1}(G − ∇g_{n}) dx + ∫_{S_{gn} ∩A} γ_{1}([g_{n}], ν(g_{n})) dH^{N−1}}

= lim inf_{n→∞} {∫_{A} W_{1}(G) dx + ∫_{S_{gn} ∩A} γ_{1}([g_{n}], ν(g_{n})) dH^{N−1}}

≤ C ∫_{A} |G| dx + lim_{n→∞} ∫_{S_{gn} ∩A} γ_{1}([g_{n}], ν(g_{n})) dH^{N−1}  (3.45)

where in the last inequality we have used (3.30) together with the fact that W_{1}(0) = 0. For m fixed by (3.45), the definition of g_{n}, (3.43) and Theorem 2.2 it follows that

I_{1}(g, G, A) ≤ C ∫_{A} |G| dx + lim_{n→∞} ∫_{∂E_{n}∩A} γ_{1}^{m}(ν(g_{n})) dH^{N−1}

≤ C ∫_{A} |G| dx + ∫_{∂E∩A} γ_{1}^{m}(ν_{g}) dH^{N−1}.

Letting now m → ∞ and using the Monotone Convergence Theorem, we obtain

I_{1}(g, G, A) ≤ C ∫_{A} |G| dx + ∫_{S_{g}∩A} γ_{1}(λ, ν_{g}) dH^{N−1}.  (3.46)

Consider x_{0} satisfying (3.36). Then from (3.46) we immediately conclude that for g defined by (3.44)

\frac{dI_{1}(g, G, .)}{dH^{N−1}⌊S_{g}} (x_{0}) = lim_{δ→0} \frac{1}{δ^{N−1}} I_{1}(g, G, Q_{ν}(x_{0}, δ))
≤ γ_{1}([g](x_{0}), ν_{g}(x_{0})).
□

4. Integral representation in BV

This section is devoted to the caractherization of the energy I(g, G) (cf. (1.3)) in the full BVsetting for (g, G) ∈ GSD(Ω; ℝ^{d}) := BV^{2}(Ω; ℝ^{d}) × BV (Ω; ℝ^{d×N}). We refer to Theorem 3.2 for the hypotheses and notations used throughout its statement that reads as follows.

Theorem 4.1. Let (g, G) ∈ GSD(Ω; ℝ^{d}). Under hypotheses (H_{1}) − (H_{7})

I(g, G) = ∫_{Ω} (W_{1}(G − ∇g) + W_{2}(G, ∇G)) dx + ∫_{S_{g}} γ_{1}([g], ν_{g})dH^{N−1}

+ ∫_{S_{G}} γ_{2}(G^{+}, G^{−}, ν_{G}) dH^{N−1} + ∫_{Ω} W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}) d|D^{c}g|

+ ∫_{Ω} W_{2}^{∞} (G, \frac{dD^{c}G}{d|D^{c}G|}) d|D^{c}G|.  (4.1)

To prove Theorem 4.1 we start by deriving two auxiliary results.

Proposition 4.2. Let ν ∈ S^{N−1} and define for all C ∈ ℝ^{d×N}

\begin{matrix}W̃_{1}(C) = inf \{∫_{Q_{ν}} W_{1}(C − ∇v(x)) dx + ∫_{Q_{ν}∩S_{v}} γ_{1}([v], ν(v)) dH^{N−1}, \\ v ∈ SBV^{2}(Q_{ν}; ℝ^{d}), v|_{∂Q_{ν}} (x) = b(x · ν), b ∈ SBV^{2}([−1/2, 1/2]; ℝ^{d}), b(1/2) = b(−1/2)\} . \end{matrix}

Then W̃_{1}(C) = W_{1}(C).

Proof. Clearly W̃_{1}(C) ≤ W_{1}(C). Let us prove the reverse inequality. Fix ϵ > 0 and let v ∈ SBV^{2}(Q_{ν}; ℝ^{d}) with v|_{∂Q_{ν}} (x) = b(x · ν) for some b ∈ SBV^{2}([−1/2, 1/2]; ℝ^{d}), with b(1/2) = b(−1/2) be such that

W̃_{1}(C) ≥ ∫_{Q_{ν}} W_{1}(C − ∇v(x)) dx + ∫_{Q_{ν}∩S_{v}} γ_{1}([v], ν(v)) dH^{N−1} − ϵ.  (4.2)

Extend v by periodicity to all of ℝ^{N} and define

w_{n}(y) = \frac{v(ny)}{n} − Cy, y ∈ Q_{ν}.

By Theorem 3.2 it follows that

I_{1}(w_{n}, 0, Q_{ν}) = ∫_{Q_{ν}} W_{1}(C − ∇v(ny)) dy + \frac{1}{n} ∫_{Q_{ν}∩\{y: ny∈S_{v}\}} γ_{1}([v](ny), ν(v)(ny)) dH^{N−1}

= ∫_{Q_{ν}} W_{1}(C − ∇v(ny)) dy + \frac{1}{n^{N}} ∫_{nQ_{ν}∩S_{v}} γ_{1}([v](y), ν(v)(y)) dH^{N−1}

= ∫_{Q_{ν}} W_{1}(C − ∇v(ny)) dy + ∫_{Q_{ν}∩S_{v}} γ_{1}([v](y), ν(v)(y)) dH^{N−1}.

Therefore, by Riemann Lebesgue’s Lemma

lim_{n→∞} I_{1}(w_{n}, 0, Q_{ν}) = ∫_{Q_{ν}} W_{1}(C − ∇v(y)) dy + ∫_{Q_{ν}∩S_{v}} γ_{1}([v](y), ν(v)(y)) dH^{N−1},

and consequently, by Proposition 3.11,

I_{1}(−C(.), 0, Q_{ν}) ≤ ∫_{Q_{ν}} W_{1}(C − ∇v(y)) dy + ∫_{Q_{ν}∩S_{v}} γ_{1}([v](y), ν(v)(y)) dH^{N−1}.

As I_{1}(−C(.), 0, Q_{ν}) = ∫_{Q_{ν}} W_{1}(C) dy = W_{1}(C), then by (4.2)

W̃_{1}(C) ≥ W_{1}(C) − ϵ

and the result follows by letting ε → 0.

□

Lemma 4.3. Let (g, G) ∈ GSD(Ω; ℝ^{d}). For all A ∈ A(Ω) define

Ĩ_{1}(g, G, A) = inf_{\begin{matrix}\{g_{n}\} ⊂ SBV^{2}(A; ℝ^{d}) \\ \{G_{n}\} ⊂ SBV (A; ℝ^{d×N} ) \end{matrix}}\{lim inf_{n→∞} I_{1}(g_{n}, G_{n}, A), g_{n} ⟶_{n→∞}^{L^{1}} g, G_{n} ⟶_{n→∞}^{L^{1}} G\} .

Then Ĩ_{1}(g, G, A) = I_{1}(g, G, A).

Proof. Let (g_{n}, ∇g_{n}) ∈ SD(Ω; ℝ^{d}) with g_{n} ⟶_{n→∞}^{L^{1}} g and ∇g_{n} ⟶_{n→∞}^{L^{1}} G. Then

I_{1}(g_{n}, ∇g_{n}, A) ≤ ∫_{S_{gn} ∩A} ψ_{1}([g_{n}], ν(g_{n})) dH^{N−1}

for all n ∈ ℕ. Hence

lim inf_{n→∞} ∫_{S_{gn} ∩A} ψ_{1}([g_{n}], ν(g_{n})) dH^{N−1} ≥ lim inf_{n→∞} I_{1}(g_{n}, ∇g_{n}, A) ≥ Ĩ_{1}(g, G, A).

By the arbitrariness of the sequence {g_{n}} it follows that

I_{1}(g, G, A) ≥ Ĩ_{1}(g, G, A).

To show that the reverse inequality is true let Ĩ_{1}(g, G, A) = lim inf_{n→∞} I_{1}(g_{n}, G_{n}, A) with g_{n} ∈ SBV^{2}(A; ℝ^{d}),

G_{n} ∈ SBV (A; ℝ^{d×N}), g_{n} ⟶_{n→∞}^{L^{1}} g and G_{n} ⟶_{n→∞}^{L^{1}} G. For each n ∈ ℕ let u_{n} ∈ SBV^{2}(A; ℝ^{d}) be such

that

I_{1}(g_{n}, G_{n}, A) + \frac{1}{n} ≥ ∫_{S_{un}} ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1},

|u_{n} − g_{n}|_{L_{1}} ≤ \frac{1}{n} and |∇u_{n} − G_{n}|_{L_{1}} ≤ \frac{1}{n}. Therefore

Ĩ_{1}(g, G, A) = lim inf_{n→∞} I_{1}(g_{n}, G_{n}, A) ≥ lim inf_{n→∞} ∫_{S_{un}} ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1} ≥ I_{1}(g, G, A).

□

We proceed now to the proof of Theorem 4.1 Proof of Theorem 4.1. Given (g, G) ∈ GSD(Ω; ℝ^{d}) by Lemma 3.4 (that still holds) we can decompose

I(g, G) = I_{1}(g, G) + I_{2}(G)

where

I_{1}(g, G) = inf_{{u_{n}}⊂SBV^{2}(Ω;ℝ^{d})} {lim inf_{n→∞} ∫_{S_{un}} Ψ_{1}([u_{n}], ν(u_{n})) dH^{N−1}, u_{n} ⟶^{L^{1}} g, ∇u_{n} ⟶^{L^{1}} G} (4.3)

and

I_{2}(G) = inf_{{v_{n}}⊂SBV (Ω;ℝ^{d×N})} {lim inf_{n→∞} ∫_{Ω} W (v_{n}, ∇v_{n}) dx + ∫_{S_{vn}} Ψ_{2}([v_{n}], ν(v_{n})) dH^{N−1}, v_{n} ⟶^{L^{1}} G} .

(4.4)

As in the proof of Theorem 3.2, by Theorem 4.2.2 in Bouchitté, Fonseca and Mascarenhas [11] we have that

I_{2}(G) = ∫_{Ω} W_{2}(G, ∇G) dx + ∫_{S_{G}∩Ω} γ_{2}(G^{+}, G^{−}, ν_{G}) dH^{N−1} + ∫_{Ω} W_{2}^{∞} (G, \frac{dD^{c}G}{d|D^{c}G|}) d|D^{c}G|,

and hence, to prove our claim (4.1), it is enough to show that

I_{1}(g, G) = ∫_{Ω} W_{1}(G − ∇g) dx + ∫_{S_{g}} γ_{1}([g], ν_{g})dH^{N−1} + ∫_{Ω} W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}) d|D^{c}g|.  (4.5)

We divide the argument in four steps.

Step 1. (Localization) As in Section 3.1 we can see that I_{1}(g, G, ·)⌊A(Ω) is a Radon measure absolutely continuous with respect to L^{N} + H^{N−1}⌊S_{g} + d|D^{c}g|.

Step 2. (Upper bound for the density energy derivation with respect to the Cantor part D^{c}g of Dg) Let us prove that for |D^{c}g|-a.e x_{0} ∈ Ω

\frac{dI_{1}(g, G, .)}{d|D^{c}g|}(x_{0}) ≤ W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}(x_{0})) .  (4.6)

Let g_{n} be a sequence of regular functions such that g_{n} ⟶_{n→∞}^{L^{1}} g, ||Dg_{n}||(Ω) ⟶_{n→∞} ||Dg||(Ω) and in

addition consider G_{n} ∈ SBV (Ω; ℝ^{d×N}) with G_{n} ⟶_{n→∞}^{L^{1}} G. Given A ∈ A(Ω) by Theorem 3.2 (see (3.7)), Lemma 4.3 and Proposition 3.10 we get that

I_{1}(g, G, A) ≤ lim inf_{n→∞} I_{1}(g_{n}, G_{n}, A)
= lim inf_{n→∞} ∫_{A} W_{1}(G_{n}(x) − ∇g_{n}(x)) dx
≤ C ∫_{A} |G(x)| dx + lim_{n→∞} ∫_{A} W_{1}(−∇g_{n}(x)) dx
= C ∫_{A} |G(x)| dx + ∫_{A} W_{1} (−\frac{dDg}{d|Dg|}) d|Dg|,  (4.7)

where the last equality follows by Theorem 2.2 since W_{1} is Lipschitz continuous and homogeneous of degree one.

Let x_{0} ∈ supp|D^{c}g| such that \frac{dI_{1}(g,G,.)}{d|D^{c}g|}(x_{0}) exists and

W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}(x_{0})) = lim_{δ→0} \frac{1}{|D^{c}g|(Q(x_{0}, δ))} ∫_{Q(x_{0},δ)} W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}) d|D^{c}g|.

Then

\frac{dI_{1}(g, G, .)}{d|D^{c}g|}(x_{0}) = lim_{δ→0} \frac{I_{1}(g, G, Q(x_{0}, δ))}{|D^{c}g|(Q(x_{0}, δ))}

≤ W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}(x_{0})) .

by (4.7) so that (4.6) holds.

Step 3. (Lower bound for the density energy derivation with respect to the Cantor part D^{c}g of Dg) Let us prove that for |D^{c}g|-a.e x_{0} ∈ Ω

\frac{dI_{1}(g, G, .)}{d|D^{c}g|}(x_{0}) ≥ W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}(x_{0})) .  (4.8)

Let x_{0} ∈ supp|D^{c}g| be such that

\frac{dD^{c}g}{d|D^{c}g|}(x_{0}) = a_{x_{0}} ⊗ ν_{g(x_{0})}

for some a_{x_{0}} ≡ a ∈ ℝ^{d} and ν_{g(x_{0})} ≡ ν ∈ S^{N−1} (see Alberti’s Rank One Theorem (cf. [3])) and with

\frac{d|G|L^{N}}{d|D^{c}g|} (x_{0}) = 0.  (4.9)

Thus showing (4.8) is equivalent to see that

\frac{dI_{1}(g, G, .)}{d|D^{c}g|}(x_{0}) ≥ W_{1} (−a ⊗ ν) .  (4.10)

A RELAXATION RESULT IN THE FRAMEWORK OF STRUCTURED DEFORMATIONS 23

To prove (4.10) let (g_{n}, G_{n}) ∈ SBV^{2}(Ω; ℝ^{d}) × SBV (Ω; ℝ^{d×N}) be a sequence with g_{n} ⟶_{n→∞}^{L^{1}(Ω;ℝ^{d})} g and G_{n} ⟶_{n→∞}^{L^{1}(Ω;ℝ^{d×N})} G, and fix δ > 0. Note that by Proposition 3.10

lim_{n→∞} [I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ)) − I_{1}(g_{n}, G, Q_{ν}(x_{0}, δ))] = 0

and by Theorem 3.2 and (4.3) we have that

I_{1}(g_{n}, G, Q_{ν}(x_{0}, δ)) = ∫_{Q_{ν}(x_{0},δ)} W_{1}(G − ∇g_{n}) dx + ∫_{Q_{ν}(x_{0},δ)∩S_{gn}} γ_{1}([g_{n}], ν(g_{n})) dH^{N−1}
= δ^{N} ∫_{Q_{ν}} W_{1}(G(x_{0} + δ y) − ∇g_{n}(x_{0} + δ y)) dy
+δ^{N−1} ∫_{Q_{ν}∩{y: x_{0}+δ y∈S_{gn} }} γ_{1}([g_{n}](x_{0} + δ y), ν(g_{n})(x_{0} + δ y)) dH^{N−1}.

Defining

t_{δ} = \frac{|D^{c}g|(Q_{ν}(x_{0}, δ))}{δ^{N}}  (4.11)

we can write that

\frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} = \frac{1}{t_{δ}} ∫_{Q_{ν}} W_{1}(G(x_{0} + δ y) − ∇g_{n}(x_{0} + δ y)) dy

+\frac{1}{δt_{δ}} ∫_{Q_{ν}∩\{y: x_{0}+δ y∈S_{gn} \}} γ_{1}([g_{n}](x_{0} + δ y), ν(g_{n})(x_{0} + δ y)) dH^{N−1}

and setting

w_{n,δ}(y) = \frac{g_{n}(x_{0} + δy) − ∫_{Q_{ν}} g_{n}(x_{0} + δy) dy}{δ t_{δ}} , y ∈ Q_{ν}

we derive that

\frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} = \frac{1}{t_{δ}} ∫_{Q_{ν}} W_{1}(G(x_{0} + δ y) − t_{δ}∇w_{n,δ}(y))) dy
+ ∫_{Q_{ν}∩S_{w_{n,δ}}} γ_{1}([w_{n,δ}](y), ν(w_{n,δ})(y)) dH^{N−1}.

Consequently

lim inf_{δ→0} lim inf_{n→∞} \frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} = lim inf_{δ→0} lim inf_{n→∞} [\frac{1}{t_{δ}} ∫_{Q_{ν}} W_{1}(G(x_{0} + δ y) − t_{δ}∇w_{n,δ}(y))) dy
+ ∫_{Q_{ν}∩S_{w_{n,δ}}} γ_{1}([w_{n,δ}](y), ν(w_{n,δ})(y)) dH^{N−1}] .

Using Alberti’s result on the blow-up of the Cantor part (see [3], Theorem 2.3 in [4] and Lemma 5.1 in [31]), there exists a nondecreasing function ζ ∈ BV [−1/2, 1/2] such that

ζ (1/2) − ζ (−1/2) = 1, ∫− \frac{1}{2}^{\frac{1}{2}} ζ(s) ds = 0  (4.12)

and

lim_{δ→0} lim_{n→∞} ∫_{Q_{ν}} |w_{n,δ}(y) − aζ(y.ν)| dy = 0.

Therefore passing to a diagonalizing sequence w_{k} ≡ w_{n(k),δ(k)}, setting δ_{k} = δ(k) and using the homogeneity property of W_{1} we have that

lim inf_{δ→0} lim inf_{n→∞} \frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} = lim_{k→∞} [∫_{Q_{ν}} W_{1} (\frac{G(x_{0} + δ_{k} y)}{t_{δ_{k}}} − ∇w_{k}(y)) dy

+ ∫_{Q_{ν}∩S_{w_{k}}} γ_{1}([w_{k}](y), ν(w_{k})(y)) dH^{N−1}] .

Now set

v_{k}(y) = \frac{a(ρ_{k} ∗ ζ)(y.ν)}{c_{k}}, y ∈ Q_{ν},  (4.13)

where ρ_{k} denotes the standard mollifier sequence, and

c_{k} = (ρ_{k} ∗ ζ)(1/2) − (ρ_{k} ∗ ζ)( − 1/2).

It is clear, by (4.12), that c_{k} → 1 as k → ∞. Since

w_{k} − v_{k} ⟶_{k→∞}^{L^{1}} 0

with a similar argument to the one used in Lemma 3.8 we can assume that w_{k}⌊_{∂Q_{ν}} = v_{k}⌊_{∂Q_{ν}}. Thus defining

w̄_{k}(y) = w_{k}(y) − (a ⊗ ν)y, y ∈ Q_{ν},

we have that

lim inf_{δ→0} lim inf_{n→∞} \frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} = lim_{k→∞} ∫_{Q_{ν}} W_{1} (\frac{G(x_{0} + δ_{k}y)}{t_{δ_{k}}} − ∇w̄_{k}(y)) − (a ⊗ ν)) dy

+ ∫_{Q_{ν}∩Sw̄_{k}} γ_{1}([w̄_{k}](y), ν(w̄_{k})(y)) dH^{N−1}.

Since by Proposition 3.10 and (4.11)

lim sup_{k→∞} ∫_{Q_{ν}} |W_{1} (\frac{G(x_{0} + δ_{k}y)}{t_{δ_{k}}} − ∇w̄_{k}(y) − (a ⊗ ν)) − W_{1} (−∇w̄_{k}(y) − (a ⊗ ν))| dy

≤ lim sup_{k→∞} C ∫_{Q_{ν}} |\frac{G(x_{0} + δ_{k}y)}{t_{δ_{k}}} | dy
= lim sup_{k→∞} \frac{C}{t_{δ_{k}}} ∫_{Q_{ν}} |G(x_{0} + δ_{k}y)| dy
= lim_{k→∞} \frac{C}{t_{δ_{k}}δ_{k}^{N}} ∫_{Q_{ν}(x_{0},δ_{k})} |G(x)| dx = 0  (4.14)

by (4.9). From (4.13) it is easy to see that for each k ∈ ℕ the function w̄_{k}(y) is admissible for W̃_{1}. Therefore from (4.14) and Proposition 4.2 we conclude that

lim inf_{δ→0} lim inf_{n→∞} \frac{I_{1}(g_{n}, G_{n}, Q_{ν}(x_{0}, δ))}{|D^{c}g|(Q_{ν}(x_{0}, δ))} ≥ lim_{k→∞} ∫_{Q_{ν}} W_{1}( − ∇w̄_{k}(y)) − (a ⊗ ν)) dy

+ ∫_{Q∩Sw̄_{k}} γ_{1}([w̄_{k}](y), ν(w̄_{k})(y)) dH^{N−1}

≥ W̃_{1}( − a ⊗ ν) = W_{1}( − a ⊗ ν).

Finally the lower bound (4.10) follows from the arbitrariness of the sequence (g_{n}, G_{n}) ∈ SBV^{2}(Ω; ℝ^{d})× SBV (Ω; ℝ^{d×N}) considered and from the characterization of I_{1} given in Lemma 4.3.

Step 4. We remark that, as in Theorem 3.2,

\frac{dI_{1}(g, G, ·)}{dL^{N}}(x_{0}) = W_{1}(G(x_{0}) − ∇g(x_{0})), L^{N} -a.e x_{0} ∈ Ω

and

\frac{dI_{1}(g, G, ·)}{dH^{N−1}⌊S_{g}} (x_{0}) = γ_{1}([g](x_{0}), ν(g)(x_{0})), H^{N−1} -a.e x_{0} ∈ S_{g}.

In fact, the proof of the upper bounds can be obtained in a similar way than (4.6). For this, it

is enough to choose sequences g_{n} ⟶_{n→∞}^{L^{1}} g with |Dg_{n}|(Ω) ⟶_{n→∞} |Dg|(Ω) (regular functions for the

case of \frac{dI_{1}(g,G,·)}{dL^{N}} and piecewise constant functions to address \frac{dI_{1}(g,G,·)}{dH^{N−1}⌊S_{g}} ). Since both W_{1} and γ_{1} are homogeneous functions of degree one, the result follows from Lemma 4.3 and Theorem 2.2. It is also easy to check that the lower bounds hold since the proof of their counterparts in Theorem 3.2 is still valid in the BV -setting.

As a consequence of Steps 1-4.

I_{1}(g, G, A) = ∫_{A} W_{1}(G − ∇g) dx + ∫_{A∩S_{g}} γ_{1}([g], ν_{g})dH^{N−1} + ∫_{A} W_{1} (−\frac{dD^{c}g}{d|D^{c}g|}) d|D^{c}g|

for all A ∈ A(Ω), from where, taking A = Ω, equality (4.5) follows ending the proof of Theorem 4.1.

□

Acknowledgements:

The authors would like to thank Irene Fonseca and David Owen for their fruitful discussions on the model. This work was partially supported by the Fundação para a Ciência e a Tecnologia (FCT/Portugal).

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(Margarida Baía) CAMGSD, Departamento de Matemática, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

   E-mail address: mbaia@math.ist.utl.pt

(José Matias) CAMGSD, Departamento de Matemática, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

   E-mail address: jmatias@math.ist.utl.pt

(Pedro M. Santos) CAMGSD, Departamento de Matemática, Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

   E-mail address: pmsantos@math.ist.utl.pt