A RELAXATION THEOREM IN THE SPACE OF FUNCTIONS OF BOUNDED DEFORMATION

ANA CRISTINA BARROSO, IRENE FONSECA, AND RODICA TOADER

Abstract. We obtain an integral representation for the relaxation, in the space of functions of bounded deformation, of the energy

∫_{Ω} f(Eu(x))dx

with respect to L^{1}-convergence. Here Eu represents the absolutely continuous part of the symmetrized distributional derivative Eu and the function f satisfies linear growth and coercivity conditions.

Keywords : functions of bounded deformation, relaxation, E-quasiconvexity

1991 Mathematics Subject Classification: 35J50, 49J45, 49Q20, 73E99.

2000 Mathematics Subject Classification: 35J50, 49J45, 49Q20, 74C15, 74G65.

1. Introduction

The analysis of variational problems that model interesting elastic and magnetic properties exhibited by certain materials often involves minimization and relaxation of nonconvex energies of the type

∫_{Ω} f(∇u)dx, (1.1)

where Ω ⊂ ℝ^{N} is an open, bounded set and the density function f satisfies certain growth and coercivity conditions. In each case the function u in (1.1) represents a specific physical entity and belongs to a space which is appropriate to describe the material phenomenon in question.

In the context of perfect plasticity the function u represents the displacement field of a body occupying the reference configuration Ω with volume energy density f. In this case u belongs to the space BD of functions of bounded deformation composed of integrable vector-valued functions u for which all components E_{ij}, i, j = 1, …, N, of the deformation tensor Eu := \frac{Du+Du^{T}}{2} are bounded Radon measures. Thus, the study of equilibria leads naturally to questions concerning lower semicontinuity properties and relaxation of

∫_{Ω} f(Eu(x))dx, (1.2)

where Eu represents the absolutely continuous part (with respect to the Lebesgue measure) of the symmetrized distributional derivative Eu.

Our goal in this paper is to show that the relaxation of the bulk energy (1.2), in the space of functions of bounded deformation, admits an integral representation where a surface energy term is naturally produced. In a forthcoming paper we will consider more general linear operators A acting on Du for which A(Du) is a bounded Radon measure (see also [11]).

We remark that a relaxation result has been proved by Braides, Defranceschi and Vitali in [6] in the case where the bulk energy density is of the form ∥Eu∥^{2} + (div u)^{2}, i.e. f(ξ) := |ξ|^{2} + |tr ξ|^{2}, and the total energy also includes a surface energy term. In our work no convexity assumptions are placed on the volume density f.

Precisely, for u ∈ BD(Ω) we consider the energy

F (u, Ω) := \{ \begin{matrix}∫_{Ω} f(Eu(x))dx & if u ∈ W^{1,1}(Ω, ℝ^{N}) , \\ +∞ & otherwise , \end{matrix}

and the localized functional

F(u, V) := inf_{{u_{n}}} { lim inf_{n→∞} ∫_{V} f(Eu_{n}(x))dx | u_{n} ∈ W^{1,1}(V, ℝ^{N}), u_{n} → u in L^{1}(V, ℝ^{N})}

defined on the set O(Ω) of all open subsets V of Ω.

Assuming that f is continuous and satisfies growth and coercivity conditions of the type

\frac{1}{C} |ξ| ≤ f(ξ) ≤ C(1 + |ξ|) for all ξ ∈ M_{sym}^{N×N},

where C > 0, we show that F(u, ·) is the restriction to O(Ω) of a Radon measure μ which is absolutely continuous with respect to L^{N} +|E^{s}u|, where Eu = EuL^{N} +E^{s}u, and L^{N} stands for the N-dimensional Lebesgue measure. By the Radon-Nikodym Theorem it follows that

F(u, Ω) = ∫_{Ω} μ_{a}(x)dx + ∫_{Ω} μ_{s}(x)d|E^{s}u|(x) ,

and the question now is to identify the densities μ_{a} and μ_{s}. The main ingredient of our approach is the blow-up method introduced by Fonseca and Müller [9] which reduces the identification of the energy densities of the relaxed functional to the characterization of F(v, Q) when Q is a unit cube and v is obtained as the blow-up of the function u around a point x_{0}. Similar ideas have been used in the context of BV -functions (see, for instance [2], [4], [7], [10]) where, in order to identify the density of the absolutely continuous part μ_{a} one chooses a point x_{0} of approximate differentiability of the function u satisfying

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. (1.3)

Using the L^{1}-convergence, as ε → 0^{+}, of the rescaled functions

v_{ε}(y) := \frac{u(x_{0} + εy) − u(x_{0})}{ε}

to the homogeneous function v_{0}(y) = ∇u(x_{0})y, which is guaranteed by (1.3), it is possible to reduce the identification of the energy density of the absolutely continuous part μ_{a}, to a relaxation problem where the target function is homogeneous.

However this reasoning cannot be applied to the BD framework where the equivalent of (1.3), replacing ∇u(x_{0}) by Eu(x_{0}), is, in general, false (see [1]). To overcome this difficulty we will use a Poincaré-type inequality

∥u − P u∥_{L^{1}(Q,ℝ^{N})} ≤ C|Eu|(Q),

where P is the orthogonal projection onto the kernel of the operator E (cf. [12], see also Theorem 3.1 in [1]). This, together with a compact embedding result, will ensure the L^{1}- convergence we need to proceed.

We organize the paper as follows: in Section 2 we recall the main properties of the space BD which will be used in the sequel. In Section 3 we state our main theorem, prove a version of De Giorgi’s Slicing Lemma and show some properties of the localized functional F(u, V). Section 4 is devoted to the characterization of the absolutely continuous density μ_{a}, whereas μ_{s} is studied in Section 5.

2. The space of functions of bounded deformation

Let Ω be a bounded, open subset of ℝ^{N} with Lipschitz boundary Γ. We denote by B(Ω), O(Ω) and O_{∞}(Ω) the family of Borel, open and open subsets of Ω with Lipschitz boundary, respectively. We use the standard notation, L^{N} and H^{N−1}, for the Lebesgue and Hausdorff measures. We recall that the symmetric product between two vectors u and v ∈ ℝ^{N}, u ⊙ v, is the symmetric N × N matrix defined by u ⊙ v = (u ⊗ v + v ⊗ u)/2, where ⊗ indicates the tensor product. We denote by B(x, ρ) the ball in ℝ^{N} of centre x and radius ρ, by Q(x, ρ) the cube of centre x and sidelength ρ, while Q_{ν}(x, ρ) will be used to indicate the cube with two of its faces perpendicular to the unit vector ν. When x = 0 and ρ = 1 we shall simply write B and Q. Let Q′ := Q ∩ {x ∈ ℝ^{N} | x_{N} = 0}, S^{N−1} := ∂B and ω_{N} := L^{N}(B).

By C we indicate a constant whose value might change from line to line. As usual, L^{q}(Ω, ℝ^{N}), W^{m,q}(Ω, ℝ^{N}) denote Lebesgue and Sobolev spaces, respectively, C_{0}^{∞}(Ω, ℝ^{N}) is the space of ℝ^{N}-valued smooth functions with compact support in Ω, and C_{per}^{∞}(Q, ℝ^{N}) are the smooth and Q-periodic functions from Q into ℝ^{N}.

For u ∈ L^{1}(Ω, ℝ^{N}) define Ω_{u} as the set of the Lebesgue points of u, i.e. the set of points x ∈ Ω such that there exists ũ(x) ∈ ℝ^{N} satisfying

lim_{ρ→0^{+}} \frac{1}{ρ^{N}} ∫_{B(x,ρ)} |u(y) − ũ(x)|dy = 0.

The Lebesgue discontinuity set S_{u} of u is defined as the set of points which are not Lebesgue points, that is S_{u} := Ω \ Ω_{u}. By Lebesgue’s Differentiation Theorem, S_{u} is L^{N}-negligible and the function ũ : Ω → ℝ^{N}, which coincides with u L^{N}-almost everywhere in Ω_{u}, is called the Lebesgue representative of u. We say that u has one sided Lebesgue limits u^{+}(x) and u^{−}(x) at x ∈ Ω with respect to a suitable direction ν_{u}(x) ∈ S^{N−1}, if

lim_{ρ→0^{+}} \frac{1}{ρ^{N}} ∫_{B_{ρ}^{±}(x,ν_{u}(x))} |u(y) − u^{±}(x)|dy = 0 ,

where B_{ρ}^{±}(x, ν_{u}(x)) := {y ∈ B(x, ρ) | (y − x) · (±ν_{u}(x)) > 0}. Then the rescaled functions u_{ρ}(y) := u(x + ρy) converge in L^{1}(B, ℝ^{N}), as ρ → 0^{+}, to

\overline{u}_{x,ν_{u}(x)}(y) := \{ \begin{matrix}u^{+}(x) & if y · ν_{u}(x) > 0, \\ u^{−}(x) & if y · ν_{u}(x) < 0. \end{matrix}

When u^{+}(x) ≠ u^{−}(x), the triplet (u^{+}(x), u^{−}(x), ν_{u}(x)) is uniquely determined up to a change of sign of ν_{u}(x) and a permutation of (u^{+}(x), u^{−}(x)). If u^{+}(x) = u^{−}(x) then x ∈ Ω_{u} and the one sided Lebesgue limits coincide with the Lebesgue representative of u. The jump set J_{u} of u is defined as the set of points in Ω where the approximate limits u^{+}, u^{−} exist and are not equal. We use [u](x) to denote the jump of u at x, i.e. [u](x) = u^{+}(x) − u^{−}(x).

Definition 2.1. ([13]) The space of functions of bounded deformation, BD(Ω), is the set of functions u ∈ L^{1}(Ω, ℝ^{N}) whose symmetric part of the distributional derivative Du, Eu := \frac{Du+Du^{T}}{2} , is a matrix-valued bounded Radon measure. We denote by LD(Ω) the subspace of functions u ∈ BD(Ω) for which Eu ∈ L^{1}(Ω, M_{sym}^{N×N}).

We recall that Korn’s Inequality holds in certain subspaces of LD(Ω), e.g. for functions u ∈ LD(Ω) such that u ∈ L^{p}(Ω, ℝ^{N}), with 1 < p < +∞, and Eu ∈ L^{p}(Ω, M_{sym}^{N×N}); precisely, there exists a constant C = C(Ω) such that

∑_{i,j=1}^{N} ∫_{Ω} |D_{i}u_{j}(x)|^{p}dx ≤ C(Ω) ∫_{Ω} (|u(x)|^{p} + |Eu(x)|^{p}) dx ,

and thus this subspace of LD(Ω) coincides with W^{1,p}(Ω, ℝ^{N}). A counterexample due to Ornstein [14] shows that Korn’s Inequality does not hold for p = 1 and that the space LD(Ω) differs from the Sobolev space W^{1,1}(Ω, ℝ^{N}).

We recall some properties of functions of bounded deformation that will be used in the sequel. For a more detailed study of BD(Ω) we refer to [1], [5], [12], [16] [17], [18] and [19]. Notice first that the space BD(Ω) endowed with the norm

∥u∥_{BD(Ω)} := ∥u∥_{L^{1}(Ω,ℝ^{N})} + |Eu|(Ω)

is a Banach space.

We cannot expect smooth functions to be dense in BD(Ω) with respect to this topology but it can be shown, see for instance [18], that such a density result is true in a weaker topology (cf. Theorem 2.6).

Given u, v ∈ BD(Ω) we define the distance

d(u, v) := ∥u − v∥_{L^{1}(Ω,ℝ^{N})} + | |Eu|(Ω) − |Ev|(Ω)|,

and we denote by intermediate topology the one determined by this distance: a sequence {u_{n}}_{n∈ℕ} in BD(Ω) converges to a function u ∈ BD(Ω) with respect to this topology, and we

write u_{n} →^{i} u, if

u_{n} → u in L^{1}(Ω, ℝ^{N}),
Eu_{n} ⇀^{∗} Eu in the sense of measures,
|Eu_{n}|(Ω) → |Eu|(Ω).

If u_{n} → u in L^{1}(Ω, ℝ^{N}) and |Eu_{n}|(Ω) ≤ C then u ∈ BD(Ω) and |Eu|(Ω) ≤ lim inf_{n→∞}|Eu_{n}|(Ω). Indeed, Eu_{n} converges weakly-∗ in the sense of measures to some measure μ, and by the linearity of the operator E, μ = Eu, so that u ∈ BD(Ω). The inequality follows from the lower semicontinuity of the total variation of Radon measures.

Proposition 2.2. If u, v ∈ BD(Ω), and ω ⊂ Ω has Lipschitz boundary, then the function w defined by

w = \{ \begin{matrix}u & in ω \\ v & in Ω \ ω \end{matrix}

belongs to BD(Ω) and

Ew = Euχ_{ω} + Evχ_{Ω\ω} + (v − u) ⊙ νH^{N−1}⌊∂ω,

where ν is the outward unit normal to ∂ω.

Lemma 2.3. (Lemma 4.5 [3]) Let u ∈ BD(Ω) and let ρ ∈ C_{0}^{∞}(ℝ^{N}) be a non-negative

function such that supp (ρ) ⊂⊂ B(0, 1), ρ(−x) = ρ(x) for every x ∈ ℝ^{N} and ∫_{ℝ^{N}} ρ(x)dx = 1.

For any n ∈ ℕ set ρ_{n}(x) := n^{N}ρ(nx) and

u_{n}(x) := (u ∗ ρ_{n})(x) = ∫_{Ω} u(y)ρ_{n}(x − y)dy, for x ∈ \{y ∈ Ω | dist (y, ∂Ω) > \frac{1}{n}\} .

Then u_{n} ∈ C^{∞} (\{y ∈ Ω | dist (y, ∂Ω) > \frac{1}{n}\} , ℝ^{N}) and i) for any non-negative Borel function h : Ω → ℝ

∫_{B(x_{0},δ)} h(x)|Eu_{n}(x)|dx ≤ ∫_{B(x_{0},δ+ \frac{1}{n})}(h ∗ ρ_{n})(x)d|Eu|(x),

whenever δ + \frac{1}{n} < dist (x_{0}, ∂Ω); ii) for any positively homogeneous of degree one, convex function θ : M_{sym}^{N×N} → [0, +∞[

and any δ ∈]0, dist (x_{0}, ∂Ω)[ such that |Eu|(∂B(x_{0}, δ)) = 0,

lim_{n→∞} ∫_{B(x_{0},δ)} θ(Eu_{n}(x))dx = ∫_{B(x_{0},δ)} θ (\frac{d Eu}{d|Eu|}) d|Eu|;

iii) lim_{n→∞} u_{n}(x) = ũ(x) and lim_{n→∞}(|u_{n} −u|∗ρ_{n})(x) = 0 for every x ∈ Ω\S_{u}, whenever u ∈ L^{∞}(Ω, ℝ^{N}).

Proof. i) In the same way as it was shown for the distributional derivative of BV functions, it can be proved that Eu_{n}(x) = (Eu ∗ ρ_{n})(x) for every x ∈ Ω with dist(x, ∂Ω) > \frac{1}{n}. Then, by Fubini’s Theorem, we get

∫_{B(x_{0},δ)} h(z)|Eu_{n}(z)|dz ≤ ∫_{B(x_{0},δ)} ∫_{Ω} h(z)ρ_{n}(z − y)d|Eu|(y)dz
≤ ∫_{B(x_{0},δ+ \frac{1}{n})} ∫_{ℝ^{N}} h(z)ρ_{n}(z − y)dzd|Eu|(y)
= ∫_{B(x_{0},δ+ \frac{1}{n})}(h ∗ ρ_{n})(y)d|Eu|(y).

ii) Letting h ≡ 1 in part i) and since |Eu|(∂B(x_{0}, δ)) = 0, we obtain

lim sup_{n→∞} ∫_{B(x_{0},δ)} |Eu_{n}(x)|dx ≤ lim sup_{n→∞} ∫_{B(x_{0},δ+ \frac{1}{n})} d|Eu|(x)
= |Eu|(B(x_{0}, δ))
≤ lim inf_{n→∞} ∫_{B(x_{0},δ)} |Eu_{n}(x)|dx.

Thus

lim_{n→∞} ∫_{B(x_{0},δ)} |Eu_{n}(x)|dx = |Eu|(B(x_{0}, δ)),

and since Eu_{n} ⇀^{∗} Eu in the sense of measures in B(x_{0}, δ), the result now follows by applying Reshetnyak’s Theorem (see [15]) (note that the hypotheses on θ imply that 0 ≤ θ(w) ≤ C∥w∥).

The proof of part iii) is carried out exactly as in [3] since it is independent of the operator E.

Theorem 2.4. Assume that Ω is a bounded, open subset of ℝ^{N} with Lipschitz boundary Γ. There exists a linear, continuous, and surjective trace operator

tr : BD(Ω) → L^{1}(Γ, ℝ^{N})

such that tr u = u if u ∈ BD(Ω) ∩ C(\overline{Ω}, ℝ^{N}). This operator is also continuous with respect to the intermediate topology.

Furthermore, the following Gauss-Green formula holds

∫_{Ω}(u ⊙ Dϕ)(x)dx + ∫_{Ω} ϕ(x)dEu(x) = ∫_{Γ} ϕ(x)(tr u ⊙ ν)(x)dH^{N−1} (2.1)

for every ϕ ∈ C^{1}(\overline{Ω}).

Adapting the proof of Lemma II.2.2 in [18] it is easy to show the following proposition (see also [1] and [12]).

Proposition 2.5. If M is a countably (H^{N−1}, N − 1) rectifiable Borel subset of Ω and u ∈ BD(Ω) then

Eu⌊M = (u^{+} − u^{−}) ⊙ ν_{M}H^{N−1}⌊M,

where ν_{M} is a unit normal to M and u^{±} are the traces of u on the sides of M determined by ν_{M}.

The following result, proved in [18], asserts that it is possible to approximate a function u ∈ BD(Ω) by a sequence of smooth functions which preserve the trace of u.

Theorem 2.6. For every u ∈ BD(Ω) there exists a sequence of smooth functions {u_{n}}_{n∈ℕ} ⊂ C^{∞}(Ω, ℝ^{N}) ∩ W^{1,1}(Ω, ℝ^{N}) such that u_{n} →^{i} u and tr u_{n} = tr u. If, in addition, u ∈ LD(Ω), then ∥Eu_{n} − Eu∥_{L^{1}(Ω,M_{sym}^{N×N})} → 0.

A Poincaré-type inequality holds, precisely

Theorem 2.7. There exists a constant C = C(Ω) such that for every u ∈ BD(Ω) with tr u = 0

∥u∥_{L^{1}(Ω,ℝ^{N})} ≤ C(Ω)|Eu|(Ω). (2.2)

For the proof see [18], Remark II.2.5.

Note that the kernel of the operator E is the class R of rigid motions in ℝ^{N}, composed of affine maps of the form Mx + b where M is a skew-symmetric N × N matrix and b ∈ ℝ^{N}, and therefore is closed and finite-dimensional. Hence it is possible to define the orthogonal projection P : BD(Ω) → R. This operator belongs to the class considered in the following Poincaré-Friedrichs type inequality for BD functions (see [1], [12] and [18]).

Theorem 2.8. Let Ω be a bounded, connected, open set with Lipschitz boundary, and let R : BD(Ω) → R be a continuous linear map which leaves the elements of R fixed. Then there exists a constant C(Ω, R) such that

(2.3)  ∫_{Ω} |u − R(u)|dx ≤ C(Ω, R)|Eu|(Ω)  for every u ∈ BD(Ω).

Remark 2.9. Let C(Ω) denote the smallest constant for which (2.2), (2.3) hold. Then, as usual, a simple homothety argument shows that C(λΩ) = λC(Ω).

The following embedding result proved in [18], Theorem II.2.4, will be used in Sections 4 and 5 to obtain strong L^{1} convergence of a sequence which is uniformly bounded in BD(Ω).

Theorem 2.10. The space BD(Ω) is compactly embedded in L^{q}(Ω, ℝ^{N}) for every 1 ≤ q < \frac{N}{N−1}. In particular, if \{u_{n}\} is bounded in BD(Ω) then there exists a subsequence \{u_{n_{k}}\} such that u_{n_{k}} → u in L^{1}(Ω, ℝ^{N}) and u ∈ BD(Ω).

The next theorem was proved in [1] (see also [12]).

Theorem 2.11. For every u ∈ BD(Ω) the jump set J_{u} is a countably (H^{N−1}, N − 1) rectifiable Borel set.

This result, together with Proposition 2.5, yields the following decomposition of the Radon measure Eu (see Definition 4.1 and Remark 4.2 in [1]):

Eu = EuL^{N} + E^{s}u = EuL^{N} + ([u] ⊙ ν_{u})H^{N−1}⌊J_{u} + E^{c}u, (2.4)

where Eu is the density of the absolutely continuous part of Eu with respect to L^{N}, E^{s}u is the singular part, E^{c}u is the so-called Cantor part and vanishes on Borel sets B with H^{N−1}(B) < +∞ (see Proposition 4.4 in [1]).

Definition 2.12. (cf. Definition 4.6 in [1]) The space of special functions of bounded deformation, denoted by SBD(Ω), is the set of functions u ∈ BD(Ω) such that the measure E^{c}u in (2.4) is zero.

The next lemma will be used in Section 5 to identify the surface term of the relaxed energy F(u, Ω).

Lemma 2.13. (Lemma 2.6 [10]) Let u ∈ BD(Ω). Then for H^{N−1}-a.e. x_{0} ∈ J_{u}

lim_{ε→0} \frac{1}{ε^{N−1}} ∫_{J_{u}∩(x_{0}+εQ_{ν(x_{0})})} |([u] ⊙ ν_{u})(x)|dH^{N−1}(x) = |([u] ⊙ ν_{u})(x_{0})|. (2.5)

Its proof is based on the definition of rectifiable set and on the version of Lebesgue’s Differentiation Theorem proved by Ambrosio and Dal Maso [2] (see [10] for details).

3. Statement of the main result

Let f : M_{sym}^{N×N} → ℝ be a continuous function satisfying

(3.1) \frac{1}{C} |ξ| ≤ f(ξ) ≤ C(1 + |ξ|) for all ξ ∈ M_{sym}^{N×N} ,

and for some C > 0. Assume also that there exist Λ, L > 0, 0 ≤ β < 1 such that

(3.2) |f^{∞}(ξ) − \frac{f(tξ)}{t}| ≤ Λ\frac{|ξ|^{1−β}}{t^{β}} for all ξ ∈ M_{sym}^{N×N} and for all t such that t|ξ| > L ,

where the recession function, f^{∞}, of f is defined as

f^{∞}(ξ) := lim sup_{t→∞} \frac{f(tξ)}{t}.

Definition 3.1. The function f : M_{sym}^{N×N} → ℝ is said to be E-quasiconvex if, for every ξ ∈ M_{sym}^{N×N},

f(ξ) ≤ ∫_{Q} f(ξ + Eu(x))dx (3.3)

whenever u ∈ C_{per}^{∞}(Q, ℝ^{N}).

Note that the E-quasiconvexity property (3.3) is independent of the size, orientation, and centre of the cube Q. Moreover, it is equivalent to the following property: for every ξ ∈ M_{sym}^{N×N},

f(ξ)L^{N}(Ω) ≤ ∫_{Ω} f(ξ + Ev(x))dx

whenever v ∈ C_{0}^{∞}(Ω, ℝ^{N}).

Given a function f : M_{sym}^{N×N} → ℝ we define its E-quasiconvex envelope Q_{E}f by

Q_{E}f(ξ) := inf {∫_{Q} f(ξ + Eu(x))dx : u ∈ C_{per}^{∞}(Q, ℝ^{N})} . (3.4)

As it was proven for the usual quasiconvex envelope (see [8] Chapter 5) it is possible to show that Q_{E}f is the greatest E-quasiconvex function that is less than or equal to f. Moreover, the definition (3.4) does not depend on the domain, i.e.

Q_{E}f(ξ) = inf \{\frac{1}{L^{N}(Ω)} ∫_{Ω} f(ξ + Eu(x))dx : u ∈ C_{0}^{∞}(Ω, ℝ^{N})\} .

Remark 3.2. If f is upper semicontinuous and locally bounded from above, it is easy to see, using Theorem 2.6 and Fatou’s Lemma, that in the definition of E-quasiconvexity C_{per}^{∞}(Q, ℝ^{N}) may be replaced by u ∈ LD_{per}(Q).

Remark 3.3. Let f : M_{sym}^{N×N} → ℝ satisfy the growth and coercivity conditions (3.1). Then Q_{E}f also satisfies (3.1) and

\frac{1}{C} |ξ| ≤ f^{∞}(ξ) ≤ C|ξ|.

Moreover, if f is E-quasiconvex then so is f^{∞}.

The first statement is easily verified. As for the second one, we have for any u ∈ C_{per}^{∞}(Q, ℝ^{N})

f^{∞}(ξ) = lim sup_{t→∞} \frac{f(tξ)}{t} ≤ lim_{k→∞} \frac{1}{t_{k}} ∫_{Q} f(t_{k}ξ + t_{k}Eu(x))dx
≤ C lim_{k→∞} ∫_{Q} \frac{1 + t_{k}|ξ + Eu(x)|}{t_{k}}dx −

− lim inf_{k→∞} ∫_{Q} [\frac{C(1 + t_{k}|ξ + Eu(x)|)}{t_{k}} − \frac{1}{t_{k}} f(t_{k}ξ + t_{k}Eu(x))dx] .

By the growth condition (3.1), we may apply Fatou’s Lemma to obtain

f^{∞}(ξ) ≤ ∫_{Q} lim sup_{k→∞} \frac{1}{t_{k}} f(t_{k}(ξ + Eu(x)))dx ≤ ∫_{Q} f^{∞}(ξ + Eu(x))dx.

The following lemma will be used in Section 4.

Lemma 3.4. Let g : ℝ^{d} → ℝ be a continuous function satisfying |g(v)| ≤ C(1+|v|) for some constant C > 0. If u_{n} → u_{0} in L^{1}(Ω, ℝ^{d}), with u_{0} ∈ L^{∞}(Ω, ℝ^{d}), and sup_{n} ∥v_{n}∥_{∞} < +∞, then

lim_{n→+∞} ∫_{Ω} |g(u_{n} + v_{n}) − g(u_{0} + v_{n})|dx = 0.

Proof. We divide the given integral into two terms

\{ \begin{matrix}J_{1} := ∫_{\{x∈Ω : |u_{n}(x)−u_{0}(x)|>1\}} |g(u_{n} + v_{n}) − g(u_{0} + v_{n})|dx , \\ J_{2} := ∫_{\{x∈Ω : |u_{n}(x)−u_{0}(x)|≤1\}} |g(u_{n} + v_{n}) − g(u_{0} + v_{n})|dx. \end{matrix}

The growth condition on g, together with the hypotheses on u_{n} and v_{n}, ensure that the sequence {|g(u_{n} + v_{n})(·) − g(u_{0} + v_{n})(·)|}_{n∈ℕ} is equi-integrable.

Thus, since u_{n} → u_{0} strongly in L^{1}(Ω, ℝ^{d}), the measure of the set {x ∈ Ω : |u_{n}(x) − u_{0}(x)| > 1} can be made arbitrarily small and thus for any ε > 0 there exists an n_{0} such that J_{1} < ε/2 for every n ≥ n_{0}.

On the other hand, using the uniform continuity of g on the closed ball of radius ∥u_{0}∥_{∞} + 1 + sup_{n} ∥v_{n}∥_{∞} centered in 0, it follows that J_{2} < ε/2 for every n ≥ n_{0}.

Let u ∈ BD(Ω) and define

F (u, Ω) := \{ \begin{matrix}∫_{Ω} f(Eu(x))dx & if u ∈ W^{1,1}(Ω, ℝ^{N}), \\ +∞ & otherwise. \end{matrix}

We consider the functional defined for every V ∈ O(Ω) by

F(u, V) := inf_{{u_{n}}} {lim inf_{n→∞} ∫_{V} f(Eu_{n}(x))dx | u_{n} ∈ W^{1,1}(V, ℝ^{N}), u_{n} → u in L^{1}(V, ℝ^{N})} .

The main result of this paper is the following integral representation for the relaxed energy F(u, Ω) of F (u, Ω) when u ∈ SBD(Ω) (see also Proposition 3.9).

Theorem 3.5. If f : M_{sym}^{N×N} → ℝ is a continuous function satisfying (3.1) and (3.2), then for every u ∈ SBD(Ω) we have

F(u, V) = ∫_{V} Q_{E}f(Eu(x))dx + ∫_{J_{u}∩V} (Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x))dH^{N−1}(x) ,

for every open subset V of Ω.

Remark 3.6. This integral representation does not provide a full description of the functional since we consider only the case where the Cantor part of the deformation tensor is zero. To extend this representation result to the whole space BD a more complete characterization of the Cantor part of the measure Eu is needed.

In this section we shall establish some properties of the relaxed functional, while the remaining sections will be devoted to the characterization of the volume and surface terms, respectively.

We shall use the following version of De Giorgi’s Slicing Lemma (see also [4] and [7]) which allows us to modify a sequence on the boundary without increasing the energy.

Proposition 3.7. Let V ⊂ ℝ^{N} be an open, bounded set and let f : M_{sym}^{N×N} → ℝ satisfy the growth condition (3.1). Let {u_{n}}, {v_{n}} be sequences of functions in BD(V) such that

u_{n} − v_{n} → 0 in L^{1}(V, ℝ^{N}), sup_{n} |Eu_{n}|(V) < +∞ and |Ev_{n}| ⇀^{∗} μ, |Ev_{n}|(V) → μ(V). Then

there exist a subsequence {v_{n_{k}}} and a sequence {w_{k}} ⊂ BD(V) such that w_{k} = v_{n_{k}} near the boundary of V, w_{k} − v_{n_{k}} → 0 in L^{1}(V, ℝ^{N}), and

lim sup_{k→∞} ∫_{V} f(Ew_{k}(x))dx ≤ lim inf_{n→∞} ∫_{V} f(Eu_{n}(x))dx.

Proof. Without loss of generality we may assume that

lim inf_{n→∞} ∫_{V} f(Eu_{n}(x))dx = lim_{n→∞} ∫_{V} f(Eu_{n}(x))dx < +∞.

Then, by (3.1), there exists a bounded Radon measure λ such that

|Eu_{n}| + |Ev_{n}| ⇀^{∗} λ.

Choose r_{k} ↗ ∞ as k → ∞ and ε_{j}^{k} ↘ 0 as j → ∞ such that

λ (\{x ∈ V | dist (x, ∂V) = \frac{1}{r_{k}} \}) = 0 ,
μ (\{x ∈ V | dist (x, ∂V) = \frac{1}{r_{k}+ε_{j}^{k}} \}) = 0
λ (\{x ∈ V | dist (x, ∂V) = \frac{1}{r_{k}±ε_{j}^{k}} \}) = 0.

(3.5)

Let ϕ_{k,j} be a smooth cut-off function such that 0 ≤ ϕ_{k,j} ≤ 1, ϕ_{k,j}(x) = 1 if x ∈ V and dist (x, ∂V) > \frac{1}{r_{k}−ε_{j}^{k}}, and ϕ_{k,j}(x) = 0 if x ∈ V and dist (x, ∂V) < \frac{1}{r_{k}+ε_{j}^{k}}, and define

w_{n,k,j} := ϕ_{k,j}u_{n} + (1 − ϕ_{k,j})v_{n}.

Then w_{n,k,j} ∈ BD(V), w_{n,k,j} = v_{n} near the boundary of V, and

∥w_{n,k,j} − v_{n}∥_{L^{1}(V,ℝ^{N})} ≤ ∥u_{n} − v_{n}∥_{L^{1}(V,ℝ^{N})}.

On the other hand, as Ew_{n,k,j} = ϕ_{k,j}Eu_{n} + (1 − ϕ_{k,j})Ev_{n} + (u_{n} − v_{n}) ⊙ Dϕ_{k,j}, from the growth condition on f it follows that

∫_{V} f(Ew_{n,k,j}(x))dx
≤ ∫_{V} f(Eu_{n}(x))dx + ∫_{V_{k,j}} f(Ev_{n}(x))dx + ∫_{L_{k,j}} f(Ew_{n,k,j}(x))dx
≤ ∫_{V} f(Eu_{n}(x))dx + CL^{N} (V_{k,j}) + C|Ev_{n}| (V_{k,j})
+C(L^{N}(L_{k,j}) + (|Eu_{n}| + |Ev_{n}|)(L_{k,j}) + ∥Dϕ_{k,j}∥_{∞}∥u_{n} − v_{n}∥_{L^{1}(V,ℝ^{N})})

where

L_{k,j} := \{x ∈ V | \frac{1}{r_{k} + ε_{j}^{k}} < dist (x, ∂V) < \frac{1}{r_{k} − ε_{j}^{k}} \} ,

V_{k,j} := \{x ∈ V | dist (x, ∂V) < \frac{1}{r_{k} + ε_{j}^{k}} \} .

Thus, by the strong convergence of u_{n} − v_{n} to zero in L^{1}(V, ℝ^{N}) and (3.5),

lim sup_{k→∞} lim sup_{j→∞} lim sup_{n→∞} ∫_{V} f(Ew_{n,k,j}(x))dx

≤ lim_{n→∞} ∫_{V} f(Eu_{n}(x))dx + C lim sup_{k→∞} lim sup_{j→∞} L^{N} \binom{}{Ṽ_{k,j}}

+C lim sup_{k→∞} lim sup_{j→∞} μ (V_{k,j}) + C lim sup_{k→∞} lim sup_{j→∞} λ(L_{k,j})

≤ lim_{n→∞} ∫_{V} f(Eu_{n}(x))dx,

where

Ṽ_{k,j} := \{x ∈ V | dist (x, ∂V) < \frac{1}{r_{k} − ε_{j}^{k}} \} .

By a standard diagonal argument, we may choose a subsequence n_{k} → ∞ and a sequence ε_{j(k)}^{k} → 0 such that w_{k} := w_{n_{k},k,ε_{j(k)}^{k}} satisfies the required properties.

Remark 3.8. If the sequences {u_{n}}, {v_{n}} are more regular, e.g. if u_{n}, v_{n} ∈ W^{1,1}(V, ℝ^{N}) or u_{n}, v_{n} ∈ C^{∞}(V, ℝ^{N}), then so is the sequence {w_{k}}.

The localized relaxed functional has the following properties

Proposition 3.9. Under hypotheses (3.1) and (3.2)

1. F(·, V) is L^{1}(V, ℝ^{N}) lower semicontinuous;

2. for every u ∈ BD(Ω),

\frac{1}{C} |Eu|(V) ≤ F(u, V) ≤ C(L^{N}(V) + |Eu|(V)); (3.6)

3. F(u, ·) is the restriction to O(Ω) of a Radon measure.

We use the same notation for F(u, ·) and its extension to the Borel subsets of Ω.

Proof. The proof of 1) follows from a standard diagonal argument, whereas 2) is an immediate consequence of the growth conditions on f, of the lower semicontinuity of the total variation of Radon measures and of Theorem 2.6.

To prove 3), we begin by showing that for every u ∈ BD(Ω) and every U, V, W ∈ O(Ω) with W ⊂⊂ V ⊂⊂ U

F(u, U) ≤ F(u, V) + F(u, U \ \overline{W}). (3.7)

Indeed, let v_{n} ∈ W^{1,1}(V, ℝ^{N}) and w_{n} ∈ W^{1,1}(U \\overline{W} , ℝ^{N}) be such that v_{n} → u in L^{1}(V, ℝ^{N}), w_{n} → u in L^{1}(U \ \overline{W} , ℝ^{N}),

F(u, V) = lim_{n→∞} ∫_{V} f(Ev_{n}(x))dx and F(u, U \ \overline{W}) = lim_{n→∞} ∫_{U\\overline{W}} f(Ew_{n}(x))dx .

Let V_{0} ∈ O_{∞}(Ω) be such that W ⊂⊂ V_{0} ⊂⊂ V and |Eu|(∂V_{0}) = 0. Applying Proposition 3.7 and Remark 3.8 to \{v_{n}\} and u in V_{0}, we obtain a sequence \{\overline{v}_{n}\} ⊂ W^{1,1}(V_{0}, ℝ^{N}) such that \overline{v}_{n} = u near the boundary of V_{0}, \overline{v}_{n} → u in L^{1}(V_{0}, ℝ^{N}) and

lim sup_{n→∞} ∫_{V_{0}} f(E\overline{v}_{n}(x))dx ≤ lim inf_{n→∞} ∫_{V_{0}} f(Ev_{n}(x))dx.

By Proposition 3.7 and Remark 3.8, now applied to \{w_{n}\} and u in U \ \overline{V}_{0}, there exists a sequence \{\overline{w}_{n}\} ⊂ W^{1,1}(U \ \overline{V}_{0}, ℝ^{N}) such that \overline{w}_{n} = u in a neighbourhood of ∂V_{0}, \overline{w}_{n} → u in L^{1}(U \ \overline{V}_{0}, ℝ^{N}) and

lim sup_{n→∞} ∫_{U\\overline{V}_{0}} f(E\overline{w}_{n}(x))dx ≤ lim inf_{n→∞} ∫_{U\\overline{V}_{0}} f(Ew_{n}(x))dx.

Define

z_{n} := \{ \begin{matrix}\overline{v}_{n} & in V_{0}, \\ \overline{w}_{n} & in U \ V_{0}. \end{matrix}

Then z_{n} ∈ W^{1,1}(U, ℝ^{N}), z_{n} → u in L^{1}(U, ℝ^{N}), and

F(u, U) ≤ lim inf_{n→∞} ∫_{U} f(Ez_{n}(x))dx
≤ lim sup_{n→∞} ∫_{V_{0}} f(E\overline{v}_{n}(x))dx + lim sup_{n→∞} ∫_{U\\overline{V}_{0}} f(E\overline{w}_{n}(x))dx
≤ lim inf_{n→∞} ∫_{V_{0}} f(Ev_{n}(x))dx + lim inf_{n→∞} ∫_{U\\overline{V}_{0}} f(Ew_{n}(x))dx
≤ lim inf_{n→∞} ∫_{V} f(Ev_{n}(x))dx + lim inf_{n→∞} ∫_{U\\overline{W}} f(Ew_{n}(x))dx
= F(u, V) + F(u, U \ \overline{W}).

Now let u_{n} ∈ W^{1,1}(Ω, ℝ^{N}) be such that u_{n} → u in L^{1}(Ω, ℝ^{N}) and

F(u, Ω) = lim_{n→∞} ∫_{Ω} f(Eu_{n}(x))dx. (3.8)

Since {f(Eu_{n})(·)}_{n∈ℕ} is bounded in L^{1}(Ω), it follows that there exists a bounded Radon measure μ on \overline{Ω} such that for a subsequence (not relabelled) f(Eu_{n})χ_{Ω}L^{N} ⇀^{∗} μ in the sense of measures and so

F(u, Ω) = μ(\overline{Ω}). (3.9)

By the definition of F(u, ·), for every U ∈ O(Ω)

F(u, U) ≤ lim inf_{n→∞} ∫_{U} f(Eu_{n}(x))dx ≤ μ(\overline{U}). (3.10)

Let now V ∈ O(Ω) and ε > 0 be fixed, and consider W ∈ O(Ω) with W ⊂⊂ V and μ(V \ W) < ε. Then

μ(V) ≤ μ(W) + ε = μ(\overline{Ω}) − μ(\overline{Ω} \ W) + ε.

By (3.9), applying (3.10) with U = Ω \ \overline{W}, and then (3.7) for U = Ω, we get

μ(V) ≤ F(u, Ω) − μ(\overline{Ω} \ W) + ε ≤ F(u, Ω) − F(u, Ω \ \overline{W}) + ε ≤ F(u, V) + ε.

Letting ε → 0^{+} we obtain μ(V) ≤ F(u, V) ≤ μ(\overline{V}) for every V ∈ O(Ω). It remains to show that F(u, U) ≤ μ(U) for every U ∈ O(Ω). In order to do this, we fix again an ε > 0 and choose V, W ∈ O(Ω) such that W ⊂⊂ V ⊂⊂ U and L^{N}(U \ \overline{W}) + |Eu|(U \ \overline{W}) < ε. By (3.6), (3.7), and (3.10), we get

F(u, U) ≤ F(u, U \ \overline{W}) + F(u, V)
≤ C(L^{N}(U \ \overline{W}) + |Eu|(U \ \overline{W})) + F(u, V) ≤ ε + μ(\overline{V}) ≤ ε + μ(U).

To conclude it suffices to let ε → 0^{+}.

In the sequel, given u ∈ BD(Ω) and V ∈ O_{∞}(Ω) we use the notation tr u or u_{|∂V} to indicate the trace of u on the boundary of V. For any u ∈ BD(Ω) and V ∈ O_{∞}(Ω) define

m(u, V) := inf {F(v, V) | v_{|∂V} = u_{|∂V} , v ∈ BD(Ω)}.

In order to identify the surface term of the relaxed energy we will follow the general method for relaxation introduced by Bouchitté, Fonseca and Mascarenhas [7]. The idea is to compare the asymptotic behaviours of m(u, Q(x_{0}, ε)) and F(u, Q(x_{0}, ε)) as ε → 0^{+}, and to show, via a blow-up argument, that relaxation reduces to solving a Dirichlet problem (see Lemma 3.12).

The following three lemmas are entirely similar to Lemmas 3.1, 3.3 and 3.5 in [7]; for completeness of the presentation we include their proofs here.

Lemma 3.10. There exists a positive constant C such that for any u_{1}, u_{2} ∈ BD(Ω) and any V ∈ O_{∞}(Ω) we have

|m(u_{1}, V) − m(u_{2}, V)| ≤ C ∫_{∂V} |tr (u_{1} − u_{2})(x)|dH^{N−1}(x).

Proof. Given δ > 0 consider the set V_{δ} := {x ∈ V | dist (x, ∂V) > δ} and let v ∈ BD(Ω) be such that v = u_{2} on ∂V. Define v_{δ} ∈ BD(Ω) by v_{δ} := v in V_{δ} and v_{δ} := u_{1} in Ω \ V_{δ}. Then, by the definition of m, and by virtue of the additivity and locality of F,

m(u_{1}, V) ≤ F(v_{δ}, V) = F(v, V_{δ}) + F(v_{δ}, V \ V_{δ}). (3.11)

By (3.6),

F(v_{δ}, V \ V_{δ}) ≤ C (L^{N}(V \ V_{δ}) + |Ev_{δ}|(V \ V_{δ}))
(3.12)  = C (L^{N}(V \ V_{δ}) + |Eu_{1}|(V \ \overline{V_{δ}}) + |E^{s}v_{δ}|(∂V_{δ})) .

The first two terms in (3.12) tend to zero as δ → 0, and as for the third one we have

|E^{s}(Dv_{δ})|(∂V_{δ}) = ∫_{∂V_{δ}} |((v_{|∂V_{δ}} − u_{1}_{|∂V_{δ}}) ⊙ ν_{δ})(x)|dH^{N−1}(x)
(3.13)  → ∫_{∂V} |((tr u_{2} − tr u_{1}) ⊙ ν)(x)|dH^{N−1}(x) as δ → 0.

Therefore from (3.11), (3.12) and (3.13) it follows that

m(u_{1}, V) ≤ F(v, V) + C ∫_{∂V} |(tr (u_{2} − u_{1}) ⊙ ν)(x)|dH^{N−1}(x).

Taking the infimum over v and using the fact that |(tr (u_{2} − u_{1}) ⊙ ν| ≤ C|tr (u_{2} − u_{1})|, we obtain

m(u_{1}, V) − m(u_{2}, V) ≤ C ∫_{∂V} |tr (u_{1} − u_{2})(x)|dH^{N−1}(x).

Interchanging the roles of u_{1} and u_{2} the proof is concluded.

As in [7], fix u ∈ BD(Ω), ν ∈ S^{N−1}, and define μ := L^{N} + |E^{s}u|. Let

O^{∗} := {Q_{ν}(x, ε) | x ∈ Ω, ν ∈ S^{N−1}, ε > 0} ,

and for δ > 0, V ∈ O(Ω), set

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

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

Since δ → m^{δ}(u, V) is a decreasing function, we define

m^{∗}(u, V) := sup {m^{δ}(u, V) : δ > 0} = lim_{δ→0^{+}} m^{δ}(u, V).

Lemma 3.11. (Lemma 3.3 [7]) Let u ∈ BD(Ω). Under conditions 1), 2), 3) of Proposition 3.9, for any V ∈ O(Ω)

F(u, V) = m^{∗}(u, V).

Proof. Since F(u, ·) is a Radon measure and m(u, V) ≤ F(u, V), the inequality m^{∗}(u, V) ≤ F(u, V) is clear. To show the reverse inequality, fix δ > 0 and let (Q_{i}^{δ}) be an admissible sequence for m^{δ}(u, V) such that

∑_{i=1}^{∞} m(u, Q_{i}^{δ}) < m^{δ}(u, V) + δ. (3.14)

Let now v_{i}^{δ} ∈ BD(Ω) be such that v_{i}^{δ}_{|∂Q_{i}^{δ}} = u_{|∂Q_{i}^{δ}} and

F(v_{i}^{δ}, Q_{i}^{δ}) ≤ m(u, Q_{i}^{δ}) + δL^{N}(Q_{i}^{δ}). (3.15)

Define

v^{δ} := ∑_{i=1}^{∞} v_{i}^{δ}χ_{Q_{i}^{δ}} + uχ_{N_{0}^{δ}},

where N_{0}^{δ} = Ω \ ∪_{i=1}^{∞}Q_{i}^{δ}. Clearly v^{δ} ∈ L^{1}(Ω, ℝ^{N}). On the other hand, given ϕ ∈ C_{0}^{∞}(Ω), using the fact that v^{δ} − u = 0 in N_{0}^{δ}, and integrating by parts (see (2.1)) over the cubes Q_{i}^{δ} we get

< E(v^{δ} − u), ϕ > = ⟨E (∑_{i=1}^{∞} χ_{Q_{i}^{δ}}(v^{δ} − u)) , ϕ⟩

= ∑_{i=1}^{∞} ⟨E (χ_{Q_{i}^{δ}}(v^{δ} − u)) , ϕ⟩

= ∑_{i=1}^{∞} ∫_{∂Ω} ϕtr (χ_{Q_{i}^{δ}}(v^{δ} − u)) ⊙ νdH^{N−1}

− ∑_{i=1}^{∞} ∫_{Ω} χ_{Q_{i}^{δ}}(v^{δ} − u) ⊙ Dϕdx

= − ∑_{i=1}^{∞} ∫_{Q_{i}^{δ}} v^{δ} ⊙ Dϕdx + ∑_{i=1}^{∞} ∫_{Q_{i}^{δ}} u ⊙ Dϕdx

= − ∑_{i=1}^{∞} ∫_{∂Q_{i}^{δ}} ϕ(x)tr v_{i}^{δ} ⊙ νdH^{N−1} + ∑_{i=1}^{∞} ∫_{Q_{i}^{δ}} ϕdEv_{i}^{δ}

+ ∑_{i=1}^{∞} ∫_{∂Q_{i}^{δ}} ϕ(x)tr v_{i}^{δ} ⊙ νdH^{N−1} − ∑_{i=1}^{∞} ∫_{Q_{i}^{δ}} ϕdEu

= ∑_{i=1}^{∞} ⟨Ev_{i}^{δ}⌊Q_{i}^{δ}, ϕ⟩ − ⟨Eu⌊(Ω \ N_{0}^{δ}), ϕ⟩ .

Hence

Ev^{δ} = ∑_{i=1}^{∞} Ev_{i}^{δ}⌊Q_{i}^{δ} + Eu⌊N_{0}^{δ} (3.16)

in the sense of distributions, and due to the lower bound in (3.6) and (3.14), (3.15), the right-hand side is a bounded Radon measure on Ω, and we conclude that v^{δ} ∈ BD(Ω).

By the choice of Q_{i}^{δ}, setting N^{δ} := V \ ∪_{i=1}^{∞}Q_{i}^{δ} we have μ(N^{δ}) = 0, and by (3.16) |Ev^{δ}|(N^{δ}) = |Eu|(N^{δ}) ≤ μ(N^{δ}) = 0. Thus, by (3.6),

F(v^{δ}, N^{δ}) ≤ C(L^{N}(N^{δ}) + |Ev^{δ}|(N^{δ})) = 0.

Since F(v^{δ}, ·) is the restriction of a measure, by (3.14) and (3.15) it follows that

F(v^{δ}, V) = ∑_{i=1}^{∞} F(v_{i}^{δ}, Q_{i}^{δ}) + F(v^{δ}, N^{δ})

(3.17)  ≤ ∑_{i=1}^{∞} m(u, Q_{i}^{δ}) + δL^{N}(V) ≤ m^{δ}(u, V) + δ + δL^{N}(V).

The result is now a consequence of the lower semicontinuity of F(·, V) provided we can show that {v^{δ}} converges to u strongly in L^{1}(V, ℝ^{N}). Indeed, if this is the case, then

F(u, V) ≤ lim inf_{δ→0} F(v^{δ}, V) ≤ lim inf_{δ→0} m^{δ}(u, V) = m^{∗}(u, V).

Since v_{i}^{δ} = u on ∂Q_{i}^{δ}, by Poincaré’s Inequality (2.2) and Remark 2.9, there exists a constant C > 0 such that

∥v_{i}^{δ} − u∥_{L^{1}(Q_{i}^{δ},ℝ^{N})} ≤ Cδ|E(v_{i}^{δ} − u)|(Q_{i}^{δ}).

Hence

∥v^{δ} − u∥_{L^{1}(V,ℝ^{N})} = ∑_{i=1}^{∞} ∥v_{i}^{δ} − u∥_{L^{1}(Q_{i}^{δ},ℝ^{N})} ≤ Cδ|E(v_{i}^{δ} − u)|(∪_{i=1}^{∞}Q_{i}^{δ})
≤ Cδ(|Ev^{δ}|(V) + |Eu|(V)).

Since the sequence {|Ev^{δ}|(V)}_{δ} is bounded by the coercivity condition (3.6) and (3.17), it follows that {v^{δ}} converges to u strongly in L^{1}(V, ℝ^{N}).

Lemma 3.12. (Lemma 3.5 [7]) If F satisfies conditions 1)-3) of Proposition 3.9 then

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

The proof is the same as in [7] since it does not depend on whether we are considering the distributional derivative D or its symmetrized part E.

Remark 3.13. The conclusion of this lemma still holds replacing Q_{ν}(x_{0}, ε) by U(x_{0}, ε) := x_{0} + εU, where U is any bounded, open, convex subset of ℝ^{N} containing the origin.

The following result shows that in the definition of the Dirichlet functional m one may consider more regular functions.

Lemma 3.14. If the function f satisfies (3.1), then for every u ∈ BD(Ω) and every V ∈ O_{∞}(Ω)

m(u, V) = m_{0}(u, V) := inf {F (v, V) : v = u on ∂V, v ∈ W^{1,1}(Ω, ℝ^{N})}.

Proof. Clearly m_{0}(u, V) ≥ m(u, V). To show the reverse inequality, fix ε > 0 and let v ∈ BD(Ω) be such that v = u on ∂V and m(u, V) ≥ F(v, V) − ε. Consider a sequence u_{n} ∈ W^{1,1}(Ω, ℝ^{N}) such that u_{n} → v in L^{1}(Ω, ℝ^{N}),

F(v, V) = lim_{n→∞} ∫_{V} f(Eu_{n}(x))dx ,

and, using Theorem 2.6, let v_{n} ∈ W^{1,1}(Ω, ℝ^{N}) satisfy v_{n} → v in L^{1}(Ω, ℝ^{N}), v_{n} = v on ∂V − thus v_{n} = u on ∂V − and |Ev_{n}|(V) → |Ev|(V). It suffices now to apply Proposition 3.7 to {u_{n}} and {v_{n}}, and, in light of Remark 3.8, there exists a sequence {w_{k}} of functions in W^{1,1}(Ω, ℝ^{N}) such that w_{k} = u on ∂V and

m_{0}(u, V) ≤ lim sup_{k→∞} ∫_{V} f(Ew_{k}(x))dx ≤ lim inf_{n→∞} ∫_{V} f(Eu_{n}(x))dx ≤ m(u, V) + ε,

and we conclude by letting ε ↘ 0^{+}.

4. The volume term

Proposition 4.1. If u ∈ BD(Ω) then

F(u, Ω) ≥ ∫_{Ω} Q_{E}f(Eu(x))dx,

where Eu = EuL^{N} + E^{s}u.

Proof. Let u_{n} ∈ W^{1,1}(Ω, ℝ^{N}) be such that u_{n} → u in L^{1}(Ω, ℝ^{N}) and

F(u, Ω) = lim_{n→∞} ∫_{Ω} f(Eu_{n}(x))dx < +∞.

By passing to a subsequence (not relabelled), there exists a finite non-negative Radon measure λ such that

f(Eu_{n})L^{N} ⇀^{∗} λ in the sense of measures.

Thus, it suffices to show that for L^{N}-a.e. x_{0} ∈ Ω

\frac{dλ}{dL^{N}} (x_{0}) ≥ Q_{E}f(Eu(x_{0})).

Consider a point x_{0} ∈ Ω such that

(4.1) \frac{dλ}{dL^{N}} (x_{0}) = lim_{ε→0} \frac{λ(Q(x_{0}, ε))}{ε^{N}} exists and is finite,

lim_{ε→0} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} |Eu(x) − Eu(x_{0})|dx = 0, (4.2)

lim_{ε→0} \frac{1}{ε^{N}} |E^{s}u|(Q(x_{0}, ε)) = 0, (4.3)

where the sequence of ε ↘ 0^{+} is chosen such that λ(∂Q(x_{0}, ε)) = 0. It is well-known that properties (4.1)-(4.3) hold for L^{N}-almost every x_{0} ∈ Ω.

Then

\frac{dλ}{dL^{N}} (x_{0}) = lim_{ε→0} lim_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f(Eu_{n}(x))dx
(4.4)  = lim_{ε→0} lim_{n→∞} ∫_{Q} f(Eu_{n,ε}(y))dy

where

u_{n,ε}(y) := \frac{u_{n}(x_{0} + εy) − u(x_{0})}{ε}.

By (4.1), (4.4) and the coercivity hypothesis on f, we obtain

sup_{n,ε} |Eu_{n,ε}|(Q) < +∞ (4.5)

and so, applying Theorem 2.8,

∥u_{n,ε} − P u_{n,ε}∥_{L^{1}(Q,ℝ^{N})} ≤ C,

where P is the projection of BD(Q) onto the kernel of the operator E. Theorem 2.10, together with (4.5) and the fact that EP u_{n,ε} = 0, implies the existence of a function v ∈ L^{1}(Q, ℝ^{N}) such that

lim_{ε→0} lim_{n→∞} ∥u_{n,ε} − P u_{n,ε} − v∥_{L^{1}(Q,ℝ^{N})} = 0, (4.6)

and this, in turn, entails

(4.7) Eu_{n,ε} ⇀^{∗} Ev in the sense of distributions.

On the other hand, given ϕ ∈ C_{0}(Q), from the strong convergence in L^{1} of u_{n} to u and the convergence of Eu_{n} to Eu in the sense of measures, we deduce that

lim_{ε→0} lim_{n→∞} ∫_{Q} ϕ(y)Eu_{n,ε}(y)dy = lim_{ε→0} lim_{n→∞} ∫_{Q} ϕ(y)Eu_{n}(x_{0} + εy)dy
= lim_{ε→0} lim_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)}ϕ (\frac{x − x_{0}}{ε} ) Eu_{n}(x)dx
= lim_{ε→0} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)}ϕ (\frac{x − x_{0}}{ε} ) dEu(x)
= lim_{ε→0} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} ϕ (\frac{x − x_{0}}{ε} ) Eu(x)dx

+ lim_{ε→0} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} ϕ (\frac{x − x_{0}}{ε} ) dE^{s}u(x)

= lim_{ε→0} ∫_{Q} ϕ(y) [Eu(x_{0} + εy) − Eu(x_{0})] dy

+ ∫_{Q} ϕ(y)Eu(x_{0})dy

+ lim_{ε→0} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} ϕ (\frac{x − x_{0}}{ε} ) dE^{s}u(x)

= ∫_{Q} ϕ(y)Eu(x_{0})dy

by (4.2) and (4.3). We have thus showed that Eu_{n,ε} ⇀^{∗} Eu(x_{0})L^{N} in the sense of measures, which, together with (4.7), yields

Ev = Eu(x_{0})L^{N}. (4.8)

Extracting a diagonal subsequence \{\overline{u}_{m}\} of \{u_{n,ε} − P u_{n,ε}\}, according to (4.6), (4.7) and (4.8), we then have, by (4.4)

\frac{dλ}{dL^{N}} (x_{0}) ≥ lim_{m→∞} ∫_{Q} f(E\overline{u}_{m}(x))dx, \overline{u}_{m} → v in L^{1}(Q, ℝ^{N}).

By virtue of Proposition 3.7, without loss of generality, we may assume that \overline{u}_{m} = v on ∂Q. Hence by the definition of Q_{E}f, Remark 3.2 and (4.8), we have

\frac{dλ}{dL^{N}} (x_{0}) ≥ lim_{m→∞} ∫_{Q} f(E\overline{u}_{m}(x))dx
= lim inf_{m→∞} ∫_{Q} f(Eu(x_{0}) + E(\overline{u}_{m} − v)(x))dx
≥ Q_{E}f(Eu(x_{0})).

Proposition 4.2. Let u ∈ BD(Ω). Then for L^{N}-almost every x_{0} ∈ Ω

\frac{dF(u, ·)}{dL^{N}}(x_{0}) ≤ Q_{E}f(Eu(x_{0})).

Proof. Choose a point x_{0} ∈ Ω such that (4.2) and (4.3) are satisfied, and

\frac{dF(u, ·)}{dL^{N}}(x_{0}) = lim_{ε→0} \frac{F(u, Q(x_{0}, ε))}{ε^{N}} exists and is finite,

where the sequence of ε ↘ 0^{+} is chosen such that |Eu|(∂Q(x_{0}, ε)) = 0.

Let ϕ_{n} ∈ C_{per}^{∞}(Q, ℝ^{N}) be such that

∫_{Q} f(Eu(x_{0}) + Eϕ_{n}(x))dx ≤ Q_{E}f(Eu(x_{0})) + \frac{1}{n}. (4.9)

Extend ϕ_{n} to ℝ^{N} by periodicity and define

ϕ_{n}^{m}(x) := \frac{1}{m}ϕ_{n}(mx).

Then Eϕ_{n}^{m}(x) = Eϕ_{n}(mx) so,

(4.10) Eϕ_{n}^{m} ⇀ ∫_{Q} Eϕ_{n}(x)dx = 0 in L^{1}(Q, M_{sym}^{N×N}) as m → ∞

and

|Eϕ_{n}^{m}|(Q) = ∫_{Q} |Eϕ_{n}(mx)|dx
= \frac{1}{m^{N}} ∫_{mQ} |Eϕ_{n}(y)|dy
= |Eϕ_{n}|(Q).

By (4.9) and the coercivity hypothesis on f, it follows that |Eϕ_{n}|(Q) is uniformly bounded. Applying Theorem 2.8, Theorem 2.10 and the argument in the previous proof, passing to a diagonal subsequence, we obtain

ϕ_{n}^{m(n)} − P ϕ_{n}^{m(n)} → ϕ in L^{1}(Q, ℝ^{N})

and Eϕ_{n}^{m(n)} ⇀^{∗} Eϕ in the sense of measures, which, together with (4.10), yields Eϕ = 0 and thus ϕ is a rigid motion. Define

\{ \begin{matrix}\overline{ϕ}_{n}(x) := ϕ_{n}^{m(n)}(x) − P ϕ_{n}^{m(n)}(x), \\ \overline{u}_{n}^{ε}(x) := (ρ_{n} ∗ u)(x) + ε [\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} ) − ϕ (\frac{x − x_{0}}{ε} )] . \end{matrix}

Then, as n → ∞, \overline{u}_{n}^{ε} → u in L^{1}(Q(x_{0}, ε), ℝ^{N}) so that \overline{u}_{n}^{ε} ∈ W^{1,1}(Q(x_{0}, ε), ℝ^{N}) is admissible for F(u, Q(x_{0}, ε)). Hence we obtain, taking into account that Eϕ = 0,

\frac{dF(u, ·)}{dL^{N}}(x_{0}) = lim_{ε→0} \frac{F(u, Q(x_{0}, ε))}{ε^{N}}
≤ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f (E(u ∗ ρ_{n})(x) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} )) dx
≤ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} f (Eu(x_{0}) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} )) dx

+ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} [f (E(u ∗ ρ_{n})(x) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} ))

−f (Eu(x_{0}) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} ))] dx

=: I_{1} + I_{2}

where, by changing variables and using the periodicity of ϕ_{n} and (4.9),

I_{1} = lim inf_{n→∞} ∫_{Q} f(Eu(x_{0}) + E\overline{ϕ}_{n}(y))dy = Q_{E}f(Eu(x_{0})).

On the other hand, by Lemma 2.3 and setting u_{0}(x) := Eu(x_{0})x,

lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} |E(u ∗ ρ_{n})(x) − Eu(x_{0})|dx

= lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε)} |E((u − u_{0}) ∗ ρ_{n})(x)|dx
≤ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε+ \frac{1}{n})} d|E(u − u_{0})|(x)
≤ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε+ \frac{1}{n})} |Eu(x) − Eu(x_{0})|dx

+ lim sup_{ε→0} lim inf_{n→∞} \frac{1}{ε^{N}} ∫_{Q(x_{0},ε+ \frac{1}{n})} d|E^{s}u|(x)

= lim sup_{ε→0} \frac{1}{ε^{N}} ∫_{\overline{Q}(x_{0},ε)} |Eu(x) − Eu(x_{0})|dx

+ lim sup_{ε→0} \frac{1}{ε^{N}} |E^{s}u|(\overline{Q}(x_{0}, ε)) = 0

where we have used (4.2), the fact that ε ↘ 0^{+} were chosen such that |Eu|(∂Q(x_{0}, ε)) = 0, and (4.3). This, together with Lemma 3.4 proves that I_{2} = 0.

5. The surface term

Proposition 5.1. Let u ∈ SBD(Ω). Then, under hypotheses (3.1), (3.2),

F(u, V ∩ J_{u}) = ∫_{V ∩J_{u}} g(x, u^{+}(x), u^{−}(x), ν_{u}(x))dH^{N−1}(x) for all V ∈ O(Ω),

where g(x, λ, θ, ν) := lim sup_{ε→0} \frac{m(u_{λ,θ,ν}, Q_{ν}(x, ε))}{ε^{N−1}}

and u_{λ,θ,ν}(y) = \{ \begin{matrix}λ & if y · ν > 0 \\ θ & if y · ν < 0. \end{matrix}

Proof. Since u^{+}, u^{−} are Borel functions and J_{u} is rectifiable, writing ν := ν_{u}(x_{0}), for H^{N−1} almost every x_{0} ∈ J_{u}

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν}^{+}(x_{0},ε)} |u(x) − u^{+}(x_{0})| dx = 0, (5.1)

lim_{ε→0^{+}} \frac{1}{ε^{N}} ∫_{Q_{ν}^{−}(x_{0},ε)} |u(x) − u^{−}(x_{0})| dx = 0, (5.2)

\frac{dF(u, ·)}{d|[u] ⊙ ν_{u}|H^{N−1}⌊J_{u}} (x_{0}) = lim_{ε→0^{+}} \frac{F(u, Q_{ν}(x_{0}, ε))}{|[u] ⊙ ν_{u}|H^{N−1}⌊J_{u}(Q_{ν}(x_{0}, ε))} (5.3)

exists and is finite.

Choose one such point x_{0} ∈ J_{u}. Let α := |[u] ⊙ ν_{u}|H^{N−1}⌊J_{u} and set μ := L^{N} + α. By Lemma 3.12 the limit in (5.3) is equal to

lim_{ε→0^{+}} \frac{m(u, Q_{ν}(x_{0}, ε))}{α(Q_{ν}(x_{0}, ε))} for α a.e. x_{0} ∈ Ω.

Recall that

m(u, V) = inf {F(v, V) | v_{|∂V} = u_{|∂V} , v ∈ BD(Ω)} .

By Lemma 2.13 the point x_{0} can be chosen to satisfy also

lim_{ε→0^{+}} \frac{1}{ε^{N−1}} |Eu|(Q_{ν}(x_{0}, ε)) = lim_{ε→0^{+}} \frac{1}{ε^{N−1}} α(Q_{ν}(x_{0}, ε)) = |([u] ⊙ ν_{u})(x_{0})|. (5.4)

Let u_{ε} : Q_{ν} → ℝ^{N} be defined by u_{ε}(y) := u(x_{0} + εy). Then, by (5.1) and (5.2) u_{ε} → \overline{u}_{x_{0},ν} strongly in L^{1}(Q_{ν}, ℝ^{N}), where

\overline{u}_{x_{0},ν}(y) := \{ \begin{matrix}u^{+}(x_{0}) & if y · ν > 0, \\ u^{−}(x_{0}) & if y · ν < 0. \end{matrix}

Also, from Lemma 2.13 it follows that

|Eu_{ε}|(Q_{ν}) = \frac{1}{ε^{N−1}} |Eu|(Q_{ν}(x_{0}, ε)) ⟶ |([u] ⊙ ν)(x_{0})| = |E\overline{u}_{x_{0},ν}|(Q_{ν}) ,

so that u_{ε} → \overline{u}_{x_{0},ν} in the intermediate topology of BD(Q_{ν}), and since the trace operator is continuous with respect to this topology (cf. Theorem 2.4), we deduce that

(5.5) lim_{ε→0} \frac{1}{ε^{N−1}} ∫_{∂Q_{ν}(x_{0},ε)} |tr (u − \overline{u}_{x_{0},ν})| dH^{N−1} = lim_{ε→0} ∫_{∂Q_{ν}} |tr (u_{ε} − \overline{u}_{x_{0},ν})| dH^{N−1} = 0.

We have, by (5.3), Lemma 3.10 and (5.4), (5.5),

\frac{dF(u, ·)}{dα}(x_{0}) = lim_{ε→0^{+}} \frac{F(u, Q_{ν}(x_{0}, ε))}{α(Q_{ν}(x_{0}, ε))}
= lim_{ε→0^{+}} \frac{m(u, Q_{ν}(x_{0}, ε))}{α(Q_{ν}(x_{0}, ε))}
= lim_{ε→0^{+}} \frac{m(u, Q_{ν}(x_{0}, ε)) − m(\overline{u}_{x_{0},ν}, Q_{ν}(x_{0}, ε)) + m(\overline{u}_{x_{0},ν}, Q_{ν}(x_{0}, ε))}{ε^{N−1}}

·\frac{ε^{N−1}}{α(Q_{ν}(x_{0}, ε))}

= lim_{ε→0^{+}} \frac{m(\overline{u}_{x_{0},ν}, Q_{ν}(x_{0}, ε))}{ε^{N−1}}(|([u] ⊙ ν)(x_{0})|)^{−1}.

Hence

F(u, V ∩ J_{u}) = ∫_{V ∩J_{u}} \frac{dF(u, ·)}{dα}(x)dα(x) = ∫_{V ∩J_{u}} g(x, u^{+}(x), u^{−}(x), ν(x))dH^{N−1}.

We will show now that the surface energy density may be more explicitly characterized.

Proposition 5.2. If (3.1), (3.2) hold and if u ∈ SBD(Ω), then for H^{N−1}-almost every x_{0} ∈ J_{u}

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) = (Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x_{0})).

Proof. By Lemma 3.14, for H^{N−1}-almost every x_{0} ∈ J_{u} the function g is given by

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν) = lim sup_{ε→0} \frac{1}{ε^{N−1}} inf \{ ∫_{Q_{ν}(x_{0},ε)} f(Ew(x))dx

w ∈ W^{1,1}(Ω, ℝ^{N}), w = \overline{u}_{x_{0},ν} on ∂Q_{ν}(x_{0}, ε)\}

where ν := ν_{u}(x_{0}). Hence, for any k ∈ ℕ there exists a function w_{k,ε}, depending also on ε, such that, setting \overline{w}_{k,ε}(y) := w_{k,ε}(x_{0} + εy),

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν) + \frac{1}{k} ≥ lim sup_{ε→0} \frac{1}{ε^{N−1}} ∫_{Q_{ν}(x_{0},ε)} f(Ew_{k,ε}(x))dx
= lim sup_{ε→0} ∫_{Q_{ν}} εf (\frac{1}{ε}E\overline{w}_{k,ε}(y)) dy
(5.6)  ≥ lim sup_{ε→0} ∫_{Q_{ν}} εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) dy,

and

∫Q_{ν} εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) dy
= ∫_{Q_{ν}} [εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) − (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))] dy + ∫_{Q_{ν}} (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))dy
=: I_{1} + I_{2}.

Using hypothesis (3.2), the growth condition of (Q_{E}f)^{∞}, and Hölder’s inequality, it follows that

|I_{1}| ≤ ∫_{Q_{ν}∩\{|E\overline{w}_{k,ε}|≤Lε\}} |εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) − (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))| dy

+ ∫_{Q_{ν}∩\{|E\overline{w}_{k,ε}|>Lε\}} |εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) − (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))| dy

≤ ∫_{Q_{ν}∩\{|E\overline{w}_{k,ε}|≤Lε\}} [εC (1 + \frac{1}{ε}|E\overline{w}_{k,ε}(y)|) + C|E\overline{w}_{k,ε}(y)|] dy

+ ∫_{Q_{ν}} Λε^{β}|E\overline{w}_{k,ε}(y)|^{1−β}dy

≤ Cε + CLε + Λε^{β} (∫_{Q_{ν}} |E\overline{w}_{k,ε}(y)|dy)^{1−β} .

By the coercivity of Q_{E}f (cf. Remark 3.3) and (5.6) it follows that

C ≥ ε ∫_{Q_{ν}} Q_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) dy ≥ ∫_{Q_{ν}} |E\overline{w}_{k,ε}(y)|dy,

where C is independent of ε, and thus

∫_{Q_{ν}} εQ_{E}f (\frac{1}{ε}E\overline{w}_{k,ε}(y)) dy = O(ε) + ∫_{Q_{ν}} (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))dy. (5.7)

Set

z_{k,ε}(y) := \overline{w}_{k,ε}(y) − [([u] ⊗ ν_{u})(x_{0})y + \frac{u^{−}(x_{0}) + u^{+}(x_{0})}{2}] .

Then z_{k,ε} ∈ W_{per}^{1,1}(Q_{ν}, ℝ^{N}) so using the E-quasiconvexity of (Q_{E}f)^{∞} (cf. Remark 3.3) we obtain

(Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x_{0})) ≤ ∫_{Q_{ν}} (Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x_{0}) + Ez_{k,ε}(y))dy
(5.8)  = ∫_{Q_{ν}} (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))dy.

From (5.6), (5.7) and (5.8) we conclude that

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν_{u}(x_{0})) + \frac{1}{k} ≥ lim sup_{ε→0} (O(ε) + ∫_{Q_{ν}} (Q_{E}f)^{∞}(E\overline{w}_{k,ε}(y))dy)
≥ (Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x_{0})).

It suffices to let k → ∞.

To show the reverse inequality, in view of Remark 3.13, consider the parallelepiped

Q_{k,ε}(x_{0}) := \{x ∈ ℝ^{N} | |(x − x_{0}) · ν_{i}| < \frac{εk}{2} , |(x − x_{0}) · ν| < \frac{ε}{2}, i = 1, …, N − 1\} ,

where {ν_{1}, …, ν_{N−1}, ν} is an orthonormal basis of ℝ^{N}, and define, for x ∈ Q_{k,ε}(x_{0}),

w_{ε}(x) := \frac{1}{ε}([u] ⊗ ν)(x_{0})(x − x_{0}) + \frac{u^{+}(x_{0}) + u^{−}(x_{0})}{2}.

Then w_{ε} has the same trace as \overline{u}_{x_{0},ν} when (x − x_{0}) · ν = ±\frac{ε}{2} but not on the remaining lateral faces of the boundary of Q_{k,ε}(x_{0}). In order to obtain an admissible sequence for m(\overline{u}_{x_{0},ν}, Q_{k,ε}(x_{0})), we consider a smooth function φ_{k,ε} ∈ C^{∞}(Q_{k,ε}(x_{0})), depending only on x · ν_{i}, i = 1, …, N − 1, such that 0 ≤ φ_{k,ε} ≤ 1, φ_{k,ε}(x) = 1 if x ∈ Q_{k−1,ε}(x_{0}), φ_{k,ε}(x) = 0 if |(x − x_{0}) · ν_{i}| = \frac{εk}{2} and ∥Dφ_{k,ε}∥_{∞} = O(\frac{1}{ε}). Thus φ_{k,ε}w_{ε} + (1 − φ_{k,ε})\overline{u}_{x_{0},ν} has the same trace as \overline{u}_{x_{0},ν} on ∂Q_{k,ε}(x_{0}), and so,

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν) = lim sup_{k→∞} lim sup_{ε→0} \frac{m(\overline{u}_{x_{0},ν}, Q_{k,ε}(x_{0})}{(kε)^{N−1}}
(5.9)  ≤ lim sup_{k→∞} lim sup_{ε→0} \frac{F(φ_{k,ε}w_{ε} + (1 − φ_{k,ε})\overline{u}_{x_{0},ν}, Q_{k,ε}(x_{0})}{(kε)^{N−1}}.

By definition of E-quasiconvexity, let ϕ_{n} ∈ C_{per}^{∞}(Q_{k,1}(0), ℝ^{N}) be such that

∫_{Q_{k,1}(0)}f (\frac{1}{ε}([u] ⊙ ν_{u})(x_{0}) + Eϕ_{n}(x)) dx ≤ k^{N−1}(Q_{E}f) (\frac{1}{ε}([u] ⊙ ν_{u})(x_{0})) + \frac{k^{N−1}}{n} . (5.10)

As in the proof of Proposition 4.2, we may find a sequence \overline{ϕ}_{n} ∈ C_{per}^{∞}(Q_{k,1}(0), ℝ^{N}) associated to ϕ_{n}, and a function ϕ, such that Eϕ = 0, |Eϕ_{n}|(Q_{k,1}(0)) = |E\overline{ϕ}_{n}|(Q_{k,1}(0)) and

\overline{ϕ}_{n} → ϕ in L^{1}(Q_{k,1}(0), ℝ^{N}).

By Theorem 2.6, let u_{n} ∈ C^{∞}(Q_{k,ε}(x_{0}), ℝ^{N}) ∩ W^{1,1}(Q_{k,ε}(x_{0}), ℝ^{N}) be such that u_{n} →^{i} \overline{u}_{x_{0},ν}, tr u_{n} = tr \overline{u}_{x_{0},ν} on ∂Q_{k,ε}(x_{0}) and define

u_{k,ε,n}(x) := (φ_{k,ε}w_{ε} + (1 − φ_{k,ε})u_{n})(x) + ε [\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} ) − ϕ (\frac{x − x_{0}}{ε} )] .

Then u_{k,ε,n} → φ_{k,ε}w_{ε}+(1−φ_{k,ε})\overline{u}_{x_{0},ν} strongly in L^{1}(Q_{k,ε}(x_{0}), ℝ^{N}) as n → ∞ and u_{k,ε,n} ∈ W^{1,1}(Q_{k,ε}(x_{0}), ℝ^{N}). Hence u_{k,ε,n} is admissible for F(φ_{k,ε}w_{ε} +(1−φ_{k,ε})\overline{u}_{x_{0},ν}, Q_{k,ε}(x_{0})) and therefore, as

Eu_{k,ε,n}(x) = φ_{k,ε}(x)(Ew_{ε} − Eu_{n})(x) + Eu_{n}(x) + ((w_{ε} − u_{n}) ⊙ Dφ_{k,ε})(x) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} ) ,

it follows from (5.9) that

g(x_{0}, u^{+}(x_{0}), u^{−}(x_{0}), ν) ≤ lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} lim inf_{n→∞} ∫_{Q_{k,ε}(x_{0})} f(Eu_{k,ε,n}(x))dx

≤ lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} lim inf_{n→∞} ∫_{Q_{k−1,ε}(x_{0})} f (Ew_{ε}(x) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} )) dx

+ lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}}

· lim inf_{n→∞} ∫_{L_{k,ε}} C (1 + |Ew_{ε}|(x) + |Eu_{n}|(x) + \frac{1}{ε}|w_{ε} − u_{n}|(x) + |E\overline{ϕ}_{n}| (\frac{x − x_{0}}{ε} )) dx

=: I_{1} + I_{2},

where L_{k,ε} := Q_{k,ε}(x_{0}) \ Q_{k−1,ε}(x_{0}). By (5.10)

I_{1} ≤ lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} lim inf_{n→∞} ∫_{Q_{k,ε}(x_{0})} f (\frac{1}{ε}([u] ⊙ ν_{u})(x_{0}) + E\overline{ϕ}_{n} (\frac{x − x_{0}}{ε} )) dx
= lim sup_{k→∞} lim sup_{ε→0} \frac{ε}{k^{N−1}} lim inf_{n→∞} ∫_{Q_{k,1}(0)} f (\frac{1}{ε}([u] ⊙ ν_{u})(x_{0}) + E\overline{ϕ}_{n}(y)) dy
≤ lim sup_{ε→0} lim inf_{n→∞} (ε(Q_{E}f) (\frac{1}{ε}([u] ⊙ ν_{u})(x_{0})) + \frac{ε}{n}) = (Q_{E}f)^{∞}(([u] ⊙ ν_{u})(x_{0})).

On the other hand, as L^{N}(L_{k,ε}) = O(ε^{N}k^{N−2}),

\frac{1}{(kε)^{N−1}} ∫_{L_{k,ε}} C (1 + \frac{1}{ε}|([u] ⊙ ν_{u})(x_{0})|) dx = O (\frac{ε}{k} ) + O (\frac{1}{k} ) .

By the coercivity condition in (3.1) and (5.10), |Eϕ_{n}|(Q_{k,1}(0)) is uniformly bounded in n, thus

lim sup_{ε→0} \frac{1}{(kε)^{N−1}} lim sup_{n→∞} ∫_{L_{k,ε}} |E\overline{ϕ}_{n}| (\frac{x − x_{0}}{ε} ) dx

≤ lim sup_{ε→0} \frac{ε^{N}}{(kε)^{N−1}} lim sup_{n→∞} ∫_{Q_{k,1}(0)} |E\overline{ϕ}_{n}|(y)dy
= lim sup_{ε→0} \frac{ε}{k^{N−1}} lim sup_{n→∞} ∫_{Q_{k,1}(0)} |Eϕ_{n}|(y)dy = 0.

From the convergence of u_{n} to \overline{u}_{x_{0},ν} in the intermediate topology and the fact that

([u] ⊙ ν_{u})(x_{0})H^{N−1}⌊J_{\overline{u}_{x_{0},ν}} (∂L_{k,ε}) = 0,

we deduce that

lim_{n→∞} ∫_{L_{k,ε}} |Eu_{n}|(x)dx = ∫_{L_{k,ε}∩J_{\overline{u}_{x_{0},ν}}} |([u] ⊙ ν_{u})(x_{0})|dH^{N−1}(x);

also

lim_{n→∞} ∫_{L_{k,ε}} |w_{ε} − u_{n}|(x)dx = ∫_{L_{k,ε}} |w_{ε} − \overline{u}_{x_{0},ν}|(x)dx,

with

lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} |([u] ⊙ ν_{u})(x_{0})|H^{N−1}(L_{k,ε} ∩ J_{\overline{u}_{x_{0},ν}})
= lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} O(ε^{N−1}k^{N−2}) = 0,

and

lim sup_{k→∞} lim sup_{ε→0} \frac{1}{(kε)^{N−1}} ∫_{L_{k,ε}} \frac{1}{ε}|w_{ε} − \overline{u}_{x_{0},ν}|(x)dx = 0,

since |(x − x_{0}) · ν| < \frac{ε}{2} in L_{k,ε}, which implies that |w_{ε} − \overline{u}_{x_{0},ν}| ≤ C. Thus we conclude that I_{2} = 0.

Acknowledgements. The authors wish to thank G. Dal Maso for many useful discussions concerning the subject of this paper. The research of Ana Cristina Barroso was partially supported by FCT, PRAXIS XXI and project PRAXIS 2/2.1/MAT/125/94. The research of Irene Fonseca was partially supported by the National Science Foundation under Grants No. DMS-9500531 and DMS-9731957 and through the Center for Nonlinear Analysis (CNA). This research was done while Rodica Toader was a Postdoctoral Research Fellow at the CMAF-University of Lisbon. The financial support of CMAF-UL, the Fundação para a Ciência e Tecnologia and PRAXIS XXI is gratefully acknowledged.

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(Ana Cristina Barroso) CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal

E-mail address, A. C. Barroso: abarroso@lmc.fc.ul.pt

(Irene Fonseca) Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

E-mail address, I. Fonseca: fonseca@andrew.cmu.edu

(Rodica Toader) CMAF, Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal

E-mail address, R. Toader: rodica@lmc.fc.ul.pt