Brittle Thin Films

Andrea Braides

SISSA Via Beirut 4

34014 Trieste, Italy

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

Abstract A 2-dimensional model for brittle thin films is obtained from a 3-dimensional fracture model for elastic material. Γ-convergence techniques are used to identify the limiting effective energy as the thickness of the sample approaches zero.

1991 Mathematics subject classification (Amer. Math. Soc.): 35E99, 35M10, 49J45, 73C50, 73M25 Key Words : Γ-limit, thin films, functions of bounded variation, fracture mechanics

Introduction

Elastic thin films may be modeled through a limit procedure by considering threedimensional structures with vanishing thickness ε. In the framework of nonlinear hyperelasticity, the elastic energy associated to these structures is of the form

∫_{Ω_{ε}} W (∇v) dx,

where Ω_{ε} := ω × (−ε/2, ε/2) represents the reference configuration, ω is an open subset of ℝ^{2}, ε > 0, and v ∈ W^{1,p}(Ω_{ε}; ℝ^{3}) is a deformation of the body. In order to describe the limiting behavior of these functionals, and upon a change of variables, it is more convenient to study the limit of the family of scaled energies

∫_{Ω} W (∇_{α}u|\frac{1}{ε}∇_{3}u) dx,

where Ω := Ω_{1} = ω × (−1/2, 1/2), ∇_{α} denotes differentiation with respect to the variable x_{α} = (x_{1}, x_{2}), thus reducing the analysis to the fixed reference configuration Ω and to u ∈ W^{1,p}(Ω; ℝ^{3}). It was shown in [30] that, under appropriate hypotheses on the bulk energy density W, the limit energy is of the form

∫_{ω} Q_{2}\overline{W} (∇_{α}u) dx_{α},

where u ∈ W^{1,p}(ω; ℝ^{3}), \overline{W} (\overline{F}) := min\{W (\overline{F} |F_{3}) : F_{3} ∈ ℝ^{3}\}, \overline{F} is a 3 × 2 matrix, and Q_{2} denotes the operation of quasiconvexification on 3 × 2 matrices. It is worth noting that, unless W is convex, the quasiconvexification operation is essential in the description of the limiting energy, and it underlines the formation of microstructure.

Here we take into account the possibility of fracture in a thin film, and, accordingly, we add to the formulation above a crack initiation energy. Precisely, we consider

∫_{Ω_{ε}\K} W (∇v) dx + ∫_{K∩Ω_{ε}} ϑ(v^{+} − v^{−}, ν(K)) dH^{2},

where K denotes an unknown crack surface, H^{2}(K) is the 2-dimensional Hausdorff measure of K, ν(K) is the normal to the crack surface in the reference configuration, v ∈ W^{1,p}(Ω_{ε} \ K; ℝ^{3}) is the deformation, defined in the complement of the crack, and v^{±} are the traces of u on both sides of K. The surface energy density ϑ is a fracture initiation term defined on ℝ^{3} ×S^{2}. In Griffith’s theory [29] ϑ is constant; we have added a possible dependence on ν(K) to account for anisotropy, and a dependence on the jump u^{+} − u^{−} to include Barenblatt’s approach [11]. Again, by performing a change of variables, and extending ϑ to ℝ^{3} × ℝ^{3} as a homogeneous function of degree 1 in the variable ν, we arrive at the re-scaled energy

∫_{Ω\K_{ε}} W (∇_{α}u|\frac{1}{ε}∇_{3}u) dx + ∫_{K_{ε}∩Ω} ϑ(u^{+} − u^{−}, ν_{α}(K_{ε}), \frac{1}{ε}ν_{3}(K_{ε})) dH^{2}. (0.1)

It is well known that free discontinuity problems formulated in terms of variable domains Ω \ K_{ε} and unknown admissible field u are very difficult to handle, and often it is more convenient to set up the problem within the De Giorgi and Ambrosio framework of special functions of bounded variation SBV (Ω; ℝ^{3}) (see [7]), by replacing K_{ε} by the set S(u) of essential discontinuity points for u and interpreting ∇u as an approximate gradient, thus reducing the simultaneous variation in the crack site and in the deformation to a variation on the discontinuous function u only.

In this paper we show that it is possible to characterize the limit of the family of energies (0.1) as

∫_{Ω} Q_{2}\overline{W} (∇_{α}u) dx + ∫_{S(u)∩Ω} R_{2}\overline{ϑ}(u^{+} − u^{−}, ν_{α}(u)) dH^{2}(x),  (0.2)

where the admissible deformations u belong to SBV (Ω; ℝ^{3}), satisfy ∇_{3}u = 0 and ν_{3}(u) = 0, and where W has superlinear growth (see Theorem 2.1). The effective surface energy density R_{2}\overline{ϑ} is obtained first by minimizing ϑ in ν_{3}, and then considering the BV -elliptic envelope on S^{1} of the resulting function. We recall that BV -ellipticity is the counterpart of quasiconvexity for surface energies, and for whose properties we refer to [5, 6, 2, 19]. The domain of the limit energy (0.2) is isomorphically contained in SBV (ω; ℝ^{3}) and (0.2) may be written equivalently as

∫_{ω} Q_{2}\overline{W} (∇_{α}u) dx_{α} + ∫_{S(u)∩ω} R_{2}\overline{ϑ}(u^{+} − u^{−}, ν_{α}(u)) dH^{1}(x_{α}),

for u ∈ SBV (ω; ℝ^{3}). Again, and as in [30], we recover a model where the admissible fields are restricted to the middle plate ω, although here they may have discontinuities due to the presence fracture.

In Theorem 3.2 we provide an integral representation for thin film relaxed energy in the case where the bulk energy density W grows linearly at infinity. As it is usual, here there is an interaction between bulk and interfacial energies.

1 Notation and Preliminaries

In the sequel 𝕄^{m×N} stands for the space of m × N matrices. Given a matrix F ∈ 𝕄^{3×3}, and following the notation introduced in [30], we write F =: (\overline{F} |F_{3}), where F_{i} denotes the i-th column of F, 1 ≤ i ≤ 3, and \overline{F} := (F_{1}F_{2}) ∈ 𝕄^{3×2} is the matrix of the first two columns of F. If x ∈ ℝ^{3} then x_{α} := (x_{1}, x_{2}) ∈ ℝ^{2} is the vector of the first two components of x. The N-dimensional Lebesgue measure in ℝ^{N} is designated as L^{N}, and when there is no possibility of confusion we may abbreviate |E| := L^{N}(E). We designate by C a positive constant whose value within one formula may vary from on line to the next.

1.1 Γ-convergence

We recall the definition of De Giorgi’s Γ-convergence in L^{p} spaces, 1 ≤ p < +∞: given a family of functionals J_{j} : L^{p}(Ω; ℝ^{m}) → [0, +∞), j ∈ ℕ, for u ∈ L^{p}(Ω; ℝ^{m}) we define

Γ(L^{p})- lim inf_{j→+∞} J_{j}(u) := inf {lim inf_{j→+∞} J_{j}(u_{j}) : u_{j} → u in L^{p}(Ω; ℝ^{m})} ,

and

Γ(L^{p})- lim sup_{j→+∞} J_{j}(u) := inf {lim sup_{j→+∞} J_{j}(u_{j}) : u_{j} → u in L^{p}(Ω; ℝ^{m})} .

If these two quantities coincide then their common value is called the Γ-limit of the sequence (J_{j}) at u, and is denoted by Γ(L^{p})- lim_{j→+∞} J_{j}(u).

We say that (J_{ε}) Γ(L^{p})-converges to l at u as ε → 0 if for every sequence of positive numbers (ε_{j}) converging to 0^{+} there exists a subsequence (ε_{j_{k}}) for which

l = Γ(L^{p})- lim_{k→+∞} J_{ε_{j_{k}}} (u).

We recall that the Γ-upper and lower limits defined above are L^{p}-lower semicontinuous functions.

For a comprehensive study of Γ-convergence we refer to the book of Dal Maso [23] (see also [18] Part II).

1.2 Special functions of bounded variation

Let Ω be an open subset of ℝ^{N}, and let M(Ω) be the set of all signed Radon measures on Ω with bounded total variation. We say that u ∈ L^{1}(Ω; ℝ^{m}) is a function of bounded variation, and we write u ∈ BV (Ω; ℝ^{m}), if every its distributional first derivatives D_{i}u_{j} belong to M(Ω). We denote by Du the 𝕄^{m×N}-valued measure whose entries are D_{i}u_{j}. S(u) is the complement of the Lebesgue set of u; i.e., x ∉ S(u) if and only if

lim_{ρ→0^{+}} ρ^{−N} ∫_{B_{ρ}(x)} |u(y) − z| dy = 0

for some z ∈ ℝ^{m}. If u ∈ L^{1}(Ω; ℝ^{m}) then the set S(u) is Lebesgue-negligible, and if, in addition, u ∈ BV (Ω; ℝ^{m}) then the Hausdorff dimension of S(u) is at most N − 1; precisely, S(u) is rectifiable in that there is a countable sequence of C^{1} hypersurfaces Γ_{i} which covers H^{N−1}-almost all of S(u); i.e., H^{N−1}(S(u) \ ⋃_{i=1}^{∞} Γ_{i}) = 0. Moreover, for H^{N−1}-almost every x ∈ S(u) it is possible to find a, b ∈ ℝ^{m} and ν ∈ S^{N−1} such that

lim_{ρ→0^{+}} ρ^{−N} ∫_{B_{ρ}^{ν}(x)} |u(y) − a| dy = 0,  lim_{ρ→0^{+}} ρ^{−N} ∫_{B_{ρ}^{−ν}(x)} |u(y) − b| dy = 0,

where B_{ρ}^{ν}(x) := {y ∈ B_{ρ}(x) : ⟨y − x, ν⟩ > 0}. The triplet (a, b, ν) is uniquely determined up to a change of sign of ν and an interchange between a and b, and in the sequel it will be denoted by (u^{+}, u^{−}, ν(u)).

The distributional derivative of a function u ∈ BV (Ω; ℝ^{m}) admits the decomposition

Du = ∇u L^{N} Ω + J_{u} + C_{u},

where ∇u L^{N} is the Lebesgue part of Du,

J_{u} := (u^{+} − u^{−}) ⊗ ν(u) H^{N−1} S(u)

is the Hausdorff part or jump part of Du, and C_{u} is the Cantor part of Du. We recall that the measure C_{u} is singular with respect to the Lebesgue measure and it is ‘diffuse’; i.e., C_{u}(S) = 0 for every set S with H^{N−1}(S) < +∞. For the general exposition of the theory of functions of bounded variation we refer the reader to Federer [27], Evans and Gariepy [26], Ambrosio, Fusco and Pallara [8] and Ziemer [32]. A quick introduction can be found in [15] Section 1.

We say that a function u ∈ BV (Ω; ℝ^{m}) is a special function of bounded variation if C_{u} ≡ 0, or, equivalently, if

Du = ∇u L_{N} Ω + (u^{+} − u^{−}) ⊗ ν(u) H^{N−1} S(u).

We denote the space of the special functions of bounded variation by SBV (Ω; ℝ^{m}). The introduction of this space is due to De Giorgi and Ambrosio [24]. GSBV (Ω) is defined as the space of scalar functions in L^{1}(Ω) such that for all T > 0 the truncations u_{T} := (−T)∧(u∨T) belong to SBV (Ω). We say that u ∈ GSBV (Ω; ℝ^{m}) if every component of u is in GSBV (Ω). For the properties of functions in SBV (Ω; ℝ^{m}) and GSBV (Ω; ℝ^{m}) we refer to the book by Ambrosio, Fusco and Pallara [8].

Let p > 1. SBV_{p}(Ω; ℝ^{m}) and GSBV_{p}(Ω; ℝ^{m}) are defined as the subspaces of functions u of SBV (Ω; ℝ^{m}) and GSBV (Ω; ℝ^{m}), respectively, such that

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

These spaces are natural domains for the treatment of energies with bulk and surface contributions in the case where the bulk energy density grows superlinearly at infinity.

We recall that a continuous function W : 𝕄^{m×N} → ℝ is said to be quasiconvex if affine functions are local minimizers for the corresponding volume integral functional on Sobolev functions with respect to periodic perturbations; precisely, if W_{#}^{1,∞}((0, 1)^{N}; ℝ^{m}) denotes the space of 1-periodic φ ∈ W^{1,∞}(ℝ^{N}; ℝ^{m}) then

W (F) ≤ ∫_{(0,1)^{N}} W (F + ∇φ(x)) dx

for all F ∈ 𝕄^{m×N} and φ ∈ W_{#}^{1,∞}((0, 1)^{N}; ℝ^{m}). The quasiconvex envelope QW of W : 𝕄^{m×N} → ℝ is the greatest quasiconvex function not larger than W. We have

QW (F) = inf {∫_{(0,1)^{N}} W (F + Dφ(x)) dx : φ ∈ W_{#}^{1,∞}((0, 1)^{n}; ℝ^{m})}

for all F ∈ 𝕄^{m×N} (see e.g. [10], [22]).

Similarly, a continuous function ϑ : ℝ^{m} × S^{N−1} → ℝ is said to be BVelliptic if functions with an hyperplane as jump set are local minimizers for the corresponding surface integral functional among piecewise constant functions with respect to periodic perturbations; precisely, if Q_{ν} is a unit cube in ℝ^{N} with center 0 and one face orthogonal to ν and GSBV_{#}(Q_{ν}; ℝ^{m}, z) denotes the space of functions u ∈ GSBV_{∞}(Q_{ν}; ℝ^{m}) which are 1-periodic in the direction orthogonal to ν, with ∇u = 0 a.e., and such that u(x) = ±z/2 if ⟨x, ν⟩ = ±1/2, then

ϑ(z, ν) ≤ ∫_{S(u)∩Q_{ν}} ϑ(u^{+} − u^{−}, ν(u)) dH^{N−1}

for all z ∈ ℝ^{m}, ν ∈ S^{N−1} and for all u ∈ GSBV_{#}(Q_{ν}; ℝ^{m}, z). The BV -elliptic envelope Rϑ of a continuous function ϑ : ℝ^{m}×S^{N−1} → ℝ is the largest BV -elliptic function not larger than ϑ. We have

Rϑ(z, ν) := inf {∫_{S(u)∩Q_{ν}} ϑ(u^{+} − u^{−}, ν(u)) dH^{N−1} : u ∈ GSBV_{#}(Q_{ν}; ℝ^{m}, z)} for all z ∈ ℝ^{m} and ν ∈ S^{N−1} (see [6], [19]).

The following closure and lower semicontinuity theorem holds on GSBV_{p}(Ω) (see [1], [8]).

Theorem 1.1 Let W be a quasiconvex function, let ϑ be a BV -elliptic function, with

0 ≤ W (F) ≤ C(1 + |F |^{p}), 0 ≤ ϑ(z, ν) ≤ C(1 + |z|)

for some C > 0, and for all F ∈ 𝕄^{m×N}, z ∈ ℝ^{m}, and ν ∈ S^{N−1}. If (u_{j}) is a sequence in GSBV_{p}(Ω; ℝ^{m}) with u_{j} → u in L^{1}(Ω; ℝ^{m}), and if

sup_{j}{∫_{Ω} |∇u_{j}|^{p} dx + H^{N−1}(S(u_{j}) ∩ Ω)} < +∞,  (1.1)

then u ∈ GSBV_{p}(Ω; ℝ^{m}),

∫_{S(u)} ϑ(u^{+} − u^{−}, ν(u)) dH^{N−1} ≤ lim inf_{j} ∫_{S(u_{j})} ϑ(u_{j}^{+} − u_{j}^{−}, ν(u_{j})) dH^{N−1},

and

∫_{Ω} W (∇u) dx ≤ lim inf_{j} ∫_{Ω} W (∇u_{j}) dx.

Definition 1.2 We say that a sequence (u_{j}) in GSBV_{p}(Ω; ℝ^{m}) converges weakly in GSBV_{p}(Ω; ℝ^{m}) to a function u ∈ GSBV_{p}(Ω; ℝ^{m}) if u_{j} → u in L^{1}(Ω; ℝ^{m}), and (1.1) holds.

In view of this definition, we may say that Theorem 1.1 provides a closure result for weak convergence in GSBV_{p}(Ω; ℝ^{m}).

The following relaxation theorem may be found in [19].

Theorem 1.3 [Relaxation in GSBV_{p}(Ω; ℝ^{m})] If W and ϑ are Borel functions satisfying the growth conditions

|F |^{p} ≤ W (F) ≤ C(1 + |F |^{p}) 1 + |z| ≤ ϑ(z, ν) ≤ C(1 + |z|)

for all F ∈ 𝕄^{m×N}, z ∈ ℝ^{m}, ν ∈ S^{n−1}, and for some C > 0, then the lower semicontinuous envelope of the functional

u ↦ ∫_{Ω} W (∇u) dx + ∫_{S(u)} ϑ(u^{+} − u^{−}, ν(u)) dH^{N−1}

on SBV_{p}(Ω; ℝ^{m}) with respect to the L^{1}(Ω; ℝ^{m}) convergence is given by

u ↦ ∫_{Ω} QW (∇u) dx + ∫_{S(u)} Rϑ(u^{+} − u^{−}, ν(u)) dH^{N−1}.

2 Brittle thin films

Let ω be a bounded open subset of ℝ^{2}, let Ω_{ε} := ω × (−ε/2, ε/2), Ω := Ω_{1} = ω ×(−1/2, 1/2), and fix p > 1. Let W : 𝕄^{3×3} → [0, +∞) and ϑ : (ℝ^{3} \{0})×S^{2} → [0, +∞) denote bulk and surface energy density, respectively. The ‘3D-energy’ of a deformation v ∈ GSBV_{p}(Ω_{ε}; ℝ^{3}) of the thin film occupying Ω_{ε} as a reference configuration is given by

∫_{Ω_{ε}} W (∇v) dx + ∫_{S(v)∩Ω_{ε}} ϑ(v^{+} − v^{−}, ν(v)) dH^{2}

for functions v ∈ GSBV_{p}(Ω_{ε}; ℝ^{3}). Note that there is a 1-1 correspondence between rank-1-matrices in 𝕄^{3×3} and (ℝ^{3} \ {0}) × S^{2}, so that we could as well consider the surface energy as defined on matrices by θ(z ⊗ ν) = ϑ(z, ν), so as to have a symmetric notation with that of the bulk part. However, we prefer to adopt the above notation, as it better describes the decoupled dependence of surface energy densities with respect to the amplitude of the jump, z, and the normal to the jump set, ν.

We extend ϑ by positive 1-homogeneity as follows:

ϑ(z, η) = \{ \begin{matrix}|η|ϑ(z, \frac{η}{|η|}) & if η ≠ 0, \\ 0 & if η = 0. \end{matrix}

Changing variables and setting

u(x_{α}, x_{3}) := v(x_{α}, εx_{3}), (ν_{α}, ν_{3}) := ν,

clearly u ∈ GSBV_{p}(Ω; ℝ^{3}) and the integral above becomes

∫_{Ω_{ε}} W (∇_{α}u (x_{α}, \frac{x_{3}}{ε} ) |\frac{1}{ε}∇_{3}u (x_{α}, \frac{x_{3}}{ε} )) dx

+ ∫_{S(v)∩Ω_{ε}} ϑ(u^{+} (x_{α}, \frac{x_{3}}{ε} ) − u^{−} (x_{α}, \frac{x_{3}}{ε} ) ,

ν_{α}(u) (x_{α}, \frac{x_{3}}{ε} ) , \frac{1}{ε}ν_{3}(u) (x_{α}, \frac{x_{3}}{ε} ))\frac{ε}{\sqrt{ε^{2}|ν_{α}|^{2}(v) + ν_{3}^{2}(v)}} dH^{2}

= ε [∫_{Ω} W (∇_{α}u(x)|\frac{1}{ε}∇_{3}u(x)) dx

+ ∫_{S(u)∩Ω} ϑ(u^{+}(x) − u^{−}(x), ν_{α}(u)(x), \frac{1}{ε}ν_{3}(u)(x)) dH^{2}] .

We will consider the ε-scaled 3D-energies J_{ε} : L^{1}(Ω; ℝ^{3}) → [0, +∞] defined by

J_{ε}(u) := \{ \begin{matrix}∫_{Ω} W (∇_{α}u|\frac{1}{ε}∇_{3}u) dx \\ + ∫_{S(u)} ϑ(u^{+} − u^{−}, ν_{α}(u), \frac{1}{ε}ν_{3}(u)) dH^{2} & if u ∈ GSBV_{p}(Ω; ℝ^{3}), \\ +∞. & otherwise. \end{matrix}

(2.2)

We introduce the space

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

Note that if u ∈ V then D_{3}u = 0 in the sense of distributions, so that V can be identified with SBV_{p}(ω; ℝ^{3}).

An energy density Ŵ : 𝕄^{3×2} → ℝ determines naturally two quasiconvex envelopes; precisely, given \overline{F} ∈ 𝕄^{2×3} we may define

Q_{2}Ŵ(\overline{F}) := inf \{∫_{(0,1)^{2}} Ŵ(\overline{F} + D_{α}φ(x)) dx_{α} : φ ∈ W_{#}^{1,∞}((0, 1)^{2}; ℝ^{3})\} ,

and also

Q_{3}Ŵ(\overline{F}) := inf \{∫_{(0,1)^{3}} Ŵ(\overline{F} + D_{α}φ(x)) dx : φ ∈ W_{#}^{1,∞}((0, 1)^{3}; ℝ^{3})\} .

Similarly, associated to a function ϑ̂: (ℝ^{3} \ {0}) × ℝ^{2} → ℝ we may introduce two BV -elliptic envelopes, namely

R_{2}ϑ̂(z, η) := inf {∫_{S(u)∩Q′_{η}} ϑ̂(u^{+} − u^{−}, ν(u)) dH^{1} : u ∈ GSBV_{#}(Q′_{η}; ℝ^{3}, z)}

for all z ∈ ℝ^{3} and η ∈ S^{1}, where Q′_{η} is a unit square on the plane, centered at zero and with two faces orthogonal to η, and

R_{3}ϑ̂(z, η) := inf {∫_{S(u)∩Q_{(η,0)}} ϑ̂(u^{+} − u^{−}, ν_{α}(u)) dH^{2} : u ∈ GSBV_{#}(Q_{(η,0)}; ℝ^{3}, z)} .

In the latter envelope Q_{(η,0)} stands for a unit cube in ℝ^{3} centered at zero and with two faces orthogonal to (η, 0).

By Fubini’s Theorem it can be easily seen that

Q_{3}Ŵ(\overline{F}) ≥ Q_{2}Ŵ(\overline{F}).  (2.3)

The main result of the paper is the following.

Theorem 2.1 Let W : 𝕄^{3×3} → [0, +∞) and ϑ : (ℝ^{3} \ {0}) × ℝ^{3} → [0, +∞) be continuous functions, ϑ is symmetric and positively homogeneous of degree 1 in the second variable; i.e.,

ϑ(z, ν) = ϑ(−z, −ν), ϑ(z, tν) = tϑ(z, ν) for all t > 0, z, ν ∈ ℝ^{3}.

Let W and ϑ satisfy the growth and coercivity conditions

|F |^{p} ≤ W (F) ≤ C(1 + |F |^{p}),  1 + |z| ≤ ϑ(z, ν) ≤ C(1 + |z|)  (2.4)

for all F ∈ 𝕄^{3×3}, z ∈ ℝ^{3}, ν ∈ S^{2}, and for some C > 0. Suppose, in addition, that ϑ satisfies the Lipschitz condition

|ϑ(z, ν) − ϑ(z′, ν)| ≤ L |z − z′|  (2.5)

for all F, F′ ∈ 𝕄^{3×3}, z, z′ ∈ ℝ^{3}, ν ∈ S^{2}, and for some C > 0. Then the functionals J_{ε} defined in (2.2) Γ-converge with respect to the L^{1}(Ω; ℝ^{3}) convergence as ε → 0^{+} to the functional J_{0} : L^{1}(Ω; ℝ^{3}) → [0, +∞] defined by

J_{0}(u) := \{ \begin{matrix}∫_{ω} Q_{2}\overline{W} (∇_{α}u) dx_{α} + ∫_{S(u)∩ω} R_{2}\overline{ϑ}(u^{+} − u^{−}, ν_{α}(u)) dH^{1} if u ∈ V, \\ +∞ otherwise, \end{matrix}

where

\overline{W} (\overline{F}) := inf\{W (F) : F = (\overline{F} |F_{3}), F_{3} ∈ ℝ^{3}\}

for F ∈ 𝕄^{3×2}, and

\overline{ϑ}(z, η) := inf \{ϑ(z, η, ξ) : ξ ∈ ℝ\}

for z ∈ ℝ^{3} and η ∈ ℝ^{2} \ {0}.

Remark 2.2 (i) The Lipschitz condition (2.5) imposed on the surface energy density ϑ is of technical nature, and will be used to comply with the hypotheses of the integral representation theorem in [16]. It can be easily weakened to a uniformcontinuity hypothesis. As the representation result in [16] is designed to deal with Carathéodory integrands of a more general form, it may be possible to completely remove it by modifying the arguments of other results which do not use continuity hypotheses such as the one in [14]. Note, however, that in general, contrary to quasiconvexity, BV -ellipticity does not imply continuity with respect to the jump argument. (ii) The result of Theorem 2.1 still holds in the case where the surface energy density satisfies a growth condition of the type 0 < c_{1} ≤ ϑ ≤ c_{2} (see Example 2.10; for details we refer to the final chapter of [19]). (iii) Theorem 2.1 may be reformulated for mappings u : Ω ⊂ ℝ^{N} → ℝ^{m} for any pair of positive integers N and m. The changes are straightforward, and the proofs are similar to the ones presented in this work. For simplicity, and having in mind the motivation for modeling thin films, we will restrict ourselves to the case where N = m = 3.

The proof of the theorem will follow from the propositions and lemmas in the rest of this section. The ‘localized’ versions of J_{ε} on cylinders U × (−1/2, 1/2) for all U open subsets of ω are defined as

J_{ε}(u, U) := \{ \begin{matrix}∫_{U×(−1/2,1/2)} W (∇_{α}u|\frac{1}{ε}∇_{3}u) dx \\ + ∫_{S(u)∩(U×(−1/2,1/2))} ϑ(u^{+} − u^{−}, ν_{α}(u), \frac{1}{ε}ν_{3}(u)) dH^{2} \\ if u ∈ GSBV_{p}(Ω; ℝ^{3}), \\ +∞ & otherwise. \end{matrix}

(2.6)

2.1 Proof of the lower bound

We begin by giving a lower bound for the Γ-limit.

Proposition 2.3 Let J_{ε} be defined as in (2.2) with W and ϑ Borel functions satisfying the growth conditions

|F |^{p} ≤ W (F), 1 ≤ ϑ(z, ν)

for all F ∈ 𝕄^{3×3}, z ∈ ℝ^{m}, ν ∈ S^{N−1}, and where ϑ(z, ·) is positively homogeneous of degree 1. Let (u_{ε}) be a family of functions such that sup_{ε} J_{ε}(u_{ε}) < +∞ and u_{ε} → u in L^{1}(Ω; ℝ^{3}). Then u ∈ V and (u_{ε}) converges weakly to u in GSBV_{p}(Ω; ℝ^{3}).

Proof. By the growth conditions on W and ϑ we have

∫_{Ω} |∇u_{ε}|^{p} dx + H^{2}(S(u_{ε})) ≤ J_{ε}(u_{ε})

if ε ≤ 1, from which we obtain that (u_{ε}) is pre-compact in GSBV_{p}(Ω; ℝ^{3}); hence, by Theorem 1.1 the function u belongs to GSBV_{p}(Ω; ℝ^{3}). Moreover, by the lower semicontinuity of

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

with respect to the weak GSBV_{p}-convergence, we obtain

∫_{Ω} |∇_{3}u|^{p} dx ≤ lim inf_{ε→0} ∫_{Ω}(1 + |u_{ε}^{+} − u_{ε}^{−}|) |∇_{3}u_{ε}|^{p} dx ≤ lim inf_{ε→0} ε^{p}J_{ε}(u_{ε}) = 0, (2.7)

and so u ∈ V.

Proposition 2.4 We have Γ-lim inf_{ε→0} J_{ε} ≥ J_{0}.

Proof. Given sequences (ε_{j}) converging to 0^{+} and (u_{j}) converging to u in L^{1}(Ω; ℝ^{3}) with lim inf_{j→+∞} J_{ε_{j}}(u_{j}) < +∞, by the previous proposition we have that u ∈ V, and we may consider weak GSBV_{p}(Ω; ℝ^{3})-convergence in place of L^{1}(Ω; ℝ^{3})-convergence.

Since

W (F) ≥ \overline{W} (\overline{F}) ≥ Q_{2}\overline{W} (\overline{F}),

and

ϑ(z, ν) ≥ \overline{ϑ}(z, ν_{α}) ≥ R_{2}\overline{ϑ}(z, ν_{α}),

we have

lim inf_{j→+∞} J_{ε_{j}}(u_{j})

≥ lim inf_{j→+∞} ∫_{−1/2}^{1/2} (∫_{ω} Q_{2}\overline{W} (∇_{α}u_{j})dx_{α}

+ ∫_{S(u_{j}(·,t))∩ω} R_{2}\overline{ϑ}(u_{j}^{+} − u_{j}^{−}, ν(u_{j}(·, t))(x_{α})) |ν_{α}(u_{j})(x_{α}, t)|dH^{1}(x_{α})) dt

≥ ∫_{−1/2}^{1/2} (lim inf_{j→+∞} ∫_{ω} Q_{2}\overline{W} (∇_{α}u_{j})dx_{α}) dt

+ ∫_{−1/2}^{1/2} (lim inf_{j→+∞} ∫_{S(u_{j}(·,t))∩ω} R_{2}\overline{ϑ}(u_{j}^{+} − u_{j}^{−}, ν(u_{j}(·, t))(x_{α}))dH^{1}(x_{α})) dt

− lim sup_{j→+∞} ∫_{S(u_{j})∩Ω}(1 + |u_{j}^{+} − u_{j}^{−}|)|ν_{3}(u_{j})|dH^{2}

≥ J_{0}(u),

where we have used Fatou’s Lemma, the homogeneity of ϑ, the facts that

ν(u_{j}(·, t))(x_{α}) = \frac{ν_{α}(u_{j})(x_{α}, t)}{|ν_{α}(u_{j})(x_{α}, t)|} for H^{1} a.e. x_{α} ∈ S(u(·, t)),

property (2.7) together with the inequality 1 − |ν_{α}| ≤ |ν_{3}|, and finally Theorem 1.1 with u ∈ V.

2.2 Proof of the upper bound

We start by establishing the existence of the Γ-limit and its integral representation.

Theorem 2.5 For all ε > 0 let W_{ε} : Ω×𝕄^{3×3} → [0, +∞) and ϑ_{ε} : Ω×ℝ^{3} ×ℝ^{3} → [0, +∞) be Borel functions satisfying

|F |^{p} ≤ W_{ε}(x, F) ≤ C(1 + |F |^{p}),

and

1 + |z| ≤ ϑ_{ε}(x, z, ν) ≤ C(1 + |z|).

Let J_{ε}(u, U) be defined by

J_{ε}(u, U) := \{ \begin{matrix}∫_{U×(−1/2,1/2)} W_{ε}(x, ∇_{α}u|\frac{1}{ε}∇_{3}u) dx \\ + ∫_{S(u)∩(U×(−1/2,1/2))} ϑ_{ε}(x, u^{+} − u^{−}, ν_{α}(u), \frac{1}{ε}ν_{3}(u)) dH^{2} \\ if u ∈ GSBV_{p}(Ω; ℝ^{3}), \\ +∞ & otherwise \end{matrix}

(2.8)

where U is an open subset of ω and u ∈ L^{p}(Ω; ℝ^{3}). Then for all (ε_{j}) converging to 0 there exists a subsequence, not relabeled, such that the Γ-limit

Γ- lim_{j} J_{ε_{j}}(u, U) =: J_{0}(u, U)

exists for all u ∈ V and U open subset of ω. Moreover, J_{0} satisfies:

(i) for all u ∈ V the set function J_{0}(u, ·) is the restriction of a Borel measure to the family of open subsets of ω;

(ii) if u = v a.e. on an open set U ⊂ ω then J_{0}(u, U) = J_{0}(v, U); (iii) for all open sets U ⊂ ω the functional J_{0}(·, U) is lower semicontinuous

on V with respect to the L^{1}(ω; ℝ^{3})-convergence;

(iv) there exists a constant C > 0 such that for all u ∈ V and every open set U ⊂ ω

0 ≤ J_{0}(u, U) ≤ C(∫_{U}(1 + |∇_{α}u|^{p}) dx_{α} + ∫_{S(u)∩U}(1 + |u^{+} − u^{−}|)dH^{1}).

The result above follows from standard Γ-convergence arguments and subadditivite properties of J_{ε}, which will be established in the proposition below.

Proposition 2.6 [Fundamental estimate] Let (J_{ε}) be the family of functionals defined in (2.8). For every η > 0 and for every U, U′, V open subsets of ω with U ⊂⊂ U′, there exists a constant M > 0 with the following property: for every ε > 0 and for every u ∈ SBV_{p}(U′ × (−1/2, 1/2); ℝ^{3}), v ∈ SBV_{p}(V × (−1/2, 1/2); ℝ^{3}) there exists a function φ ∈ C_{0}^{∞}(U′) with φ = 1 in a neighborhood of \overline{U} and 0 ≤ φ ≤ 1 such that

J_{ε}(φu + (1 − φ)v, U ∪ V) ≤ (1 + η)(J_{ε}(u, U′) + J_{ε}(v, V)) + M∥u − v∥_{L^{p}(S;ℝ^{3})}^{p} + η ,

where S := ((U′ \ U) ∩ V) × (−1/2, 1/2).

Proof. Let η > 0, and fix open sets U, U′ and V as in the statement. Fix k ∈ ℕ and let U_{1}, …, U_{k+1} be open subsets of ℝ^{3} such that U ⊂⊂ U_{1} ⊂⊂ U_{2} ⊂⊂ ⋯ ⊂⊂ U_{k+1} ⊂⊂ U′. For every i = 1, …, k, let φ_{i} be a cut-off function in C_{0}^{∞}(U_{i+1}) with φ_{i} = 1 on a neighborhood V_{i} of \overline{U_{i}}. Fixed ε > 0, u ∈ SBV_{p}(U′ × (−1/2, 1/2); ℝ^{3}), and v ∈ SBV_{p}(V × (−1/2, 1/2); ℝ^{3}), define on (U′ ∪ V) × (−1/2, 1/2) the function

w_{i}(x) := φ_{i}(x_{α})u(x) + (1 − φ_{i}(x_{α}))v(x),

where u and v are extended arbitrarily outside the sets U′ × (−1/2, 1/2) and V × (−1/2, 1/2), respectively. Then for i = 1, …, k

J_{ε}(w_{i}, U ∪ V) ≤ J_{ε}(u, (U ∪ V) ∩ V_{i}) + J_{ε}(v, V \ sptφ_{i})
+J_{ε}(w_{i}, V ∩ (U_{i+1} \ \overline{U_{i}}))
≤ J_{ε}(u, U′) + J_{ε}(v, V) + J_{ε}(w_{i}, V ∩ (U_{i+1} \ \overline{U_{i}})) . (2.9)

Set T_{i} := V ∩ (U_{i+1} \ \overline{U_{i}}). We estimate the last term in (2.9) as

J_{ε}(w_{i}, T_{i}) ≤ C [∫_{T_{i}×(−1/2,1/2)}(1 + |∇w_{i}|^{p}) dx

+ ∫_{S(w_{i})∩(T_{i}×(−1/2,1/2))}(1 + |w_{i}^{+} − w_{i}^{−}|) dH^{2}]

≤ C [|T_{i}| + ∫_{T_{i}×(−1/2,1/2)} |φ_{i}∇u + (1 − φ_{i})∇v + (∇_{α}φ_{i} ⊗ (u − v)|0)|^{p}dx

+ ∫_{S(u)∩(T_{i}×(−1/2,1/2))}(1 + |u^{+} − u^{−}|) dH^{2}

+ ∫_{S(v)∩(T_{i}×(−1/2,1/2))}(1 + |v^{+} − v^{−}|) dH^{2}]

≤ C [|T_{i}| + ∫_{T_{i}×(−1/2,1/2)}(|∇u|^{p} + |∇v|^{p} + |∇_{α}φ_{i}|^{p}|u − v|^{p}) dx)

+ ∫_{S(u)∩(T_{i}×(−1/2,1/2))}(1 + |u^{+} − u^{−}|) dH^{2}

+ ∫_{S(v)∩(T_{i}×(−1/2,1/2))}(1 + |v^{+} − v^{−}|) dH^{2}]

≤ C [∫_{T_{i}×(−1/2,1/2)} W_{ε}(x, ∇u) dx + ∫_{T_{i}×(−1/2,1/2)} W_{ε}(x, ∇v) dx

+ ∫_{S(u)∩(T_{i}×(−1/2,1/2))} ϑ_{ε}(x, u^{+} − u^{−}, ν(u)) dH^{2}

+ ∫_{S(v)∩(T_{i}×(−1/2,1/2))} ϑ_{ε}(x, v^{+} − v^{−}, ν(v)) dH^{2}]

+C (|T_{i}| + ∥∇_{α}φ_{i}∥_{L^{∞}}^{p} ∫_{T_{i}×(−1/2,1/2)} |u − v|^{p} dx)

≤ C(J_{ε}(u, T_{i}) + J_{ε}(v, T_{i}))

+C (|T_{i}| + max_{1≤i≤k} ∥∇_{α}φ_{i}∥_{L^{∞}}^{p} ∫_{T_{i}×(−1/2,1/2)} |u − v|^{p} dx) .

Hence there exists i_{0} ∈ {1, …, k} such that

J_{ε}(w_{i_{0}}, T_{i_{0}}) ≤ \frac{1}{k} ∑_{i=0}^{k} J_{ε}(w_{i}, T_{i})

≤ \frac{C}{k} (J_{ε}(u, U′) + J_{ε}(v, V)) + \frac{C}{k} |S| + M ∫_{S} |u − v|^{p} dx

where M = O(k^{p}) and S := ((U′ \ U) ∩ V) × (−1/2, 1/2). From (2.9) it follows that

J_{ε}(w_{i_{0}}, U ∪ V) ≤ (1 + \frac{C}{k} ) (J_{ε}(u, U′) + J_{ε}(v, V)) + \frac{C}{k} |S| + M ∫_{S} |u − v|^{p} dx .

It suffices to choose k large enough.

Proof of Theorem 2.5. The proof follows by using the standard ‘direct methods’ of Γ-convergence (described in their general features in [23] (Chapters 16-19) and [18] (Chapters 10-11)), with obvious adaptations to the SBV_{p} setting as in [19], and to the thin films framework (see [18] Section 24.3 and [20]). We just outline the necessary steps. First, using the previous proposition the thesis of Theorem 2.5 is established on V ∩L^{p}(Ω; ℝ^{3}) considering the L^{p}(Ω; ℝ^{3}) convergence in place of the L^{1}(Ω; ℝ^{3}) convergence (see [19] Proposition 3.3 and [18] Theorem 24.20). Then, a cut-off argument (see [19] Lemma 3.5) allows to obtain the thesis using the L^{1}(Ω; ℝ^{3}) convergence (see [19] Proposition 3.4 and the last steps in the proof of Theorem 2.3). Note that property (ii) follows directly from the definition of Γ-limit, property (iii) is a general property of Γ-limits, and (iv) is obtained from the growth condition taking u_{ε} := u in the definition of Γ-limsup.

We can apply the compactness result above and the integral representation theorems by Braides and Chiadò Piat [16] to our functionals, obtaining the following proposition.

Proposition 2.7 [Integral representation] Let W and ϑ satisfy the hypotheses of Theorem 2.1, and let J_{ε} be defined by (2.2). Suppose, in addition, that W satisfies the Lipschitz continuity condition

|W (F) − W (F′)| ≤ L(1 + |F |^{p−1} + |F′|^{p−1})|F − F′|,  (2.10)

for all F, F′ ∈ 𝕄^{3×3}, and for some L > 0. Then every sequence (ε_{j}) converging to 0 admits a a subsequence, not relabeled, such that the Γ-limit

Γ- lim_{j} J_{ε_{j}}(u, U) =: J_{0}(u, U)

exists for all u ∈ V and U open subset of ω. Moreover, there exist W_{0} and ϑ_{0} such that

J_{0}(u, U) = ∫_{U×(−1/2,1/2)} W_{0}(∇_{α}u) dx+∫_{S(u)∩(U×(−1/2,1/2))} ϑ_{0}(u^{+} −u^{−}, ν(u))dH^{2}

for all u ∈ V and for all U open subset of ω.

Proof. We apply Theorem 2.5 with

W_{ε}(x, F) := W (F), ϑ_{ε}(x, z, ν) := W (z, ν),

to find a corresponding Γ-limit functional J_{0}(u, U), for all u ∈ SBV_{p}(ω; ℝ^{3}). We denote by J_{0}(u, ·) its extension as a Borel measure on ω. Then J_{0} satisfies properties (i)–(iv) of Theorem 2.5, and, in addition, we have

(v) J_{0}(u + s, U) = J_{0}(u, U) for all s ∈ ℝ^{3};

(vi) for all u ∈ SBV_{p}(ω; ℝ^{3}) and Φ affine transformation in ℝ^{3}

|J_{0}(u, S(u)∩U)−J_{0}(Φ(u), S(u)∩U)| ≤ CL∥Φ−Id∥_{1,∞} ∫_{S(u)∩U}(1+|u^{+}−u^{−}|) dH^{1}.

Property (v) follows immediately from the translation invariance of J_{ε}. To prove (iv) we will exploit the Lipschitz continuity of ϑ (see (2.5)). In fact, let V be an open subset of ω, let u ∈ SBV_{p}(ω; ℝ^{3}), and let u_{j} → u in L^{1}(V × (−1/2, 1/2); ℝ^{3}) be such that

lim_{j} J_{ε_{j}}(u_{j}, V) = J_{0}(u, V).

By definition of Γ-liminf, we have

J_{0}(Φ(u), V) ≤ lim inf_{j} J_{ε_{j}}(Φ(u_{j}), V)

= lim inf_{j} (∫_{V ×(−1/2,1/2)} W (∇Φ (∇_{α}u_{j}|\frac{1}{ε_{j}} ∇_{3}u_{j}))dx

+ ∫_{S(u_{j})∩(V ×(−1/2,1/2))} ϑ(Φ(u_{j}^{+}) − Φ(u_{j}^{−}), ν_{α}(u_{j}), \frac{1}{ε_{j}} ν_{3}(u_{j})) dH^{2}]

≤ lim inf_{j} [∫_{V ×(−1/2,1/2)} W (∇_{α}u_{j}|\frac{1}{ε_{j}} ∇_{3}u_{j}) dx

+L∥∇Φ − I∥_{∞} ∫_{V ×(−1/2,1/2)}(1 + |∇_{α}u_{j}|^{p} + |\frac{1}{ε_{j}} ∇_{3}u_{j}|^{p}) dx

+ ∫_{S(u_{j})∩(V ×(−1/2,1/2))} ϑ(u_{j}^{+} − u_{j}^{−}, ν_{α}(u_{j}), \frac{1}{ε_{j}} ν_{3}(u_{j})) dH^{2})

+ L∥Φ − Id∥_{∞} ∫_{S(u_{j})∩(V ×(−1/2,1/2))} |u_{j}^{+} − u_{j}^{−}|\sqrt{|ν_{α}(u_{j})|^{2} + \frac{1}{ε_{j}^{2}} |ν_{3}(u_{j})|^{2}} dH^{2}]

≤ lim_{j} J_{ε_{j}}(u_{j}, V) + L∥Φ − Id∥_{1,∞}(J_{ε_{j}}(u_{j}, V) + |V |)

≤ J_{0}(u, V) + CL∥Φ − Id∥_{1,∞}(∫_{V} (1 + |∇_{α}u|^{p}) dx_{α}

+ ∫_{S(u)∩V} (1 + |u^{+} − u^{−}|) dH^{1}),

and where we have used (2.5) and (2.10). Letting V → S(u) ∩ U, and using the fact that J_{0}(u, ·) is the trace of a Radon measure on the open sets of ω, we obtain

J_{0}(Φ(u), S(u) ∩ U) − J_{0}(u, S(u) ∩ U) ≤ L∥Φ − Id∥_{1,∞} ∫_{S(u)∩U}(1 + |u^{+} − u^{−}|) dH^{1},

and (vi) follows by a symmetric argument.

Conditions (i)–(vi) above assert hypotheses (i)–(v) and (vi)’ of Corollary 2.8, Remark 2.5 and Lemma 6.2 in [16], whose thesis is the existence of Carathéodory functions W_{0} and ϑ_{0} such that

J_{0}(u, B) = ∫_{B} W_{0}(x_{α}, u, ∇_{α}u) dx_{α} + ∫_{B∩S(u)} ϑ_{0}(x_{α}, u^{+}, u^{−}, ν(u)) dH^{1}

for all u ∈ SBV_{p}(ω; ℝ^{3}) and for all Borel subsets B of ω. Condition (v) further implies that

W_{0}(x, z, F) = W_{0}(x, F), ϑ_{0}(x, u^{+}, u^{−}, ν) = ϑ_{0}(x, u^{+} − u^{−}, ν).

Finally, as W and ϑ are independent of x, we have

W_{0}(x, z, F) = W_{0}(F), ϑ_{0}(x, u^{+}, u^{−}, ν) = ϑ_{0}(u^{+} − u^{−}, ν),

and the proof of the proposition is concluded.

We now provide some upper bound estimates for the value of the Γ-limit on affine functions and on piecewise-constant functions jumping across a hyperplane.

Proposition 2.8 [Estimate for the bulk energy density] We have

Γ- lim sup_{ε→0} J_{ε}(\overline{F} x_{α}, U) ≤ |U| \overline{W} (\overline{F})

for all U bounded open subsets of ω and \overline{F} ∈ 𝕄^{3×2}.

Proof. Let η > 0 and let F_{3} ∈ ℝ^{3} be such that W (\overline{F} |F_{3}) ≤ \overline{W} (\overline{F}) + η. The claim follows trivially by taking u_{ε}(x) = \overline{F} x_{α} + εF_{3}x_{3} as a test function.

To estimate the surface energy density, we test the Γ-limit on piecewise constant functions which jump across a line. Define the function u_{y}^{z,ν} by

u_{y}^{z,ν}(x) = \{ \begin{matrix}z/2 & if ⟨x − y, ν⟩ > 0, \\ −z/2 & if ⟨x − y, ν⟩ ≤ 0, \end{matrix}

and let Π_{y}^{ν} = {x ∈ ℝ^{2} : ⟨x − y, ν⟩ = 0}.

Proposition 2.9 [Estimate for the surface energy density] We have

Γ- lim sup_{ε→0} J_{ε}(u_{y}^{z,ν}, U) ≤ H^{1}(U ∩ Π_{y}^{ν}) \overline{ϑ}(z, ν) + C|U|

for all bounded open subsets U of ω, y ∈ ω, z ∈ ℝ^{3}, and ν ∈ S^{1}.

Proof. Up to a translation, we can suppose that y = 0. Let Π^{ν} := Π_{0}^{ν}. Let η > 0, and let \overline{ν}_{3} ∈ ℝ be such that

ϑ(z, ν, \overline{ν}_{3}) ≤ \overline{ϑ}(z, ν) + η.

The estimate with C = W (0) then follows by taking

u_{ε}(x) := \{ \begin{matrix}z/2 & if ⟨x, (ν, ε\overline{ν}_{3})⟩ > 0, \\ −z/2 & if ⟨x, (ν, ε\overline{ν}_{3})⟩ ≤ 0, \end{matrix}

and by the arbitrariness of η > 0.

2.3 Proof of Theorem 2.1

In view of Propositions 2.7, 2.8 and 2.9, from every sequence (ε_{j}) of positive numbers converging to 0 we can extract a further subsequence (not relabelled), and find a functional J_{0} with densities W_{0} and ϑ_{0} such that for all u ∈ V

Γ- lim sup_{j→+∞} J_{ε_{j}}(u) = ∫_{Ω} W_{0}(∇_{α}u)dx + ∫_{S(u)∩Ω} ϑ_{0}(u^{+} − u^{−}, ν(u))dH^{2}
≤ ∫_{ω} \overline{W} (∇_{α}u)dx_{α} + ∫_{S(u)∩ω} \overline{ϑ}(u^{+} − u^{−}, ν_{α}(u))dH^{1}.

This, together with Proposition 2.4, yields

Q_{2}\overline{W} ≤ W_{0} ≤ \overline{W} , R_{2}\overline{ϑ} ≤ ϑ_{0} ≤ \overline{ϑ}.

By virtue of Theorem 1.3, applied to the functional J_{0}, we have

Q_{2}W_{0} = W_{0}, R_{2}ϑ_{0} = ϑ_{0},

so we may deduce that

W_{0} ≤ Q_{2}\overline{W} , and similarly ϑ_{0} = R_{2}\overline{ϑ}.

The result of the theorem follows by definition of Γ-limit and by virtue of the arbitrariness of the sequence (ε_{j}).

Example 2.10 We may consider thin films of a hyperelastic material obeying to Griffith theory, with underlying elastic energy given by

∫_{Ω_{ε}} W (∇u) dx + λH^{2}(S(u) ∩ Ω_{ε}),

where λ is a positive constant. In this case we have

ϑ(z, ν) = λ|ν| = λ\sqrt{|ν_{α}|^{2} + ν_{3}^{2}},

so that \overline{ϑ}(ν) = ϑ(ν) = λ|ν|, and the thin-film limit is simply given by

∫_{ω} Q\overline{W} (∇_{α}u) dx_{α} + λH^{1}(S(u) ∩ ω).

This result does not follow immediately from Theorem 2.1, but can be easily deduced from it. In fact, a lower estimate for the Γ-limit is still valid by Proposition 2.4, while the upper estimate can be deduced by considering the auxiliary energies

∫_{Ω_{ε}} W (∇u) dx + λ ∫_{S(u)∩Ω_{ε}} ϑ_{M}(u^{+} − u^{−}) dH^{2},

where ϑ_{M}(z) = (|z| − M) ∨ 1, with arbitrary M > 0, and applying Theorem 2.1 to the scaled versions of these energies (a similar approach is considered in detail in [19]).

3 The case p = 1

In the case where p = 1 it is well known that the analogues of Theorem 1.1 and 1.3 do not hold. Γ-limit and relaxation problems must be described in the whole space of functions of bounded variation, and in general there is an interaction phenomenon between the bulk and interfacial energy densities. In this section we show that we can still deal with the thin-film limit in this case by combining the description given for p = 1 and a relaxation result for functionals in BV, established by Bouchitté, Fonseca and Mascarenhas [14] Section 4.2 (see also [12], [17]).

3.1 Relaxation in BV

Let W : 𝕄^{m×N} → [0, +∞) and ϑ : ℝ^{m}×S^{N−1} → [0, +∞) be continuous functions. We recall that the recession function of f is defined by

W^{∞}(F) := lim sup_{t→+∞} \frac{W (tF)}{t}.

If W is Lipschitz (respectively, quasiconvex) and has linear growth (see (H1) below), then W^{∞} is a Lipschitz (respectively, quasiconvex), positively 1-homogeneous function (see [28]). Moreover, we define the positively 1-homogeneous function

ϑ_{0}(z, ν) := lim sup_{t→0^{+}} \frac{ϑ(tz, ν)}{t}.

We suppose that W and ϑ satisfy the following hypotheses:

(H1) there exists C > 0 such that

\frac{1}{C} |F | ≤ W (x, F) ≤ C(1 + |F |)

for all F ∈ 𝕄^{m×N};

(H2) there exist constants C, L > 0, 0 < r < 1, such that

|W^{∞}(F) − \frac{W (tF)}{t}| ≤ C \frac{1}{t^{r}}

for every F ∈ 𝕄^{m×N} with |F | = 1 and for all t > 0 such that t > L;

(H3) there exists C > 0 such that

\frac{1}{C} |z| ≤ ϑ(z, ν) ≤ C|z|

for all (z, ν) ∈ ℝ^{m}S^{N−1};

(H4) there exist constants C, l, s > 0 such that

|ϑ_{0}(z, ν) − \frac{ϑ(tz, ν)}{t}| ≤ Ct^{s}

for every (z, ν) ∈ ℝ^{m} × S^{N−1} with |z| = 1 and for all t such that 0 < t < l.

Let (z, ν) ∈ ℝ^{m} × S^{N−1}, let {ν_{1}, …, ν_{N−1}, ν} form an orthonormal basis of ℝ^{N}, and define the class of admissible functions

A(z, ν) :=\{v ∈ SBV_{loc}(S_{ν}; ℝ^{m}) : v(y) = 0 if ⟨y, ν⟩ = −\frac{1}{2}, v(y) = z if ⟨y, ν⟩ = \frac{1}{2},

v is 1-periodic in the directions of ν_{1}, …, ν_{N−1}},

where S_{ν} is the strip

S_{ν} := \{y ∈ ℝ^{N} : |⟨y, ν⟩| < \frac{1}{2}\} ,

and where the boundary values of v are understood in the sense of traces. We use the notation

∫_{Ω} W (dC_{u}) := ∫_{Ω} W (\frac{dC_{u}}{d|C_{u}|})d|C_{u}|,

where |C_{u}| is the total variation measure of C_{u}.

Theorem 3.1 Under hypotheses (H1)–(H4), with W Lipschitz continuous, if u ∈ BV (Ω; ℝ^{m}) then the lower semicontinuous envelope F with respect to the L^{1} convergence of the functional

u ∈ SBV (Ω; ℝ^{m}) ↦ ∫_{Ω} W (∇u) dx + ∫_{S(u)∩Ω} ϑ(u^{+} − u^{−}, ν)dH^{N−1}, (3.11)

is given by

F(u) = ∫_{Ω} g(∇u)dx + ∫_{S(u)∩Ω} h(u^{+} − u^{−}, ν)dH^{N−1} + ∫_{Ω} g^{∞}(dC_{u}),

where

g(A) := inf {∫_{(0,1)^{N}} W (∇u)dx + ∫_{S(u)∩(0,1)^{N}} ϑ_{0}(u^{+} − u^{−}, ν)dH^{N−1} :
u ∈ SBV ((0, 1)^{N}; ℝ^{m}), u |_{∂(0,1)^{N}} (x) = Ax},  (3.12)

and

h(z, ν) := inf {∫_{Q_{ν}} W^{∞}(∇u(x))dx + ∫_{S(u)∩Q_{ν}} ϑ(u^{+} − u^{−}, ν)dH^{N−1} : u ∈ A(z, ν)} .

(3.13)

3.2 Thin-film limit

Now we state the thin-film approximation result, where \overline{W} and \overline{ϑ} are as defined in Theorem 2.1,

g_{2}(\overline{F}) := inf \{∫_{(0,1)^{2}} \overline{W} (∇_{α}u) dx_{α} + ∫_{S(u)∩(0,1)^{2}}(\overline{ϑ})_{0}(u^{+} − u^{−}, ν_{α})dH^{1} :
u ∈ SBV ((0, 1)^{2}; ℝ^{3}), u |_{∂(0,1)^{2}} (x) = \overline{F} x\},

g_{3}(\overline{F}) := inf \{∫_{(0,1)^{3}} \overline{W} (∇_{α}u) dx + ∫_{S(u)∩(0,1)^{3}}(\overline{ϑ})_{0}(u^{+} − u^{−}, ν_{α})dH^{2} :
u ∈ SBV ((0, 1)^{3}; ℝ^{3}), u |_{∂(0,1)^{3}} (x) = \overline{F} x\}.

Similarly,

h_{2}(z, η) := inf \{∫_{Q′_{η}} \overline{W}^{∞}(∇_{α}u(x))dx_{α}

+ ∫_{S(u)∩Q′_{η}} \overline{ϑ}(u^{+} − u^{−}, η)dH^{1} : u ∈ A′(z, η)\} ,

h_{3}(z, η) := inf \{∫_{Q_{(η,0)}} \overline{W}^{∞}(∇_{α}u(x))dx

+ ∫_{S(u)∩Q_{(η,0)}} \overline{ϑ}(u^{+} − u^{−}, η)dH^{2} : u ∈ A(z, (η, 0))\} ,

where

A′(z, η) := \{v ∈ SBV_{loc}(S_{ν} ∩ ω; ℝ^{m}) : v(y) = 0 if ⟨y, η⟩ = −\frac{1}{2},

v(y) = z if ⟨y, η⟩ = \frac{1}{2}, v is 1-periodic in the direction of ρ\},

with ρ orthogonal to η on the plane.

Theorem 3.2 Let W : 𝕄^{3×3} → [0, +∞) and ϑ : (ℝ^{3} \ {0}) × ℝ^{3} → [0, +∞) be continuous functions, and let ϑ be symmetric and positively homogeneous of degree 1 in the second variable; i.e.,

ϑ(z, ν) = ϑ(−z, −ν), ϑ(z, tν) = tϑ(z, ν) for all t > 0, z, ν ∈ ℝ^{3}.

Suppose that W and ϑ satisfy the growth and coercivity conditions

|F | ≤ W (F) ≤ C(1 + |F |), |z| ≤ ϑ(z, ν) ≤ C|z|

for all F ∈ 𝕄^{3×3}, z ∈ ℝ^{3}, ν ∈ S^{2}, and for some C > 0. Let W and ϑ satisfy (H2) and (H4), respectively. The functionals J_{ε} defined in (2.2), with GSBV_{p}(Ω; ℝ^{m}) replaced by SBV (Ω; ℝ^{m}), Γ-converge with respect to the L^{1}(Ω; ℝ^{3}) convergence as ε → 0^{+} to the functional J_{0} : L^{1}(Ω; ℝ^{3}) → [0, +∞] defined by

J_{0}(u) = \{ \begin{matrix}∫_{ω} g_{2}(∇_{α}u) dx_{α} + ∫_{S(u)∩ω} h_{2}(u^{+} − u^{−}, ν_{α}(u)) dH^{1} + ∫_{Ω} g_{2}^{∞}(dC_{u}) \\ if u ∈ BV (Ω; ℝ^{m}), D_{3}u = 0 \\ +∞ & otherwise. \end{matrix}

(3.14)

Remark 3.3 Note that, contrary to the case p > 1, we do not assume any Lipschitz continuity on ϑ. This discrepancy is not surprising, since BV -ellipticity combined with the growth conditions at 0 does imply the Lipschitz continuity of ϑ, and hence this condition can be overcome by a relaxation argument, which is implicit in [12]. In some special cases it is possible to apply other relaxation results, which do not assume hypotheses (H2) and (H4) (see [13] for the scalar isotropic case and [17] when ϑ is positively homogeneous of degree 1).

Proof. The reasoning of the proof is close to the one used for p > 1. We briefly indicate some of the necessary changes.

Note that, since J_{ε}(u) ≥ |D_{α}u|(Ω)+\frac{1}{ε}|D_{3}u|(Ω), an argument similar to that of Proposition 2.3 yields trivially that J_{0}(u) is finite only on functions u ∈ BV (Ω; ℝ^{m}) with D_{3}u = 0.

Also, the proof of Proposition 2.4 can be reproduced in this context as follows: by Theorem 3.1 and for every δ > 0 the functional in BV (Ω; ℝ^{m})

u ↦ ∫_{Ω} g_{δ}(∇_{α}u) dx + ∫_{S(u)∩Ω} h_{δ}(u^{+} − u^{−}, ν_{α})dH^{2} + ∫_{Ω} g_{δ}^{∞}(dC_{u})

is lower semicontinuous in L^{1}, where g_{δ} and h_{δ} are, respectively, the bulk and interfacial densities of the relaxed energy according to Theorem 3.1 and corresponding to

F ↦ \overline{W} (\overline{F}) + δ|F_{3}|, (z, ν) ↦ \overline{ϑ}(z, ν_{α}) + δ|ν_{3}|.

Therefore, if (ε_{j}) is a sequence converging to zero and (u_{j}) converges to u in L^{1} with

lim inf_{ε_{j}} J_{ε_{j}}(u_{j}) < +∞,

without loss of generality we may assume that

lim_{ε_{j}} J_{ε_{j}}(u_{j}) = lim inf_{ε_{j}} J_{ε_{j}}(u_{j}), sup_{j} |D_{3}u_{j}|(Ω) ≤ C < +∞,

and we have

J_{0}(u) ≤ ∫_{Ω} g_{3}(∇_{α}u) dx + ∫_{S(u)∩Ω} h_{3}(u^{+} − u^{−}, ν_{α})dH^{2} + ∫_{Ω} g_{3}^{∞}(dC_{u})

≤ lim inf_{δ→0} [∫_{Ω} g_{δ}(∇u) dx + ∫_{S(u)∩Ω} h_{δ}(u^{+} − u^{−}, ν)dH^{2} + ∫_{Ω} g_{δ}^{∞}(dC_{u})]

≤ lim inf_{δ→0} lim inf_{ε_{j}→0} [∫_{Ω} g_{δ}(∇u_{j}) dx + ∫_{S(u_{j})∩Ω} h_{δ}(u_{j}^{+} − u_{j}^{−}, (ν_{j}))dH^{2}]
≤ lim inf_{δ→0} lim inf_{ε_{j}→0} (J_{ε_{j}}(u_{j}) + δ |D_{3}u_{j}|(Ω))
= lim inf_{ε_{j}→0} J_{ε_{j}}(u_{j}).  (3.15)

The next step is aimed at proving the existence of Γ-limits J_{0} for subsequences of arbitrarily chosen subsequences. As before, this relies on the fundamental estimate which may be obtained exactly as in Proposition 2.6. The argument of the proof of Theorem 2.5 is now somewhat simplified as there is no need to reduce from L^{p} to L^{1} convergence. We obtain that J_{0} satisfies, upon identifying {u ∈ BV (Ω; ℝ^{m}), D_{3}u = 0} with BV (ω; ℝ^{3}),

(i) for all v ∈ BV (ω; ℝ^{3}) the set function J_{0}(u, ·) is the restriction of a Borel measure to the family of open subsets of ω;

(ii) if u = v a.e. on an open set U ⊂ ω then J_{0}(u, U) = J_{0}(v, U); (iii) for all open sets U ⊂ ω the functional J_{0}(·, U) is lower semicontinuous

on BV (ω; ℝ^{3}) with respect to the L^{1}(ω; ℝ^{3})-convergence; (iv) for all v ∈ BV (ω; ℝ^{3}) and open sets U ⊂ ω

|Dv|(U) ≤ J_{0}(u, U) ≤ C(|U| + |Dv|(U)).

Furthermore, after assuming a Lipschitz condition on ϑ (which is not restrictive up to the relaxation argument in the proof of Theorem 2.1), the same argument as in the proof of Theorem 2.7 shows that J_{0} depends continuously on vertical translations, and is independent of horizontal translations; more precisely,

(v) J_{0}(u + s, U) = J_{0}(u, U) for all s ∈ R^{3};

(vi) for all u ∈ BV (ω; ℝ^{3}) and Φ affine transformation in ℝ^{3}

|J_{0}(u, U) − J_{0}(Φ(u), ∩U)| ≤ CL∥Φ − Id∥_{1,∞} (|U| + |Du|(U)) .

Conditions (i)–(vi) imply that J_{0} satisfies the hypotheses of the integral representation Theorem 3.12 in [14], so that J_{0} can be represented on BV (ω; ℝ^{m}) as

J_{0}(u, U) = ∫_{U} W_{0}(∇_{α}u)dx_{α} + ∫_{S(u)∩U} ϑ_{0}(u^{+} − u^{−}, ν_{α})dH^{2} + ∫_{U} W_{0}^{∞}(dC_{u}),

for some W_{0} and ϑ_{0}.

Clearly Propositions 2.8 and 2.9 still hold, so that we get W_{0} ≤ \overline{W} and ϑ_{0} ≤ \overline{ϑ}. As \overline{W} and \overline{ϑ} satisfy the hypotheses of Theorem 3.1, and J_{0} is equal to its own relaxation on BV (ω; ℝ^{3}), we conclude that

W_{0} ≤ g_{2}, ϑ_{0} ≤ h_{2},

which, together with (3.15), concludes the proof.

Remark 3.4 Since g_{2} ≤ Q_{2}\overline{W} ≤ \overline{W} and h_{2} ≤ R_{2}\overline{ϑ} ≤ \overline{ϑ}, then J_{0} is the lower semicontinuous envelope of the functional defined by

∫_{ω} Q_{2}\overline{W} (∇_{α}u) dx_{α} + ∫_{S(u)∩ω} R_{2}\overline{ϑ}(u^{+} − u^{−}, ν_{α}(u)) dH^{1} if u ∈ SBV (ω; ℝ^{3})
+∞  otherwise;

(3.16)

that is, the one obtained in Theorem 2.1. Hence, the case p = 1 may be seen as giving the same result, up to a final relaxation.

Acknowledgments. The research of I. Fonseca was partially supported by the National Science Foundation under Grants No. DMS–9500531 and DMS–9731957, and through the Center for Nonlinear Analysis. The research of A. Braides was partially supported by Marie-Curie fellowship ERBFMBICT972023 of the European Union program ‘Training and Mobility of Researchers’. The research of both authors was partially supported by the Max–Planck Institute for Mathematics in the Sciences, Leipzig (Germany).

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