2 Omar Anza Hafsa and Jean - Philippe

where ε > 0 is very small and Σ ⊂ R is Lipschitz, open and bounded. A point of Σε is denoted by (x, x3) with x ∈ Σ and x3 ∈]− ε 2 , ε 2 [. Let W : M 3×3 → [0,+∞] be the stored-energy function supposed to be continuous and coercive, i.e., W (F ) ≥ C|F | for all F ∈ M and some C > 0. In order to take into account the important physical properties that the interpenetration of matter does not occur and that an infinite amount of energy is required to compress a finite volume into zero volume, i.e., W (F ) → +∞ as detF → 0, where detF denotes the determinant of the 3× 3 matrix F , we assume that: (C1) W (F ) = +∞ if and only if detF ≤ 0; (C2) for every δ > 0, there exists cδ > 0 such that for all F ∈ M, if detF ≥ δ then W (F ) ≤ cδ(1 + |F | ). Our goal is to show that as ε → 0 the three-dimensional free energy functional Eε : W (Σε;R ) → [0,+∞] (with p > 1) defined by