On the Γ-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part II: The lower bound

Abstract

We construct the lower bound, in the spirit of Γ-convergence for some general classes of singular perturbation problems, with or without a prescribed differential constraint, of the form

$$E_\varepsilon (v): = \int_\Omega {\frac{1} {\varepsilon }F(\varepsilon ^n \nabla ^n v, \ldots ,\varepsilon \nabla v,v)dx} for v:\Omega \subset \mathbb{R}^N \to \mathbb{R}^k such that A \cdot \nabla v = 0, $$

where the function F is nonnegative and A: ℝ^k×N → ℝ^m is a prescribed linear operator (for example, A:≡ 0, A · ▿v:= curl v and A · ▿v = divv). Furthermore, we study the cases where we can easily prove that this lower bound coincides with the upper bound obtained in [18]. In particular, we find the formula for the Γ-limit for a general class of anisotropic problems without a differential constraint (i.e., in the case A:≡ 0).