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

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 (Formula Presented.) for (Formula Presented.) such that (Formula Presented.), 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).