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

Abstract

In part II we constructed the lower bound, in the spirit of $\Gamma$- $\liminf$ for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form E_\e(v):=\int_\Omega \frac{1}{\e}F\Big(\e^n\nabla^n v,...,\e\nabla v,v\Big)dx\quad\text{for}\;\; v:\Omega\subset\R^N\to\R^k\;\;\text{such that}\;\; A\cdot\nabla v=0, where the function $F\geq 0$ and $A:\R^{k\times N}\to\R^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\text{curl}\, v$ and $A\cdot\nabla v=\text{div} v$). Furthermore, we studied the cases where we can easy prove the coinciding of this lower bound and the upper bound obtained in [33]. In particular we find the formula for the $\Gamma$-limit for the general class of anisotropic problems without a differential constraint (i.e., in the case $A:\equiv 0$).