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

TL;DR

This paper establishes the lower bound for the $\Gamma$-limit of certain singular perturbation problems, including cases with differential constraints, and identifies conditions where this bound matches the upper bound, providing explicit formulas for the limit.

Contribution

It constructs the $\Gamma$-liminf lower bound for a broad class of singular perturbation problems, extending previous results to include differential constraints and anisotropic cases.

Findings

Derived the $\Gamma$-limit formula for problems without differential constraints.

Identified conditions where lower and upper bounds coincide.

Extended analysis to anisotropic problems with constraints.

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 $A:\equiv 0$).