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

Abstract

In Part I we construct the upper bound, in the spirit of $\Gamma$- $\limsup$, achieved by multidimensional profiles, 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 $A\cdot\nabla v=\text{div}\,v$) which includes, in particular, the problems considered in [27]. This bound is in general sharper then one obtained in [27].