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 $\Gamma$- $\mathrm{limsup}$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form $$E_\e(v):=\int_Ω\frac{1}{\e}F\Big(\e^n\nabla^n v,...,\e\nabla v,v\Big)dx\quad\text{for} v:Ω\subset\R^N\to\R^k \text{such that} A\cdot\nabla v=0,$$ where the function $F\ge 0$ and $A:{\mathbb{R}}^{k\times N}\to {\mathbb{R}}^m$ 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].

arXiv commentarXiv admin note: substantial text overlap with arXiv:1112.2968