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 an 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 \begin{equation*} \begin{split} E_{\varepsilon}(v):=\int_\Omega \frac{1}{{\varepsilon}}F\big({\varepsilon}^n\nabla^n v,\dots, {\varepsilon}\nabla v,v\big)\,dx\; \forall v:\Omega\subset \mathbb R ^N\to \mathbb R ^k\; \text{s.t.}\; A\cdot\nabla v=0, \end{split} \end{equation*} where the function $F\geq 0$ and $A: \mathbb R ^{k\times N}\to \mathbb R ^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\operatorname{curl} v$, and $A\cdot\nabla v=\text{div } v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17]. v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17].