Variational models for phase separation

Abstract

The paper deals with the asymptotic behaviour (as $\epsilon \to 0$) of a family $F_{\epsilon }(u,v)$ of integral functionals in the framework of phase separation. In order to obtain a selection criterion for the minima of the usual double-well, non-convex free energy involving the phase-variable $u,$ we add a gradient term in a new variable $v$ which is related to $u$ through the $L^2$-distance between $u$ and $v,$ weighted by a coefficient $\alpha .$ We prove that the limit as $\epsilon \to 0$ is a minimal area model with a surface tension of non-local form. The well-known Modica–Mortola constant can be recovered in this setting as a limit case when $\alpha \to +\infty .$