# Variational models for phase separation

created on 05 Jan 2002
modified on 09 Dec 2002

[BibTeX]

Accepted Paper

Inserted: 5 jan 2002
Last Updated: 9 dec 2002

Journal: Interfaces and free boundaries
Volume: 5
Pages: 1-20
Year: 2003

Abstract:

The paper deals with the asymptotic behaviour (as $\epsilon\to0$) of a family 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 $\epsilon\to0$ is a minimal area model with a surface tension of nonlocal form. The well-known Modica-Mortola constant can be recovered in this setting as a limit case when $\alpha\to+\infty$.