Variational models for phase separation
Margherita Solci, Enrico Vitali · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2003
The paper deals with the asymptotic behaviour (as $\varepsilon\to 0$) of a family $F\_{\varepsilon}(u,v)$ 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 $\varepsilon\to 0$ 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 $\alpha\to +\infty.$