JournalsifbVol. 5 , No. 1DOI 10.4171/ifb/70

Variational models for phase separation

  • Margherita Solci

    Università di Pavia, Italy
  • Enrico Vitali

    Università di Pavia, Italy
Variational models for phase separation cover

Abstract

The paper deals with the asymptotic behaviour (as ε0\varepsilon\to 0) of a family Fε(u,v)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,u, we add a gradient term in a new variable vv which is related to uu through the L2L^2-distance between uu and v,v, weighted by a coefficient α.\alpha. We prove that the limit as ε0\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 α+.\alpha\to +\infty.