Periodic phase separation: the periodic Cahn-Hilliard and isoperimetric problems

Abstract

We consider here two well-known variational problems associated with the phenomenon of phase separation: the isoperimetric problem and minimization of the Cahn-Hilliard energy. The two problems are related through a classical result in $\Gamma$ -convergence and we explore the behavior of global and local minimizers for these problems in the periodic setting. More precisely, we investigate these variational problems for competitors defined on the flat $2$ or $3$-torus. We view these two problems as prototypes for periodic phase separation. We give here a complete analysis of stable critical points of the $2$-d periodic isoperimetric problem and also obtain stable solutions to the $2$-d and $3$-d periodic Cahn-Hilliard problem. We also discuss some intriguing open questions regarding triply periodic constant mean curvature surfaces in $3$d and possible counterparts in the Cahn-Hilliard setting.