A Variational Principle for Pulsating Standing Waves and an Einstein Relation in the Sharp Interface Limit

Abstract

This paper investigates the connection between the effective, large scale behavior of Allen–Cahn functionals with periodic coefficients and the sharp interface limit of the associated \(L^{2}\) gradient flows. By introducing a Percival-type Lagrangian in the cylinder \({\mathbb {R}} \times {\mathbb {T}}^{d}\), we establish a link between the \(\Gamma \)-convergence results of Anisini, Braides, and Chiadò Piat and the sharp interface limit results of Barles and Souganidis. In laminar media, we prove a sharp interface limit in a graphical setting, making no assumptions other than sufficient smoothness of the coefficients, and we prove that the effective interface velocity and surface tension satisfy an Einstein relation. A number of pathologies are presented to highlight difficulties that do not arise in the spatially homogeneous setting.