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

Peter Morfe

Published 2020-03-16Version 1

This paper investigates the connection between the effective, large scale behavior of Allen-Cahn-type energy functionals in periodic media and the sharp interface limit of the associated $L^{2}$ gradient flows. In the first part of the paper, we introduce a Percival-type Lagrangian defined in the cylinder $\mathbb{R} \times \mathbb{T}^{d}$ and prove that minimizers, which we call pulsating standing waves, exist under very weak assumptions on the coefficients. We use the pulsating standing waves to give new proofs of the existence of plane-like minimizers of the energy and to study the differentiability properties of the surface tension in the particular case of laminar media. In the final part of the paper, we prove a sharp interface limit for a restricted class of initial data in laminar media, assuming only smoothness of the coefficients, extending previous work of Barles and Souganidis. Using what we proved about the surface tension, we show that the effective interface velocity we obtain is consistent with the Einstein relation posited by Spohn. Through the analysis of a specific class of examples, we demonstrate a number of pathologies that can arise, including cases where the coefficients of the effective equation degenerate as the angle between the laminations and the normal vector tends to zero.