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arXiv · 2003.07298

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

Peter Morfe
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Abstract

This paper investigates the connection between the effective, large scale behavior of Allen-Cahn energy functionals in periodic media 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 Chiad\`{o} 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.

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Peter Morfe. 2020-03-16. A Variational Principle for Pulsating Standing Waves and an Einstein Relation in the Sharp Interface Limit. https://arxiv.org/abs/2003.07298

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