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.
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