Homogenization of the Allen-Cahn equation with periodic mobility

Peter S. Morfe

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arXiv · 2011.15012

Homogenization of the Allen-Cahn equation with periodic mobility

Peter S. Morfe

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Abstract

We analyze the sharp interface limit for the Allen-Cahn equation with an anisotropic, spatially periodic mobility coefficient and prove that the large-scale behavior of interfaces is determined by mean curvature flow with an effective mobility. Formally, the result follows from the asymptotics developed by Barles and Souganidis for bistable reaction-diffusion equations with periodic coefficients. However, we show that the corresponding cell problem is actually ill-posed when the normal direction is rational. To circumvent this issue, a number of new ideas are needed, both in the construction of mesoscopic sub- and supersolutions controlling the large-scale behavior of interfaces and in the proof that the interfaces obtained in the limit are actually described by the effective equation.

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Peter S. Morfe. 2020-11-30. Homogenization of the Allen-Cahn equation with periodic mobility. https://arxiv.org/abs/2011.15012

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