Anisotropic Surface Tensions for Phase Transitions in Periodic Media

TL;DR

This paper derives bounds on the homogenized surface tension for heterogeneous Allen-Cahn energies in periodic media, linking it to geometric variational problems and proving homogenization results for the signed distance function.

Contribution

It introduces new bounds for surface tension in periodic media and establishes homogenization results for the signed distance function in periodic and almost periodic media.

Findings

Bounds on homogenized surface tension are established.

Homogenization results for the signed distance function are proven.

The approach relates energy to geometric variational problems.

Abstract

This paper establishes bounds on the homogenized surface tension for a heterogeneous Allen-Cahn energy functional in a periodic medium. The approach is based on relating the homogenized energy to a purely geometric variational problem involving the large scale behaviour of the signed distance function to a hyperplane in periodic media. Motivated by this, a homogenization result for the signed distance function to a hyperplane in both periodic and almost periodic media is proven.