Homogenization and Phase Separation with Fixed Wells -- The Supercritical Case

TL;DR

This paper investigates a variational model describing the interaction between homogenization and phase separation, demonstrating a scale separation in the first-order Gamma-limit in the supercritical regime, confirming prior conjectures.

Contribution

It proves the first-order Gamma-limit exhibits a scale separation between homogenization and phase separation, validating previous conjectures in the supercritical case.

Findings

First-order Gamma-limit shows scale separation

Confirms conjectured behavior in supercritical regime

Advances understanding of multiscale phase interactions

Abstract

A variational model for the interaction between homogenization and phase separation is considered in the regime where the former happens at a finer scale than the latter. The first order $\Gamma-$limit is proven to exhibit a separation of scales which has been previously conjectured in a paper by Bradies and Zeppieri and in the PhD thesis of Hagerty.