Regularity of two-phase free boundary minimizers in periodic media

TL;DR

This paper investigates the regularity properties of minimizers in a two-phase free boundary problem within periodic media, establishing large-scale Lipschitz estimates and improvement-of-flatness results.

Contribution

It introduces new regularity results for two-phase free boundary minimizers in periodic media, including Lipschitz estimates and Liouville properties.

Findings

Large-scale Lipschitz estimate for minimizers

Improvement-of-flatness for non-degenerate minimizers

Liouville property for entire minimizers

Abstract

We study the regularity of minimizers of a two-phase energy functional in periodic media. Our main result is a large scale Lipschitz estimate. We also establish improvement-of-flatness for non-degenerate minimizers, which is a key ingredient in the proof of the Lipschitz estimate. As a consequence, we obtain a Liouville property for entire non-degenerate minimizers.