Plane-like minimizers for a non-local Ginzburg-Landau-type energy in a periodic medium

TL;DR

This paper constructs plane-like solutions for a non-local Ginzburg-Landau energy in a periodic medium, demonstrating their existence, geometric properties, and local minimality, advancing understanding of phase transitions in complex environments.

Contribution

It introduces a method to construct plane-like minimizers for a non-local energy in periodic media, highlighting their geometric and minimality properties.

Findings

Existence of plane-like solutions in periodic media

Solutions maintain a prescribed directional interface

Solutions exhibit local minimality with respect to a non-local energy

Abstract

We consider a non-local phase transition equation set in a periodic medium and we construct solutions whose interface stays in a slab of prescribed direction and universal width. The solutions constructed also enjoy a local minimality property with respect to a suitable non-local energy functional.