Non-local non-homogeneous phase transitions: regularity of optimal profiles and sharp-interface limit

TL;DR

This paper develops a new Gamma-convergence analysis for a complex non-local, non-homogeneous phase transition model, revealing how interaction kernel singularities affect optimal profile regularity and establishing a sharp-interface limit.

Contribution

It introduces novel strategies for Gamma-convergence in non-local, non-homogeneous models and analyzes the regularity dependence of optimal profiles on kernel singularities.

Findings

Established Gamma-liminf lower bound using asymptotic calibration

Linked kernel singularity to regularity of optimal profiles

Provided a sharp-interface limit for the diffuse-interface model

Abstract

We provide a novel sharp-interface analysis via Gamma-convergence for a non-local and non-homogeneous diffuse-interface model for phase transitions, featuring an interplay between a non-local interaction kernel and a spatially dependent double-well potential. This interaction requires the development of new strategies both for the Gamma-liminf inequality and for the construction of recovery sequences. A key element of our approach is an asymptotic calibration, used to establish the Gamma-liminf lower bound. The study of the optimality of the lower bound hinges upon a novel analysis of the regularity dependence of one-dimensional optimal profiles on a family of parameters. In particular, we show how such regularity is influenced by the singularity of the interaction kernel at the origin, providing a precise and previously unexplored link between the two. Our results rely solely on the assumption of H\"older continuity for the moving wells, and also account for the compactness of sequences with equibounded energies.