Long-range phase coexistence models with degenerate potentials

TL;DR

This survey reviews recent progress in nonlocal phase transition models with degenerate potentials, focusing on the qualitative analysis of minimizers and critical points of the associated energies.

Contribution

It provides an overview of advances in understanding the qualitative properties of solutions in nonlocal phase transition problems with degenerate potentials.

Findings

Analysis of minimizers' regularity and structure

Characterization of phase coexistence in nonlocal models

Impact of degenerate potentials on phase transition behavior

Abstract

This survey offers an overview of recent advances in nonlocal phase transition problems, modeled by Ginzburg--Landau type energies of the form \[ \frac{1}{4}\iint_{\R^{2n}\setminus (\R^n \setminus \Omega)^2} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy \;+\; \int_\Omega W(u(x))\,dx. \] Here,~$W$ is a smooth and possibly \textit{degenerate} double well potential, with a polynomial control on its second derivatives near the wells. The emphasis is on qualitative properties of minimizers and critical points of the energy functional.