Non-local singular perturbations of non-convex functionals -- recent results

TL;DR

This paper reviews recent advances in the study of singular perturbations of non-convex functionals, focusing on fractional and higher-order seminorms, and their connections to phase transitions and free-discontinuity problems.

Contribution

It provides an overview of new results relating fractional and higher-order perturbations to classical Gamma-convergence and limit analysis in variational problems.

Findings

Connections with Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis

Extensions of phase transition models with fractional seminorms

Insights into Gamma-expansions for non-convex functionals

Abstract

Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized some earlier Gamma-convergence results by himself and S. Mortola contained in a seminal paper, validating the so-called minimal-interface criterion. I will give an overview of some recent results on perturbations with fractional and higher-order seminorms both in the framework of phase transitions and of free-discontinuity problems, relating these results with the Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.