Non-local singular perturbations of variational problems

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 Modica, who specialized some earlier

Gamma-convergence results by himself and 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/or higher-order seminorms both in the framework

of phase transitions and of free-discontinuity problems, connecting these results with the Bourgain-Brezis-Mironescu and

Maz’ya-Shaposhnikova limit analysis for fractional Sobolev seminorms, and with the theory of Gamma-expansions.

The recent and on-going results I am going to talk about were obtained by a number of authors including

O. Savin and E. Valdinoci, G. Palatucci and S. Vincini, M. Solci, G.C. Brusca, D. Donati, M. Picerni, C. Trifone, E. Vitali, G .Dal Maso, L. Kreutz, and myself.

Prof. Niola Fusco