Compactness and dichotomy in nonlocal shape optimization

TL;DR

This paper investigates the behavior of minimizing sequences in nonlocal shape optimization problems, establishing conditions for existence of optimal shapes or their dichotomous splitting, inspired by local case results.

Contribution

It introduces a nonlocal concentration-compactness principle to analyze the compactness or dichotomy of minimizing sequences in nonlocal shape functionals.

Findings

Either an optimal shape exists or minimizing sequences split into two distant parts.

The work extends local shape optimization results to nonlocal settings.

Provides a framework for analyzing nonlocal eigenvalue problems.

Abstract

We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists, or there exists a minimizing sequence consisting of two "pieces" whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.