1. Aix Marseille Univ, CNRS, Centrale Marseille, I2M, 39 Rue Frédéric Joliot Curie Marseille, CEDEX 13, 13453, France;2. Departamento de Matemática, FCEN – Universidad de Buenos Aires, Argentina
CONICET, Buenos Aires, Argentina
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.