Regularity of shape optimizers for some spectral fractional problems

This paper is dedicated to the spectral optimization problem min{λ1s(Ω)+⋯+λms(Ω)+ΛLn(Ω):Ω⊂D s-quasi-open} where Λ>0,D⊂Rn is a bounded open set and λis(Ω) is the i-th eigenvalue of the fractional Laplacian on Ω with Dirichlet boundary condition on Rn∖Ω. We first prove that the first m eigenfunctions on an optimal set are locally Hölder continuous in the class C0,s and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss type monotonicity formula, we prove that the topological boundary in D of a minimizer Ω is composed of a relatively open regular part and a closed singular part of Hausdorff dimension at most n−n⁎, for some n⁎≥3. Finally we use a viscosity approach to prove C1,α-regularity of the regular part of the boundary.