This paper is dedicated to the spectral optimization problem min {lambda(s)(1)(Omega) + ... + lambda(s)(m) (Omega) + Lambda L-n(Omega): Omega subset of D s-quasi-open} where Lambda > 0, D subset of R-n is a bounded open set and lambda(s)(i)(Omega) is the i-th eigenvalue of the fractional Laplacian on Omega with Dirichlet boundary condition on R-n Omega. We first prove that the first m eigenfunctions on an optimal set are locally Holder continuous in the class C-0,C-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 Omega 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 C-1,C-alpha-regularity of the regular part of the boundary. (C) 2021 Elsevier Inc. All rights reserved.
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