Regularity of the Optimal Sets for some Spectral Functionals
Résumé
In this paper we study the regularity of the optimal sets for the shape optimization problem min λ 1 (Ω) + · · · + λ k (Ω) : Ω ⊂ R d open , |Ω| = 1 , where λ 1 (·),. .. , λ k (·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ω * k is composed of a relatively open regular part which is locally a graph of a C ∞ function and a closed singular part, which is empty if d < d * , contains at most a finite number of isolated points if d = d * and has Hausdorff dimension smaller than (d − d *) if d > d * , where the natural number d * ∈ [5, 7] is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case.
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