Regularity of the optimal sets for some spectral functionals

Abstract

In this paper we study the regularity of the optimal sets for the shape optimization problem

$$\min\Big\{\lambda_{1}(\Omega)+\dots+\lambda_{k}(\Omega) : \Omega \subset {\mathbb {R}}^{d} {\rm open},\quad |\Omega| = 1\Big\},$$

where \({\lambda_{1}(\cdot),\ldots,\lambda_{k}(\cdot)}\) denote the eigenvalues of the Dirichlet Laplacian and \({|\cdot|}\) the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer \({\Omega_{k}^{*}}\) is composed of a relatively open regular part which is locally a graph of a \({C^{\infty}}\) 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 \({d^{*} \in [5,7]}\) 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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Correspondence to Susanna Terracini.

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Mazzoleni, D., Terracini, S. & Velichkov, B. Regularity of the optimal sets for some spectral functionals. Geom. Funct. Anal. 27, 373–426 (2017). https://doi.org/10.1007/s00039-017-0402-2

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Keywords and phrases

  • Shape optimization
  • Dirichlet eigenvalues
  • optimality conditions
  • regularity of free boundaries
  • viscosity solutions

Mathematics Subject Classification

  • 49Q10 (35R35, 47A75, 49R05)