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Geometric and Functional Analysis

, Volume 27, Issue 2, pp 373–426 | Cite as

Regularity of the optimal sets for some spectral functionals

  • Dario Mazzoleni
  • Susanna Terracini
  • Bozhidar Velichkov
Article

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.

Keywords and phrases

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

Mathematics Subject Classification

49Q10 (35R35, 47A75, 49R05) 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Dario Mazzoleni
    • 1
  • Susanna Terracini
    • 1
  • Bozhidar Velichkov
    • 2
  1. 1.Dipartimento di Matematica “Giuseppe Peano”Università di TorinoTurinItaly
  2. 2.Laboratoire Jean Kuntzmann (LJK)Université Grenoble Alpes Bâtiment IMAGSaint-Martin-d’HèresFrance

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