Regularity of the optimal sets for some spectral functionals

Dario Cesare Severo Mazzoleni, Susanna Terracini, Bozhidar Velichkov

Research output: Contribution to journalArticlepeer-review

16 Citations (Scopus)


In this paper we study the regularity of the optimal sets for the sum of the first k eigenvalues of the Dirichlet Laplacian among sets of finite measure. We prove that the topological boundary of a minimizer is composed of a relatively open regular part which is locally a graph of a C^{1,s} 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* 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.
Original languageEnglish
Pages (from-to)373-426
Number of pages54
JournalGeometric and Functional Analysis
Publication statusPublished - 2017


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

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