Existence and regularity of minimizers for some spectral functionals with perimeter constraint

Guido De Philippis, Bozhidar Velichkov

Research output: Contribution to journalArticle

Abstract

In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

Original languageEnglish (US)
Pages (from-to)199-231
Number of pages33
JournalApplied Mathematics and Optimization
Volume69
Issue number2
DOIs
StatePublished - Apr 2014

Keywords

  • Concentration-compactness
  • Eigenvalues
  • Free boundary
  • Shape optimization

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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