Regularity of minimizers of shape optimization problems involving perimeter

Guido De Philippis, Jimmy Lamboley, Michel Pierre, Bozhidar Velichkov

Research output: Contribution to journalArticlepeer-review


We prove the existence and regularity of optimal shapes for the problem min⁡{P(Ω)+G(Ω):Ω⊂D,|Ω|=m}, where P denotes the perimeter, |⋅| is the volume, and the functional G is either one of the following: • the Dirichlet energy Ef, with respect to a (possibly sign-changing) function f∈Lp;• a spectral functional of the form F(λ1,…,λk), where λk is the kth eigenvalue of the Dirichlet Laplacian and F:Rk→R is locally Lipschitz continuous and increasing in each variable.The domain D is the whole space Rd or a bounded domain. We also give general assumptions on the functional G so that the result remains valid.

Original languageEnglish (US)
Pages (from-to)147-181
Number of pages35
JournalJournal des Mathematiques Pures et Appliquees
StatePublished - Jan 2018


  • Dirichlet energy
  • Regularity
  • Shape optimization
  • Spectral functional

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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