Regularity for Shape Optimizers: The Degenerate Case

Dennis Kriventsov, Fanghua Lin

Research output: Contribution to journalArticlepeer-review


We consider minimizers of F=(λ1(Ω),..λN(Ω))+ |Ω| where F is a function nondecreasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λN. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler-Lagrange equation for the minimizers that is not otherwise available.

Original languageEnglish (US)
Pages (from-to)1678-1721
Number of pages44
JournalCommunications on Pure and Applied Mathematics
Issue number8
StatePublished - Aug 2019

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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