Regularity for Shape Optimizers: The Degenerate Case

Dennis Kriventsov, Fang-Hua Lin

Research output: Contribution to journalArticle

Abstract

We consider minimizers of F.(λ 1 (Ω).....,(λ N )(Ω)+|Ω| where F is a function nondecreasing in each parameter, and λ k (Ω) is the k th 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=λ N . 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)
JournalCommunications on Pure and Applied Mathematics
DOIs
StatePublished - Jan 1 2019

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Minimizer
Regularity
Viscosity
Dirichlet Eigenvalues
Vanishing Viscosity
Euler-Lagrange Equations
Bounded Set
Perimeter
Singular Point
Eigenvalue
Approximation
Graph in graph theory

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Regularity for Shape Optimizers : The Degenerate Case. / Kriventsov, Dennis; Lin, Fang-Hua.

In: Communications on Pure and Applied Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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