Premium
Regularity for Shape Optimizers: The Degenerate Case
Author(s) -
Kriventsov Dennis,
Lin Fanghua
Publication year - 2019
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21810
Subject(s) - mathematics , eigenvalues and eigenvectors , degenerate energy levels , bounded function , dirichlet distribution , boundary (topology) , function (biology) , viscosity solution , mathematical analysis , set (abstract data type) , pure mathematics , dirichlet boundary condition , boundary value problem , physics , quantum mechanics , evolutionary biology , biology , computer science , programming language
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. © 2019 Wiley Periodicals, Inc.