Regularity for Shape Optimizers: The Degenerate Case

2017/10/02 by Kriventsov, Dennis, Lin, Fanghua · 2 citations
#35R35 #49N60 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.00451

Abstract

We consider minimizers of F(λ₁(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 which 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 which is not otherwise available.

Cited by

• Vectorial Bernoulli Problems and Free Boundary Systems
• Quantitative stability control of the full spectrum of the Dirichlet Laplacian by the second eigenvalue

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