TY - JOUR

T1 - Regularity for Shape Optimizers

T2 - The Degenerate Case

AU - Kriventsov, Dennis

AU - Lin, Fanghua

N1 - Funding Information:
Properties (W2,4) are immediate, while property (W5) is easily passed to the limit. Property (W3) may then be checked as in the proof of Lemma 8.5 by using (W5) and the fact that W .0; 0C/ < 1. This implies the conclusion. □ Acknowledgment. DK was supported by National Science Foundation MSPRF Fellowship DMS-1502852. FL was supported by National Science Foundation Grant DMS-1501000.
Publisher Copyright:
© 2019 Wiley Periodicals, Inc.

PY - 2019/8

Y1 - 2019/8

N2 - 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.

AB - 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.

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U2 - 10.1002/cpa.21810

DO - 10.1002/cpa.21810

M3 - Article

AN - SCOPUS:85064051240

SN - 0010-3640

VL - 72

SP - 1678

EP - 1721

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 8

ER -