### 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 language | English (US) |
---|---|

Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Published - Jan 1 2019 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Regularity for Shape Optimizers

T2 - The Degenerate Case

AU - Kriventsov, Dennis

AU - Lin, Fang-Hua

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85064051240&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064051240&partnerID=8YFLogxK

U2 - 10.1002/cpa.21810

DO - 10.1002/cpa.21810

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -