### Abstract

We consider minimizers of F(λ1(Ω),..,λN(Ω))+| Ω |, where F is a function strictly increasing in each parameter, and λk(Ω) is the k^{th} Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C^{1,α} graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n - 3. We also obtain a new regularity result for vector-valued Bernoulli-type free boundary problems.

Original language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21743

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

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Regularity for Shape Optimizers

T2 - The Nondegenerate Case

AU - Kriventsov, Dennis

AU - Lin, Fang-Hua

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We consider minimizers of F(λ1(Ω),..,λN(Ω))+| Ω |, where F is a function strictly increasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n - 3. We also obtain a new regularity result for vector-valued Bernoulli-type free boundary problems.

AB - We consider minimizers of F(λ1(Ω),..,λN(Ω))+| Ω |, where F is a function strictly increasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of Ω. Our main result is that the reduced boundary of the minimizer is composed of C1,α graphs and exhausts the topological boundary except for a set of Hausdorff dimension at most n - 3. We also obtain a new regularity result for vector-valued Bernoulli-type free boundary problems.

UR - http://www.scopus.com/inward/record.url?scp=85041816492&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041816492&partnerID=8YFLogxK

U2 - 10.1002/cpa.21743

DO - 10.1002/cpa.21743

M3 - Article

AN - SCOPUS:85041816492

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -