Regularity for Shape Optimizers: The Nondegenerate Case

Dennis Kriventsov; Fanghua Lin

doi:10.1002/cpa.21743

Regularity for Shape Optimizers: The Nondegenerate Case

Dennis Kriventsov, Fanghua Lin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider minimizers of F(λ₁(Ω),...,λ_n(Ω))+|Ω| where F is a function strictly increasing in each parameter, and λ_k(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)
Pages (from-to)	1535-1596
Number of pages	62
Journal	Communications on Pure and Applied Mathematics
Volume	71
Issue number	8
DOIs	https://doi.org/10.1002/cpa.21743
State	Published - Aug 2018

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Mathematics(all)
Applied Mathematics

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title = "Regularity for Shape Optimizers: The Nondegenerate Case",

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 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.",

author = "Dennis Kriventsov and Fanghua Lin",

note = "Funding Information: Acknowledgment. DK was supported by National Science Foundation MSPRF Fellowship DMS-1502852. FL was supported by National Science Foundation Grant DMS-1501000. Publisher Copyright: {\textcopyright} 2018 Wiley Periodicals, Inc.",

year = "2018",

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T2 - The Nondegenerate Case

AU - Kriventsov, Dennis

AU - Lin, Fanghua

N1 - Funding Information: Acknowledgment. DK was supported by National Science Foundation MSPRF Fellowship DMS-1502852. FL was supported by National Science Foundation Grant DMS-1501000. Publisher Copyright: © 2018 Wiley Periodicals, Inc.

PY - 2018/8

Y1 - 2018/8

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.

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Regularity for Shape Optimizers: The Nondegenerate Case

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