Existence and regularity of minimizers for some spectral functionals with perimeter constraint

Guido De Philippis; Bozhidar Velichkov

doi:10.1007/s00245-013-9222-4

Existence and regularity of minimizers for some spectral functionals with perimeter constraint

Guido De Philippis, Bozhidar Velichkov

Mathematics

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

Abstract

In this paper we prove that the shape optimization problem {λ_k (Ω) : Ω ⊂ ℝ^d, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C ^1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λ_k1 (Ω)⋯ λ_kp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

Original language	English (US)
Pages (from-to)	199-231
Number of pages	33
Journal	Applied Mathematics and Optimization
Volume	69
Issue number	2
DOIs	https://doi.org/10.1007/s00245-013-9222-4
State	Published - Apr 2014

Keywords

Concentration-compactness
Eigenvalues
Free boundary
Shape optimization

ASJC Scopus subject areas

Control and Optimization
Applied Mathematics

Access to Document

10.1007/s00245-013-9222-4

Cite this

@article{93dff87d7568433db14aad85d2acb578,

title = "Existence and regularity of minimizers for some spectral functionals with perimeter constraint",

abstract = "In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.",

keywords = "Concentration-compactness, Eigenvalues, Free boundary, Shape optimization",

author = "{De Philippis}, Guido and Bozhidar Velichkov",

year = "2014",

month = apr,

doi = "10.1007/s00245-013-9222-4",

language = "English (US)",

volume = "69",

pages = "199--231",

journal = "Applied Mathematics and Optimization",

issn = "0095-4616",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - Existence and regularity of minimizers for some spectral functionals with perimeter constraint

AU - De Philippis, Guido

AU - Velichkov, Bozhidar

PY - 2014/4

Y1 - 2014/4

N2 - In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

AB - In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

KW - Concentration-compactness

KW - Eigenvalues

KW - Free boundary

KW - Shape optimization

UR - http://www.scopus.com/inward/record.url?scp=84896395044&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896395044&partnerID=8YFLogxK

U2 - 10.1007/s00245-013-9222-4

DO - 10.1007/s00245-013-9222-4

M3 - Article

AN - SCOPUS:84896395044

SN - 0095-4616

VL - 69

SP - 199

EP - 231

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

IS - 2

ER -

Existence and regularity of minimizers for some spectral functionals with perimeter constraint

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this