Regularity of the minimizers in the composite membrane problem in R2

Sagun Chanillo; Carlos E. Kenig; Tung To

doi:10.1016/j.jfa.2008.04.015

Regularity of the minimizers in the composite membrane problem in R²

Sagun Chanillo, Carlos E. Kenig, Tung To

School of Arts and Sciences, Mathematics

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

24 Scopus citations

Abstract

We study the regularity of the minimizers to the problem:λ (α, A) = under(inf, u ∈ H₀¹ (Ω), {norm of matrix} u {norm of matrix}₂ = 1, | D | = A) under(∫, Ω) | D u |² + α under(∫, D) u² . We prove that in the physical case α < λ in R², any minimizer u is locally C^{1, 1} and the boundary of the set {u > c} is analytic where c is the constant such that D = {u < c} (up to a zero measure set).

Original language	English (US)
Pages (from-to)	2299-2320
Number of pages	22
Journal	Journal of Functional Analysis
Volume	255
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2008.04.015
State	Published - Nov 1 2008

All Science Journal Classification (ASJC) codes

Analysis

Keywords

Composite membrane
Domain variation
Free-boundary
Minimizer
Regularity

Access to Document

10.1016/j.jfa.2008.04.015

Cite this

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title = "Regularity of the minimizers in the composite membrane problem in R2",

abstract = "We study the regularity of the minimizers to the problem:λ (α, A) = under(inf, u ∈ H01 (Ω), {norm of matrix} u {norm of matrix}2 = 1, | D | = A) under(∫, Ω) | D u |2 + α under(∫, D) u2 . We prove that in the physical case α < λ in R2, any minimizer u is locally C1, 1 and the boundary of the set {u > c} is analytic where c is the constant such that D = {u < c} (up to a zero measure set).",

keywords = "Composite membrane, Domain variation, Free-boundary, Minimizer, Regularity",

author = "Sagun Chanillo and Kenig, {Carlos E.} and Tung To",

note = "Funding Information: ✩ The first and second authors are supported in part by NSF. * Corresponding author. E-mail addresses: chanillo@math.rutgers.edu (S. Chanillo), cek@math.uchicago.edu (C.E. Kenig), totung@math.uchicago.edu (T. To).",

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T1 - Regularity of the minimizers in the composite membrane problem in R2

AU - Chanillo, Sagun

AU - Kenig, Carlos E.

AU - To, Tung

N1 - Funding Information: ✩ The first and second authors are supported in part by NSF. * Corresponding author. E-mail addresses: chanillo@math.rutgers.edu (S. Chanillo), cek@math.uchicago.edu (C.E. Kenig), totung@math.uchicago.edu (T. To).

PY - 2008/11/1

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N2 - We study the regularity of the minimizers to the problem:λ (α, A) = under(inf, u ∈ H01 (Ω), {norm of matrix} u {norm of matrix}2 = 1, | D | = A) under(∫, Ω) | D u |2 + α under(∫, D) u2 . We prove that in the physical case α < λ in R2, any minimizer u is locally C1, 1 and the boundary of the set {u > c} is analytic where c is the constant such that D = {u < c} (up to a zero measure set).

AB - We study the regularity of the minimizers to the problem:λ (α, A) = under(inf, u ∈ H01 (Ω), {norm of matrix} u {norm of matrix}2 = 1, | D | = A) under(∫, Ω) | D u |2 + α under(∫, D) u2 . We prove that in the physical case α < λ in R2, any minimizer u is locally C1, 1 and the boundary of the set {u > c} is analytic where c is the constant such that D = {u < c} (up to a zero measure set).

KW - Composite membrane

KW - Domain variation

KW - Free-boundary

KW - Minimizer

KW - Regularity

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