Convexity of the free boundary for an exterior free boundary problem involving the perimeter

Hayk Mikayelyan; Henrik Shahgholian

doi:10.3934/cpaa.2013.12.1431

Convexity of the free boundary for an exterior free boundary problem involving the perimeter

Hayk Mikayelyan^*
, Henrik Shahgholian

^*Corresponding author for this work

School of Mathematics and Physics

KTH Royal Institute of Technology

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

1 Citation (Scopus)

Abstract

We prove that if the given compact set K is convex then a minimizer of the functional I(v) = ∫B_R |∇_v|^pdx + Per({v > 0}), 1 < p < ∞ over the set {v ∈ W₀ ^1,p(B_R)|v ≡ 1 on K ⊂ B_R} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

Original language	English
Pages (from-to)	1431-1443
Number of pages	13
Journal	Communications on Pure and Applied Analysis
Volume	12
Issue number	3
DOIs	https://doi.org/10.3934/cpaa.2013.12.1431
Publication status	Published - May 2013

Keywords

Free boundary problems
Mean curvature

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10.3934/cpaa.2013.12.1431

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title = "Convexity of the free boundary for an exterior free boundary problem involving the perimeter",

abstract = "We prove that if the given compact set K is convex then a minimizer of the functional I(v) = ∫BR |∇v|pdx + Per(\{v > 0\}), 1 < p < ∞ over the set \{v ∈ W0 1,p(BR)|v ≡ 1 on K ⊂ BR\} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.",

keywords = "Free boundary problems, Mean curvature",

author = "Hayk Mikayelyan and Henrik Shahgholian",

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N2 - We prove that if the given compact set K is convex then a minimizer of the functional I(v) = ∫BR |∇v|pdx + Per({v > 0}), 1 < p < ∞ over the set {v ∈ W0 1,p(BR)|v ≡ 1 on K ⊂ BR} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

AB - We prove that if the given compact set K is convex then a minimizer of the functional I(v) = ∫BR |∇v|pdx + Per({v > 0}), 1 < p < ∞ over the set {v ∈ W0 1,p(BR)|v ≡ 1 on K ⊂ BR} has a convex support, and as a result all its level sets are convex as well. We derive the free boundary condition for the minimizers and prove that the free boundary is analytic and the minimizer is unique.

KW - Free boundary problems

KW - Mean curvature

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EP - 1443

JO - Communications on Pure and Applied Analysis

JF - Communications on Pure and Applied Analysis

IS - 3

ER -

Convexity of the free boundary for an exterior free boundary problem involving the perimeter

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Keywords

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