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

Hayk Mikayelyan; Henrik Shahgholian

doi:10.3934/cpaa.2013.12.1431

Article Contents

2013, Volume 12, Issue 3: 1431-1443. Doi: 10.3934/cpaa.2013.12.1431

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

Hayk Mikayelyan^1, and
Henrik Shahgholian^2,

1.
Xi'an Jiaotong-Liverpool University, Mathematical Sciences, 111 Ren'ai Road, Suzhou 215123, Jiangsu Prov.
2.
Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm

Received: November 2011

Revised: March 2012

Published: May 2013

Abstract / Introduction Related Papers Cited by

Abstract

We prove that if the given compact set $K$ is convex then a minimizer of the functional \begin{eqnarray*} I(v)=\int_{B_R} |\nabla v|^p dx+ Per(\{v>0\}), 1 < p < \infty, \end{eqnarray*} over the set $\{v\in W^{1,p}_0 (B_R)| v\equiv 1 \ \text{on} \ K\subset B_R\}$ 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.

Keywords:

Free boundary problems,
mean curvature.

Mathematics Subject Classification: Primary: 35R35; Secondary: 49K20.

Citation:

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