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MiS Preprint
52/2006

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

Hayk Mikayelyan and Henrik Shahgholian

Abstract

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

Received:
May 18, 2006
Published:
May 18, 2006
Keywords:
free boundary problems, perimeter, convexity

Related publications

inJournal
2013 Repository Open Access
Hayk Mikayelyan and Henrik Shahgholian

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

In: Communications on pure and applied analysis, 12 (2013) 3, pp. 1431-1443