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arXiv · 1010.2960

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

Hayk Mikayelyan·Henrik Shahgholian
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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.

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Hayk Mikayelyan, Henrik Shahgholian. 2010-10-14. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. https://arxiv.org/abs/1010.2960

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