Convexity of the free boundary for an exterior free boundary problem involving the perimeter
Hayk Mikayelyan and Henrik Shahgholian
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Submission date: 18. May. 2006 (revised version: August 2006)
Keywords and phrases: free boundary problems, perimeter, convexity
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We prove that if the given compact set K is convex then a minimizer of the functional
over the set 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.