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

TL;DR

This paper proves that for a convex set K, the minimizer of a perimeter-penalized p-Laplacian functional has a convex support, with an analytic free boundary, and establishes the uniqueness of the minimizer.

Contribution

It demonstrates the convexity of the free boundary and level sets for minimizers in a perimeter-penalized free boundary problem involving the p-Laplacian.

Findings

Minimizers have convex support when K is convex.

The free boundary is analytic.

The minimizer is unique.

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.