Regularity of the Free Boundary in an Optimization Problem Related to the Best Sobolev Trace Constant

In this paper we study the regularity properties of a free boundary problem arising in the optimization of the best Sobolev trace constant in the immersion $H^1(\Omega)\hookrightarrow L^q(\partial\Omega)$ for functions that vanish in a subset of $\Omega$. This problem is also related to a minimization problem for Steklov eigenvalues.

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