An optimization problem with volume constrain for a degenerate quasilinear operator

Abstract

We consider the optimization problem of minimizing ${\int }_{\Omega }|\nabla u{|}^{p}dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial \{u>0\}\cap \Omega$, is smooth.

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• A minimum problem with free boundary in Orlicz spaces