A Free boundary problem for the $p(x)$- Laplacian

Abstract

We consider the optimization problem of minimizing ${\int }_{\Omega }|\nabla u{|}^{p(x)}+\lambda {\chi }_{\{u>0\}}dx$ in the class of functions $W^{1,p(\cdot )}(\Omega )$ with $u-{\varphi }_0\in W_0^{1,p(\cdot )}(\Omega )$, for a given ${\varphi }_0\ge 0$ and bounded. $W^{1,p(\cdot )}(\Omega )$ is the class of weakly differentiable functions with ${\int }_{\Omega }|\nabla u{|}^{p(x)}dx<\infty$. We prove that every solution $u$ is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, $\Omega \cap \partial \{u>0\}$, is a regular surface.

35 pages, submitted