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 $u-\phi_0\in W_0^{ 1,p(\cdot)}(\Omega)$, for a given $\phi_0\geq 0$ 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.
Categories: math.AP