  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.
