2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/109015We consider the optimization problem of minimizing $\int_Ω|\nabla u|^p dx$ 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, $\partial\{u>0\}\cap Ω$, is smooth.Analysis of PDEs35J20, 35P30, 49K20text