An optimization problem with volume constrain in Orlicz spaces

We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|) dx$ in the class of functions $W^{1,G}(Ω)$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\infty$. 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.

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