A minimum problem with free boundary in Orlicz spaces

Abstract

We consider the optimization problem of minimizing $\int_ΩG(|\nabla u|)+λχ_{\{u>0\}} dx$ in the class of functions $W^{1,G}(Ω)$ with $u-ϕ_0\in W_0^{1,G}(Ω)$, for a given $ϕ_0\geq 0$ and bounded. $W^{1,G}(Ω)$ is the class of weakly differentiable functions with $\int_ΩG(|\nabla u|) dx<\infty$. The conditions on the function G allow for a different behavior at 0 and at $\infty$. We prove that every solution u is locally Lipschitz continuous, that they are solution to a free boundary problem and that the free boundary, $\partial\{u>0\}\cap Ω$, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the $C^{1,α}$ regularity of their free boundaries near ``flat'' free boundary points.