A minimum problem with two-phase free boundary in Orlicz spaces

Abstract

We consider the optimization problem of minimizing \(\mathcal{J }_{\gamma }(u)=\int _{\Omega }(G(|\nabla u|)+\lambda _{+}(u^{+})^{\gamma }+\lambda _{-}(u^{-})^{\gamma }+fu)\,\text {d}x\) in the class of functions \(W^{1,G}(\Omega )\) with \(u-\varphi \in W^{1,G}_{0}(\Omega )\) for a given function \(\varphi \), where \(W^{1,G}(\Omega )\) is the class of weakly differentiable functions with \(\int _{\Omega }G(|\nabla u|)\,\text {d}x<\infty \). The conditions on the function \(G\) allow for a different behavior at \(0\) and at \(\infty \). We give a rather complete description of regularity theory for a family of two-phase variational free boundary problems. For \(0<\gamma \le 1\), we prove that every minimizer \(u_{\gamma }\) of \(\mathcal{J }_{\gamma }(u)\) is \(C^{1,\alpha }_{loc}\) continuous. For \(\gamma =0\), we obtain local Lipschitz continuity for any minimizer \(u_{0}\) of \(\mathcal{J }_{0}(u)\). Here we also consider an asymptotic problem as \(\gamma \rightarrow 0\). At last, we establish sharp geometric estimates for the free boundary corresponding to the minimizer \(u_{0}\) of \(\mathcal J _{0}(u)\).