Regularity of Minimizers in the Two-Phase Free Boundary Problems in Orlicz–Sobolev Spaces

Abstract

In this paper, we consider the optimization problem of minimizing $\mathcal{J}(u)={\int }_{\Omega }(G(|\nabla u|)+{\lambda }_{+}(u^{+}{)}^{\gamma }+{\lambda }_{-}(u^{-}{)}^{\gamma }+fu)\text{d}x$ in the class of functions $W^{1,G}(\Omega )$ with $u-\phi \in W_0^{1,G}(\Omega )$ for a given function $\phi$, 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 $\infty$. For $0<\gamma \le 1$, we prove that every minimizer $u$ of $\mathcal{J}(u)$ is $C_{loc}^{1,\alpha }$-continuous.