A two-phase problem with degenerate operator in Orlicz-Sobolev spaces

Abstract

In this paper we are interested in the study of a two-phase problem equipped with the $\hat{I}¦$-Laplacian operator $$ Δ_Φu \coloneqq \mbox{div} \left(ϕ(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right), $$ where $\hat{I}¦(s)=e^{s^2}-1$ and $\ddot{I}•=\hat{I}{¦}^{\prime }$. We obtain the existence, boundedness, and Log-Lipschitz regularity of the minimizers of the energy functional associated to the two-phase problem. Furthermore, we also prove that the phase change free boundaries of the minimizers possess a finite perimeter.