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

In this paper we are interested in the study of a two-phase problem equipped with the $\Phi$-Laplacian operator $$\Delta_\Phi u \coloneqq \mbox{div} \left(\phi(|\nabla u|)\dfrac{\nabla u}{|\nabla u|}\right),$$ where $\Phi(s)=e^{s^2}-1$ and $\phi=\Phi'$. 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.