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 $\mathrm{\Phi }$-Laplacian operator $${\mathrm{\Delta }}_{\mathrm{\Phi }}udiv(\phi (\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }){\displaystyle \frac{\mathrm{\nabla }u}{\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }}}),$$ where $\mathrm{\Phi }(s)=e^{s^2}-1$ and $\phi ={\mathrm{\Phi }}^{\mathrm{\prime }}$. 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.