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

TL;DR

This paper investigates a two-phase problem involving a degenerate operator in Orlicz-Sobolev spaces, establishing existence, regularity, and geometric properties of minimizers and their free boundaries.

Contribution

It introduces a novel analysis of a two-phase problem with the $ ext{Phi}$-Laplacian operator in Orlicz-Sobolev spaces, proving regularity and free boundary finiteness.

Findings

Existence of minimizers established.

Minimizers are bounded and Log-Lipschitz continuous.

Free boundaries have finite perimeter.

Abstract

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.