A Free boundary problem for the $p(x)$- Laplacian

TL;DR

This paper studies a free boundary problem involving the variable exponent p(x)-Laplacian, proving regularity of solutions and the free boundary in an optimization setting.

Contribution

It establishes Lipschitz continuity of solutions, characterizes them as free boundary problem solutions, and proves the regularity of the free boundary surface.

Findings

Solutions are locally Lipschitz continuous.

The solutions solve a free boundary problem.

The free boundary is a regular surface.

Abstract

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^{p(x)}+ \lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(\Omega)$ with $u-\phi_0\in W_0^{1,p(\cdot)}(\Omega)$, for a given $\phi_0\geq 0$ and bounded. $W^{1,p(\cdot)}(\Omega)$ is the class of weakly differentiable functions with $\int_\Omega |\nabla u|^{p(x)} dx<\infty$. We prove that every solution $u$ is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, $\Omega\cap\partial\{u>0\}$, is a regular surface.