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

Bonder, Julián Fernández; Martínez, Sandra; Wolanski, Noemi

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

Authors

Bonder, Julián Fernández

Martínez, Sandra

Wolanski, Noemi

Description

We consider the optimization problem of minimizing $\int_Ω|\nabla u|^{p(x)}+ λχ_{\{u>0\}} dx$ in the class of functions $W^{1,p(\cdot)}(Ω)$ with $u-ϕ_0\in W_0^{1,p(\cdot)}(Ω)$, for a given $ϕ_0\geq 0$ and bounded. $W^{1,p(\cdot)}(Ω)$ is the class of weakly differentiable functions with $\int_Ω|\nabla u|^{p(x)} dx<\infty$. 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, $Ω\cap\partial\{u>0\}$, is a regular surface.
35 pages, submitted