A minimization problem for the p(x)-Laplacian involving area

Abstract

In the present article we study a minimization problem in $\mathbb R^N$ involving the perimeter of the positivity set of the solution u and the integral of $|\nabla u|^{p(x)}$. Here p(x) is a Lipschitz continuous function such that $1<p_\mathrm{min}\le p(x)\le p_\mathrm{max}<\infty$. We prove that such a minimizing function exists and that it is a classical solution to a free boundary problem. In particular, the reduced free boundary is a $C^2$ surface and the dimension of the singular set is at most $N-8$. Under further regularity assumptions on the exponent p(x) we get more regularity of the free boundary. In particular, if $p\in C^\infty$ we have that $\partial _\mathrm{red}\{u>0\}$ is a $C^\infty$ surface.