Inhomogeneous minimization problems for the $p(x)$-Laplacian

Abstract

We study an inhomogeneous minimization problems associated to the $p(x)$-Laplacian. We make a thorough analysis of the essential properties of their minimizers and we establish a relationship with a suitable free boundary problem. On the one hand, we study the problem of minimizing the functional $J(v)={\int }_{\Omega }(\frac{|\nabla v{|}^{p(x)}}{p(x)}+\lambda (x){\chi }_{\{v>0\}}+fv)dx$. We show that nonnegative local minimizers $u$ are solutions to the free boundary problem: $u\ge 0$ and \begin{equation} \label{fbp-px}\tag{$P(f,p,{\lambda }^{\ast })$} \begin{cases} Δ_{p(x)}u:=\mbox{div}(|\nabla u(x)|^{p(x)-2}\nabla u)= f & \mbox{in }\{u>0\}\\ u=0,\ |\nabla u| = λ^*(x) & \mbox{on }\partial\{u>0\} \end{cases} \end{equation} with ${\lambda }^{\ast }(x)=(\frac{p(x)}{p(x)-1}\lambda (x){)}^{1/p(x)}$ and that the free boundary is a $C^{1,\alpha }$ surface. On the other hand, we study the problem of minimizing the functional $J_{\epsilon }(v)={\int }_{\Omega }(\frac{|\nabla v{|}^{p_{\epsilon }(x)}}{p_{\epsilon }(x)}+B_{\epsilon }(v)+f_{\epsilon }v)dx$, where $B_{\epsilon }(s)={\int }_0^s{\beta }_{\epsilon }(\tau )d\tau$, $\epsilon >0$, ${\beta }_{\epsilon }(s)=\frac{1}{\epsilon }\beta (\frac{s}{\epsilon })$, with $\beta$ a Lipschitz function satisfying $\beta >0$ in $(0,1)$, $\beta \equiv 0$ outside $(0,1)$. We prove that if $u_{\epsilon }$ are nonnegative local minimizers, then any limit function $u$ ($\epsilon \to 0$) is a solution to the free boundary problem $P(f,p,{\lambda }^{\ast })$ with ${\lambda }^{\ast }(x)=(\frac{p(x)}{p(x)-1}M{)}^{1/p(x)}$, $M=\int \beta (s)ds$, $p=\lim p_{\epsilon }$, $f=\lim f_{\epsilon }$, and that the free boundary is a $C^{1,\alpha }$ surface. In order to obtain our results we need to overcome deep technical difficulties and develop new strategies, not present in the previous literature for this type of problems.