An Inhomogeneous Minimization Problem for the p(x)-Laplacian

LEDERMAN, CLAUDIA; WOLANSKI, NOEMÍ

Valparaíso

Congreso; Congreso SUMA 2016 - Primer Encuentro Conjunto entre la Sociedad de Matemática de Chile y la Unión Matemática Argentina; 2016

Sociedad de Matemática de Chile y la Unión Matemática Argentina

We willpresent the results we obtained on the problem of minimizing the functional $J_{\varepsilon}(v)=\displaystyle\int_\Omega \Big(\frac{|\nablav|^{{p_{\varepsilon}}(x)}}{{p_{\varepsilon}}(x)}+B_{\varepsilon}(v)+{f^{{\varepsilon}}}v\Big)\, dx$, where $\Omega\subset{\mathbb R}^N$,$B_{\varepsilon}(s)=\int _0^s\beta_{\varepsilon}(\tau) \,d\tau$, ${\varepsilon}>0$,${\beta}_{\varepsilon}(s)={1 \over \varepsilon} \beta({s\over\varepsilon})$, with $\beta$ a Lipschitz function satisfying$\beta>0$ in $(0,1)$, $\beta\equiv 0$ outside$(0,1)$.   We proved that if ${u^\varepsilon}$ are nonnegative localminimizers, then they are solutions of \begin{equation}\label{(1)}\Delta_{p_{\varepsilon}(x)}{u^\varepsilon}:=\mbox{div}(|\nabla{u^\varepsilon}(x)|^{p_{\varepsilon}(x)-2}\nabla{u^\varepsilon})={\beta}_{\varepsilon}({u^\varepsilon})+{f^{{\varepsilon}}},\quadu^{\varepsilon}\geq 0.\end{equation}Moreover, if the functions ${u^\varepsilon}$,${f^{{\varepsilon}}}$ y $p_{\varepsilon}$ are uniformly bounded, we proved thatlimit functions $u$ (${\varepsilon}\to 0$) are solutions tothe inhomogeneous free boundary problem for the $p(x)$-Laplacian: $u\ge0$ and$$        \left\{            \begin{array}{l}            \Delta_{p(x)}u=f \qquad \qquad \qquad \mbox{in }\{u>0\}\\            u=0,\ |\nablau| = \lambda^*(x) \qquad \mbox{on }\partial\{u>0\}            \end{array}            \right.$$with$\lambda^*(x)=\Big(\frac{p(x)}{p(x)-1}\,M\Big)^{1/p(x)}$, $M=\int\beta(s)\, ds$, $p=\lim p_{\varepsilon}$, $f=\lim{f^{{\varepsilon}}}$,  and that the freeboundary $\partial\{u>0\}$ is a $C^{1,\alpha}$ surface with exception of a subset of zero${\mathcal H}^{N-1}$ measure. See \cite{LW1, LW2, LW3}. When $p_{\varepsilon}(x)\equiv 2$ y ${f^{{\varepsilon}}}\equiv 0$, problem\eqref{(1)} apears in combustion theory, in the description ofdeflagrationflames (\cite{BCN, CLW}). The inhomogeneous case,  ${f^{{\varepsilon}}}\not\equiv 0$, allowsmore general combustion models with nonlocal difusion and / or  transport. The case$p_{\varepsilon}(x)\not\equiv 2$ allows the consideration of models withelectromagnetic effects.