An ultracontractivity property for semigroups generated by the $p$-Laplacian with nonlinear Wentzell-Robin boundary conditions

Abstract

Let ${\Omega}\subset{\mathbb{R}}^N$ be a bounded domain with a Lipschitz boundary and let ${\sigma}$ be the restriction to ${\partial \Omega}$ of the $(N-1)$-dimensional Hausdorff measure. Let $B:\;{\partial \Omega}\times{\mathbb{R}}\to [0,+\infty]$ be ${\sigma}$-measurable in the first variable and assume that for ${\sigma}$-almost every $x\in{\partial \Omega}$, $B(x,\cdot)$ is a proper, convex, lower semicontinuous functional. We prove that, for every $p\in [2, N)$, the nonlinear submarkovian semigroup $S_{B,p}$ on $L^2({\Omega})\times L^2({\partial \Omega})$ generated by the operator $\Delta_p:=\mbox{div}(|\nabla u|^{p-2}\nabla u)$ with the nonlinear Wentzell-Robin type boundary conditions \[\Delta_pu+|\nabla u|^{p-2}\frac{\partial u}{\partial n} +\beta(\cdot,u)\ni 0\;\;\mbox{ on }\;{\partial \Omega},\] satisfies the following H\"older-continuity type property: for every $q\ge 2$, and for all $U_0, V_0\in L^q({\Omega})\times L^q({\partial \Omega})$ the following holds: \[\|S_{B,p}(t)U_0-S_{B,p}(t)V_0\|_{L^\infty({\Omega})\times L^\infty({\partial \Omega})}\le C_1e^{C_2t}t^{-\alpha}\|U_0-V_0\|_{L^q({\Omega})\times L^q({\partial \Omega})} ^{\gamma},\] where $C_1$ and $C_2$ are positive constants depending only on $p, q, N$ and ${\Omega}$ and $\alpha, \gamma$ are suitable positive constants depending only on $p, q$ and $N$.