On a global gradient estimate in $p$-Laplacian problems

Abstract

We make explicit the $p$-dependence of $C$ in the gradient estimate ${\Vert \nabla u\Vert }_{\infty }^{p-1}\le C{\Vert f\Vert }_{N,1}$ by Cianchi and Maz'ya (2011). In such inequality, the constant $C$ is uniform with respect to $f\in L^{N,1}(\Omega ),$ and $u$ is the weak solution to the Poisson equation $-\mathrm{div}({\left|\nabla u\right|}^{p-2}\nabla u)=f$ in a bounded domain $\Omega \subset {\mathbb{R}}^N,$ $N\ge 3,$ coupled with either Neumann or Dirichlet homogeneous boundary conditions. The case $N=2$ with $f\in L^q(\Omega ),$ for some $q>2,$ is also considered .

19 pages