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

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