We make explicit the $p$-dependence of $C$ in the gradient estimate $\left\Vert \nabla u\right\Vert _{ \infty}^{ p-1}\leq C\left\Vert f\right\Vert _{ N,1}$ 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 $-\operatorname{ div}(\left\vert \nabla u\right\vert ^{ p-2}\nabla u)=f$ in a bounded domain $\Omega\subset\mathbb{ R}^{ N},$ $N\geq3,$ 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 .
Categories: math.AP