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

TL;DR

This paper explicitly determines how the constant in a gradient estimate for p-Laplacian problems depends on p, extending known inequalities to include explicit p-dependence and considering different dimensions and boundary conditions.

Contribution

It makes explicit the p-dependence of the constant in a key gradient estimate for p-Laplacian problems, extending previous results to include this dependence and different boundary conditions.

Findings

Explicit p-dependence of the gradient estimate constant is established.

The estimate is uniform with respect to the source term in L^{N,1}().

The case for N=2 with f in L^{q}(), q>2, is also analyzed.

Abstract

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 .