Sharp and quantitative estimates for the $p-$Torsion of convex sets

Abstract

Let $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, be a bounded, open and convex set and let $f$ be a positive and non-increasing function depending only on the distance from the boundary of $\Omega$. We consider the $p-$torsional rigidity associated to $\Omega$ for the Poisson problem with Dirichlet boundary conditions, denoted by $T_{f,p}(\Omega )$. Firstly, we prove a Pólya type lower bound for $T_{f,p}(\Omega )$ in any dimension; then, we consider the planar case and we provide two quantitative estimates in the case $f\equiv 1$.

18 pages, 4 figures