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

TL;DR

This paper establishes sharp, quantitative bounds for the $p$-torsional rigidity of convex sets, including a Pólya type lower bound in any dimension and specific estimates for planar cases with constant density.

Contribution

It introduces new sharp bounds for the $p$-torsional rigidity of convex sets, extending Pólya type inequalities and providing quantitative estimates in planar cases.

Findings

Proved a Pólya type lower bound for $T_{f,p}( ext{Omega})$ in any dimension.

Derived two quantitative estimates for the planar case with $f \\equiv 1$.

Enhanced understanding of the $p$-torsional rigidity for convex sets.

Abstract

Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$, 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 P\'olya 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 $.