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

Abstract

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