Sharp bounds for the p-torsion of convex planar domains

TL;DR

This paper derives sharp bounds for the p-torsion of convex planar domains using geometric properties and web functions, and investigates a shape optimization problem related to maximizing p-torsion among polygons.

Contribution

It introduces new sharp estimates for p-torsion based on geometric measures and confirms the Pólya-Szegő conjecture for polygons with sufficient asymmetry.

Findings

Established sharp bounds for p-torsion in convex domains.

Proved the Pólya-Szegő conjecture for polygons exceeding a certain asymmetry threshold.

Applied isoperimetric inequalities to shape optimization problems.

Abstract

We obtain some sharp estimates for the $p$-torsion of convex planar domains in terms of their area, perimeter, and inradius. The approach we adopt relies on the use of web functions (i.e. functions depending only on the distance from the boundary), and on the behaviour of the inner parallel sets of convex polygons. As an application of our isoperimetric inequalities, we consider the shape optimization problem which consists in maximizing the $p$-torsion among polygons having a given number of vertices and a given area. A long-standing conjecture by P\'olya-Szeg\"o states that the solution is the regular polygon. We show that such conjecture is true within the subclass of polygons for which a suitable notion of "asymmetry measure" exceeds a critical threshold.