On location of maximum of gradient of torsion function

TL;DR

This paper investigates the location of the maximum gradient of the torsion function in convex domains, revealing that it does not always relate to inscribed circle contact points or boundary curvature, and provides explicit formulas and counterexamples.

Contribution

It derives a precise formula for the maximum gradient location in nearly ball domains and shows it has a nonlocal nature, challenging previous assumptions.

Findings

Maximum occurs at face centers in rectangular domains.

Counterexamples show deviation from traditional beliefs.

Derived explicit formulas for nearly ball domains.

Abstract

It has been a widely belief that for a planar convex domain with two coordinate axes of symmetry, the location of maximal norm of gradient of torsion function is either linked to contact points of largest inscribed circle or connected to points on boundary of minimal curvature. However, we show that this is not quite true in general. Actually, we derive the precise formula for the location of maximal norm of gradient of torsion function on nearly ball domains in $\mathbb{R}^n$, which displays nonlocal nature and thus does not inherently establish a connection to the aforementioned two types of points. Consequently, explicit counterexamples can be straightforwardly constructed to illustrate this deviation from conventional understanding. We also prove that for a rectangular domain, the maximum of the norm of gradient of torsion function exactly occurs at the centers of the faces of largest $(n-1)$-volume.