On the location of the maximal gradient of the torsion function over some non-symmetric planar domains

TL;DR

This paper studies where the maximum gradient of the torsion function occurs in various non-symmetric planar domains, confirming conjectures and revealing geometric influences on the location of these maxima.

Contribution

It provides new results on the location of the maximal gradient in narrow convex domains, triangles, and annuli, extending understanding of Saint-Venant's conjecture and geometric effects.

Findings

Maximal gradient converges to endpoints in narrow convex domains.

In triangles, the maximum occurs on the longest side, near the midpoint.

In non-concentric annuli, the maximum is on the inner boundary closest to the outer center.

Abstract

We investigate the location of the maximal gradient of the torsion function on certain non-symmetric planar domains. First, by establishing uniform estimates for convex narrow domains, we show that as a planar domain bounded by two graphs becomes increasingly narrow, the location of the maximal gradient of its torsion function converges to the endpoints of the longest vertical segment, with smaller curvature among them. This result confirms that Saint-Venant's conjecture on the location of fail points holds for asymptotically narrow domains. Second, for triangles, we prove that the maximal gradient of the torsion function always occurs on the longest side, lying between the foot of the altitude and the midpoint of that side. Moreover, via nodal line analysis, we show that, restricted to each side, the critical point of the gradient is unique and non-degenerate. Additionally, by perturbation and barrier arguments, we establish that for a class of nearly equilateral triangles, this critical point is closer to the midpoint than to the foot of the altitude, and the maximal gradient at the midpoint exceeds that at the foot of the altitude. Third, employing the reflection method, we demonstrate that for a non-concentric annulus, the maximal gradient of the torsion function is always attained at the point on the inner boundary closest to the center of the outer boundary.