Preprint

Inserted: 13 dec 2025
Last Updated: 13 dec 2025

Year: 2024

Abstract:

We investigate the location of the maximal gradient of the torsion function on some non-symmetric planar domains. First, by establishing uniform estimates for narrow domains, we prove that as a planar domain bounded by two graphs of functions becomes increasingly narrow, the location of the maximal gradient of its torsion function tends toward the endpoint of the longest vertical line segment, with smaller curvature among them. This shows that the Saint-Venant's conjecture on the location of fail points is valid for asymptotically narrow domains. Second, for triangles, we show that the maximal gradient of the torsion function always occurs on the longest sides, lying between the foot of the altitude and the midpoint of that side. Moreover, via nodal line analysis and the continuity method, we demonstrate that restricted on each side, the critical point of the gradient of the torsion function is unique and non-degenerate. Furthermore, by perturbation and barrier argument, we prove that for a class of nearly equilateral triangles, the critical point is closer to the midpoint than to the foot of the altitude, and the norm of the gradient of the torsion function has a larger value at the midpoint than at the foot of the altitude. Third, using the reflection method, we prove that for a non-concentric annulus, the maximal gradient of torsion always occurs at the point on the inner ring closest to the center of the outer ring.