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

Qinfeng Li, Shuangquan Xie, Hang Yang, Ruofei Yao

TL;DR

The paper analyzes the location of maximum boundary gradients of the torsion function $u$ across three non-symmetric planar settings. It obtains an explicit asymptotic expansion for narrow, graph-bounded domains that pins fail points to the ends of the longest vertical cross-section, with curvature entering in second-order terms. For triangles, it proves fail points lie on the longest side along a midpoint–altitude-foot segment, with uniqueness and non-degeneracy of the restricted critical points; perturbation and barrier arguments show nearly equilateral cases push the fail point toward the midpoint and increase the gradient there. In non-concentric annuli, the maximal gradient occurs at the inner boundary point closest to the outer center, established via a reflection method. Together, these results advance the understanding of Saint-Venant-type fail points beyond symmetric or convex domains, using a mix of asymptotic analysis, moving-plane techniques, nodal-line arguments, and barrier methods.

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.

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

TL;DR

The paper analyzes the location of maximum boundary gradients of the torsion function across three non-symmetric planar settings. It obtains an explicit asymptotic expansion for narrow, graph-bounded domains that pins fail points to the ends of the longest vertical cross-section, with curvature entering in second-order terms. For triangles, it proves fail points lie on the longest side along a midpoint–altitude-foot segment, with uniqueness and non-degeneracy of the restricted critical points; perturbation and barrier arguments show nearly equilateral cases push the fail point toward the midpoint and increase the gradient there. In non-concentric annuli, the maximal gradient occurs at the inner boundary point closest to the outer center, established via a reflection method. Together, these results advance the understanding of Saint-Venant-type fail points beyond symmetric or convex domains, using a mix of asymptotic analysis, moving-plane techniques, nodal-line arguments, and barrier methods.

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.
Paper Structure (8 sections, 23 theorems, 130 equations, 9 figures)

This paper contains 8 sections, 23 theorems, 130 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Outline of the paper
  5. Fail points on narrow domains
  6. location and uniqueness of fail points on triangles
  7. fail points on nearly equilateral triangles
  8. Fail points on non-concentric annuli

Key Result

Proposition 1.1

Let $[a, b]$ be a closed interval, $f_{1}\in C^{4}([a, b])$ be convex and $f_{2}\in C^{4}([a, b])$ be concave satisfying $f_{1}(x)=f_{2}(x)=0$ for $x=a, b$. Let and $u_{\epsilon}$ be the torsion function in $\Omega_{\epsilon}$, then we have where $\lambda_{k, 2}(x)$ is a quantity relying on second derivatives of $f_1$ and $f_2$, and its explicit formula is given in section 3.

Figures (9)

  • Figure 1.1: Planar domain bounded by two graphs
  • Figure 1.2: Stretched equilateral triangles
  • Figure 3.1: Fail point on the longest side
  • Figure 3.2: Location of fail point
  • Figure 3.3: The moving plane method for the semilinear case
  • ...and 4 more figures

Theorems & Definitions (52)

  • Conjecture 1: Saint-Venant's Conjecture
  • Conjecture 2: Ramaswamy's Conjecture
  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • Theorem 1.8
  • Remark
  • ...and 42 more