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On convex domains maximizing the gradient of the torsion function

Linhang Huang

TL;DR

The paper investigates how large the gradient of the torsion function can be on convex planar domains by recasting the PDE problem via conformal maps $f:\mathbb{D}\to\Omega$ and formulating two shape-functionals $\mathbb{L}_1$ and $\mathbb{L}_2$ that are scale-invariant. Through domain-restriction and curvature-signature perturbations, the authors derive two sharp structural conclusions about extremizers: (i) if the boundary around a gradient-maximum point is $C^{1+\alpha}$, no adjacent subarcs can meet at an angle sharper than $\pi/2$, and (ii) extremizers must exhibit either non-$C^{2+\varepsilon}$ regularity or have infinitely many zero-curvature points accumulating near the maximizing boundary point. The core contribution is an Euler-Lagrange framework expressed entirely in conformal geometry, linking the PDE problem to a rich geometric-analytic structure via $v_f(z)=\mathrm{Re}(1+z f''(z)/f'(z))$ and the curvature signature $k_f$. Numerical results based on a Sweers-type family indicate near-optimal domains resemble a rounded 'D', corroborating the theoretical bounds and illustrating the practical impact for shape optimization in elasticity contexts.

Abstract

We consider the solution of $-Δu = 1$ on convex domains $Ω\subset \mathbb{R}^2$ subject to Dirichlet boundary conditions $u =0$ on $\partial Ω$. Our main concern is the behavior of $\|\nabla u\|_{L^{\infty}}$, also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems $\| \nabla u\|_{L^{\infty}}/ |Ω|^{1/2}$ and $\| \nabla u\|_{L^{\infty}}/ H^1( \partial Ω)$. Numerically, the extremal domain for each functional looks a bit like the rounded letter `D'. We prove that (1) either the extremal domain does not have a $C^{2 + \varepsilon}$ boundary or (2) there exists an infinite set of points on $\partial Ω$ where the curvature vanishes. Either scenario seems curious and is rarely encountered for such problems. The techniques are based on finding a representation of the functional using only conformal geometry and classic perturbation arguments.

On convex domains maximizing the gradient of the torsion function

TL;DR

The paper investigates how large the gradient of the torsion function can be on convex planar domains by recasting the PDE problem via conformal maps and formulating two shape-functionals and that are scale-invariant. Through domain-restriction and curvature-signature perturbations, the authors derive two sharp structural conclusions about extremizers: (i) if the boundary around a gradient-maximum point is , no adjacent subarcs can meet at an angle sharper than , and (ii) extremizers must exhibit either non- regularity or have infinitely many zero-curvature points accumulating near the maximizing boundary point. The core contribution is an Euler-Lagrange framework expressed entirely in conformal geometry, linking the PDE problem to a rich geometric-analytic structure via and the curvature signature . Numerical results based on a Sweers-type family indicate near-optimal domains resemble a rounded 'D', corroborating the theoretical bounds and illustrating the practical impact for shape optimization in elasticity contexts.

Abstract

We consider the solution of on convex domains subject to Dirichlet boundary conditions on . Our main concern is the behavior of , also known as the maximum shear stress in Elasticity Theory and first investigated by Saint Venant in 1856. We consider the two shape optimization problems and . Numerically, the extremal domain for each functional looks a bit like the rounded letter `D'. We prove that (1) either the extremal domain does not have a boundary or (2) there exists an infinite set of points on where the curvature vanishes. Either scenario seems curious and is rarely encountered for such problems. The techniques are based on finding a representation of the functional using only conformal geometry and classic perturbation arguments.

Paper Structure

This paper contains 27 sections, 30 theorems, 156 equations, 8 figures.

Table of Contents

  1. Introduction and Results
  2. Introduction
  3. The Problem
  4. Statement of Results
  5. The techniques
  6. Preliminaries
  7. Conformal Geometry on Convex Domains
  8. Two Types of Perturbations
  9. Setup for Torsion Functions
  10. $L_i$ is well-defined.
  11. Structure of the proofs
  12. Domain Restriction
  13. Analytic Domains
  14. Family of Shrinking Disks
  15. Local Square Integrability on $\partial\mathbb{D}$
  16. ...and 12 more sections

Key Result

Theorem 1.1

If $\Omega$ is a convex maximizer of $\|\nabla u\|_{L^{\infty}}/|\Omega|^{1/2}$ and if the boundary is $\partial \Omega$ is $C^{1+\alpha}$ around any maximizing point for some $\alpha>0$, then $\partial \Omega$ cannot admit two adjacent $C^{1+\beta}$-smooth subarcs that meet at an angle sharper than

Figures (8)

  • Figure 1: Left: an example of a convex domain for which $\|\nabla u\|_{L^{\infty}}/|\Omega|^{1/2} \sim 0.8926/\sqrt{2\pi}$. Right: am example of a domain where one has $\|\nabla u\|_{L^{\infty}}/\mathcal{H}^1(\Omega) \sim 0.8723/(\sqrt{8}\pi)$. Both are from a family of domains described by Sweers (see hoskins2021towards and Section 6).
  • Figure 2: Image of concentric circles under $\varphi$
  • Figure 3: Domain restriction.
  • Figure 4: Changing the curvature signature.
  • Figure 5: Cone $C_{1.2}(\pi/3)$ highlighted in blue
  • ...and 3 more figures

Theorems & Definitions (51)

  • Theorem 1.1: No Sharp Corners
  • Theorem 1.2: Main Result
  • Lemma 2.1: Characterization of Convexity, see duren2001univalent
  • Lemma 2.2: Study's Theorem, emch1912study
  • proof
  • Lemma 2.3: Radial Increase, pascu2002scaling
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5: Characterization by the curvature signature
  • ...and 41 more