Towards Optimal Gradient Bounds for the Torsion Function in the Plane

TL;DR

This paper establishes a universal upper bound for the gradient of the torsion function in convex planar domains, improving classical estimates using advanced mathematical tools and numerical methods.

Contribution

It introduces a new bound for the gradient of the torsion function in convex domains, refining previous estimates with a universal constant less than (2π)^{-1/2}.

Findings

Proves the gradient bound with a universal constant c < (2π)^{-1/2}.

Numerical construction shows the optimal constant c ≥ 0.358.

Highlights applications in elasticity, Brownian motion, and inequalities for subharmonic functions.

Abstract

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex domain and let $u$ be the solution of $-\Delta u = 1$ vanishing on the boundary $\partial \Omega$. The estimate $$ \| \nabla u\|_{L^{\infty}(\Omega)} \leq c |\Omega|^{1/2}$$ is classical. We use the P-functional, the stability theory of the torsion function and Brownian motion to establish the estimate for a universal $c < (2\pi)^{-1/2}$. We also give a numerical construction showing that the optimal constant satisfies $c \geq 0.358$. The problem is important in different settings: (1) as the maximum shear stress in Saint Venant Elasticity Theory, (2) as an optimal control problem for the constrained maximization of the lifetime of Brownian motion started close to the boundary and (3) and optimal Hermite-Hadamard inequalities for subharmonic functions on convex domains.