Geometric properties of optimizers for the maximum gradient of the torsion function

TL;DR

This paper investigates the geometric characteristics of convex domains that optimize certain gradient-based functionals related to the torsion function, revealing that maximizers have smooth boundaries with line segments where the maximum gradient is attained.

Contribution

It proves the existence of convex domains that maximize specific gradient functionals and characterizes their boundary regularity and structure.

Findings

Existence of maximizers for the functionals J and J_P.

Maximizers have a C^1 boundary with a line segment where | abla u_\Omega| attains its maximum.

Maximizers are convex domains with specific geometric properties.

Abstract

Consider $J(\Omega):= \|\nabla u_\Omega\|_\infty/\sqrt{|\Omega|} $ and $J_P(\Omega):= \|\nabla u_\Omega\|_\infty/P(\Omega) $, where $\Omega$ is a planar convex domain, $u_\Omega$ is the torsion function, $P(\Omega)$ is the perimeter of $\Omega$ and $|\Omega|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_\Omega|$ attains its maximum.