Mean-to-max ratio of the torsion function and honeycomb structures

TL;DR

This paper investigates the extremal behavior of the mean-to-max ratio of the p-torsion function, revealing geometric bounds and asymptotic behaviors related to honeycomb-like structures in specific cases.

Contribution

It establishes uniform upper bounds for the mean-to-max ratio of the p-torsion function for p > N and characterizes asymptotic attainment by hexagonal tilings in two dimensions for p=+ ∞.

Findings

Upper bound below 1 for p > N

Asymptotic attainment by hexagonal tilings in 2D for p=+ ∞

Distinct behaviors depending on the relation between p and the dimension N

Abstract

In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.