Calculus of Variations and Geometric Measure Theory

L. Briani - D. Bucur

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

created by briani1 on 23 Jan 2023



Inserted: 23 jan 2023
Last Updated: 23 jan 2023

Year: 2023


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.

Keywords: $p$-Laplacian, Torsion function, principal eigenvalue, honeycomb