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

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 ∈ (1, N ]. For p = +∞, 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.