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 $$p \in (1,N]$$ p ∈ ( 1 , N ] . For $$p=+\infty $$ 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.