Inequalities between torsional rigidity and principal eigenvalue of the p-Laplacian

Abstract

We consider the torsional rigidity and the principal eigenvalue related to the p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $$p=1$$ p = 1 and $$p=\infty $$ p = ∞ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.

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

“…Such a sequence can be obtained, for instance, by removing from a fixed open set Ω a network of points given by the vertices of a tiling of the plane with regular hexagons of vanishing size. This behavior was conjectured in [13,Open Problem 4]. The second main result of our paper is the proof of this assertion.…”

“…As a byproduct of our analysis we obtain some information concerning a shape optimization problem that recently has been investigated : those of comparing the torsional rigidity T p (Ω), defined as This supremum above was proved to be equal to 1 in [11] for p = 2 and in [13] for p ≤ N . The following Corollary gives a positive answer to the Open problem 2 of [13].…”

“…We finish with some remarks and applications in Section 5, in particular we give a positive answer to Open problem 2 of [13] concerning the shape optimization of a functional involving a competition between the first eigenvalue of the p-Laplacian and the p-torsion energy.…”

“…This paper is devoted to the study of some extremal behavior of the p-torsion function and, in particular, to answer some questions left open in [13]. We consider the mean-to-max ratio of the p-torsion function…”

“…, where ρ(Ω) stands for the inradius of Ω, that is the radius of the largest ball which can be inscribed in Ω. For convex sets, this quantity has been investigated in [13] and it has been proved that 1…”