Torsional Rigidity and Expected Lifetime of Brownian Motion

Abstract

Let D be an open set in euclidean space Rm with non‐empty boundary ∂D, and let pD : D × D × [0,∞) → R be the Dirichlet heat kernel for the parabolic operator −Δ + ∂/∂t, where −Δ is the Dirichlet laplacian on L2(D). Since the Dirichlet heat kernel is non‐negative, we may define the (open) set function PD=∫0∞∫D∫DpD(x,y;t)dxdydt. (1.1) We say that D has finite torsional rigidity if PD < ∞. It is well known that if D has finite volume, then D has finite torsional rigidity [11]. As we shall see, the converse is not true. The main purpose of this paper is to obtain necessary and sufficient conditions on the geometry of D to guarantee finite torsional rigidity and to gain some understanding of the behaviour of the expected lifetime of brownian motion in a certain natural class of domains that do not have finite torsional rigidity.

Torsional rigidity of submanifolds with controlled geometry

“…[15], with the purpose to give some idea about the volume-relative swiftness of the Brownian motion defined on the submanifold P at infinity, in connection with the more classical properties like transience and recurrence. As alluded to in the Introduction, we have been inspired partially by the works [2,3], where the authors find upper bounds for the torsional rigidity of domains in Euclidean spaces which satisfy Hardy inequalities. These inequalities guarantee that the boundaries of the domains are not too thin so that the Brownian diffusion is guaranteed sufficient room for escape.…”

“…In the work [2] the existence of regions in R m with finite torsional rigidity and yet infinite volume were considered. To get to such regions, the authors assume Hardy inequalities for these domains.…”

The best constant in a fractional Hardy inequality

“…By Lemma 2 below, (8), and (3), we obtain (1) for u ∈ C ∞ c (D) ⊂ Dom(C), with κ d,α given by (2). The case of general u ∈ C c (D) is obtained by an approximation.…”

“…However, A d,−α κ d,α → 1/4 and Γ 2 ( 1+α 2 )/π → 1/4 as α → 2, an agreement with the classical Hardy inequality for Laplacian ( [11]) related to the fact that for u ∈ C ∞ c (D), ∆ α/2 D u → ∆u and C(u, u) → − ∆u(x)u(x)dx = |∇u(x)| 2 dx as α → 2 (the latter holds by Taylor's expansion of order 2, and a similar result is valid for K). For the vast literature concerning optimal weights and constants in the classical Hardy inequalities, and their applications we refer to [5,15,11,20,23,18,2].…”

On the torsion function with Robin or Dirichlet boundary conditions

“…The question of the sharp constant in the upper bound in the right hand side of (2) for these sets was addressed in [3,4]. In this paper we consider the torsion function u b for the Laplacian with Robin boundary conditions.…”