Torsional Rigidity and Expected Lifetime of Brownian Motion
Rodrigo Bañuelos, M. van den Berg, Tom Carroll · Journal of the London Mathematical Society · 2002
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.