Torsional rigidity for cylinders with a Brownian fracture

Michiel van den Berg, Frank den Hollander

Research output: Contribution to journalArticle (Academic Journal)peer-review

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Abstract

We obtain bounds for the expected loss of torsional rigidity of a cylinder CL of length L and planar cross-section Ω due to a Brownian fracture that starts at a random point in CL and runs until the first time it exits CL. These bounds are expressed in terms of the geometry of the cross-section Ω⊂R2. It is shown that if Ω is a disc with radius R, then in the limit as L→∞ the expected loss of torsional rigidity equals cR5 for some c∈(0,∞). We derive bounds for c in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in R3 with radius 1, and runs until the first time it exits this ball.

Original languageEnglish
Pages (from-to)321-339
Number of pages19
JournalBulletin of the London Mathematical Society
Volume50
Early online date25 Jan 2018
DOIs
Publication statusPublished - Apr 2018

Keywords

  • Brownian motion
  • torsional rigidity
  • heat kernel
  • capacity

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