Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function

Journal Article (Journal Article)

We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE−Δu+b(x)⋅∇u=1inΩ subject to Dirichlet boundary conditions for ∥b∥L∞ fixed. We show that, in any given C −domain Ω, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the solution and satisfies b=−∥b∥L∞∇u/|∇u| which reduces the problem to the study of the nonlinear PDE− Δ u− b⋅ | ∇ u| = 1 , where b=∥b∥L∞ is a constant. We believe that this PDE is a natural and interesting nonlinear analogue of the torsion function (b = 0). We prove that, for fixed volume, ∥∇u∥L1 and ∥Δu∥L1 are maximized if Ω is the ball (the ball is also known to maximize ∥u∥Lp for p ≥ 1 from a result of Hamel & Russ). 2

Full Text

Duke Authors

Cited Authors

  • Lu, J; Steinerberger, S

Published Date

  • April 1, 2021

Published In

Volume / Issue

  • 54 / 4

Start / End Page

  • 687 - 698

Electronic International Standard Serial Number (EISSN)

  • 1572-929X

International Standard Serial Number (ISSN)

  • 0926-2601

Digital Object Identifier (DOI)

  • 10.1007/s11118-020-09845-5

Citation Source

  • Scopus