Math @ Duke

Publications [#349489] of Jianfeng Lu
Papers Published
 Lu, J; Steinerberger, S, Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function,
Potential Analysis, vol. 54 no. 4
(April, 2021),
pp. 687698 [doi]
(last updated on 2021/05/12)
Abstract: 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


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