Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function
© 2020, Springer Nature B.V. 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 C2 −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).
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