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- W3017638463 abstract "We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE $$ - {Delta} u + b(x) cdot nabla u = 1 qquad text{in}~{Omega} $$ subject to Dirichlet boundary conditions for $|b|_{L^{infty }}$ 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^{infty }} nabla u/ lvert nabla urvert $ which reduces the problem to the study of the nonlinear PDE $$ -{Delta} u - b cdot left| nabla u right| = 1, $$ where $b = |b|_{L^{infty }}$ 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, $| nabla u|_{L^{1}}$ and $|{Delta } u|_{L^{1}}$ are maximized if Ω is the ball (the ball is also known to maximize $|u|_{L^{p}}$ for p ≥ 1 from a result of Hamel & Russ)." @default.
- W3017638463 created "2020-05-01" @default.
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- W3017638463 date "2020-04-22" @default.
- W3017638463 modified "2023-10-12" @default.
- W3017638463 title "Optimal Trapping for Brownian Motion: a Nonlinear Analogue of the Torsion Function" @default.
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- W3017638463 doi "https://doi.org/10.1007/s11118-020-09845-5" @default.
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