We study the problem of maximizing the expected lifetime of drift diffusion in a bounded domain. More formally, we consider the PDE

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 \(b = -\|b\|_{L^{\infty }} \nabla u/ \lvert \nabla u\rvert \) which reduces the problem to the study of the nonlinear PDE

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).

我们研究最大化有界域中漂移扩散的预期寿命的问题。更正式地说，我们考虑PDE

固定为\（\ | b \ | __ {L ^ {\ infty}} \）的Dirichlet边界条件。我们证明，在任何给定的C ^2-区域Ω中，使期望寿命最大化的矢量场（非线性）耦合到解，并且满足\（b =-\ | b \ | _ {L ^ {\ infty}} \ nabla u / \ lvert \ nabla u \ rvert \）将问题简化为非线性PDE的研究

其中\（b = \ | b \ | _ {L ^ {\ infty}} \）是一个常数。我们认为该PDE是扭转函数的自然且有趣的非线性模拟（b = 0）。我们证明了，对于固定体积，\（\ | \ nabla U \ | _ {L ^ {1}} \）和\（\ | {\三角洲} U \ | _ {L ^ {1}} \）是最大化如果Ω是球（球还已知最大化\（\ | U \ | _ {L ^ {p}} \）为p从哈默＆拉斯的结果≥1）。