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

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - {\Delta} u + b(x) \cdot \nabla u = 1 \qquad \text{in}~{\Omega} $$\end{document} subject to Dirichlet boundary conditions for ∥b∥L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|b\|_{L^{\infty }}$\end{document} 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|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b = -\|b\|_{L^{\infty }} \nabla u/ \lvert \nabla u\rvert $\end{document} which reduces the problem to the study of the nonlinear PDE −Δu−b⋅∇u=1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -{\Delta} u - b \cdot \left| \nabla u \right| = 1, $$\end{document} where b=∥b∥L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b = \|b\|_{L^{\infty }}$\end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\| \nabla u\|_{L^{1}}$\end{document} and ∥Δu∥L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|{\Delta } u\|_{L^{1}}$\end{document} are maximized if Ω is the ball (the ball is also known to maximize ∥u∥Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\|u\|_{L^{p}}$\end{document} for p ≥ 1 from a result of Hamel & Russ).