Optimal Trapping of 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 $$- \Delta u + b(x) \cdot \nabla u = 1 \qquad \mbox{in}~\Omega$$ subject to Dirichlet boundary conditions for $\|b\|_{L^{\infty}}$ fixed. We show that, in any given $C^2-$domain $\Omega$, the vector field maximizing the expected lifetime is (nonlinearly) coupled to the solution and satisfies $b = -\|b\|_{L^{\infty}} \nabla u/ |\nabla u|$ 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. We prove that, for fixed volume, $\| \nabla u\|_{L^1}$ and $\|\Delta u\|_{L^1}$ are maximized if $\Omega$ is the ball (the ball is also known to maximize $\|u\|_{L^p}$ for $p \geq 1$ from a result of Hamel \& Russ).