English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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).

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