{
  "id": "1908.06273",
  "version": "v1",
  "published": "2019-08-17T09:33:15.000Z",
  "updated": "2019-08-17T09:33:15.000Z",
  "title": "Optimal Trapping of Brownian Motion: A Nonlinear Analogue of the Torsion Function",
  "authors": [
    "Jianfeng Lu",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "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 \\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).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-17T09:33:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "torsion function",
      "brownian motion",
      "optimal trapping",
      "expected lifetime",
      "dirichlet boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}