{
  "id": "2011.04083",
  "version": "v1",
  "published": "2020-11-08T21:25:18.000Z",
  "updated": "2020-11-08T21:25:18.000Z",
  "title": "Positive solutions and harmonic measure for Schrödinger operators in uniform domains",
  "authors": [
    "Michael W. Frazier",
    "Igor E. Verbitsky"
  ],
  "comment": "38 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We give bilateral pointwise estimates for positive solutions of the equation \\begin{equation*} \\left\\{ \\begin{aligned} -\\triangle u & = \\omega u \\, \\,& & \\mbox{in} \\, \\, \\Omega, \\quad u \\ge 0, \\\\ u & = f \\, \\, & &\\mbox{on} \\, \\, \\partial \\Omega , \\end{aligned} \\right. \\end{equation*} in a bounded uniform domain $\\Omega\\subset {\\bf R}^n$, where $\\omega$ is a locally finite Borel measure in $\\Omega$, and $f\\ge 0$ is integrable with respect to harmonic measure $d H^{x}$ on $\\partial\\Omega$. We also give sufficient and matching necessary conditions for the existence of a positive solution in terms of the exponential integrability of $M^{*} (m \\omega)(z)=\\int_\\Omega M(x, z) m(x)\\, d \\omega (x)$ on $\\partial\\Omega$ with respect to $f \\, d H^{x_0}$, where $M(x, \\cdot)$ is Martin's function with pole at $x_0\\in \\Omega, m(x)=\\min (1, G(x, x_0))$, and $G$ is Green's function. These results give bilateral bounds for the harmonic measure associated with the Schr\\\"{o}dinger operator $-\\triangle - \\omega $ on $\\Omega$, and in the case $f=1$, a criterion for the existence of the gauge function. Applications to elliptic equations of Riccati type with quadratic growth in the gradient are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-08T21:25:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "31B35",
      "35J10"
    ],
    "keywords": [
      "harmonic measure",
      "positive solution",
      "uniform domain",
      "schrödinger operators",
      "locally finite borel measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}