{
  "id": "1506.08041",
  "version": "v1",
  "published": "2015-06-26T12:07:44.000Z",
  "updated": "2015-06-26T12:07:44.000Z",
  "title": "On ratios of harmonic functions II",
  "authors": [
    "Alexander Logunov",
    "Eugenia Malinnikova"
  ],
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We study the ratio of harmonic functions $u,v$ which have the same zero set $Z$ in the unit ball $B\\subset \\mathbb{R}^n$. The ratio $f=u/v$ can be extended to a real analytic, nowhere vanishing function in $B$. We prove the Harnack inequality and the gradient estimate for such ratios: for a given compact set $K\\subset B$ we show that $\\sup_K|f|\\le C_1\\inf_K|f|$ and $\\sup_K\\left|\\nabla f\\right|\\le C_2 \\inf_K|f|$, where $C_1$ and $C_2$ depend only on $K,Z$. In dimension two we specify these estimates by showing that only the number of nodal domains of $u$ plays a role.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-26T12:07:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "harmonic functions",
      "unit ball",
      "real analytic",
      "zero set",
      "harnack inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}