{
  "id": "1601.02325",
  "version": "v1",
  "published": "2016-01-11T05:26:54.000Z",
  "updated": "2016-01-11T05:26:54.000Z",
  "title": "Regularity of Weak Solutions of Elliptic and Parabolic Equations with Some Critical or Supercritical Potentials",
  "authors": [
    "Zijin Li",
    "Qi S. Zhang"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove H\\\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\\Delta u-\\frac{A}{|x|^{2+\\beta}}u=0,\\,\\,(\\beta\\geq 0)$, and variable second order term coefficients case $%% \\begin{equation}\\label{01} \\partial_{i} (a_{ij}(x) \\partial_{j}u(x)) - \\frac{A}{|x|^{2+\\beta}} u(x) =0\\quad (A>0,\\quad\\beta\\geq 0), \\end{equation} \\begin{equation}\\label{02} \\partial_{i} (a_{ij}(x,t) \\partial_{j}u(x,t)) - \\frac{A}{|x|^{2+\\beta}} u(x,t)-\\partial_{t}u(x,t) =0\\quad (A>0,\\quad\\beta\\geq 0), \\end{equation} with critical or supercritical 0-order term coefficients which are beyond De Giorgi-Nash-Moser's Theory. We also prove, in some special cases, weak solutions are even differentiable. Previously P. Baras and J. A. Goldstein \\cite{Baras1984} treated the case when $A<0$, $(a_{ij})=I$ and $\\beta=0$ for which they show that there does not exist any regular positive solution or singular positive solutions, depending on the size of $|A|$. When $A>0$, $\\beta=0$ and $(a_{ij})=I$, P. D. Milman and Y. A. Semenov \\cite{Milman2003}\\cite{Milman2004} obtain bounds for the heat kernel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-11T05:26:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "35J15",
      "35K10"
    ],
    "keywords": [
      "parabolic equations",
      "weak solutions",
      "supercritical potentials",
      "second order term coefficients case",
      "regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102325L"
    }
  }
}