{
  "id": "2012.14779",
  "version": "v1",
  "published": "2020-12-29T14:45:12.000Z",
  "updated": "2020-12-29T14:45:12.000Z",
  "title": "Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality",
  "authors": [
    "P. R. Stinga",
    "M. Vaughan"
  ],
  "comment": "54 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We define the fractional powers $L^s=(-a^{ij}(x)\\partial_{ij})^s$, $0 < s < 1$, of nondivergence form elliptic operators $L=-a^{ij}(x)\\partial_{ij}$ in bounded domains $\\Omega\\subset\\mathbb{R}^n$, under minimal regularity assumptions on the coefficients $a^{ij}(x)$ and on the boundary $\\partial\\Omega$. We show that these fractional operators appear in several applications such as fractional Monge--Amp\\`ere equations, elasticity, and finance. The solution $u$ to the nonlocal Poisson problem $$\\begin{cases} (-a^{ij}(x) \\partial_{ij})^su = f&\\hbox{in}~\\Omega\\\\ u=0&\\hbox{on}~\\partial\\Omega \\end{cases}$$ is characterized with a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge--Amp\\`ere geometry and prove the interior Harnack inequality and H\\\"older estimates for solutions to the extension problem when the coefficients $a^{ij}(x)$ are bounded, measurable functions. This in turn implies the interior Harnack inequality and H\\\"older estimates for solutions $u$ to the fractional problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-29T14:45:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional elliptic equations",
      "interior harnack inequality",
      "applications",
      "definition",
      "nondivergence form elliptic operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 54,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}