{
  "id": "1111.2728",
  "version": "v2",
  "published": "2011-11-11T13:03:54.000Z",
  "updated": "2011-12-14T14:08:12.000Z",
  "title": "Universal moduli of continuity for solutions to fully nonlinear elliptic equations",
  "authors": [
    "Eduardo V. Teixeira"
  ],
  "comment": "16 pages - improvement of C^{1,Log-Lip} regularity theory, comments on BMO estimates on Du and D^2u",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper provides universal, optimal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations $F(X, D^2u) = f(X)$, based on weakest integrability properties of $f$ in different scenarios. The primary result established in this work is a sharp Log-Lipschitz estimate on $u$ based on the $L^n$ norm of $f$, which corresponds to optimal regularity bounds for the critical threshold case. Optimal $C^{1,\\alpha}$ regularity estimates are delivered when $f\\in L^{n+\\epsilon}$. The limiting upper borderline case, $f\\in L^\\infty$, also has transcendental importance to elliptic regularity theory and its applications. In this paper we show, under convexity assumption on $F$, that $u \\in C^{1,\\mathrm{Log-Lip}}$, provided $f$ has bounded mean oscillation. Once more, such an estimate is optimal. For the lower borderline integrability condition allowed by the theory, we establish interior \\textit{a priori} estimates on the $C^{0,\\frac{n-2\\epsilon}{n-\\epsilon}}$ norm of $u$ based on the $L^{n-\\epsilon}$ norm of $f$, where $\\epsilon$ is the Escauriaza universal constant. The exponent $\\frac{n-2\\epsilon}{n-\\epsilon}$ is optimal. When the source function $f$ lies in $L^q$, $n > q > n-\\epsilon$, we also obtain the exact, improved sharp H\\\"older exponent of continuity.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-12-14T14:08:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35J60"
    ],
    "keywords": [
      "fully nonlinear elliptic equations",
      "universal moduli",
      "continuity",
      "lower borderline integrability condition",
      "sharp log-lipschitz estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.2728T"
    }
  }
}