{
  "id": "1206.4089",
  "version": "v3",
  "published": "2012-06-18T22:08:27.000Z",
  "updated": "2013-08-21T17:45:53.000Z",
  "title": "Optimal gradient continuity for degenerate elliptic equations",
  "authors": [
    "Damião J. Araújo",
    "Gleydson C. Ricarte",
    "Eduardo V. Teixeira"
  ],
  "comment": "Fixed few typos",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\\mathscr{L}(X, \\nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \\textit{a priori} unknown singular set of an existing solution, $\\mathscr{S}(u) := \\{X : \\nabla u(X) = 0 \\}$. The innovative feature of our main result concerns its optimality -- the sharp, encoded smoothness aftereffects of the operator. Such a quantitative information usually plays a decisive role in the analysis of a number of analytic and geometric problems. Our result is new even for the classical equation $|\\nabla u | \\cdot \\Delta u = 1$. We further apply these new estimates in the study of some well known problems in the theory of elliptic PDEs.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-08-21T17:45:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J70",
      "35B45",
      "35B65"
    ],
    "keywords": [
      "degenerate elliptic equations",
      "2nd order partial differential equations",
      "optimal gradient continuity estimates",
      "unknown singular set",
      "main result concerns"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.4089A"
    }
  }
}