{
  "id": "1705.01236",
  "version": "v1",
  "published": "2017-05-03T02:18:33.000Z",
  "updated": "2017-05-03T02:18:33.000Z",
  "title": "Constant scalar curvature equation and the regularity of its weak solution",
  "authors": [
    "Yu Zeng",
    "Weiyong He"
  ],
  "comment": "22 pages. Comments are welcome",
  "categories": [
    "math.DG",
    "math.AP"
  ],
  "abstract": "In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on K\\\"ahler manifolds. We first define a notion of weak solution of CSCK for an $L^\\infty$ K\\\"ahler metric. The main result is to show that such a weak solution (with uniform $L^\\infty$ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of $K$-energy minimizers. The new technical ingredient is a $W^{2, 2}$ regularity result for the Laplacian equation $\\Delta_g u=f$ on K\\\"ahler manifolds, where the metric has only $L^\\infty$ coefficients. It is well-known that such a $W^{2, 2}$ regularity ($W^{2, p}$ regularity for any $p>1$) fails in general (except for dimension two) for uniform elliptic equations of the form $a^{ij}\\partial^2_{ij}u=f$ for $a^{ij}\\in L^\\infty$, without certain smallness assumptions on the local oscillation of $a^{ij}$. We observe that the K\\\"ahler condition plays an essential role to obtain a $W^{2, 2}$ regularity for elliptic equations with only $L^\\infty$ elliptic coefficients on compact manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-03T02:18:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weak solution",
      "regularity",
      "study constant scalar curvature equation",
      "nonlinear fourth order elliptic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}