{
  "id": "math/0509606",
  "version": "v1",
  "published": "2005-09-26T19:31:52.000Z",
  "updated": "2005-09-26T19:31:52.000Z",
  "title": "Necessary and Sufficient Conditions for the Solvability of the $L^p$ Dirichlet Problem On Lipschitz Domains",
  "authors": [
    "Zhongwei Shen"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the homogeneous elliptic systems of order $2\\ell$ with real constant coefficients on Lipschitz domains in $R^n$, $n\\ge 4$. For any fixed $p>2$, we show that a reverse H\\\"older condition with exponent $p$ is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in $L^p$. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the $L^p$ Dirichlet problem for $n\\ge 4$ and $2-\\e< p<\\frac{2(n-1)}{n-3} +\\e$. The range of $p$ is known to be sharp if $\\ell\\ge 2$ and $4\\le n\\le 2\\ell +1$. For the polyharmonic equation, the sharp range of $p$ is also found in the case $n=6$, 7 if $\\ell=2$, and $n=2\\ell+2$ if $\\ell\\ge 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-09-26T19:31:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J40",
      "35J55"
    ],
    "keywords": [
      "dirichlet problem",
      "lipschitz domains",
      "solvability",
      "simple sufficient condition",
      "real constant coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......9606S"
    }
  }
}