{
  "id": "2401.11486",
  "version": "v1",
  "published": "2024-01-21T13:18:17.000Z",
  "updated": "2024-01-21T13:18:17.000Z",
  "title": "Expansion of Green's function and regularity of Robin's function for elliptic operators in divergence form",
  "authors": [
    "Daomin Cao",
    "Jie Wan"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \\mathcal{L}_K=-\\text{div}(K(x)\\nabla ) $ on a bounded smooth domain $ \\Omega\\subseteq\\mathbb{R}^n (n\\geq 2) $ with zero Dirichlet boundary condition, where $ K $ is a smooth positively definite matrix-valued function on $ \\Omega $. We obtain a high-order asymptotic expansion of $ G_K(x, y) $, which defines uniquely a regular part $ H_K(x, y) $. Moreover, we prove that the associated Robin's function $ R_K(x) = H_K(x, x) $ is smooth in $ \\Omega $, despite the regular part $ H_K\\notin C^1(\\Omega\\times\\Omega) $ in general.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-21T13:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "divergence form",
      "robins function",
      "greens function",
      "elliptic operator",
      "regular part"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}