{
  "id": "2110.15104",
  "version": "v2",
  "published": "2021-10-28T13:34:05.000Z",
  "updated": "2022-03-25T16:53:47.000Z",
  "title": "Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients",
  "authors": [
    "Federico Franceschini",
    "Federico Glaudo"
  ],
  "comment": "Added reference to companion software, included a section on the covariance of the construction",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\\subseteq\\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\\to\\mathbb R$ such that $Lu=\\delta_0$. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the \"complexity\" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by $\\mathbb Z^2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-25T16:53:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A08",
      "33C55",
      "35C20",
      "05C22"
    ],
    "keywords": [
      "second-order elliptic operator",
      "fundamental solution",
      "analytic coefficients",
      "analytic riemannian manifold",
      "homogeneous term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}