{
  "id": "2104.02341",
  "version": "v1",
  "published": "2021-04-06T07:53:24.000Z",
  "updated": "2021-04-06T07:53:24.000Z",
  "title": "Weyl formula for the eigenvalues of the dissipative acoustic operator",
  "authors": [
    "Vesselin Petkov"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\\partial_{\\nu} u - \\gamma(x) u = 0$ on the boundary $\\Gamma$ and $\\gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \\: t \\geq 0.$ The eigenvalues $\\lambda_k$ of $G$ with ${\\rm Re}\\: \\lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{\\lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $\\min_{x\\in \\Gamma} \\gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-06T07:53:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20",
      "35P25",
      "47A40",
      "58J50"
    ],
    "keywords": [
      "dissipative acoustic operator",
      "weyl formula",
      "eigenvalues",
      "formula concerns",
      "wave equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}