{
  "id": "1708.00859",
  "version": "v1",
  "published": "2017-08-02T07:22:23.000Z",
  "updated": "2017-08-02T07:22:23.000Z",
  "title": "Spectral approach to homogenization of hyperbolic equations with periodic coefficients",
  "authors": [
    "Mark Dorodnyi",
    "Tatiana Suslina"
  ],
  "comment": "41 pages. arXiv admin note: substantial text overlap with arXiv:1606.05868, arXiv:1508.07641",
  "categories": [
    "math.AP"
  ],
  "abstract": "In $L_2(\\mathbb{R}^d;\\mathbb{C}^n)$, we consider selfadjoint strongly elliptic second order differential operators ${\\mathcal A}_\\varepsilon$ with periodic coefficients depending on ${\\mathbf x}/ \\varepsilon$, $\\varepsilon>0$. We study the behavior of the operators $\\cos( {\\mathcal A}^{1/2}_\\varepsilon \\tau)$ and ${\\mathcal A}^{-1/2}_\\varepsilon \\sin( {\\mathcal A}^{1/2}_\\varepsilon \\tau)$, $\\tau \\in \\mathbb{R}$, for small $\\varepsilon$. Approximations for these operators in the $(H^s\\to L_2)$-operator norm with a suitable $s$ are obtained. The results are used to study the behavior of the solution ${\\mathbf v}_\\varepsilon$ of the Cauchy problem for the hyperbolic equation $\\partial^2_\\tau {\\mathbf v}_\\varepsilon = - \\mathcal{A}_\\varepsilon {\\mathbf v}_\\varepsilon +\\mathbf{F}$. General results are applied to the acoustics equation and the system of elasticity theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-02T07:22:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "periodic coefficients",
      "hyperbolic equation",
      "spectral approach",
      "elliptic second order differential operators",
      "strongly elliptic second order differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}