{
  "id": "2007.13177",
  "version": "v1",
  "published": "2020-07-26T16:48:39.000Z",
  "updated": "2020-07-26T16:48:39.000Z",
  "title": "Homogenization of hyperbolic equations with periodic coefficients in ${\\mathbb R}^d$: sharpness of the results",
  "authors": [
    "Mark Dorodnyi",
    "Tatiana Suslina"
  ],
  "comment": "95 pages",
  "journal": "Algebra I Analiz 32 (2020), no. 4 (Russian); English transl., St. Petersburg Math. J., 32 (2021), no. 4",
  "categories": [
    "math.AP"
  ],
  "abstract": "In $L_2({\\mathbb R}^d;{\\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\\mathcal A}_\\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\\mathcal A}_\\varepsilon$ are periodic and depend on ${\\mathbf x}/\\varepsilon$, where $\\varepsilon >0$ is a small parameter. We find approximations for the operators $\\cos ( {\\mathcal A}_\\varepsilon^{1/2}\\tau)$ and ${\\mathcal A}_\\varepsilon^{-1/2}\\sin ( {\\mathcal A}_\\varepsilon^{1/2}\\tau)$ in the norm of operators acting from the Sobolev space $H^s({\\mathbb R}^d)$ to $L_2({\\mathbb R}^d)$ (with suitable $s$). We also find approximation with corrector for the operator ${\\mathcal A}_\\varepsilon^{-1/2}\\sin ( {\\mathcal A}_\\varepsilon^{1/2}\\tau)$ in the $(H^s \\to H^1)$-norm. The question about the sharpness of the results with respect to the type of the operator norm and with respect to the dependence of estimates on $\\tau$ is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation $\\partial_\\tau^2 {\\mathbf u}_\\varepsilon = - {\\mathcal A}_\\varepsilon {\\mathbf u}_\\varepsilon + {\\mathbf F}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-26T16:48:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic equation",
      "periodic coefficients",
      "elliptic second order differential operator",
      "strongly elliptic second order differential",
      "homogenization"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 95,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}