{
  "id": "1511.04260",
  "version": "v1",
  "published": "2015-11-13T12:31:51.000Z",
  "updated": "2015-11-13T12:31:51.000Z",
  "title": "Homogenization of high order elliptic operators with periodic coefficients",
  "authors": [
    "Andrey Kukushkin",
    "Tatiana Suslina"
  ],
  "comment": "53 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In $L_2({\\mathbb R}^d;{\\mathbb C}^n)$, we study a selfadjoint strongly elliptic operator $A_\\varepsilon$ of order $2p$ given by the expression $b({\\mathbf D})^* g({\\mathbf x}/\\varepsilon) b({\\mathbf D})$, $\\varepsilon >0$. Here $g({\\mathbf x})$ is a bounded and positive definite $(m\\times m)$-matrix-valued function in ${\\mathbb R}^d$; it is assumed that $g({\\mathbf x})$ is periodic with respect to some lattice. Next, $b({\\mathbf D})=\\sum_{|\\alpha|=p}^d b_\\alpha {\\mathbf D}^\\alpha$ is a differential operator of order $p$ with constant coefficients; $b_\\alpha$ are constant $(m\\times n)$-matrices. It is assumed that $m\\ge n$ and that the symbol $b({\\boldsymbol \\xi})$ has maximal rank. For the resolvent $(A_\\varepsilon - \\zeta I)^{-1}$ with $\\zeta \\in {\\mathbb C} \\setminus [0,\\infty)$, we obtain approximations in the norm of operators in $L_2({\\mathbb R}^d;{\\mathbb C}^n)$ and in the norm of operators acting from $L_2({\\mathbb R}^d;{\\mathbb C}^n)$ to the Sobolev space $H^p({\\mathbb R}^d;{\\mathbb C}^n)$, with error estimates depending on $\\varepsilon$ and $\\zeta$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-13T12:31:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "high order elliptic operators",
      "periodic coefficients",
      "homogenization",
      "selfadjoint strongly elliptic operator",
      "differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151104260K"
    }
  }
}