{
  "id": "0812.4788",
  "version": "v2",
  "published": "2008-12-28T03:03:53.000Z",
  "updated": "2010-08-04T20:23:34.000Z",
  "title": "Asymptotic Analysis of Boundary Layer Correctors and Applications",
  "authors": [
    "D. Onofrei",
    "B. Vernescu"
  ],
  "comment": "29 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we extend the ideas presented in Onofrei and Vernescu [\\textit{Asymptotic Analysis, 54, 2007, 103-123}] and introduce suitable second order boundary layer correctors, to study the $H^1$-norm error estimate for the classical problem in homogenization. Previous second order boundary layer results assume either smooth enough coefficients (which is equivalent to assuming smooth enough correctors $\\chi_j,\\chi_{ij}\\in W^{1,\\infty}$), or smooth homogenized solution $u_0$, to obtain an estimate of order $\\displaystyle O(\\epsilon^{\\frac{3}{2}})$. For this we use the periodic unfolding method developed by Cioranescu, Damlamian and Griso [\\textit{C. R. Acad. Sci. Paris, Ser. I 335, 2002, 99-104}]. We prove that in two dimensions, for nonsmooth coefficients and general data, one obtains an estimate of order $\\displaystyle O(\\epsilon^\\frac{3}{2})$. In three dimenssions the same estimate is obtained assuming $\\chi_j,\\chi_{ij}\\in W^{1,p}$, with $p>3$. We also discuss how our results extend, in the case of nonsmooth coefficients, the convergence proof for the finite element multiscale method proposed by T.Hou et al. [\\textit{ J. of Comp. Phys., 134, 1997, 169-189}] and the first order correctoranalysis for the first eigenvalue of a composite media obtained by Vogelius et al.[\\textit{Proc. Royal Soc. Edinburgh, 127A, 1997, 1263-1299}].",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-08-04T20:23:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27",
      "35R05"
    ],
    "keywords": [
      "asymptotic analysis",
      "second order boundary layer results",
      "second order boundary layer correctors",
      "order boundary layer results assume",
      "suitable second order boundary layer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.4788O"
    }
  }
}