{
  "id": "1201.6683",
  "version": "v2",
  "published": "2012-01-31T20:53:12.000Z",
  "updated": "2017-01-14T18:10:18.000Z",
  "title": "Homogenization of the boundary value for the Dirichlet Problem",
  "authors": [
    "Sunghan Kim",
    "Ki-ahm Lee",
    "Henrik Shahgholian"
  ],
  "categories": [
    "math.AP",
    "math.DS"
  ],
  "abstract": "In this paper,we give a mathematically rigorous proof of the averaging behavior of oscillatory surface integrals, which has been a well-known yet unsolved problem. Based on ergodic theory, we find a sharp geometric condition called IDDC under which the averaging takes place. It should be stressed that this condition does not imply any control on the curvature of the given surface. As an application to partial differential equations, we prove boundary layer homogenization for elliptic systems with Dirichlet boundary data, by characterizing the oscillatory Poisson kernels in a periodic manner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-31T20:53:12.000Z",
      "abstract": "The Dirichlet problem with oscillating boundary data is the subject of study in this paper. It turns out that due to integral representation of such problems we can reduce the study to the case of surface integrals of rapidly oscillating functions, and their limit behavior: $$ \\lim_{\\e \\to 0} \\int_{\\Gamma} g(y,\\frac{y}{\\e}) d\\sigma_y, $$ where $g(x,y)$, represents the boundary value in the Dirichlet problem. The lower dimensional character of the surface $\\Gamma$ produces unexpected and surprising effective limits. In general, the limit of the integral depends strongly on the sequence $\\e=\\e_j$ chosen. Notwithstanding this, when the surface does not have flat portions with \\emph{rational directions} a full averaging takes place and we obtain a unique effective limit in the above integral. The results here are connected to recent works of D. G\\'erard-Varet and N. Masmoudi, where they study this problem in combination with homogenization of the operator, in convex domains.",
      "comment": null,
      "journal": null,
      "doi": null,
      "authors": [
        "Ki-ahm Lee",
        "Henrik Shahgholian"
      ]
    },
    {
      "version": "v2",
      "updated": "2017-01-14T18:10:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35J25",
      "35J57",
      "35J65",
      "28D99",
      "37A25",
      "47A35"
    ],
    "keywords": [
      "dirichlet problem",
      "boundary value",
      "homogenization",
      "lower dimensional character",
      "flat portions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.6683L"
    }
  }
}