{
  "id": "1209.0483",
  "version": "v2",
  "published": "2012-09-03T20:45:49.000Z",
  "updated": "2013-10-19T14:31:09.000Z",
  "title": "Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates",
  "authors": [
    "Hayk Aleksanyan",
    "Per Sjölin",
    "Henrik Shahgholian"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $u_\\e$ be a solution to the system $$ \\mathrm{div}(A_\\e(x) \\nabla u_{\\e}(x))=0 \\text{\\ in} D, \\qquad u_{\\e}(x)=g(x,x/\\e) \\text{\\ on}\\partial D, $$ where $D \\subset \\R^d $ ($d \\geq 2$), is a smooth uniformly convex domain, and $g$ is 1-periodic in its second variable, and both $A_\\e$ and $g$ reasonably smooth. Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\\e(x) =A(x)$, with the following convergence rate for all $1\\leq p <\\infty$ $$ \\|u_\\e - u_0\\|_{L^p(D)} \\leq C_p \\begin{cases} \\e^{1/2p} ,&\\text{$d=2$}, (\\e |\\ln \\e |)^{1/p}, &\\text{$d = 3$}, \\e^{1/p} ,&\\text{$d \\geq 4$,} \\end{cases} $$ which we prove is (generically) sharp for $d\\geq 4$. Here $u_0$ is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen \\cite{KLS1}, we prove (for certain class of operators and when $d\\geq 3$) $$ || u_\\e - u_0 ||_{L^p(D)} \\leq C_p [ \\e (\\ln(1/ \\e))^2 ]^{1/p}. $$ for both oscillating operator and boundary data. For this case, we take $A_\\e=A(x/\\e)$, where $A$ is 1-periodic as well. Some further applications of the method to the homogenization of Neumann problem with oscillating boundary data are also considered.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-19T14:31:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B27"
    ],
    "keywords": [
      "dirichlet problem",
      "fourier analysis",
      "applications",
      "homogenization",
      "boundary data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.0483A"
    }
  }
}