{
  "id": "2210.06329",
  "version": "v1",
  "published": "2022-10-12T15:49:37.000Z",
  "updated": "2022-10-12T15:49:37.000Z",
  "title": "Homogenization Theory of Elliptic System with Lower Order Terms for Dimension Two",
  "authors": [
    "Wei Wang",
    "Ting Zhang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the homogenization problem for generalized elliptic systems $$ \\mathcal{L}_{\\va}=-\\operatorname{div}(A(x/\\va)\\nabla+V(x/\\va))+B(x/\\va)\\nabla+c(x/\\va)+\\lambda I $$ with dimension two. Precisely, we will establish the $ W^{1,p} $ estimates, H\\\"{o}lder estimates, Lipschitz estimates and $ L^p $ convergence results for $ \\mathcal{L}_{\\va} $ with dimension two. The operator $ \\mathcal{L}_{\\va} $ has been studied by Qiang Xu with dimension $ d\\geq 3 $ in \\cite{Xu1,Xu2} and the case $ d=2 $ is remained unsolved. As a byproduct, we will construct the Green functions for $ \\mathcal{L}_{\\va} $ with $ d=2 $ and their convergence rates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-12T15:49:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lower order terms",
      "homogenization theory",
      "homogenization problem",
      "generalized elliptic systems",
      "lipschitz estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}