{
  "id": "2204.02532",
  "version": "v1",
  "published": "2022-04-06T01:16:07.000Z",
  "updated": "2022-04-06T01:16:07.000Z",
  "title": "Critical Sets of Elliptic Equations with Rapidly Oscillating Coefficients in Two Dimensions",
  "authors": [
    "Fanghua Lin",
    "Zhongwei Shen"
  ],
  "comment": "12 pages. arXiv admin note: text overlap with arXiv:2203.13393",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we continue the study of critical sets of solutions $u_\\e$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. In \\cite{Lin-Shen-3d}, by controling the \"turning\" of approximate tangent planes, we show that the $(d-2)$-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period $\\e$, provided that doubling indices for solutions are bounded. In this paper we use a different approach, based on the reduction of the doubling indices of $u_\\e$, to study the two-dimensional case. The proof relies on the fact that the critical set of a homogeneous harmonic polynomial of degree two or higher in dimension two contains only one point.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-06T01:16:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "35B27"
    ],
    "keywords": [
      "critical set",
      "rapidly oscillating coefficients",
      "dimensional hausdorff measures",
      "doubling indices",
      "approximate tangent planes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}