{
  "id": "2101.04841",
  "version": "v1",
  "published": "2021-01-13T02:44:27.000Z",
  "updated": "2021-01-13T02:44:27.000Z",
  "title": "Doubling inequalities and nodal sets in periodic elliptic homogenization",
  "authors": [
    "Carlos E. Kenig",
    "Jiuyi Zhu",
    "Jinping Zhuge"
  ],
  "comment": "Comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove explicit doubling inequalities and obtain uniform upper bounds (under $(d-1)$-dimensional Hausdorff measure) of nodal sets of weak solutions for a family of linear elliptic equations with rapidly oscillating periodic coefficients. The doubling inequalities, explicitly depending on the doubling index, are proved at different scales by a combination of convergence rates, a three-ball inequality from large-scale analyticity, and a monotonicity formula of a frequency function. The upper bounds of nodal sets are shown by using the doubling inequalities, approximations by harmonic functions and an iteration argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-13T02:44:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A02",
      "35B27",
      "35J15"
    ],
    "keywords": [
      "nodal sets",
      "periodic elliptic homogenization",
      "inequality",
      "uniform upper bounds",
      "dimensional hausdorff measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}