{
  "id": "2309.14977",
  "version": "v1",
  "published": "2023-09-26T14:53:01.000Z",
  "updated": "2023-09-26T14:53:01.000Z",
  "title": "Weak equals strong L2 regularity for partial tangential traces on Lipschitz domains",
  "authors": [
    "Nathanael Skrepek",
    "Dirk Pauly"
  ],
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "We investigate the boundary trace operators that naturally correspond to $\\mathrm{H}(\\operatorname{curl},\\Omega)$, namely the tangential and twisted tangential trace, where $\\Omega \\subseteq \\mathbb{R}^{3}$. In particular we regard partial tangential traces, i.e., we look only on a subset $\\Gamma$ of the boundary $\\partial\\Omega$. We assume both $\\Omega$ and $\\Gamma$ to be strongly Lipschitz. We define the space of all $\\mathrm{H}(\\operatorname{curl},\\Omega)$ fields that possess a $\\mathrm{L}^{2}$ tangential trace in a weak sense and show that the set of all smooth fields is dense in that space, which is a generalization of \\cite{BeBeCoDa97}. This is especially important for Maxwell's equation with mixed boundary condition as we answer the open problem by Weiss and Staffans in \\cite[Sec.~5]{WeSt13} for strongly Lipschitz pairs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-26T14:53:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "35Q61"
    ],
    "keywords": [
      "weak equals strong l2 regularity",
      "lipschitz domains",
      "regard partial tangential traces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}