{
  "id": "2401.10066",
  "version": "v2",
  "published": "2024-01-18T15:32:41.000Z",
  "updated": "2024-03-01T09:50:41.000Z",
  "title": "$L^p$ continuity of eigenprojections for 2-d Dirichlet Laplacians under perturbations of the domain",
  "authors": [
    "Ryan L. Acosta Babb",
    "James C. Robinson"
  ],
  "comment": "27 pages, 2 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We generalise results by Lamberti and Lanza de Cristoforis (2005) concerning the continuity of projections onto eigenspaces of self-adjoint differential operators with compact inverses as the (spatial) domain of the functions is perturbed in $\\mathbb{R}^2$. Our main case of interest is the Dirichlet Laplacian. We extend these results from bounds from $H_0^1$ to $H_0^1$ to bounds from $L^p$ to $L^p$, under the assumption that $(-\\Delta^{-1}-z)^{-1}$ is $L^p$ bounded when $z$ lies outside of the spectrum of $-\\Delta^{-1}$. We show that this assumption is met if the initial domain is a square or a rectangle.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-01T09:50:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A70",
      "35P10",
      "42B08"
    ],
    "keywords": [
      "dirichlet laplacian",
      "continuity",
      "perturbations",
      "eigenprojections",
      "self-adjoint differential operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}