{
  "id": "1505.02453",
  "version": "v1",
  "published": "2015-05-10T23:17:10.000Z",
  "updated": "2015-05-10T23:17:10.000Z",
  "title": "The smoothness problem of eigenvalues of the Laplace operator on the plane",
  "authors": [
    "Julian Haddad",
    "Marcos Montenegro"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "A classical open problem involving the Laplace operator on symmetric domains in Rn, n >= 2, such as balls and cubes, is whether all its Dirichlet eigenvalues vary smoothly upon one-parameter C1 perturbations of the domain. We provide a fairly complete answer to this question in dimension n = 2 on disks and squares. We also give a positive answer to the problem for the second eigenvalue on balls in Rn. The proofs are based on a suitable framework of problem and on a new degenerate implicit function theorem on Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-10T23:17:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35Pxx",
      "47A75"
    ],
    "keywords": [
      "laplace operator",
      "smoothness problem",
      "degenerate implicit function theorem",
      "one-parameter c1 perturbations",
      "banach spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}