{
  "id": "2210.16878",
  "version": "v1",
  "published": "2022-10-30T16:39:46.000Z",
  "updated": "2022-10-30T16:39:46.000Z",
  "title": "Interpolation inequalities on the sphere: rigidity, branches of solutions, and symmetry breaking",
  "authors": [
    "Esther Bou Dagher",
    "Jean Dolbeault"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is devoted to three Gagliardo-Nirenberg-Sobolev interpolation inequalities on the sphere. We are interested in branches of optimal functions when a scale parameter varies and investigate whether optimal functions are constant, or not. In the latter case, a symmetry breaking phenomenon occurs and our goal is to decide whether the threshold between symmetry and symmetry breaking is determined by a spectral criterion or not, that is, whether it appears as a perturbation of the constants or not. The first inequality is classical while the two other inequalities are variants which reproduce patterns similar to those observed in Caffarelli-Kohn-Nirenberg inequalities, for weighted inequalities on the Euclidean space. In the simpler setting of the sphere, it is possible to implement a parabolic version of the entropy methods associated to nonlinear diffusion equations, which is so far an open question on weighted Euclidean spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-30T16:39:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J35",
      "35B06",
      "26D10",
      "35K55",
      "53C21"
    ],
    "keywords": [
      "inequality",
      "optimal functions",
      "euclidean space",
      "nonlinear diffusion equations",
      "scale parameter varies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}