{
  "id": "1307.8039",
  "version": "v2",
  "published": "2013-07-30T16:43:18.000Z",
  "updated": "2015-02-08T17:18:22.000Z",
  "title": "Sharp constant in Riemannian L^p-Gagliardo-Nirenberg inequalities",
  "authors": [
    "Jurandir Ceccon",
    "Carlos Duran"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let (M,g) be a smooth compact Riemannian manifold of dimension n \\geq 2, 1 < p < n and 1 \\leq q < r < p^\\ast = \\frac{np}{n-p} be real parameters. This paper concerns to the validity of the optimal Gagliardo-Nirenberg inequality (\\int_M |u|^r\\; dv_g)^{\\frac{\\tau}{r \\theta}} \\leq (A_{opt} (\\int_M |\\nabla_g u|^p\\; dv_g)^{\\frac{\\tau}{p}} + B_{opt} (\\int_M |u|^p\\; dv_g)^{\\frac{\\tau}{p}}) (int_M |u|^q\\; dv_g)^{\\frac{\\tau(1 - \\theta)}{\\theta q}} \\; . This kind of inequality is studied in Chen and Sun (Nonlinear Analysis 72 (2010), pp. 3159-3172) where the authors established its validity when 2 < p < r < p^\\ast and (implicitly) \\tau = 2. Here we solve the case p \\geq r and introduce one more parameter 1 \\leq \\tau \\leq \\min\\{p,2\\}. Moreover, we prove the existence of extremal function for the optimal inequality above.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-30T16:43:18.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-08T17:18:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp constant",
      "smooth compact riemannian manifold",
      "optimal gagliardo-nirenberg inequality",
      "nonlinear analysis",
      "optimal inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.8039C"
    }
  }
}