{
  "id": "1701.08249",
  "version": "v1",
  "published": "2017-01-28T04:55:09.000Z",
  "updated": "2017-01-28T04:55:09.000Z",
  "title": "A sharp Adams inequality in dimension four and its extremal functions",
  "authors": [
    "Van Hoang Nguyen"
  ],
  "comment": "33 pages, comment are welcome",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a smooth oriented bounded domain in $\\mathbb R^4$, $H_0^2(\\Omega)$ be the Sobolev space, and $\\lambda_1(\\Omega)= \\inf \\{\\|\\Delta u\\|_2^2 : u\\in H_0^2(\\Omega), \\|u\\|_2 =1\\}$ be the first eigenvalue of the bi-Laplacian operator $\\Delta^2$ on $\\Omega$. For $\\alpha \\in [0,\\lambda_1(\\Omega))$, we define $\\|u\\|_{2,\\alpha}^2 = \\|\\Delta u\\|_2^2 - \\alpha \\|u\\|_2^2$, for $u \\in H_0^2(\\Omega)$. In this paper, we will prove the following inequality \\[ \\sup_{u\\in H_0^2(\\Omega),\\, \\|u\\|_{2,\\alpha} \\leq 1} \\int_{\\Omega} e^{32 \\pi^2 u(x)^2} dx < \\infty. \\] This strengthens a recent result of Lu and Yang \\cite{LuYang}. We also show that there exists a function $u^*\\in H_0^2(\\Omega)\\cap C^4(\\overline{\\Omega})$ such that $\\|u^*\\|_{2,\\alpha} =1$ and the supremum above is attained by $u^*$. Our proofs are based on the blow-up analysis method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-28T04:55:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35"
    ],
    "keywords": [
      "sharp adams inequality",
      "extremal functions",
      "blow-up analysis method",
      "bi-laplacian operator",
      "sobolev space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}