{
  "id": "1807.01073",
  "version": "v1",
  "published": "2018-07-03T10:40:09.000Z",
  "updated": "2018-07-03T10:40:09.000Z",
  "title": "Extremal functions for Adams' inequalities in dimension four",
  "authors": [
    "Xiaomeng Li"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega\\subset \\mathbb{R}^4$ be a smooth bounded domain, $W_0^{2,2}(\\Omega)$ be the usual Sobolev space. For any positive integer $\\ell$, $\\lambda_{\\ell}(\\Omega)$ is the $\\ell$-th eigenvalue of the bi-Laplacian operator. Define $E_{\\ell}=E_{\\lambda_1(\\Omega)}\\oplus E_{\\lambda_2(\\Omega)}\\oplus\\cdots\\oplus E_{\\lambda_{\\ell}(\\Omega)}$, where $E_{\\lambda_i(\\Omega)}$ is eigenfunction space associated with $\\lambda_i(\\Omega)$. $E^{\\bot}_{\\ell}$ denotes the orthogonal complement of $E_\\ell$ in $W_0^{2,2}(\\Omega)$. For $0\\leq\\alpha<\\lambda_{\\ell+1}(\\Omega)$, we define a norm by $\\|u\\|_{2,\\alpha}^{2}=\\|\\Delta u\\|^2_2-\\alpha \\|u\\|^2_2$ for $u\\in E^\\bot_{\\ell}$. In this paper, using the blow-up analysis, we prove the following Adams inequalities $$\\sup_{u\\in E_{\\ell}^{\\bot},\\,\\| u\\|_{2,\\alpha}\\leq 1}\\int_{\\Omega}e^{32\\pi^2u^2}dx<+\\infty;$$ moreover, the above supremum can be attained by a function $u_0\\in E_{\\ell}^{\\bot}\\cap C^4(\\overline{\\Omega})$ with $\\|u_0\\|_{2,\\alpha}=1$. This result extends that of Yang (J. Differential Equations, 2015), and complements that of Lu and Yang (Adv. Math. 2009) and Nguyen (arXiv: 1701.08249, 2017).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-03T10:40:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal functions",
      "usual sobolev space",
      "th eigenvalue",
      "bi-laplacian operator",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}