{
  "id": "1708.03028",
  "version": "v1",
  "published": "2017-08-09T22:37:40.000Z",
  "updated": "2017-08-09T22:37:40.000Z",
  "title": "Improved Moser--Trudinger inequality for functions with mean value zero in $\\mathbb R^n$ and its extremal functions",
  "authors": [
    "Van Hoang Nguyen"
  ],
  "comment": "24 pages, to appear in Nonlinear Analysis",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a bounded smooth domain in $\\mathbb R^n$, $W^{1,n}(\\Omega)$ be the Sobolev space on $\\Omega$, and $\\lambda(\\Omega) = \\inf\\{\\|\\nabla u\\|_n^n: \\int_\\Omega u dx =0, \\|u\\|_n =1\\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-\\Delta_n$ on $\\Omega$. For $0 \\leq \\alpha < \\lambda(\\Omega)$, let us define $\\|u\\|_{1,\\alpha}^n =\\|\\nabla u\\|_n^n -\\alpha \\|u\\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\\Omega$, \\[ \\sup_{u\\in W^{1,n}(\\Omega), \\int_\\Omega u dx =0, \\|u\\|_{1,\\alpha} =1} \\int_{\\Omega} e^{\\beta_n |u|^{\\frac n{n-1}}} dx < \\infty, \\] where $\\beta_n = n (\\omega_{n-1}/2)^{1/(n-1)}$, and $\\omega_{n-1}$ denotes the surface area of unit sphere in $\\mathbb R^n$. We also show that this supremum is attained by some function $u^*\\in W^{1,n}(\\Omega)$ such that $\\int_\\Omega u^* dx =0$ and $\\|u^*\\|_{1,\\alpha} =1$. This generalizes a result of Ngo and Nguyen \\cite{NN17} in dimension two and a result of Yang \\cite{Yang07} for $\\alpha=0$, and improves a result of Cianchi \\cite{Cianchi05}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-09T22:37:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "26D10"
    ],
    "keywords": [
      "mean value zero",
      "moser-trudinger inequality",
      "extremal functions",
      "first nonzero neumann eigenvalue",
      "laplace operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}