{
  "id": "1305.1065",
  "version": "v2",
  "published": "2013-05-06T00:10:22.000Z",
  "updated": "2013-07-24T16:05:30.000Z",
  "title": "Geometric Properties of Gelfand's Problems with Parabolic Approach",
  "authors": [
    "Sunghoon Kim",
    "Ki-Ahm Lee"
  ],
  "comment": "16 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the asymptotic profiles of the nonlinear parabolic flows $$(e^{u})_{t}= \\La u+\\lambda e^u$$ to show the geometric properties of the following elliptic nonlinear eigenvalue problems known as a Gelfand's problem: \\begin{equation*} \\begin{split} \\La \\vp &+ \\lambda e^{\\vp}=0, \\quad \\vp>0\\quad\\text{in $\\Omega$}\\\\ \\vp&=0\\quad\\text{on $\\Omega$} \\end{split} \\end{equation*} posed in a strictly convex domain $\\Omega\\subset\\re^n$. In this work, we show that there is a strictly increasing function $f(s)$ such that $f^{-1}(\\vp(x))$ is convex for $0<\\lambda\\leq\\lambda^{\\ast}$, i.e., we prove that level set of $\\vp$ is convex. Moreover, we also present the boundary condition of $\\vp$ which guarantee the $f$-convexity of solution $\\vp$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-24T16:05:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gelfands problem",
      "geometric properties",
      "parabolic approach",
      "elliptic nonlinear eigenvalue problems",
      "nonlinear parabolic flows"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.1065K"
    }
  }
}