{
  "id": "1508.00708",
  "version": "v1",
  "published": "2015-08-04T09:14:20.000Z",
  "updated": "2015-08-04T09:14:20.000Z",
  "title": "Symmetry and spectral properties for viscosity solutions of fully nonlinear equations",
  "authors": [
    "Isabeau Birindelli",
    "Fabiana Leoni",
    "Filomena Pacella"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study symmetry properties of viscosity solutions of fully nonlinear uniformly elliptic equations. We show that if $u$ is a viscosity solution of a rotationally invariant equation of the form $F(x,D^2u)+f(x,u)=0$, then the operator $\\mathcal{L}_u=\\mathcal{M}^++\\frac{\\partial f}{\\partial u}(x,u)$, where $\\mathcal{M}^+$ is the Pucci's sup--operator, plays the role of the linearized operator at $u$. In particular, we prove that if $u$ is a solution in a radial bounded domain, if $f$ is convex in $u$ and if the principal eigenvalue of $\\mathcal{L}_u$ (associated with positive eigenfunctions) in any half domain is nonnegative, then $u$ is foliated Schwarz symmetric. We apply our symmetry results to obtain bounds on the spectrum and to deduce properties of possible nodal eigenfunctions for the operator $\\mathcal{M}^+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-04T09:14:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60"
    ],
    "keywords": [
      "viscosity solution",
      "fully nonlinear equations",
      "spectral properties",
      "fully nonlinear uniformly elliptic equations",
      "study symmetry properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150800708B"
    }
  }
}