{
  "id": "0912.5220",
  "version": "v2",
  "published": "2009-12-29T15:46:07.000Z",
  "updated": "2010-03-19T15:45:45.000Z",
  "title": "Hyperbolic polynomials and the Dirichlet problem",
  "authors": [
    "F. Reese Harvey",
    "H. Blaine Lawson Jr"
  ],
  "comment": "Minor modifications in the exposition have been made.",
  "categories": [
    "math.AP",
    "math.CO",
    "math.DG",
    "math.OC"
  ],
  "abstract": "This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. In a second, independent part of the paper, the relationship of Garding's theory to the authors' recent work (arXiv:0710.3991) on the Dirichlet problem for fully nonlinear partial differential equations is investigated. Let p be a homogeneous polynomial of degree m on S^2(R^n) which is hyperbolic with respect to the all positive directions A \\geq 0. Then p has an associated eigenvalue map lambda:S^2(R^n) \\to R^m, defined modulo the permutation group acting on R^m. Consequently, each closed symmetric set E of R^m induces a second-order p.d.e. by requiring, for a C^2-function u in n-variables, that (D^2 u)(x) lie in the boundary of E for all x. Assume that E + (R_+)^m is contained in E. A main result is that for smooth domains in R^n whose boundary is suitably (p,E)-pseudo-convex, the Dirichlet problem has a unique continuous solution for all continuous boundary data. This applies to a vast collection of examples the most basic of which are the m distinct branches of the equation p(D^2 u) =0. In the authors' recent extension of results from euclidean domains to domains in riemannian manifolds (arXiv:0907.1981), a new global ingredient, called a monotonicity subequation, was introduced. It is shown in this paper that for every polynomial $p$ as above, the associated Garding cone is a monotonicity cone for all branches of the the equation p(Hess u) = 0 where Hess u denotes the riemannian Hessian of u.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-03-19T15:45:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "58J32",
      "35J70",
      "90C05",
      "52A41",
      "15A45"
    ],
    "keywords": [
      "dirichlet problem",
      "hyperbolic polynomials",
      "gardings theory",
      "fully nonlinear partial differential equations",
      "real analytic arrangement"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.5220R"
    }
  }
}