{
  "id": "0907.1981",
  "version": "v4",
  "published": "2009-07-13T13:50:44.000Z",
  "updated": "2011-11-09T20:18:41.000Z",
  "title": "Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds",
  "authors": [
    "F. Reese Harvey",
    "H. Blaine Lawson Jr"
  ],
  "comment": "Final version, ready for publication",
  "categories": [
    "math.AP",
    "math.CV",
    "math.DG"
  ],
  "abstract": "In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting with a constant coefficient equation as a model, which then universally determines an equation on every riemannian manifold which is equipped with a topological reduction of the structure group to the invariance group of the model. For example, this covers all branches of the homogeneous complex Monge-Ampere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain D in X, we give geometric conditions which imply that the Dirichlet problem on D is uniquely solvable for all continuous boundary functions. We begin by introducing a weakened form of comparison which has the advantage that local implies global. We then associate to F two natural \"conical subequations\": a monotonicity subequation M and the asymptotic interior of F. If X carries a global M-subharmonic function, then weak comparison implies full comparison. The asymptotic interior of F is used to formulate boundary convexity and provide barriers. In combination the Dirichlet problem becomes uniquely solvable as claimed. A considerable portion of the paper is concerned with specific examples. They include a wide variety of equations which make sense on any riemannian manifold, and many which hold universally on almost complex or quaternionic hermitian manifolds, or topologically calibrated manifolds.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2011-11-09T20:18:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J32",
      "53C99",
      "35J25",
      "35J70",
      "32W20"
    ],
    "keywords": [
      "riemannian manifold",
      "nonlinear dirichlet problem",
      "dirichlet duality",
      "weak comparison implies full comparison",
      "wide variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.1981R"
    }
  }
}