{
  "id": "0710.3991",
  "version": "v1",
  "published": "2007-10-22T08:11:34.000Z",
  "updated": "2007-10-22T08:11:34.000Z",
  "title": "Dirichlet Duality and the Nonlinear Dirichlet Problem",
  "authors": [
    "F. Reese Harvey",
    "H. Blaine Lawson, Jr"
  ],
  "categories": [
    "math.AP",
    "math.CV",
    "math.DG"
  ],
  "abstract": "We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess, u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices, with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric ``F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of the Special Lagrangian potential equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-10-22T08:11:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "35J70",
      "32W20"
    ],
    "keywords": [
      "nonlinear dirichlet problem",
      "dirichlet duality",
      "special lagrangian potential equation",
      "degenerate elliptic equations",
      "dual dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.3991R"
    }
  }
}