Dirichlet Duality and the Nonlinear Dirichlet Problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F.Hessu/ D 0 on a smoothly bounded domain  b Rn. In our approach the equation is replaced by a subset F Sym.Rn/ of the symmetric n nmatrices with @F fF D 0g. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F -convexity” assumption on the boundary @. We also study the topological structure of F -convex domains and prove a theorem of Andreotti-Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set z F that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of z F is F , and in the analysis the roles of F and z 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 MongeAmpère 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. c 2008 Wiley Periodicals, Inc.