Dirichlet duality and the nonlinear dirichlet problem

We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F(Hess u) = 0 on a smoothly bounded domain Ω ∈ Rdbl; ⁿ. In our approach the equation is replaced by a subset F ⊂ Sym ²(ℝ ⁿ) of the symmetric n ×n matrices with ∂F ⊆ {F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric "F-convexity" assumption on the boundary ∂ Ω. 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 F∼ that 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- Ampére 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.