Dirichlet Duality and the Nonlinear Dirichlet Problem

TL;DR

This paper introduces a geometric duality framework for solving fully nonlinear degenerate elliptic Dirichlet problems, establishing existence and uniqueness of solutions under F-convexity assumptions and exploring the structure of F-convex domains.

Contribution

It develops a duality approach using subaffine functions and introduces the concept of F-convexity, providing a unified method to analyze a broad class of nonlinear elliptic equations.

Findings

Proved existence and uniqueness of solutions under F-convexity.

Established a duality between F and F* sets for the Dirichlet problem.

Applied results to equations in calibrated, Lagrangian, and p-convex geometries.

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.