Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemanninan Manifolds

Abstract

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in our previous paper, "Dirichlet duality and the non-linear Dirichlet problem," Comm. on Pure and Applied Math., 62 (2009), 396–443, we define equations via closed subsets of the 2-jet bundle where each equation has a natural dual equation. 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-Ampère equation on an almost complex hermitian manifold $X$.

In general, for an equation $F$ on a manifold $X$ and a smooth domain $\Omega \subset\subset X$, we give geometric conditions which imply that the Dirichlet problem on $\Omega$ 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 introduce two fundamental concepts. The first is the notion of a monotonicity cone $M$ for $F$. If $X$ carries a global $M$-subharmonic function, then weak comparison implies full comparison. The second notion is that of boundary $F$-convexity, which is defined in terms of the asymptotics of $F$ and is used to define barriers. In combining these notions the Dirichlet problem becomes uniquely solvable as claimed.

This article also introduces the notion of local affine jet-equivalence for subequations. It is used in treating the cases above, but gives results for a much broader spectrum of equations on manifolds, including inhomogeneous equations and the Calabi-Yau equation on almost complex hermitian manifolds.

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.