Journal of Differential Geometry
- J. Differential Geom.
- Volume 88, Number 3 (2011), 395-482.
Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemanninan Manifolds
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.
J. Differential Geom., Volume 88, Number 3 (2011), 395-482.
First available in Project Euclid: 15 November 2011
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Harvey, F. Reese; Lawson, H. Blaine. Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemanninan Manifolds. J. Differential Geom. 88 (2011), no. 3, 395--482. doi:10.4310/jdg/1321366356. https://projecteuclid.org/euclid.jdg/1321366356