Dirichlet Duality and the Nonlinear Dirichlet Problem on Riemannian Manifolds

By F. R. Harvey, H. Lawson. 2009

145 citations recorded by the source corpus.

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. 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-Ampere equation on an almost complex hermitian manifold X. In general, for an equation F on a manifold X and a smooth domain D in X, we give geometric conditions which imply that the Dirichlet problem on D 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 associate to F two natural "conical subequations": a monotonicity subequation M and the asymptotic interior of F. If X carries a global M-subharmonic function, then weak comparison implies full comparison. The asymptotic interior of F is used to formulate boundary convexity and provide barriers. In combination the Dirichlet problem becomes uniquely solvable as claimed. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/eb595e525579df776d5a22925ba2cd91d8239c43

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

The Monge-Ampère equation for (n − 1)-plurisubharmonic functions on a compact Kähler manifold (2013, background): “Harvey-Lawson [27, 28] introduced a more general notion of k-plurisubharmonicity, where the complex lines in this second definition are replaced by complex k-planes.”
The Monge–Ampère equation for non-integrable almost complex structures (2016, unknown): “1) is replaced by a Monge-Ampère type equation for (n − 1)-plurisubharmonic functions [26, 17, 18, 38, 50, 51].”
Gauduchon metrics with prescribed volume form (2015, methodology): “As an aside, if α is the Euclidean metric on Cn then the condition Pα(uij) > 0 is equivalent to saying that u is (n− 1)- plurisubharmonic, in the sense of Harvey-Lawson [17].”
Open problems in pluripotential theory (2015, background): “A viscosity approach complementary to the pluripotential one has been proposed in [59,97,98].”
$$C^{2,\alpha }$$C2,α estimates for nonlinear elliptic equations in complex and almost complex geometry (2014, background): “Harvey– Lawson introduced this equation and the notion of (n −1)-plurisubharmonic functions in Cn [32].”