Dirichlet duality and the nonlinear Dirichlet problem on Riemannian manifolds. (English) Zbl 1235.53042

The paper is devoted to the Dirichlet problem for fully nonlinear second-order equations on a Riemannian manifold. The equations are defined by means of closed subsets of the \(2\)-jet bundle where each equation has a natural dual definition. 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 homogeneous complex Monge-Ampère equations on an almost complex Hermitian manifold \(X\). In general, for an equation \(F\) on a manifold \(X\) and a smooth domain \(\Omega\subset\subset X\), geometric conditions are given which imply that the Dirichlet problem on \(\Omega\) is uniquely solvable for all continuous boundary functions. The article also introduces the notion of local affine jet-equivalence for subequations.


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58A20 Jets in global analysis
35G30 Boundary value problems for nonlinear higher-order PDEs
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