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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.

##### MSC:
 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
##### Keywords:
Dirichlet problem; Riemannian manifold; jet-equivalence
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