The inhomogeneous dirichlet problem for natural operators on manifolds

We discuss the inhomogeneous Dirichlet problem written locally as: f(x, u,Du,D²u) = φ(x) where f is a "natural" differential operator on a manifold X, with a restricted domain F in the space of 2-jets. "Naturality" refers to operators that arise intrinsically from a given geometry on X. Importantly, the equation need not be convex and can be highly degenerate. Furthermore, φ can take the values of f on @F. A main new tool is the idea of local jet-equivalence, which gives rise to local weak comparison, and then to comparison under a natural and necessary global assumption. The main theorem covers many geometric equations, for example: orthogonally invariant operators on a riemannian manifold, G-invariant operators on manifolds with G-structure, operators on almost complex and symplectic manifolds. It also applies to all branches of these operators. Complete existence and uniqueness results are established. There are also results where φ is a delta function.