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We shall discuss the inhomogeneous Dirichlet problem for: $f(x,u, Du, D^2u) =\psi(x)$ where $f$ is a "natural" differential operator, with a restricteddomain $F$, on a manifold $X$. By "natural" we mean operators that ariseintrinsically from a given geometry on $X$. An important point is that theequation need not be convex and can be highly degenerate. Furthermore, theinhomogeneous term can take values at the boundary of the restricted domain $F$of the operator $f$. A simple example is the real Monge-Amp\`ere operator ${\rmdet}({\rm Hess}\,u) = \psi(x)$ on a riemannian manifold $X$, where ${\rm Hess}$is the riemannian Hessian, the restricted domain is $F = \{{\rm Hess} \geq0\}$, and $\psi$ is continuous with $\psi\geq0$.  A main new tool is the idea of local jet-equivalence, which gives rise tolocal weak comparison, and then to comparison under a natural and necessaryglobal assumption.  The main theorem applies to pairs $(F,f)$, which are locally jet-equivalentto a given constant coefficient pair $({\bf F}, {\bf f})$. This covers a largefamily of geometric equations on manifolds: orthogonally invariant operators ona riemannian manifold, G-invariant operators on manifolds with G-structure,operators on almost complex manifolds, and operators, such as the LagrangianMonge-Amp\`ere operator, on symplectic manifolds. It also applies to allbranches of these operators.  Complete existence and uniqueness results are established with existencerequiring the same boundary assumptions as in the homogeneous case [10]. Wealso have results where the inhomogeneous term $\psi$ is a delta function.

The inhomogeneous Dirichlet Problem for natural operators on manifolds