  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 restricted
domain $F$, on a manifold $X$. By "natural" we mean operators that arise
intrinsically from a given geometry on $X$. An important point is that the
equation need not be convex and can be highly degenerate. Furthermore, the
inhomogeneous 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 ${\rm
det}({\rm Hess}\,u) = \psi(x)$ on a riemannian manifold $X$, where ${\rm Hess}$
is the riemannian Hessian, the restricted domain is $F = \{{\rm Hess} \geq
0\}$, and $\psi$ is continuous with $\psi\geq0$.
  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 applies to pairs $(F,f)$, which are locally jet-equivalent
to a given constant coefficient pair $({\bf F}, {\bf f})$. This covers a large
family of geometric equations on manifolds: orthogonally invariant operators on
a riemannian manifold, G-invariant operators on manifolds with G-structure,
operators on almost complex manifolds, and operators, such as the Lagrangian
Monge-Amp\`ere operator, on symplectic manifolds. It also applies to all
branches of these operators.
  Complete existence and uniqueness results are established with existence
requiring the same boundary assumptions as in the homogeneous case [10]. We
also have results where the inhomogeneous term $\psi$ is a delta function.
