  This article gives an existence theory for weak solutions of second order
non-elliptic linear Dirichlet problems of the form {eqnarray} \nabla'P(x)\nabla
u +{\bf HR}u+{\bf S'G}u +Fu &=& f+{\bf T'g} \textrm{in}\Theta
u&=&\phi\textrm{on}\partial \Theta.{eqnarray} The principal part $\xi'P(x)\xi$
of the above equation is assumed to be comparable to a quadratic form ${\cal
Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$.
This is achieved using techniques of functional analysis applied to the
degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Omega,Q)$ and
$QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in recent work of E. Sawyer and
R. L. Wheeden. The aforementioned authors in referenced work give a regularity
theory for a subset of the class of equations dealt with here.
