Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients

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} $\text{in}$\Theta u&=&\phi$\text{on}$\partial \Theta.{eqnarray} The principal part ${\xi }^{\mathrm{\prime }}P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\cal Q}(x,\xi) = \xi’Q(x)\xi$thatmayvanishfornon-zero$\xi\in$\mathbb{R}$^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $Q{H}^1(\mathrm{\Theta })=W^{1,2}(\mathrm{\Omega },Q)$ and $Q{H}_0^1(\mathrm{\Theta })=W_0^{1,2}(\mathrm{\Theta },Q)$ 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.