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} ablaP(x) abla u +{bf HR}u+{bf SG}u +Fu &=& f+{bf Tg} textrm{in}Theta u&=&phitextrm{on}partial Theta.{eqnarray} The principal part $xiP(x)xi$ of the above equation is assumed to be comparable to a quadratic form ${cal Q}(x,xi) = xiQ(x)xi$ that may vanish for non-zero $xiinmathbb{R}^n$. 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.