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

Abstract

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}Θu&=&ϕ\textrm{on}\partial Θ.{eqnarray} The principal part ${\xi }^{\prime }P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form $\mathcal{Q}(x,\xi )={\xi }^{\prime }Q(x)\xi$ 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 $Q{H}^1(\Theta )=W^{1,2}(\Omega ,Q)$ and $Q{H}_0^1(\Theta )=W_0^{1,2}(\Theta ,Q)$ 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.