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}\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.