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 del'P(x)del u + HRu + S'Gu + Fu = f + T'g in Theta u = phi on partial derivative circle minus The principal part 'P(x) of the above equation is assumed to be comparable to a quadratic form Q(x, xi) = xi'Q(x)xi that may vanish for non-zero xi is an element of R-n. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH(1)(circle minus) = W-1,W-2(circle minus, Q) and QH(0)(1)(circle minus) = W-0(1,2)(circle minus,Q) as defined in previous works. E. T. Sawyer and R. L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.