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

Abstract

Abstract This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the formThe principal part ξ'P(x)ξ of the above equation is assumed to be comparable to a quadratic form Q(x,ξ)=ξ'Q(x)ξ that may vanish for non-zero ξ ∊ ℝn. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ)=W1,2(Θ,Q) and QH10(Θ)= W1,20 (Θ,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.