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 ∇ ′P(x)∇u + HRu + S ′Gu + F u = f + T ′ g in Θ u = ϕ on ∂Θ. The 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 ∈ R n . This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ) = W1,2 (Ω, Q) and QH1 0 (Θ) = W1,2 0 (Θ, Q) as defined in [R1],[SW2],[MRW] and [CRW]. In [SW1], with generalizations in [SW2], the authors give a regularity theory for a subset of the class of equations dealt with here.