Existence and spectral theory for weak solutions of Neumann and Dirichlet problems for linear degenerate elliptic operators with rough coefficients

In this paper we study existence and spectral properties for weak solutions of Neumann and Dirichlet problems associated with second order linear degenerate elliptic partial differential operators X   with rough coefficients, of the form X=−div(P∇)+HR+S′G+FX=−div(P∇)+HR+S′G+F, where the n×nn×n matrix function P=P(x)P=P(x) is nonnegative definite and allowed to degenerate, R, S are families of subunit vector fields, G, H are vector valued functions and F   is a scalar function. We operate in a geometric homogeneous space setting and we assume the validity of certain Sobolev and Poincaré inequalities related to a symmetric nonnegative definite matrix of weights Q=Q(x)Q=Q(x) that is comparable to P  ; we do not assume that the underlying measure is doubling. We give a maximum principle for weak solutions of Xu≤0Xu≤0, and we follow this with a result describing a relationship between compact projection of the degenerate Sobolev space QH1,pQH1,p, related to the matrix of weights Q  , into LqLq and a Poincaré inequality with gain adapted to Q.