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In this paper we study existence and spectral properties for weak solutionsof Neumann and Dirichlet problems associated to second order linear degenerateelliptic partial differential operators $X$, with rough coefficients of theform $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometrichomogeneous space setting where the $n\times n$ matrix function $P=P(x)$ isallowed to degenerate. We give a maximum principle for weak solutions of$Xu\leq 0$ and follow this with a result describing a relationship betweencompact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and aPoincar\'e inequality with gain adapted to $Q$.

Existence and Spectral Theory for Weak Solutions of Neumann and Dirichlet Problems for Linear Degenerate Elliptic Operators with Rough Coefficients