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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 to second order linear degenerate elliptic partial differential operators $X$, with rough coefficients of the form $$X=-\text{div}(P\nabla )+{\bf HR}+{\bf S^\prime G} +F$$ in a geometric homogeneous space setting where the $n\times n$ matrix function $P=P(x)$ is allowed to degenerate. We give a maximum principle for weak solutions of $Xu\leq 0$ and follow this with a result describing a relationship between compact projection of the degenerate Sobolev space $QH^{1,p}$ into $L^q$ and a Poincaré inequality with gain adapted to $Q$.