Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L2-based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.