Sharp Geometric Maximum Principles for Semi-Elliptic Operators with Singular Drift

We discuss a sharp generalization of the Hopf–Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with singular drift. In turn, this result is used to derive a version of the strong maximum principle under optimal pointwise blow-up conditions for the coefficients of the differential operator involved. We also explain how a uniform two-sided pseudo-ball condition may be used to provide a purely geometric characterization of Lyapunov domains, and clarify the role this class of domains plays vis-à-vis to the BPP.

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Published 19 August 2011