An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

Abstract

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.