We make a further advance concerning the maximum principle for second-order elliptic operators. We investigate in particular a geometric condition, firstly considered by Berestycki–Nirenberg–Varadhan, that seems to be natural in view of the application of the boundary weak Harnack inequality, on which our argument is based. Setting it free from some technical assumptions, apparently needed in earlier papers, we significantly enlarge the class of unbounded domains where the maximum principle holds, compatibly with the first-order term.
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