On the maximum principle for second-order elliptic operators in unbounded domains

We are concerned with the maximum principle for second-order elliptic operators in unbounded domains. Using a geometric condition, already considered by Berestycki, Nirenberg and Varadhan, and a weak boundary Harnack inequality due to Trudinger, Cabré was able to prove the ABP (Alexandroff–Bakelman–Pucci) estimate for a large class of unbounded domains, obtaining as a consequence the maximum principle for general elliptic operators. In this Note we introduce a weak form of the above geometric condition showing that this is enough to obtain the maximum principle for a larger class of domains.