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

Abstract

We are concerned with the maximum principle for second-order elliptic operators of the kind Lu= a ij ( x) u x i x j + c( x) u in unbounded domains of R n . Using a geometric condition, already considered by Berestycki, Nirenberg and Varadhan in [2] and a weak boundary Harnack inequality due to Trudinger, Cabré [3] 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 and we show that in the case c⩽0 this is enough to obtain the maximum principle for a larger class of domains. To cite this article: V. Cafagna, A. Vitolo, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 359–363.