A sharp Liouville theorem for elliptic operators

Abstract

We introduce a new condition on elliptic operators L = 1/2 Δ + b∙ ∇ which ensures the validity of the Liouville property, i.e., all smooth bounded solutions to Lu = 0 on ℝ_d_ are constant. Such condition is sharp when d=1. We extend our Liouville theorem to more general second order operators in non-divergence form assuming a Cordes type condition.