A Sharp Liouville Theorem for Elliptic Operators

Abstract

We introduce a new condition on elliptic operators $L=1/2△+b\cdot \nabla$ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on ${\mathbb{R}}^d$. 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.