A Sharp Liouville Theorem for Elliptic Operators

TL;DR

This paper establishes a new sharp condition on elliptic operators that guarantees bounded solutions are constant, extending Liouville's theorem to broader classes of second order operators in non-divergence form.

Contribution

It introduces a novel condition on elliptic operators ensuring Liouville property, extending classical results to more general operators in non-divergence form.

Findings

New sharp condition for elliptic operators ensuring Liouville property

Extension of Liouville theorem to non-divergence form operators

Condition is optimal in one dimension

Abstract

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