We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂t-Δ+b-∇ resp. -Δ+b-∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b∈L∞(BMO-1) resp. b∈BMO-1 and provide a counterexample demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm ||b||L1. In three dimensions, on the other hand, bounded solutions with L1 drifts may be discontinuous.
Bibliographical noteFunding Information:
G.S. was partially supported by the RFFI grant 08-01-00372-a. The other authors were supported in part by NSF grants DMS-1001629 (L.S.), DMS-0800908 (V.Š.), DMS-1113017 and DMS-1056327 (A.Z.). L.S. and A.Z. also acknowledge partial support by Alfred P. Sloan Research Fellowships.