Author
Seregin, G
Silvestre, L
Šverák, V
Zlatoš, A
Journal title
Journal of Differential Equations
DOI
10.1016/j.jde.2011.08.039
Issue
1
Volume
252
Last updated
2023-12-18T02:57:25.04+00:00
Page
505-540
Abstract
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. © 2011 Elsevier Inc.
Symplectic ID
196153
Favourite
On
Publication type
Journal Article
Publication date
01 Jan 2012
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