On divergence-free drifts |
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Authors: | Gregory Seregin Luis Silvestre Vladimír Šverák Andrej Zlatoš |
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Affiliation: | 1. University of Oxford, 24-29 St Giles?, Oxford OX1 3LB, UK;2. University of Chicago, 5374 University Ave., Chicago, IL 60637, USA;3. University of Minnesota, 206 Church St. SE, Minneapolis, MN 55455, USA;4. University of Wisconsin–Madison, 480 Lincoln Dr., Madison, WI 53706, USA |
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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 resp. with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when resp. 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 . In three dimensions, on the other hand, bounded solutions with drifts may be discontinuous. |
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