Abstract
We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker—Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Hölder continuity, an elliptic Harnack inequality and a Liouville theorem.
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Acknowledgements
S. Mosconi is partially supported by project PIACERI - Linea 2 and 3 of the University of Catania. Part of the paper was developed during a visit of the first and third authors at the Department of Mathematics and Computer Science of the University of Catania, Italy. We gratefully thank Matias Vestberg of the Uppsala University, Sweden, for his suggestions, which greatly improved a preliminary version of the manuscript.
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Ciani, S., Mosconi, S. & Vespri, V. Parabolic Harnack Estimates for anisotropic slow diffusion. JAMA 149, 611–642 (2023). https://doi.org/10.1007/s11854-022-0261-0
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DOI: https://doi.org/10.1007/s11854-022-0261-0