Quantitative De Giorgi methods in kinetic theory
Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1159-1181.

We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies Hölder continuity with quantitative estimates. The paper is self-contained.

Nous considérons des équations hypoelliptiques de type Fokker-Planck cinétique, également appelées équations de Kolmogorov ou ultraparaboliques, avec des coefficients sans régularité dans l’opérateur de dérive-diffusion. Nous donnons de nouvelles preuves quantitatives du lemme des valeurs intermédiaires de De Giorgi ainsi que des inégalités de Harnack faibles et fortes. Cela implique la continuité höldérienne avec bornes explicites. L’article ne fait pas appel à des résultats précédents.

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DOI: 10.5802/jep.203
Classification: 35K70, 35Q84, 35R09, 35B45, 35B65
Keywords: Hypoelliptic equations, kinetic theory, Fokker-Planck equation, ultraparabolic equations, Kolmogorov equation, Hölder continuity, De Giorgi method, Moser iteration, averaging lemma, weak Harnack inequality, trajectories
Mot clés : Équations hypoelliptiques, théorie cinétique, équation de Fokker-Planck, équations ultraparaboliques, équation de Kolmogorov, continuité höldérienne, méthode de De Giorgi, itération de Moser, lemme de moyenne, inégalité de Harnack faible, trajectoires
Jessica Guerand 1; Clément Mouhot 2

1 Université de Montpellier, IMAG 499-554 rue du Truel, 34090 Montpellier, France
2 University of Cambridge, Department of Pure Mathematics and Mathematical Statistics Wilberforce Road, Cambridge CB3 0WA, United Kingdom
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Jessica Guerand; Clément Mouhot. Quantitative De Giorgi methods in kinetic theory. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 1159-1181. doi : 10.5802/jep.203. https://jep.centre-mersenne.org/articles/10.5802/jep.203/

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