[Méthodes à la De Giorgi quantitatives en théorie cinétique]
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.
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.
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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
CC-BY 4.0
@article{JEP_2022__9__1159_0,
author = {Jessica Guerand and Cl\'ement Mouhot},
title = {Quantitative {De~Giorgi} methods in kinetic theory},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1159--1181},
publisher = {\'Ecole polytechnique},
volume = {9},
year = {2022},
doi = {10.5802/jep.203},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.203/}
}
TY - JOUR AU - Jessica Guerand AU - Clément Mouhot TI - Quantitative De Giorgi methods in kinetic theory JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 1159 EP - 1181 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.203/ DO - 10.5802/jep.203 LA - en ID - JEP_2022__9__1159_0 ER -
%0 Journal Article %A Jessica Guerand %A Clément Mouhot %T Quantitative De Giorgi methods in kinetic theory %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 1159-1181 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.203/ %R 10.5802/jep.203 %G en %F JEP_2022__9__1159_0
Jessica Guerand; Clément Mouhot. Quantitative De Giorgi methods in kinetic theory. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 1159-1181. doi : 10.5802/jep.203. https://jep.centre-mersenne.org/articles/10.5802/jep.203/
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