# HÖLDER CONTINUITY OF SOLUTIONS TO HYPOELLIPTIC EQUATIONS WITH BOUNDED MEASURABLE COEFFICIENTS

Abstract : We prove that L 2 weak solutions to hypoelliptic equations with bounded measurable coefficients are Hölder continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions; second in proving that the gradient with respect to the velocity variable is L 2+ε loc ; third, in the proof of an " hypoelliptic isoperimetric De Giorgi lemma ". To get such a lemma, we develop a new method which combines the classical isoperimetric inequality on the diffusive variable with the structure of the integral curves of the first-order part of the operator. It also uses that the gradient of solutions w.r.t. v is L 2+ε loc .
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Type de document :
Pré-publication, Document de travail
The identification of $\bar h$ in Lemma 15 is due to F. Golse. Several typos fixed. This paper wi.. 2015

Littérature citée [17 références]

https://hal.archives-ouvertes.fr/hal-01152145
Contributeur : Cyril Imbert <>
Soumis le : vendredi 19 juin 2015 - 13:50:40
Dernière modification le : jeudi 11 janvier 2018 - 06:12:31
Document(s) archivé(s) le : mardi 15 septembre 2015 - 19:31:45

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• HAL Id : hal-01152145, version 5
• ARXIV : 1505.04608

### Citation

Cyril Imbert, Clément Mouhot. HÖLDER CONTINUITY OF SOLUTIONS TO HYPOELLIPTIC EQUATIONS WITH BOUNDED MEASURABLE COEFFICIENTS. The identification of $\bar h$ in Lemma 15 is due to F. Golse. Several typos fixed. This paper wi.. 2015. 〈hal-01152145v5〉

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