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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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https://hal.archives-ouvertes.fr/hal-01152145
Contributor : Cyril Imbert <>
Submitted on : Friday, June 19, 2015 - 1:50:40 PM
Last modification on : Thursday, March 19, 2020 - 12:26:02 PM
Document(s) archivé(s) le : Tuesday, September 15, 2015 - 7:31:45 PM

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

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Cyril Imbert, Clément Mouhot. HÖLDER CONTINUITY OF SOLUTIONS TO HYPOELLIPTIC EQUATIONS WITH BOUNDED MEASURABLE COEFFICIENTS. 2015. ⟨hal-01152145v5⟩

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