Hôlder continuity for quasilinear hypoelliptic equations

Cyril Imbert (1) , Clément Mouhot (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We prove that $L^2$ weak solutions to a quasilinear hypoelliptic equations with bounded measurable coefficients are H\"older 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 speed variable is $L^{2+\epsilon}_{\mathrm{loc}}$; 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 $L^{2+\epsilon}_{\mathrm{loc}}$.

Mots clés: Moser iteration; Hypoelliptic equations; Hölder continuity; averaging lemma; De Giorgi method

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Clément Mouhot. Hôlder continuity for quasilinear hypoelliptic equations. 2015. ⟨hal-01152145v3⟩