Title: De Giorgi methods for kinetic hypoelliptic equations
Abstract: We present new results that extend the De Giorgi-Nash-Moser theory to a class of hypoelliptic equations arising in kinetic theory. These operators are mathematically characterized by two parts: a diffusion part in the velocity variables governed by the (fractional) Laplace operator, and a first order differential operator which propagates smoothness in the remaining spatial variables. A key ingredient to prove our results is a Poincaré inequality adapted to the hypoelliptic geometry. This allows us to treat equations with an arbitrary number of Hörmander commutators and with either local or nonlocal diffusion. Combined with an L² −L∞ estimate and a covering argument, this yields Harnack inequalities and Hölder regularity.
This talk is based on a series of papers in collaboration with F. Anceschi, H. Dietert, J. Guerand, A. Loher and C. Mouhot.