The theory of De Giorgi (1958) and Nash (1959) solved Hilbert's 19th problem and was a major contribution to 20th century PDE analysis. It is concerned with the Hölder regularity of solutions to elliptic and parabolic PDEs with rough (merely measurable) coefficients; it was developed by Moser (1960-1964) to include the Harnack inequality. These lectures are an introduction to a recent active research area (Pascucci-Polidoro, Wang-Zhang, Golse-Imbert-M-Vasseur, Imbert-Silvestre, Imbert-Guerand, Guerand-M, Anceschi-Rebucci, Loher, Niebel-Zacher...): the extension of this theory to the hypoelliptic PDEs, local and nonlocal, that appear naturally in kinetic theory. The simpler prototypical case is the Kolmogorov equation (aka kinetic Fokker-Planck equation) with a rough matrix of coefficients in the kinetic diffusion. The course will in particular emphasize the recent quantitative robust methods based on the construction of trajectories and their connexions to control theory and hypocoercivity (works with Dieter, Hérau, Hutridurga, Niebel, Zacher).
Thierry Bodineau
