Quantitative De Giorgi methods in kinetic theory

. We consider hypoelliptic equations of kinetic Fokker-Planck type, also known as Kolmogorov or ultraparabolic equations, with rough coefficients in the drift-diffusion operator. We give novel short quantitative proofs of the De Giorgi intermediate-value Lemma as well as weak Harnack and Harnack inequalities. This implies Hölder continuity with quantitative estimates. The paper is self-contained.

[1]  Cyril Imbert,et al.  LOG-TRANSFORM AND THE WEAK HARNACK INEQUALITY FOR KINETIC FOKKER-PLANCK EQUATIONS , 2021, Journal of the Institute of Mathematics of Jussieu.

[2]  S. Polidoro,et al.  A survey on the classical theory for Kolmogorov equation , 2019, 1907.05155.

[3]  Jessica Guerand.  Quantitative regularity for parabolic De Giorgi classes , 2019, 1903.07421.

[4]  Christian Schmeiser,et al.  Hypocoercivity without confinement , 2017, Pure and Applied Analysis.

[5]  L. Silvestre,et al.  The weak Harnack inequality for the Boltzmann equation without cut-off , 2016, Journal of the European Mathematical Society.

[6]  C. Mouhot,et al.  Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation , 2016, ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE.

[7]  Dongsheng Li,et al.  A note on the Harnack inequality for elliptic equations in divergence form , 2016, 1901.06128.

[8]  F. Golse,et al.  H\"{o}lder regularity for hypoelliptic kinetic equations with rough diffusion coefficients , 2015, 1506.01908.

[9]  Wendong Wang,et al.  The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations , 2010 .

[10]  Wendong Wang,et al.  The Cα regularity of a class of non-homogeneous ultraparabolic equations , 2007, 0711.3411.

[11]  A. Pascucci,et al.  THE MOSER'S ITERATIVE METHOD FOR A CLASS OF ULTRAPARABOLIC EQUATIONS , 2004 .

[12]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[13]  J. Moser.  A Harnack inequality for parabolic di2erential equations , 1964 .

[14]  A. Vasseur.  THE DE GIORGI METHOD FOR ELLIPTIC AND PARABOLIC EQUATIONS AND SOME APPLICATIONS , 2014 .

[15]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[16]  A. Kolmogoroff,et al.  Zufallige Bewegungen (Zur Theorie der Brownschen Bewegung) , 1934 .

[17]  Ronald F. Gariepy,et al.  Measure Theory and Fine Properties of Functions, Revised Edition , 1865 .