Quantitative De Giorgi methods in kinetic theory

TL;DR

This paper develops new quantitative De Giorgi methods for hypoelliptic kinetic equations, establishing regularity results like Hölder continuity with explicit estimates, using simplified proofs of classical inequalities.

Contribution

It introduces novel, concise proofs of De Giorgi's lemma and Harnack inequalities for kinetic Fokker-Planck equations with rough coefficients, advancing regularity theory.

Findings

Hölder continuity with explicit estimates established

New short proofs of De Giorgi lemma and Harnack inequalities

Applicable to equations with rough coefficients

Abstract

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 H{\"o}lder continuity with quantitative estimates. The paper is self-contained.