Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Introduction to quantitative De Giorgi methods

Giovanni Brigati, Clément Mouhot

TL;DR

The paper develops a quantitative De Giorgi–Nash–Moser regularity theory for kinetic equations with rough coefficients, unifying elliptic, parabolic, and hypoelliptic (kinetic) settings. It advances both classical and constructive approaches by introducing trajectory-based methods and quantitative oscillation control across elliptic, parabolic, and Kolmogorov-type equations, yielding local Hölder regularity under uniform ellipticity-like bounds. It also discusses conditional regularity questions for Boltzmann and Landau equations in non-cutoff regimes, linking DGNM theory to kinetic theory and nonlocal diffusion. Overall, the work provides a cohesive, quantitative framework that extends DGNM to hypoelliptic, kinetic contexts, with potential implications for the regularity theory of fundamental kinetic models.

Abstract

The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the Hölder regularity of solutions to elliptic and parabolic equations with non-regular coefficients, and it was extended by Moser (1960) to include the Harnack inequality. This course reviews the classical De Giorgi method in the elliptic and parabolic cases and introduces its recent extension to hypoelliptic equations which appear naturally in kinetic theory. The simplest case is the Kolmogorov equation with a rough diffusion coefficients matrix in the kinetic variable. We present compactness arguments but emphasize the recently developed quantitative methods based on the construction of trajectories. These lecture notes are self-contained and can be used as a general introduction to the topic.

Introduction to quantitative De Giorgi methods

TL;DR

The paper develops a quantitative De Giorgi–Nash–Moser regularity theory for kinetic equations with rough coefficients, unifying elliptic, parabolic, and hypoelliptic (kinetic) settings. It advances both classical and constructive approaches by introducing trajectory-based methods and quantitative oscillation control across elliptic, parabolic, and Kolmogorov-type equations, yielding local Hölder regularity under uniform ellipticity-like bounds. It also discusses conditional regularity questions for Boltzmann and Landau equations in non-cutoff regimes, linking DGNM theory to kinetic theory and nonlocal diffusion. Overall, the work provides a cohesive, quantitative framework that extends DGNM to hypoelliptic, kinetic contexts, with potential implications for the regularity theory of fundamental kinetic models.

Abstract

The theory of De Giorgi (1958) and Nash (1959) solves Hilbert's 19th problem and constitutes a major advance in the analysis of PDEs in the 20th century. This theory concerns the Hölder regularity of solutions to elliptic and parabolic equations with non-regular coefficients, and it was extended by Moser (1960) to include the Harnack inequality. This course reviews the classical De Giorgi method in the elliptic and parabolic cases and introduces its recent extension to hypoelliptic equations which appear naturally in kinetic theory. The simplest case is the Kolmogorov equation with a rough diffusion coefficients matrix in the kinetic variable. We present compactness arguments but emphasize the recently developed quantitative methods based on the construction of trajectories. These lecture notes are self-contained and can be used as a general introduction to the topic.

Paper Structure

This paper contains 26 sections, 26 theorems, 447 equations, 13 figures.

Table of Contents

  1. Introduction
  2. The solution to Hilbert's $19$th Problem
  3. Hörmander's hypoellipticity theory
  4. Mathematical kinetic theory
  5. Notation
  6. Acknowledgements
  7. The elliptic case
  8. The elliptic case
  9. The result of De Giorgi
  10. The first De Giorgi Lemma
  11. The improved first De Giorgi lemma
  12. The second De Giorgi Lemma
  13. The method of Moser for controlling the oscillations
  14. The method of Kruzhkov for controlling the oscillations
  15. The parabolic case
  16. ...and 11 more sections

Key Result

Lemma 3

The following two steps reduce Hilbert's $19$th problem to the regularity theory of elliptic PDEs with rough coefficients:

Figures (13)

  • Figure 1: Noise in $v$ propagates to the $x$ variable thanks to the transport drift $v \cdot \nabla_x$.
  • Figure 2: A collision between two particles.
  • Figure 3: The sequence of balls in the De Giorgi--Moser iteration.
  • Figure 4: The setting of the decrease of maximum lemma.
  • Figure 5: Trajectories in the elliptic case.
  • ...and 8 more figures

Theorems & Definitions (55)

  • Lemma 3
  • proof
  • Theorem 4: Elliptic De Giorgi--Nash--Moser
  • Lemma 6: First De Giorgi Lemma: gain of integrability
  • proof : Proof of Lemma \ref{['lem:dg1']} (De Giorgi's version)
  • proof : Proof of Lemma \ref{['lem:dg1']} (Moser's version)
  • Lemma 8: Improved first De Giorgi Lemma
  • proof
  • Remark 9
  • Lemma 10: Second De Giorgi Lemma: reduction of oscillation
  • ...and 45 more