Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A trajectorial interpretation of Moser's proof of the Harnack inequality

Lukas Niebel, Rico Zacher

TL;DR

The note revisits Moser's parabolic Harnack proof and provides a trajectorial interpretation that avoids reliance on Poincaré inequalities. By propagating along parabolic trajectories and using the Bombieri–Giusti lemma twice, it yields a novel weak $L^1$-estimate for $\log u$ of supersolutions and leads to the Harnack inequality with explicit ellipticity-dependent constants. The approach extends to the elliptic setting, producing analogous weak logarithmic estimates with optimal dependence on the ellipticity ratio. It also outlines potential applications to kinetic and hypoelliptic equations, highlighting a geometric, trajectory-based perspective on regularity theory.

Abstract

In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an $L^p-L^\infty$-estimate with a weak $L^1$-estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak $L^1$-estimate. The presented argument uses parabolic trajectories and does not use any Poincaré inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations.

A trajectorial interpretation of Moser's proof of the Harnack inequality

TL;DR

The note revisits Moser's parabolic Harnack proof and provides a trajectorial interpretation that avoids reliance on Poincaré inequalities. By propagating along parabolic trajectories and using the Bombieri–Giusti lemma twice, it yields a novel weak -estimate for of supersolutions and leads to the Harnack inequality with explicit ellipticity-dependent constants. The approach extends to the elliptic setting, producing analogous weak logarithmic estimates with optimal dependence on the ellipticity ratio. It also outlines potential applications to kinetic and hypoelliptic equations, highlighting a geometric, trajectory-based perspective on regularity theory.

Abstract

In 1971 Moser published a simplified version of his proof of the parabolic Harnack inequality. The core new ingredient is a fundamental lemma due to Bombieri and Giusti, which combines an -estimate with a weak -estimate for the logarithm of supersolutions. In this note, we give a novel proof of this weak -estimate. The presented argument uses parabolic trajectories and does not use any Poincaré inequality. Moreover, the proposed argument gives a geometric interpretation of Moser's result and could allow transferring Moser's method to other equations.
Paper Structure (4 sections, 6 theorems, 31 equations, 6 figures)

This paper contains 4 sections, 6 theorems, 31 equations, 6 figures.

Table of Contents

  1. Moser's proof of the parabolic Harnack inequality
  2. Employing parabolic trajectories to prove a weak $L^1$-estimate for the logarithm of supersolutions
  3. Elliptic equations
  4. Further comments

Key Result

Theorem 1.1

Let $\delta \in (0,1)$, $\tau >0$. There exists $C = C(d,\delta,\tau)>0$ such that for any cylinder $\tilde{Q} = (t_0-\delta \tau r^2,t_0+2 \tau r^2) \times B_{ r}(x_0) \subset \Omega_T$ with $r>0$ and any nonnegative weak solution $u$ of equation eq:par in $\tilde{Q}$ satisfies where $Q_- = (t_0,t_0+\delta \tau r^2) \times B_{\delta r}(x_0)$ and $Q_+ = (t_0+(2-\delta) \tau r^2,t_0+2\tau r^2) \ti

Figures (6)

  • Figure 1: The sets $\tilde{Q},Q_-,Q_+$ in the Harnack inequality of Theorem \ref{['thm:harnack']}.
  • Figure 2: The cylinders in Lemma \ref{['lem:lplinf']}.
  • Figure 3: The cylinders $K_-,K_+$ and the spatial balls $\delta B,B$ in Lemma \ref{['lem:weakl1log']}.
  • Figure 4: The cylinders $Q_-,Q_+,\bar{U}_\sigma,U_\sigma,K_-,K_+$ and the spatial balls $B_\delta,B_\sigma,B_1,B_\beta$ in the proof of Theorem \ref{['thm:harnack']} for $(t_0,x_0) = 0$ and $r = 1$.
  • Figure 5: The cylinders $K_-,K_+$ and the spatial balls $\delta B,B$ in the proof of Lemma \ref{['lem:weakl1log2']} for $\tau = 1$, $(t_0,x_0) = 0$ and $r = 1$. The dotted line depicts the parabolic trajectory connecting a point $(t,x) \in K_-$ to a point $(\eta,y)$ with $y \in B$.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • proof : Proof of Theorem \ref{['thm:harnack']}
  • Lemma 2.1
  • proof
  • ...and 3 more