Parabolic trajectories and the Harnack inequality

Date: Wednesday March 15, 2023
Organized by: FAU DCN-AvH, Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship at FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

Title: Parabolic trajectories and the Harnack inequality
Speaker: Lukas Niebel
Affiliation: Ulm University

Abstract. In this talk, we will study the Harnack inequality for weak solutions to a parabolic diffusion problem with rough coefficients. This is often referred to as De Giorgi-Nash-Moser theory.

In the first part, I will present the proof of the Harnack inequality due to Moser (1971). He combines a weak L1-estimate for the logarithm of supersolutions with Lp −L∞-estimates and a lemma due to Bombieri and Giusti. His method has been applied to nonlocal parabolic problems (Kassmann and Felsinger 2013), time-fractional diffusion equations (Zacher 2013), discrete problems (Delmotte 1999) and many more. In each of these works, the proof of the weak L1-estimate follows more or less the strategy of Moser and is based on a Poincaré inequality.

In the second part, I will present a novel proof of this weak L1-estimate, based on parabolic trajectories, which does not rely on any Poincaré inequality. The approach is entirely different from Moser’s proof and gives a very nice geometric interpretation of the result. The argument does not treat the temporal and spatial variables separately but considers both variables simultaneously.

In the end, I will draw some connections to Li-Yau inequalities and kinetic equations.

This is based on joint work with Rico Zacher (Ulm University).


On-site / Online

Room Übung 4
1st. floor. Department Mathematik. Friedrich-Alexander-Universität Erlangen-Nürnberg
Cauerstraße 11, 91058 Erlangen
GPS-Koord. Raum: 49.573572N, 11.030394E

Zoom meeting link
Meeting ID: 614 4658 159 | PIN code: 914397

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