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

 

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.

 

I will present a novel proof of this weak L1-estimate, based on parabolic trajectories. The approach differs entirely from Moser’s proof, does not rely on any Poincaré inequality 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. Moreover, I will draw some connections to Li-Yau inequalities and kinetic equations. 

 

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