Harnack inequality for $p$-harmonic functions: probabilistic and analytic approaches

Abstract

We survey analytic and probabilistic approaches to the Harnack inequality for $p$-harmonic functions, $p>1$, with particular attention to the dependence of the constants on $p$ and the dimension. We explain the ideas of Moser iteration, the tug-of-war approach of Luiro, Parviainen and Saksman, and the log-gradient estimate of Kotschwar and Ni. Along the way, we give a quantitative refinement of the probabilistic argument and a direct planar proof.

23 pages, 2 figures