September 19, 2018
1:00 PM - 2:00 PM
Math Tower P-131
|Jack Burkart, Stony Brook University|
Improving Liouville's Theorem for Harmonic Functions
Liouville's theorem says that the only harmonic functions on $R^n$ that are bounded above are actually constant. We will discuss the proof of this fact using Harnack's inequality, which is a primitive example of extremely useful "3-Ball inequalities" that show up in harmonic analysis. We will compare this continuous version to the discrete version of the Liouville theorem after defining harmonic functions on $Z^2$ in terms of the mean value property. In particular, we will discuss a recent advancement from 2017 due to Buhovsky, Logunov, Malinnikova, and Sodin, which says that, in a way that we will make precise, a harmonic function bounded on 99.999% of $Z^2$ must actually be constant.