0,01% improvement of Liouville property for discrete harmonic functions

Alexander Logunov
Saint Petersburg Univ and Tel Aviv University

Date(s) : 06/03/2017   iCal
10 h 00 min - 11 h 00 min

Let u be a harmonic function on the plane. The Liouville theorem claims that if |u| is bounded on the whole plane, then u is identically constant. At the same moment for any angle on the plane R^2, there exist a harmonic function that is non-constant and is bounded outside the angle. It appears that if u is a harmonic function on a lattice Z^2, and |u| < 1 on 99,99% of Z^2, then u is a constant function. In particular there are no (non-constant) discrete harmonic functions bounded outside a sufficiently small angle. Based on a joint work (in progress) with L. Buhovsky, Eu. Malinnikova and M. Sodin.


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