A discrete harmonic function bounded on a large portion of ℤ² is constant

2017/12/21 by Buhovsky, Lev, Logunov, Alexander, Malinnikova, Eugenia +1 · 2 citations
#35J05 #65N22 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.07902

Abstract

An improvement of the Liouville theorem for discrete harmonic functions on ℤ² is obtained. More precisely, we prove that there exists a positive constant ε such that if u is discrete harmonic on ℤ² and for each sufficiently large square Q centered at the origin |u|≤ 1 on a (1-ε) portion of Q then u is constant.

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