A DISCRETE HARMONIC FUNCTION BOUNDED ON A LARGE PORTION OF Z2 IS CONSTANT

Lev Buhovsky; Alexander Logunov; Eugenia Malinnikova; Mikhail Sodin

doi:10.1215/00127094-2021-0037

A DISCRETE HARMONIC FUNCTION BOUNDED ON A LARGE PORTION OF Z² IS CONSTANT

Lev Buhovsky, Alexander Logunov, Eugenia Malinnikova, Mikhail Sodin

School of Mathematical Sciences

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

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 " 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.

Original language	English
Pages (from-to)	1349-1378
Number of pages	30
Journal	Duke Mathematical Journal
Volume	171
Issue number	6
DOIs	https://doi.org/10.1215/00127094-2021-0037
State	Published - 15 Apr 2022

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1215/00127094-2021-0037

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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 {"} 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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AU - Sodin, Mikhail

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AB - An improvement of the Liouville theorem for discrete harmonic functions on Z2 is obtained. More precisely, we prove that there exists a positive constant " 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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JF - Duke Mathematical Journal

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A DISCRETE HARMONIC FUNCTION BOUNDED ON A LARGE PORTION OF Z² IS CONSTANT

Abstract

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