Stability estimates for discrete harmonic functions on product domains

TL;DR

This paper develops stability estimates for discrete harmonic functions on product domains, extending classical theorems and employing spectral inequalities to analyze the Dirichlet problem in unbounded lattice domains.

Contribution

It introduces discrete analogs of classical complex analysis theorems and derives stability estimates for the Dirichlet problem on multidimensional lattice domains.

Findings

Discrete Phragmén-Lindelöf theorems established

A discrete three-line theorem for harmonic gradients proved

Stability estimates for Dirichlet problems in cylinder domains derived

Abstract

We study the Dirichlet problem for discrete harmonic functions in unbounded product domains on multidimensional lattices. First we prove some versions of the Phragm\'en-Lindel\"of theorem and use Fourier series to obtain a discrete analog of the three-line theorem for the gradients of harmonic functions in a strip. Then we derive estimates for the discrete harmonic measure and use elementary spectral inequalities to obtain stability estimates for Dirichlet problem in cylinder domains.