Constructing discrete harmonic functions in wedges
Viet Hung Hoang, Kilian Raschel, Pierre Tarrago
TL;DR
This paper introduces a systematic method for constructing signed harmonic functions for discrete Laplacians with Dirichlet boundary conditions in a quarter plane, revealing an algebraic structure generated by a single element.
Contribution
It demonstrates that the set of harmonic functions forms an algebra generated by one element, conjecturally the unique positive harmonic function.
Findings
Harmonic functions form an algebra generated by a single element
The construction applies to discrete Laplacians with Dirichlet conditions in wedges
Conjecture that the generator corresponds to the unique positive harmonic function
Abstract
We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.
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