Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics

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Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics 55
2022 発行

POTENTIAL THEORY OF THE DISCRETE EQUATION Δu = qu

KURATA, HISAYASU
YAMASAKI, MARETSUGU
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Description
We develop a discrete potential theory for the equation Δu = qu on an infinite network similar to the classical potential theory on Riemannian surfaces. The q-Green function for the Schrödinger operator - Δ + q plays the role of the Green function for the Laplace operator. We study some properties of q-Green potential whose kernel is the q-Green function. As an application, we give a classification of infinite networks by the classes of q-harmonic functions.
We also focus on the role of the q-elliptic measure of the ideal boundary of the network.