Estimates for sums of logarithmic potentials associated with the Green function

Abstract

This article presents a study on the analytical solution of the Dirichlet problem for the Laplace equation in two-dimensional space. The primary focus is on the Green's function, which is a key tool for solving such problems. We considered cases where the sources are located on geometrically simple sets: on a straight line, a unit circle, and a line segment.
We developed and applied an effective method for calculating the sum of harmonic functions $$$h_k(z,a_k)$$$, which are the correcting terms in the Green's function expression. This allowed us to obtain analytical formulas for the potential generated by a system of point sources located in the specified configurations. Specifically, for each case, we found an estimate for the total potential.
The findings are of significant value to theoretical physics and engineering applications, particularly in electrostatics, heat conduction, and hydrodynamics, where similar boundary value problems arise. The proposed approach can serve as a basis for further research aimed at solving more complex problems with sources located on curved or higher-dimensional manifolds.