Boundary Integrals and Approximations of Harmonic Functions

TL;DR

This paper derives boundary integral formulas to accurately approximate the value of harmonic functions at the center of rectangles, utilizing Steklov spectral representations for exponential convergence, and extends these methods to Robin and Neumann problems.

Contribution

It introduces explicit boundary integral formulas for harmonic function values at rectangle centers based on Steklov eigenvalues and eigenfunctions, with extensions to Robin and Neumann boundary conditions.

Findings

Central harmonic function values are well approximated by boundary means.

Approximation formulas converge exponentially due to spectral properties.

Explicit expressions for Steklov eigenvalues and eigenfunctions are provided.

Abstract

Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic function is shown to be well approximated by the mean value of the function on the boundary plus a very small number (often just 1 or 2) of additional boundary integrals. The formulae are consequences of Steklov (spectral) representations of the functions that converge exponentially at the center. Similar approximation are found for the central values of solutions of Robin and Neumann boundary value problems. The results are based on explicit expressions for the Steklov eigenvalues and eigenfunctions.