Giles Auchmuty, Manki Cho
Published 2014-11-05Version 1
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.
Note on an elementary inequality and its application to the regularity of $p$-harmonic functions
Curvature Estimate of Nodal Sets of Harmonic Functions in the Plane
Approximation of the spectrum of a manifold by discretization
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