Steklov Eigenproblems and Approximations of Harmonic Functions
Cho, Manki 1984-
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In this work, we provide explicit formulae for harmonic Steklov eigenvalues and associated Steklov eigenfunctions by solving a Steklov eigenproblem on bounded rectangles. This allows the description of all the Steklov eigenvalues and their corresponding Steklov eigenfunctions. Solutions of Laplace's equation subject to inhomogeneous Dirichlet, Neumann, Robin, or other boundary data are approximated by Steklov expansions. These representations involve boundary conditions and explicit spectral approximations are found. In the end, we provide the general formulae for u(0,0) where harmonic functions u(x, y) are solutions of boundary value problems on a rectangle. These harmonic functions are determined precisely in terms of their respective boundary data.