2021 UNC Greensboro PDE Conference. Electron. J. Diff. Eqns., Conference 26 (2022), pp. 45-58.

On the L^2-orthogonality of Steklov eigenfunctions

Manki Cho, Mauricio A. Rivas

This article analyzes the interior $L^2$ -orthogonality of the Steklov eigenfunctions on rectangles $\Omega_{1\alpha}$. It is shown that most Steklov eigenfunctions are, indeed, pairwise orthogonal in $L^2(\Omega_{1\alpha})$, and pairs that are not orthogonal are nearly orthogonal. Explicit formulae for exact inner products in $L^2(\Omega_{1\alpha})$ of the eigenfunctions are found, and to elucidate the intricate formulae obtained, accompanying numerics are provided. Then envelopes that bound the calculated inner products are constructed that simplify the convoluted formulae. This leads to a straightforward description of the nearly orthogonal Steklov eigenfunctions. A consequence of the calculations is a tabulation of the mean value of Steklov eigenfunctions over $\Omega_{1\alpha}$.

Published August 25, 2022.
Math Subject Classifications: 35P05, 31A20, 35J05.
Key Words: Harmonic functions; Steklov eigenfunctions; Laplacian eigenfunctions.

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Manki Cho
Department of Mathematics and Statistics
University of Houston - Clear Lake
2700 Bay Area Blvd
Houston, TX 77058, USA
email: cho@uhcl.edu
Mauricio A. Rivas
Department of Mathematics and Statistics
North Carolina A&T State University
1601 East Market Street
Greensboro, NC 27411, USA
email: marivas@ncat.edu

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