On the L^2-orthogonality of Steklov eigenfunctions

Abstract

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  \(\Omega_{1\alpha}\).  For more information see https://ejde.math.txstate.edu/conf-proc/26/c1/abstr.html