The Steklov Problem on Rectangles and Cuboids

TL;DR

This paper investigates the Steklov eigenvalue problem on rectangles and cuboids, providing analytic results for rectangles, listing eigenfunctions for cuboids, and discussing conjectures for 3D cases, with special focus on squares and cubes.

Contribution

It offers a detailed analysis of Steklov eigenvalues on rectangles and cuboids, including existence proofs, eigenfunction classifications, and conjectural extensions to three dimensions.

Findings

Existence of eigenfunction classes for rectangles

Complete listing of eigenfunctions on cuboids

Conjectured 3D eigenvalue analogues

Abstract

This paper is a brief account of the Steklov eigenvalue problem on a 2-dimensional rectangular domain, and then on a 3-dimensional rectangular box. It is divided into four sections. Section 1 relies heavily on real analytic methods to show the existence of an eigenfunction class which always produces the first non-trivial Steklov eigenvalue on a rectangle. Section 2 lists all possible Steklov eigenfunctions on a cuboid. The very brief section 3 gives the 3-dimensional analogue of the analytic results in Section 1. The analogue is given as conjecture, but is expected to derivate from standard (albeit tedious) real analysis methods, should one wish to expound on these calculations. Section 4 deals with the special cases of the square and the cube.