Comparison of the first positive Neumann eigenvalues for rectangles and special parallelograms

TL;DR

This paper compares the first positive Neumann eigenvalues of rectangles and parallelograms with the same base and area, revealing geometric insights and applications to eigenvalue normalization and a related 3D problem.

Contribution

It provides a novel comparison of Neumann eigenvalues for specific parallelograms and rectangles, extending geometric spectral analysis.

Findings

Neumann eigenvalues are larger for parallelograms with greater height

Normalized eigenvalues relate to geometric restrictions and the square

The results are inspired by classical geometric theorems

Abstract

First non-zero Neumann eigenvalues of a rectangle and a parallelogram with the same base and area are compared in case when the height of the parallelogram is greater than the base. This result is applied to compare first non-zero Neumann eigenvalue normalized by the square of the perimeter on the parallelograms with a geometrical restriction and the square. The result is inspired by Wallace--Bolyai--Gerwien theorem. An interesting three-dimensional problem related to this theorem is proposed.