Eigenfunctions for rectangles with Neumann boundary conditions

TL;DR

This paper proves that positive eigenfunctions for a rectangular membrane with Neumann boundary conditions are constant, providing a characterization of such eigenfunctions in the context of the Laplace operator.

Contribution

It establishes that any positive eigenfunction with Neumann boundary conditions on a rectangle must be constant, offering a new insight into eigenfunction behavior under these conditions.

Findings

Positive eigenfunctions are constant on rectangles with Neumann boundary conditions.

Non-constant positive eigenfunctions do not exist under these conditions.

The result characterizes the structure of eigenfunctions for Neumann problems on rectangles.

Abstract

Consider the eigenfunctions $u$ for a ree rectangular membrane wo that $-\Delta u=\lambda u$ on $\mathcal R(c,d)=(0,d)\times(0,d)$. In this note we show that if if $u>0$ on $\partial \mathcal R(c,d)) then $u\equiv C$ for some positive constant.