The Payne conjecture for Dirichlet and Buckling eigenvalues

TL;DR

This paper proves the Payne conjecture, establishing that the eigenvalues of the buckling problem are always greater than or equal to the subsequent eigenvalues of the membrane problem for the same shape, extending to higher dimensions.

Contribution

It provides a rigorous proof of the Payne conjecture for Dirichlet and buckling eigenvalues, including the n-dimensional case, which was previously unresolved.

Findings

The eigenvalue inequality holds for 2D shapes.

The conjecture is valid in n-dimensional spaces (n ≥ 2).

The proof confirms the eigenvalue ordering for buckling and membrane problems.

Abstract

We prove the long-standing Payne conjecture that the $k^{\text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{\text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover, we show that the Payne conjecture is still true for $n$-dimensional case ($n\ge 2)$.