# On the behaviour of clamped plates under large compression

## TL;DR

This paper analyzes how the eigenvalues of clamped plates behave under large compression, relating the problem to Laplacian eigenvalues with Robin boundary conditions, and studies extremal domain shapes numerically.

## Contribution

It establishes the asymptotic behavior of eigenvalues under compression and explores the shape and nodal domain changes of extremal domains numerically.

## Key findings

- Eigenvalues relate to Laplacian with Robin boundary conditions under large compression.
- Extremal domain shapes depend on compression level and develop boundary structures.
- Number of nodal domains increases with compression.

## Abstract

We determine the asymptotic behaviour of eigenvalues of clamped plates under large compression, by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.