Minimising Dirichlet eigenvalues on cuboids of unit measure

TL;DR

This paper investigates the minimization of Dirichlet eigenvalues on three-dimensional cuboids with fixed volume, proving that as the eigenvalue index increases, the optimal shapes approach a perfect cube.

Contribution

It establishes the convergence of optimal cuboids to a cube in Hausdorff sense for large eigenvalue indices, advancing understanding of shape optimization for Laplacian eigenvalues.

Findings

Optimal cuboids converge to a cube as k increases

Convergence is in the Hausdorff sense

Results apply to eigenvalues of the Laplacian on cuboids

Abstract

We consider the minimisation of Dirichlet eigenvalues $\lambda_k$, $k \in \N$, of the Laplacian on cuboids of unit measure in $\R^3$. We prove that any sequence of optimal cuboids in $\R^3$ converges to a cube of unit measure in the sense of Hausdorff as $k \rightarrow \infty$.