# On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian

## TL;DR

This paper studies how to minimize eigenvalues of the Laplacian under constraints on the shape of domains, showing that the optimal shape tends to a ball as the eigenvalue index increases.

## Contribution

The paper extends known results for Dirichlet eigenvalues to Neumann and Robin eigenvalues under specific geometric constraints.

## Key findings

- Minimisers of Neumann eigenvalues under diameter constraints converge to a ball in any dimension.
- For perimeter constraints in 2D, Neumann eigenvalue minimisers also converge to a ball as the eigenvalue index increases.
- Similar results are shown for Robin and mixed Dirichlet–Neumann eigenvalues with additional geometric constraints.

## Abstract

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension \documentclass[12pt]{minimal}
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\begin{document}$$d\ge 2$$\end{document}d≥2 and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as \documentclass[12pt]{minimal}
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\begin{document}$$k\rightarrow +\infty $$\end{document}k→+∞. In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension \documentclass[12pt]{minimal}
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\begin{document}$$d=2$$\end{document}d=2. We also consider these problems for Robin eigenvalues and mixed Dirichlet–Neumann eigenvalues, under an additional geometric constraint.