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.

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.

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