On the isoperimetric and isodiametric inequalities and the minimisation of eigenvalues of the Laplacian

TL;DR

This paper investigates the minimisation of Laplacian eigenvalues for convex domains under perimeter or diameter constraints, extending known results to Neumann and mixed eigenvalues and exploring geometric applications.

Contribution

It extends the minimisation results for Laplacian eigenvalues to Neumann cases under diameter constraints and to perimeter constraints in 2D, including mixed eigenvalues with additional geometric constraints.

Findings

Minimisation of Neumann eigenvalues under diameter constraint holds in any dimension.

Minimisation of eigenvalues under perimeter constraint is confirmed in 2D.

Applications of the techniques to geometric problems are discussed.

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 $d\geq 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$. 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 $d=2$. We also consider these problems for mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques.