Maximal spectral surfaces of revolution converge to a catenoid

Abstract

We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in ${\mathbb{R}}^3$ with two prescribed boundary components. For every $j$, we show that there is a surface ${\Sigma }_j$ which maximizes the $j$-th Dirichlet eigenvalue. The maximizing surface has a meridian which is a rectifiable curve. If there is a catenoid which is the unique area minimizing surface with the prescribed boundary, then the eigenvalue maximizing surfaces of revolution converge to this catenoid.

References in corpus (3)

• Extremal Eigenvalues of the Laplacian on Euclidean domains and closed surfaces
• Maximising Neumann eigenvalues on rectangles
• On the minimization of Dirichlet eigenvalues