Maximising Neumann eigenvalues on rectangles

Abstract

We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in ${\mathbb{R}}^2$ with a measure or perimeter constraint. We show that the rectangle with measure $1$ which maximises the $k$'th Neumann eigenvalue converges to the unit square in the Hausdorff metric as $k\to \infty$. Furthermore, we determine the unique maximiser of the $k$'th Neumann eigenvalue on a rectangle with given perimeter.

17 pages

References in corpus (2)

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

Cited by in corpus (4)

• Stretching convex domains to capture many lattice points
• Triangles capturing many lattice points
• Lattice points in stretched model domains of finite type in ${\mathbb{R}}^d$
• Maximal spectral surfaces of revolution converge to a catenoid