Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane

TL;DR

This paper proves that for certain symmetric shapes, the sum of the first Laplace eigenvalues is maximized when the shape is the most symmetric, such as the equilateral triangle, square, or disk, under a fixed ratio involving area and moment of inertia.

Contribution

It establishes maximization results for eigenvalue sums among symmetric shapes and extends the results to various boundary conditions and potentials, using the roots of unity's tight frame property.

Findings

Maximal sums of eigenvalues for equilateral triangle, square, and disk.

Results hold for Dirichlet, Neumann, Robin, and Schrödinger eigenvalues.

Conjecture that the disk maximizes sums of Neumann eigenvalues among convex domains.

Abstract

The sum of the first $n \geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio $\text{(area)}^3/\text{(moment of inertia)}$ for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schr\"odinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.