Global minimality of critical balls for finite eigenvalue sums

Determine whether the ball minimizes the volume-constrained sum of the first n Dirichlet eigenvalues for every n≥2 for which the ball is a critical shape, thereby resolving the finite-index shape-optimization problem beyond the principal eigenvalue.

Background

The paper studies the shape optimization problem of minimizing the sum of the first n Dirichlet eigenvalues among domains of fixed volume. The ball is critical only when n is the last index in an eigenvalue cluster. The authors derive perturbative criteria near the disk and ball that distinguish weak local minimizers from saddle points, but these criteria do not settle global minimality in all critical cases.

References

However, to our knowledge, in the cases where the ball is a critical shape it is far from understood whether it is a minimizer of eq:inf_problem for $n\geq 2$.

— Sums of Dirichlet eigenvalues in proximity of the disk and the ball  (2610.13256 - Baur, 9 Oct 2026) in Section 1, Introduction

For eq:inf_problem, Freitas proved asymptotics of the attained minimal eigenvalue sums, but it remains a conjecture that the ball is the asymptotic minimizer of eq:inf_problem, i.e.~minimizers of eq:inf_problem converge to the ball for $n\to\infty$.

— Sums of Dirichlet eigenvalues in proximity of the disk and the ball  (2610.13256 - Baur, 9 Oct 2026) in Section 1, Introduction

Regarding conjectures about eq:inf_problem, we are only aware of Henrot's conjecture that $\lambda_2(\Omega)+\lambda_3(\Omega)$ and thus, by Faber-Krahn, eq:inf_problem for $n=3$ is minimized by the disk .

— Sums of Dirichlet eigenvalues in proximity of the disk and the ball  (2610.13256 - Baur, 9 Oct 2026) in Section 1, Introduction

We are not able to decide this question in this work, but numerical computations (see Appendix \ref{app:numerical_addendum}) speak in favor of the conjecture that the number of weak local minimizer cases is also infinite.

— Sums of Dirichlet eigenvalues in proximity of the disk and the ball  (2610.13256 - Baur, 9 Oct 2026) in Section 1, subsection Main results, immediately after Theorem 2

An analogue of Theorem \ref{thm:saddle_point_infinitely_many} for three dimensions thus remains an open problem. We conjecture that as in two dimensions both categories, weak local minimizers and saddle points, give infinite sequences.

— Sums of Dirichlet eigenvalues in proximity of the disk and the ball  (2610.13256 - Baur, 9 Oct 2026) in Section 1, subsection Main results, immediately after Theorem 2