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Sums of Dirichlet eigenvalues in proximity of the disk and the ball
Abstract
We derive second order perturbation expansions of volume-normalized sums of the first n Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when n is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.
Related
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BibTeX
@misc{baur2026sums,
title = {{Sums of Dirichlet eigenvalues in proximity of the disk and the ball}},
author = {Matthias Baur},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13256},
eprint = {2610.13256},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13256},
}
RIS
TY - UNPB
TI - Sums of Dirichlet eigenvalues in proximity of the disk and the ball
AU - Matthias Baur
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13256
UR - https://arxiv.org/abs/2610.13256
AB - We derive second order perturbation expansions of volume-normalized sums of the first n Dirichlet eigenvalues for near-circular and near-spherical domains. The disk (resp. ball) is only a critical shape when n is the last index of an eigenvalue cluster, and in dimension two and three our expansions yield a finitely checkable criterion deciding whether the disk or ball is a weak local minimizer or a saddle point. In two dimensions, this allows us to show that there is an infinite number of cases in which the disk is a saddle point for finite eigenvalue sums.
ER -