Generic boundary behavior of Dirichlet eigenfunctions
Abstract
We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of Ck domains in ℝn, k > n, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a Ck−1 embedded submanifold, whenever the intersection is non-empty. This is achieved by analyzing the boundary asymptotics of Dirichlet eigenfunctions via homogeneous harmonic polynomials and comparing their respective nodal sets. Along the way, we give a complete proof that generic Euclidean domains have simple Dirichlet eigenfunctions with zero as a regular value in the interior and zero as a regular value of the boundary normal derivative. In the special case of planar domains, a different argument provides one-parameter families of perturbed domains with the desired properties.
Related
- Generic properties of the Laplace spectrum and eigenfunctions
- Strict Log-Concavity of First Dirichlet Eigenfunctions under Symmetry in ℂℙn
- Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators
- Log-Concavity of First Dirichlet Eigenfunctions on ℂℙ2
- Cherednik integrable system: eigenfunctions at generic eigenvalues
BibTeX
@misc{payne2026generic,
title = {{Generic boundary behavior of Dirichlet eigenfunctions}},
author = {Alec Payne and Zihui Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18619},
eprint = {2610.18619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18619},
}
RIS
TY - UNPB
TI - Generic boundary behavior of Dirichlet eigenfunctions
AU - Alec Payne
AU - Zihui Zhao
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18619
UR - https://arxiv.org/abs/2610.18619
AB - We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of C^k domains in ℝ^n, k > n, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a C^k−1 embedded submanifold, whenever the intersection is non-empty. This is achieved by analyzing the boundary asymptotics of Dirichlet eigenfunctions via homogeneous harmonic polynomials and comparing their respective nodal sets. Along the way, we give a complete proof that generic Euclidean domains have simple Dirichlet eigenfunctions with zero as a regular value in the interior and zero as a regular value of the boundary normal derivative. In the special case of planar domains, a different argument provides one-parameter families of perturbed domains with the desired properties.
ER -