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arXiv · 2610.18619

Generic boundary behavior of Dirichlet eigenfunctions

Alec Payne·Zihui Zhao
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Abstract

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 $\mathbb{R}^n$, $k > n$, 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.

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Alec Payne, Zihui Zhao. 2026-10-13. Generic boundary behavior of Dirichlet eigenfunctions. https://arxiv.org/abs/2610.18619

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