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arXiv · 2104.09012

The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions

A. Logunov·E. Malinnikova·N. Nadirashvili·F. Nazarov
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Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_\lambda$ be a Dirichlet Laplace eigenfunction in $\Omega$ with eigenvalue $\lambda$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_\lambda$ does not exceed $C(\Omega)\sqrt{\lambda}$. This result is new even for the case of domains with $C^\infty$-smooth boundary.

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A. Logunov, E. Malinnikova, N. Nadirashvili, F. Nazarov. 2021-04-19. The sharp upper bound for the area of the nodal sets of Dirichlet Laplace eigenfunctions. https://arxiv.org/abs/2104.09012

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