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High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology

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Torres de Lizaur,  Francisco
Max Planck Institute for Mathematics, Max Planck Society;

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1810.09277.pdf
(Preprint), 201KB

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Citation

Enciso, A., Peralta-Salas, D., & Torres de Lizaur, F. (2021). High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology. In S. Koike (Ed.), Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October 2–6, 2017 (pp. 245-261). Singapore: Springer.


Cite as: http://hdl.handle.net/21.11116/0000-0008-B537-E
Abstract
Let $\Sigma$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $\Sigma$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $\psi$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $\Delta \psi=-\lambda \psi$ (with $\lambda=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $\psi$ given by~$\Sigma$, up to an ambient diffeomorphism.