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Nodal sets of Steklov eigenfunctions

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Abstract

We study the nodal set of the Steklov eigenfunctions on the boundary of a smooth bounded domain in \(\mathbb {R}^n\)- the eigenfunctions of the Dirichlet-to-Neumann map. Under the assumption that the domain \(\Omega \) is \(C^2\), we prove a doubling property for the eigenfunction \(u\). We estimate the Hausdorff \(\mathcal H^{n-2}\)-measure of the nodal set of \(u|_{\partial \Omega }\) in terms of the eigenvalue \(\lambda \) as \(\lambda \) grows to infinity. In case that the domain \(\Omega \) is analytic, we prove a polynomial bound O(\(\lambda ^6\)). Our arguments, which make heavy use of Almgren’s frequency functions, are built on the previous works [Garofalo and Lin, Commun Pure Appl Math 40(3):347–366, 1987; Lin, Commun Pure Appl Math 44(3):287–308, 1991].

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Acknowledgments

After our paper was submitted to Calculus of Variations and Partial Differential Equations and posted on arXiv, there has been recent progress on the topic. By very different methods, Steve Zelditch [Measure of nodal sets of analytic Steklov eigenfunctions, arXiv:1403.0647] obtained the optimal upper bound of \(C \lambda \) for the size of the \(\mathcal H^{n-2}\)-measure of the nodal set of Steklov eigenfunctions corresponding to \(\lambda \). Later Jiuyi Zhu [Doubling property and vanishing order of Steklov eigenfunctions, Commun. Partial Differ. Equ., to appear (doi:10.1080/03605302.2015.1025980)] improved the constant in our doubling condition to the optimal one and Angkana Rüland even removed the smallness assumption on the size of the balls on which it is valid, and hence also proved the optimal upper bound of the nodal set, even for more general problems [On some quantitative unique continuation properties of fractional Schrödinger equations: doubling, vanishing order and nodal domain estimates, arXiv:1407.0817]. Furthermore, Xing Wang and Jiuyi Zhu proved a polynomial lower bound of the nodal set under the assumption that 0 is a regular value for the Steklov eigenfunction [A Lower bound for the nodal sets of Steklov eigenfunctions, arXiv:1411.0708]. Also, it was brought to our attention that in 1994, Giovanni Alessandrini and Rolando Magnanini considered the Steklov problem in two dimensions in [Elliptic equations in divergence form, geometric critical points of solutions and Stekloff eigenfunctions, SIAM J. Math. Anal., 25 (5) (1994), 1259–1268]. It follows from their work that the \(n\)-th eigenfunction on the boundary of a simply-connected planar domain has at most \(2n\) nodal domains.

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  1. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Katarína Bellová

  2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Fang-Hua Lin

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  1. Katarína Bellová
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Correspondence to Katarína Bellová.

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Communicated by L. Ambrosio.

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Bellová, K., Lin, FH. Nodal sets of Steklov eigenfunctions. Calc. Var. 54, 2239–2268 (2015). https://doi.org/10.1007/s00526-015-0864-8

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