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Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

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Abstract

The aim of this article is to provide a simple and unified way to obtain the sharp upper bounds of nodal sets of eigenfunctions for different types of eigenvalue problems on real analytic domains. The examples include biharmonic Steklov eigenvalue problems, buckling eigenvalue problems and champed-plate eigenvalue problems. The geometric measure of nodal sets are derived from doubling inequalities and growth estimates for eigenfunctions. It is done through analytic estimates of Morrey–Nirenberg and Carleman estimates.

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Authors and Affiliations

  1. Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Fanghua Lin

  2. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Jiuyi Zhu

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  1. Fanghua Lin
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  2. Jiuyi Zhu
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Correspondence to Jiuyi Zhu.

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Communicated by Y. Giga.

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Lin is supported in part by NSF Grant DMS-1955249, Zhu is supported in part by NSF Grant OIA-1832961.

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Lin, F., Zhu, J. Upper bounds of nodal sets for eigenfunctions of eigenvalue problems. Math. Ann. 382, 1957–1984 (2022). https://doi.org/10.1007/s00208-020-02098-y

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  • DOI: https://doi.org/10.1007/s00208-020-02098-y

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