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Doubling Inequality and Nodal Sets for Solutions of Bi-Laplace Equations

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Abstract

We investigate the doubling inequality and nodal sets for the solutions of bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper bound for the nodal sets of solutions. We show two types of doubling inequalities for the solutions of bi-Laplace equations. As a consequence, the rate of vanishing is given for the solutions.

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Authors and Affiliations

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

    Jiuyi Zhu

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  1. Jiuyi Zhu
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Correspondence to Jiuyi Zhu.

Additional information

Communicated by F. Lin

Zhu is supported in part by NSF grant DMS-1656845 and OIA-1832961

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Zhu, J. Doubling Inequality and Nodal Sets for Solutions of Bi-Laplace Equations. Arch Rational Mech Anal 232, 1543–1595 (2019). https://doi.org/10.1007/s00205-018-01349-2

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