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Lower bounds for interior nodal sets of Steklov eigenfunctions


Authors: Christopher D. Sogge, Xing Wang and Jiuyi Zhu
Journal: Proc. Amer. Math. Soc. 144 (2016), 4715-4722
MSC (2010): Primary 35-xx
DOI: https://doi.org/10.1090/proc/13067
Published electronically: July 22, 2016
MathSciNet review: 3544523
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Abstract: We study the interior nodal sets, $ Z_\lambda $ of Steklov eigenfunctions in an $ n$-dimensional relatively compact manifold $ M$ with boundary and show that one has the lower bounds $ \vert Z_\lambda \vert\ge c\lambda ^{\frac {2-n}2}$ for the size of its $ (n-1)$-dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in previous works of the first author and Zelditch.


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Additional Information

Christopher D. Sogge
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: sogge@jhu.edu

Xing Wang
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Wayne State University, Detroit, MI 48202
Email: fz1316@wayne.edu

Jiuyi Zhu
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803
Email: zhu@math.isu.edu

DOI: https://doi.org/10.1090/proc/13067
Received by editor(s): March 16, 2015
Published electronically: July 22, 2016
Additional Notes: The first two authors were supported in part by the NSF grant DMS-1361476
The third author was supported in part by the NSF grant DMS-1500468
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society

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