1 November 2018 Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues
Alex H. Barnett, Andrew Hassell, Melissa Tacy
Duke Math. J. 167(16): 3059-3114 (1 November 2018). DOI: 10.1215/00127094-2018-0031


For smooth bounded domains in Rn, we prove upper and lower L2 bounds on the boundary data of Neumann eigenfunctions, and we prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of the eigenvalues; this is achieved by working with an appropriate norm for boundary functions, which includes a spectral weight, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery-type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter E>0 to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the Helmholtz equation (ΔE)u=0. This inclusion bound improves over previously known bounds by a factor of E5/6, analogously to a recently improved inclusion bound in the Dirichlet case due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the 42000th) from nine to fourteen digits, with negligible extra numerical effort.


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Alex H. Barnett. Andrew Hassell. Melissa Tacy. "Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues." Duke Math. J. 167 (16) 3059 - 3114, 1 November 2018. https://doi.org/10.1215/00127094-2018-0031


Received: 16 December 2015; Revised: 8 May 2018; Published: 1 November 2018
First available in Project Euclid: 10 October 2018

zbMATH: 06985301
MathSciNet: MR3870081
Digital Object Identifier: 10.1215/00127094-2018-0031

Primary: 35J67
Secondary: 35J05 , 58J50 , 65N25

Keywords: boundary values , eigenfunction estimates , inclusion bounds , Neumann eigenfunctions , quasiorthogonality , spectral weight

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 16 • 1 November 2018
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