## Duke Mathematical Journal

### Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

#### Abstract

For smooth bounded domains in $\mathbb{R}^{n}$, we prove upper and lower $L^{2}$ 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 $E\gt 0$ to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function $u$ solving the Helmholtz equation $(\Delta-E)u=0$. This inclusion bound improves over previously known bounds by a factor of $E^{5/6}$, 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 $42000$th) from nine to fourteen digits, with negligible extra numerical effort.

#### Article information

Source
Duke Math. J., Volume 167, Number 16 (2018), 3059-3114.

Dates
Received: 16 December 2015
Revised: 8 May 2018
First available in Project Euclid: 10 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1539137164

Digital Object Identifier
doi:10.1215/00127094-2018-0031

Mathematical Reviews number (MathSciNet)
MR3870081

Zentralblatt MATH identifier
06985301

#### Citation

Barnett, Alex H.; Hassell, Andrew; Tacy, Melissa. Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues. Duke Math. J. 167 (2018), no. 16, 3059--3114. doi:10.1215/00127094-2018-0031. https://projecteuclid.org/euclid.dmj/1539137164

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