A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate
Publication Date
2022
Description
Among all $C^{\infty}$ bounded domains with equal volume, we show that the second eigenvalue of the Robin plate is uniquely maximized by an open ball, so long as the Robin parameter lies within a particular range of negative values. Our methodology combines recent techniques introduced by Freitas and Laugesen to study the second eigenvalue of the Robin membrane problem and techniques employed by Chasman to study the free plate problem. In particular, we choose eigenfunctions of the ball as trial functions in the Rayleigh quotient for a general domain; such eigenfunctions are comprised of ultraspherical Bessel and modified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.
Journal
Journal of Spectral Theory
Volume
12
Issue
2
First Page
617
Last Page
657
Department
Mathematics
Link to Published Version
https://ems.press/content/serial-article-files/33879
DOI
10.4171/JST/413
Recommended Citation
Langford, Jeffrey J. and Chasman, L. Mercredi. "A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate." (2022) : 617-657.