Among all C∞ 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.
Bibliographical noteFunding Information:
Chasman thanks the Bucknell University Mathematics Department for providing housing during a visit during the spring of 2018, when conversations about this project first began. Both authors thank Richard Laugesen for encouraging us to pursue this problem and for useful discussions.
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- Bessel functions
- Robin boundary conditions
- isoperimetric inequality