A Sharp Isoperimetric Inequality for the Second Eigenvalue of the Robin Plate

TL;DR

This paper proves that among smooth bounded domains of equal volume, the second eigenvalue of the Robin plate is maximized by a ball for certain negative Robin parameters, using advanced eigenfunction techniques.

Contribution

It establishes a sharp isoperimetric inequality for the second eigenvalue of the Robin plate, extending methods from Robin membrane and free plate problems.

Findings

The second eigenvalue is maximized by a ball for negative Robin parameters.

Eigenfunctions of the ball involving Bessel functions serve as effective trial functions.

The work develops new insights into properties of ultraspherical Bessel functions.

Abstract

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.