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Among all $C^{\infty}$ bounded domains with equal volume, we show that thesecond eigenvalue of the Robin plate is uniquely maximized by an open ball, solong as the Robin parameter lies within a particular range of negative values.Our methodology combines recent techniques introduced by Freitas and Laugesento study the second eigenvalue of the Robin membrane problem and techniquesemployed by Chasman to study the free plate problem. In particular, we chooseeigenfunctions of the ball as trial functions in the Rayleigh quotient for ageneral domain; such eigenfunctions are comprised of ultraspherical Bessel andmodified Bessel functions. Much of our work hinges on developing anunderstanding of delicate properties of these special functions, which may beof independent interest.

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