A sharp isoperimetric inequality for the second eigenvalue of the Robin plate

  • Laura Mercredi Chasman

    University of Minnesota - Morris, USA
  • Jeffrey J. Langford

    Bucknell University, Lewisburg, USA
A sharp isoperimetric inequality for the second eigenvalue of the Robin plate cover
Download PDF

This article is published open access under our Subscribe to Open model.

Abstract

Among all 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 andmodified Bessel functions. Much of our work hinges on developing an understanding of delicate properties of these special functions, which may be of independent interest.

Cite this article

Laura Mercredi Chasman, Jeffrey J. Langford, A sharp isoperimetric inequality for the second eigenvalue of the Robin plate. J. Spectr. Theory 12 (2022), no. 2, pp. 617–657

DOI 10.4171/JST/413