Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes

Jeffrey J. Langford, Richard S. Laugesen

Research output: Contribution to journalArticlepeer-review

Abstract

The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.

Original languageEnglish (US)
JournalComputational Methods and Function Theory
DOIs
StateAccepted/In press - 2023

Keywords

  • Curvature bound
  • Hyperbolic disk
  • Isoperimetric inequality
  • Laplace–Beltrami
  • Laplacian eigenfunction
  • Simply connected surface
  • Spherical cap
  • Vibrating membrane

ASJC Scopus subject areas

  • Analysis
  • Computational Theory and Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes'. Together they form a unique fingerprint.

Cite this