TY - JOUR
T1 - Maximizing the Second Robin Eigenvalue of Simply Connected Curved Membranes
AU - Langford, Jeffrey J.
AU - Laugesen, Richard S.
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2023
Y1 - 2023
N2 - 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.
AB - 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.
KW - Curvature bound
KW - Hyperbolic disk
KW - Isoperimetric inequality
KW - Laplace–Beltrami
KW - Laplacian eigenfunction
KW - Simply connected surface
KW - Spherical cap
KW - Vibrating membrane
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U2 - 10.1007/s40315-023-00516-1
DO - 10.1007/s40315-023-00516-1
M3 - Article
AN - SCOPUS:85180496619
SN - 1617-9447
JO - Computational Methods and Function Theory
JF - Computational Methods and Function Theory
ER -