Scaling inequalities and limits for Robin and Dirichlet eigenvalues

TL;DR

This paper investigates scaling inequalities for Robin and Dirichlet eigenvalues of the Laplacian on spherical and hyperbolic spaces, revealing new behaviors and extending previous results to higher dimensions and different boundary conditions.

Contribution

It extends existing scaling inequalities to Robin problems in two dimensions and Dirichlet problems in higher dimensions, and uncovers exotic asymptotic behaviors of Robin eigenvalues.

Findings

Scaling inequalities analogous to Euclidean case are established.

Robin eigenvalues tend to the spectrum of an exterior Robin problem as domains expand.

Results extend prior work of Langford and Laugesen to new settings.

Abstract

For the Laplacian in spherical and hyperbolic spaces, Robin eigenvalues in two dimensions and Dirichlet eigenvalues in higher dimensions are shown to satisfy scaling inequalities analogous to the standard scale invariance of the Euclidean Laplacian. These results extend work of Langford and Laugesen to Robin problems and to Dirichlet problems in higher dimensions. In addition, scaled Robin eigenvalues behave exotically as the domain expands to a 2-sphere, tending to the spectrum of an exterior Robin problem.