Scaling inequalities and limits for clamped plate eigenvalues on geodesic disks

Abstract

For the bilaplacian in spherical and hyperbolic spaces, clamped vibrating and buckling eigenvalues on geodesic disks are shown to satisfy scaling inequalities analogous to the standard scale invariance of Euclidean plate eigenvalues. These results extend curved-space scaling results from the Laplacian to fourth-order eigenvalue problems. In addition, the limiting behavior of the scaled eigenvalues is determined as the disks expand to fill the ambient space. Depending on the geometry and the choice of scaling, the eigenvalues tend to zero, positive or negative infinity, or finite values associated with the bottom of the hyperbolic spectrum. The negative divergence in one spherical buckling limit is related by conformal inversion to an exterior buckling problem.