The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Journal of Spectral Theory

Full-Text PDF (230 KB) | Metadata | Table of Contents | JST summary
Volume 8, Issue 4, 2018, pp. 1529–1550
DOI: 10.4171/JST/234

Published online: 2018-10-22

Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

Evans M. Harrell II[1] and Joachim Stubbe[2]

(1) Georgia Institute of Technology, Atlanta, USA
(2) EPFL, Lausanne, Switzerland

We present asymptotically sharp inequalities for the eigenvalues $\mu_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in [14]. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$.

In addition, we obtain two-sided bounds for individual $\mu_k$, which are semiclassically sharp, and we obtain a Neumann version of Laptev’s result that the Pólya conjecture is valid for domains that are Cartesian products of a generic domain with one for which Pólya’s conjecture holds. In a final section, we remark upon the Dirichlet case with the same methods.

Keywords: Neumann Laplacian, Dirichlet Laplcian, semiclassical bounds for eigenvalues

Harrell II Evans, Stubbe Joachim: Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian. J. Spectr. Theory 8 (2018), 1529-1550. doi: 10.4171/JST/234