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We present asymptotically sharp inequalities for the eigenvalues of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in . For the Riesz mean 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 .
In addition, we obtain two-sided bounds for individual , 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.
Cite this article
Evans M. Harrell II, Joachim Stubbe, Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian. J. Spectr. Theory 8 (2018), no. 4, pp. 1529–1550DOI 10.4171/JST/234