Complementary asymptotically sharp estimates for eigenvalue means of Laplacians
Abstract
We present asymptotically sharp inequalities, containing a second term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kr\"oger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean R1(z) of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.
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