This article is concerned with the improvements of certain eigenvalue inequalities of Stokes operator and Dirichlet Laplacian related to the Berezin-Li-Yau type inequalities. The formulas proved extend the earlier works of Melas [“A lower bound for sums of eigenvalues of the Laplacian,” Proc. Am. Math. Soc. 131(2), 631–636 (2002)] https://doi.org/10.1090/S0002-9939-02-06834-X on Dirichlet Laplacian and of Ilyin [“Lower bounds for the spectrum of the Laplace and Stokes operators,” Discrete. Contin. Dyn. Syst. 28(1), 131–146 (2010)] https://doi.org/10.3934/dcds.2010.28.131 on Stokes operator for any dimension d ⩾ 2 and they are asymptotically sharp as are the earlier inequalities of Berezin-Li-Yau, Melas, and Ilyin.
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2012
American Institute of Physics
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