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On extending the inequalities of Payne, Pólya, and Weinberger using spherical harmonics. (English) Zbl 1184.35232

Summary: Using spherical harmonics, rearrangement techniques, the Sobolev inequality and Chiti’s reverse Hölder inequality, we obtain extensions of a classical result of Payne, Pólya, and Weinberger bounding the gap between consecutive eigenvalues of the Dirichlet Laplacian in terms of moments of the preceding ones. The extensions yield domain-dependent inequalities.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
33C55 Spherical harmonics
49R05 Variational methods for eigenvalues of operators
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