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Sums of reciprocal Stekloff eigenvalues. (English) Zbl 1054.35041

The author considers bounds for the reciprocal \(\sum_2^\infty \lambda_j^{-2}\), where \(0 = \lambda_1 < \lambda_2 \leq \lambda_3 \leq \dots\) are the Stekloff eigenvalues of a plane domain \(D\).
Using the theory of univalent conformal mappings, if \(D\) is simply connected, then view \(D\) as the image of the unit disk. Further by the properties of the Neumann function, \(\sum_2^\infty \lambda_j^{-2}\) has an integral representation which has the Neumann function as the kernel. Through a sophisticated manipulation of the Neumann function and the conformal mapping, the author gives a lower bound for the reciprocal and proves that the estimate is sharp when \(D\) is a disk.
This result appeared in the author’s article [Pitman Res. Notes Math. Ser. 383, 78–87 (1998; Zbl 0940.35148)], here the proof is different.
Some similar statement about doubly connected planar domains is also given.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
30C20 Conformal mappings of special domains
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Citations:

Zbl 0940.35148
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