## 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

Zbl 0940.35148
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