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Sums of free membrane eigenvalues

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Abstract

LetD be a simply connected domain in the plane which is the image of the unit disk under a normalized conformal mapping, and let μ1=0<μ2≤μ3 be the free membrane eigenvalues. We prove that for anyn≥2,

$$A\sum\limits_2^n {\frac{1}{{\mu _j }} \geqslant \pi } \sum\limits_2^n {\frac{1}{{\mu _j ^{(o)} }}} $$
(1)

whereA is the area of the domainD and μ (o)j are the free membrane eigenvalues of the unit disk.

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Authors and Affiliations

  1. Fachbereich Mathematik und Informatik Institut für Analysis, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Str. 5, D-06120, Halle, Germany

    Bodo Dittmar

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  1. Bodo Dittmar
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Correspondence to Bodo Dittmar.

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Dittmar, B. Sums of free membrane eigenvalues. J. Anal. Math. 95, 323–332 (2005). https://doi.org/10.1007/BF02791506

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