About a Pólya-Schiffer inequality

Bodo Dittmar, Maren Hantke

Abstract


For simply connected planar domains with the maximal conformal radius 1 it was proven in 1954 by G. Pólya and M. Schiffer that for the eigenvalues \(\lambda\) of the fixed membrane for any \(n\) the following inequality holds \[\sum_{k=1}^n\frac{1}{\lambda_k}\geq \sum_{k=1}^n\frac{1}{\lambda_k^{(\sigma)}},\] where \(\lambda_k^{(\sigma)}\) are the eigenvalues of the unit disk. The aim of the paper is to give a sharper version of this inequality and for the sum of all reciprocals to derive formulas which allow in some cases to calculate exactly this sum.

Keywords


Membrane eigenvalues; sums of reciprocal eigenvalues

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References


Bandle, C., Isoperimetric Inequalities and Applications, Pitman Publ., London, 1980.

Dittmar, B., Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr. 237 (2002), 45-61.

Dittmar, B., Sums of free membrane eigenvalues, J. Anal. Math. 95 (2005), 323-332.

Dittmar, B., Eigenvalue problems and conformal mapping, R. K¨uhnau (ed.), Handbook of Complex Analysis: Geometric Function Theory. Vol. 2, Elsevier, Amsterdam,

, pp. 669-686.

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Hantke, M., Summen reziproker Eigenwerte, Dissertation Martin-Luther-Universitat, Halle-Wittenberg, 2006.

Henrot, A., Extremum problems for eigenvalues of elliptic operators, Birkauser, Basel-Boston-Berlin, 2006.

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Pólya, G., Schiffer, M., Convexity of functionals by transplantation, J. Analyse Math. 3 (1954), 245-345.

Pólya, G., Szego, G., Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, N. J., 1951.




DOI: http://dx.doi.org/10.2478/v10062-011-0011-8
Date of publication: 2016-07-27 21:54:08
Date of submission: 2016-07-26 07:54:18


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