About a Pólya-Schiffer inequality
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)
- Volume: 65, Issue: 2
- ISSN: 0365-1029
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topBodo Dittmar, and Maren Hantke. "About a Pólya-Schiffer inequality." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.2 (2011): null. <http://eudml.org/doc/289719>.
@article{BodoDittmar2011,
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 \[\sum \_\{k=1\}^n\frac\{1\}\{\lambda \_k\}\ge \sum \_\{k=1\}^n\frac\{1\}\{\lambda \_k^\{(\sigma )\}\},\]
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.},
author = {Bodo Dittmar, Maren Hantke},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Membrane eigenvalues; sums of reciprocal eigenvalues},
language = {eng},
number = {2},
pages = {null},
title = {About a Pólya-Schiffer inequality},
url = {http://eudml.org/doc/289719},
volume = {65},
year = {2011},
}
TY - JOUR
AU - Bodo Dittmar
AU - Maren Hantke
TI - About a Pólya-Schiffer inequality
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 2
SP - null
AB - 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 \[\sum _{k=1}^n\frac{1}{\lambda _k}\ge \sum _{k=1}^n\frac{1}{\lambda _k^{(\sigma )}},\]
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.
LA - eng
KW - Membrane eigenvalues; sums of reciprocal eigenvalues
UR - http://eudml.org/doc/289719
ER -
References
top- 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,
- 2005, pp. 669-686.
- Dittmar, B., Free membrane eigenvalues, Z. Angew. Math. Phys. 60 (2009), 565-568.
- 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.
- Luttinger, J. M., Generalized isoperimetric inequalities, J. Mathematical Phys. 14 (1973), 586-593, ibid. 14 (1973), 1444-1447, ibid. 14 (1973), 1448-1450.
- 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.
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