About a Pólya-Schiffer inequality
Bodo Dittmar ; Maren Hantke
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011), / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

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 λ of the fixed membrane for any n the following inequality holds ∑k=1n1λk≥∑k=1n1λk(σ), where λk(σ) 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.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:289719 
@article{bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_29-44,
     author = {Bodo Dittmar and Maren Hantke},
     title = {About a P\'olya-Schiffer inequality},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {65},
     year = {2011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_29-44}
}

Bodo Dittmar; Maren Hantke. About a Pólya-Schiffer inequality. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_29-44/

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.