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Sums of free membrane eigenvalues. (English) Zbl 1077.30008

Summary: Let \(D\) be a simply connected domain in the plane which is the image of the unit disk under a normalized conformal mapping, and let \(\mu_1 = 0 < \mu_2\leq \mu_3\dots\) be the free membrane eigenvalues. We prove that for any \(n\geq 2\) where \(A\) is the area of the domain \(D\) and \(\mu^{(o)}_j\) are the free membrane eigenvalues of the unit disk.

MSC:

30C35 General theory of conformal mappings
35P15 Estimates of eigenvalues in context of PDEs
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[1] Bandle, C., Extensions of an inequality by Pólya and Schiffer for vibrating membranes, Pacific J. Math., 42, 543-555 (1972) · Zbl 0217.13301
[2] Bandle, C., Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane, SIAM J. Appl. Math., 22, 142-147 (1972) · Zbl 0237.35069 · doi:10.1137/0122016
[3] Bandle, C., Isoperimetric Inequalities and Applications (1980), London: Pitman Publ., London · Zbl 0436.35063
[4] Courant, R.; Hilbert, D., Methods of Mathematical Physics (1953), New York: Wiley, New York · Zbl 0729.00007
[5] Dittmar, B., Sums of reciprocal eigenvalues of the Laplacian, Math. Nachr., 237, 45-61 (2002) · Zbl 1200.35199 · doi:10.1002/1522-2616(200204)237:1<45::AID-MANA45>3.0.CO;2-M
[6] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1934), London and New York: Cambridge Univ. Press, London and New York · JFM 60.0169.01
[7] Henrici, P., Applied and Computational Complex Analysis (1986), New York: Wiley, New York · Zbl 0578.30001
[8] Hersch, J., On symmetric membranes and conformal radius: some complements to Pólya’s and Szegö’s inequalities, Arch. Rational Mech. Anal., 20, 378-395 (1965) · Zbl 0188.17202 · doi:10.1007/BF00282359
[9] Kröger, P., Upper bounds for the Neumann eigenvalues on a bounded domain in euclidean space, J. Funct. Anal., 106, 353-357 (1992) · Zbl 0777.35044 · doi:10.1016/0022-1236(92)90052-K
[10] Kröger, P., On upper bounds for high order Neumann eigenvalues of convex domains in euclidean space, Proc. Amer. Math. Soc., 127, 1665-1669 (1999) · Zbl 0911.35079 · doi:10.1090/S0002-9939-99-04804-2
[11] Laugesen, R. S.; Morpurgo, C., Extremals for eigenvalues of Laplacians under conformal mapping, J. Funct. Anal., 155, 64-108 (1998) · Zbl 0917.47018 · doi:10.1006/jfan.1997.3222
[12] Marshall, A. M.; Olkin, I., Inequalities: Theory of Majorization and its Applications (1979), New York: Academic Press, New York · Zbl 0437.26007
[13] Nadirashvili, N., Conformal maps and isoperimetric inequalities for eigenvalues of the Neumann problem, Israel Math. Conf. Proc., 11, 197-201 (1997) · Zbl 0890.35097
[14] Pólya, G., On the characteristic frequencies of symmetric membranes, Math. Z., 63, 331-337 (1955) · Zbl 0065.08703 · doi:10.1007/BF01187944
[15] Pólya, G.; Schiffer, M., Convexity of functionals by transplantation, J. Analyse Math., 3, 245-345 (1954) · Zbl 0056.32701 · doi:10.1007/BF02803593
[16] Szegö, G., Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal., 3, 343-356 (1954) · Zbl 0055.08802
[17] Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal., 5, 633-636 (1956) · Zbl 0071.09902
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