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Proof of the Payne-Pólya-Weinberger conjecture
Author(s):
Mark S.
Ashbaugh;
Rafael D.
Benguria
Journal:
Bull. Amer. Math. Soc.
25
(1991),
19-29.
MSC (1985):
Primary 35P15, 49Gxx;
Secondary 35J05, 33A40
MathSciNet review:
1085824
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Additional information
References:
- [AB1] M. S. Ashbaugh and R. D. Benguria, On the ratio of the first two eigenvalues of Schrödinger operators with positive potentials, Differential Equations and Mathematical Physics (I. W. Knowles and Y. Saitō, eds.), Lecture Notes in Math., vol. 1285, Springer-Verlag, Berlin, 1987, pp. 16-25. MR 921249
- [AB2] M. S. Ashbaugh and R. D. Benguria, Sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions (in preparation).
- [Ba] C. Bandle, Isoperimetric inequalities and applications, Pitman, Boston, 1980. MR 572958
- [Br] J. J. A. M. Brands, Bounds for the ratios of the first three membrane eigenvalues, Arch. Rational Mech. Anal. 16 (1964), 265-268. MR 163068
- [BLL] H. J. Brascamp, E. H. Lieb, and J. M. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal. 17 (1974), 227-237. MR 346109
- [Ch1] G. Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators, J. Appl. Math. and Phys. (ZAMP) 33 (1982), 143-148. MR 652928
- [Ch2] G. Chiti, A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math. Anal. 14 (1983), 1163-1167. MR 718816
- [dV] H. L. de Vries, On the upper bound for the ratio of the first two membrane eigenvalues, Z. Natur. 22A (1967), 152-153. MR 209664
- [HLP] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd ed., Cambridge Univ. Press, Cambridge, 1952. MR 46395
- [H] J. Hersch, On symmetric membranes and conformal radius: Some complements to Pólya's and Szegö's inequalities, Arch. Rational Mech. Anal. 20 (1965), 378-390. MR 186929
- [HR] J. Hersch and G.-C. Rota (eds.), George Pólya: Collected Papers, vol. III: Analysis, MIT Press, Cambridge, MA, 1984. MR 758989
- [K] B. Kawohl, Rearrangements and convexity of level sets in PDE, Lecture Notes in Math., vol. 1150, Springer-Verlag, Berlin, 1985. MR 810619
- [L1] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math. 57 (1977), 93-105. MR 471785
- [L2] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349-374. MR 717827
- [PPW1] L. E. Payne, G. Pólya, and H. F. Weinberger, Sur le quotient de deux fréquences propres consécutives, C. R. Acad. Sci. Paris 241 (1955), 917—919 (reprinted as pp. 410-412 of [HR]).
- [PPW2] L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298 (reprinted as pp. 420-429 of [HR] with comments by J. Hersch on pp. 521-522). MR 84696
- [Th] C. J. Thompson, On the ratio of consecutive eigenvalues in N-dimensions, Stud. Appl. Math. 48 (1969), 281-283. MR 257592
- [W] H. F. Weinberger, An isoperimetric inequality for the N-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636. MR 79286
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Additional Information:
DOI:
10.1090/S0273-0979-1991-16016-7
PII:
S 0273-0979(1991)16016-7
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