A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of
Authors:
Mark S. Ashbaugh and Rafael D. Benguria
Journal:
Trans. Amer. Math. Soc. 353 (2001), 1055-1087
MSC (1991):
Primary 58G25; Secondary 35P15, 49Rxx, 33C55
DOI:
https://doi.org/10.1090/S0002-9947-00-02605-2
Published electronically:
November 8, 2000
MathSciNet review:
1707696
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: For a domain contained in a hemisphere of the
-dimensional sphere
we prove the optimal result
for the ratio of its first two Dirichlet eigenvalues where
, the symmetric rearrangement of
in
, is a geodesic ball in
having the same
-volume as
. We also show that
for geodesic balls of geodesic radius
less than or equal to
is an increasing function of
which runs between the value
for
(this is the Euclidean value) and
for
. Here
denotes the
th positive zero of the Bessel function
. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of
and having a fixed value of
the one with the maximal value of
is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for
. Various other results for
and
of geodesic balls in
are proved in the course of our work.
- 1. Handbook of mathematical functions, with formulas, graphs and mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 20402), 1966. MR 0208798
- 2. Mark Anthony Armstrong, Basic topology, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1979 original. MR 705632
- 3. Mark S. Ashbaugh and Rafael D. Benguria, Log-concavity of the ground state of Schrödinger operators: a new proof of the Baumgartner-Grosse-Martin inequality, Phys. Lett. A 131 (1988), no. 4-5, 273–276. MR 954970, https://doi.org/10.1016/0375-9601(88)90026-6
- 4. M. S. Ashbaugh and R. Benguria, Optimal lower bounds for eigenvalue gaps for Schrödinger operators with symmetric single-well potentials and related results, Maximum principles and eigenvalue problems in partial differential equations (Knoxville, TN, 1987) Pitman Res. Notes Math. Ser., vol. 175, Longman Sci. Tech., Harlow, 1988, pp. 134–145. MR 963464
- 5. Mark S. Ashbaugh and Rafael D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 1, 19–29. MR 1085824, https://doi.org/10.1090/S0273-0979-1991-16016-7
- 6. Mark S. Ashbaugh and Rafael D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Ann. of Math. (2) 135 (1992), no. 3, 601–628. MR 1166646, https://doi.org/10.2307/2946578
- 7. Mark S. Ashbaugh and Rafael D. Benguria, A second proof of the Payne-Pólya-Weinberger conjecture, Comm. Math. Phys. 147 (1992), no. 1, 181–190. MR 1171765
- 8. Mark S. Ashbaugh and Rafael D. Benguria, Isoperimetric inequalities for eigenvalue ratios, Partial differential equations of elliptic type (Cortona, 1992) Sympos. Math., XXXV, Cambridge Univ. Press, Cambridge, 1994, pp. 1–36. MR 1297771
- 9. Mark S. Ashbaugh and Rafael D. Benguria, Sharp upper bound to the first nonzero Neumann eigenvalue for bounded domains in spaces of constant curvature, J. London Math. Soc. (2) 52 (1995), no. 2, 402–416. MR 1356151, https://doi.org/10.1112/jlms/52.2.402
- 10. Mark S. Ashbaugh and Rafael D. Benguria, On the Payne-Pólya-Weinberger conjecture on the 𝑛-dimensional sphere, General inequalities, 7 (Oberwolfach, 1995) Internat. Ser. Numer. Math., vol. 123, Birkhäuser, Basel, 1997, pp. 111–128. MR 1457273, https://doi.org/10.1007/978-3-0348-8942-1_10
- 11. M. S. Ashbaugh and H. A. Levine, Inequalities for the Dirichlet and Neumann eigenvalues of the Laplacian for domains on spheres, Journées ``Équations aux Dérivées Partielles'' (Saint-Jean-de-Monts, 1997), Exp. No. 1, 15 pp., École Polytechnique, Palaiseau, 1997.
- 12. Frank E. Baginski, Ordering the zeroes of Legendre functions 𝑃^{𝑚}_{𝜈}(𝑧₀) when considered as a function of 𝜈, J. Math. Anal. Appl. 147 (1990), no. 1, 296–308. MR 1044702, https://doi.org/10.1016/0022-247X(90)90400-A
- 13. Frank E. Baginski, Comparison theorems for the 𝜈-zeroes of Legendre functions 𝑃^{𝑚}_{𝜈}(𝑧₀) when -1<𝑧₀<1, Proc. Amer. Math. Soc. 111 (1991), no. 2, 395–402. MR 1043402, https://doi.org/10.1090/S0002-9939-1991-1043402-X
- 14. Catherine Bandle, Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980. MR 572958
- 15. C. Bandle and M. Flucher, Table of inequalities in elliptic boundary value problems, Recent progress in inequalities (Niš, 1996) Math. Appl., vol. 430, Kluwer Acad. Publ., Dordrecht, 1998, pp. 97–125. MR 1609927
- 16. Bernhard Baumgartner, Level comparison theorems, Ann. Physics 168 (1986), no. 2, 484–526. MR 844889, https://doi.org/10.1016/0003-4916(86)90041-2
- 17. B. Baumgartner, Relative concavity of ground state energies as functions of a coupling constant, Phys. Lett. A 170 (1992), no. 1, 1–4. MR 1185617, https://doi.org/10.1016/0375-9601(92)90381-U
- 18. B. Baumgartner, H. Grosse, and A. Martin, The Laplacian of the potential and the order of energy levels, Phys. Lett. B 146 (1984), no. 5, 363–366. MR 763860, https://doi.org/10.1016/0370-2693(84)91715-5
- 19. B. Baumgartner, H. Grosse, and A. Martin, Order of levels in potential models, Nuclear Phys. B 254 (1985), no. 3-4, 528–542. MR 793699, https://doi.org/10.1016/0550-3213(85)90231-7
- 20. Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 4th ed., John Wiley & Sons, Inc., New York, 1989. MR 972977
- 21. Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
- 22. Isaac Chavel, Lowest-eigenvalue inequalities, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 79–89. MR 573429
- 23. Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- 24. Giuseppe Chiti, Orlicz norms of the solutions of a class of elliptic equations, Boll. Un. Mat. Ital. A (5) 16 (1979), no. 1, 178–185 (Italian, with English summary). MR 530146
- 25. Giuseppe Chiti, A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators, Z. Angew. Math. Phys. 33 (1982), no. 1, 143–148. MR 652928, https://doi.org/10.1007/BF00948319
- 26. Giuseppe Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital. A (6) 1 (1982), no. 1, 145–151 (English, with Italian summary). MR 652376
- 27. Giuseppe Chiti, A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math. Anal. 14 (1983), no. 6, 1163–1167. MR 718816, https://doi.org/10.1137/0514090
- 28. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. MR 16:426a
- 29. James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
- 30. G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungsberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München Jahrgang, 1923, pp. 169-172.
- 31. S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, 133–161. MR 412442, https://doi.org/10.1007/BF02568147
- 32. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge University Press, Cambridge, 1952. MR 13:727a
- 33. Evans M. Harrell II and Patricia L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Comm. Partial Differential Equations 19 (1994), no. 11-12, 2037–2055. MR 1301181, https://doi.org/10.1080/03605309408821081
- 34. P. D. Hislop and I. M. Sigal, Introduction to spectral theory, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. With applications to Schrödinger operators. MR 1361167
- 35. John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
- 36. Tosio Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, Band 132. MR 0407617
- 37. E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97-100.
- 38. E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1-44. [English translation: Minimal properties of the sphere in three and more dimensions, Edgar Krahn 1894-1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, The Netherlands, 1994, pp. 139-174.]
- 39. James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- 40. James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- 41. Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, https://doi.org/10.1090/S0002-9904-1978-14553-4
- 42. L. E. Payne, G. Pólya, and H. F. Weinberger, Sur le quotient de deux fréquences propres consécutives, Comptes Rendus Acad. Sci. Paris 241 (1955), 917-919. MR 17:372d
- 43. L. E. Payne, G. Pólya, and H. F. Weinberger, On the ratio of consecutive eigenvalues, J. Math. and Phys. 35 (1956), 289-298. MR 18:905c
- 44. J. W. S. Rayleigh, The Theory of Sound, second edition revised and enlarged (in two volumes), Dover Publications, New York, 1945 (republication of the 1894/1896 edition). MR 7:500e
- 45. Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
- 46. Robert D. Richtmyer, Principles of advanced mathematical physics. Vol. I, Springer-Verlag, New York-Heidelberg, 1978. Texts and Monographs in Physics. MR 517399
- 47. K. T. Smith, Primer of Modern Analysis, Bogden and Quigley, Tarrytown-on-Hudson, New York, 1971.
- 48. Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- 49. Emanuel Sperner Jr., Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317–327 (German). MR 340558, https://doi.org/10.1007/BF01214695
- 50. G. Szego, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356. MR 15:877c
- 51. Giorgio Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4, 697–718. MR 601601
- 52.
H. F. Weinberger, An isoperimetric inequality for the
-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636. MR 18:63c
- 53. E. F. Whittlesey, Fixed points and antipodal points, Amer. Math. Monthly 70 (1963), 807–821. MR 161324, https://doi.org/10.2307/2312660
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Additional Information
Mark S. Ashbaugh
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211–0001
Email:
mark@math.missouri.edu
Rafael D. Benguria
Affiliation:
Departamento de Física, P. Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
Email:
rbenguri@fis.puc.cl
DOI:
https://doi.org/10.1090/S0002-9947-00-02605-2
Keywords:
Eigenvalues of the Laplacian,
Dirichlet problem for domains on spheres,
Payne--P\'{o}lya--Weinberger conjecture,
Sperner's inequality,
ratios of eigenvalues,
isoperimetric inequalities for eigenvalues
Received by editor(s):
January 6, 1999
Received by editor(s) in revised form:
June 9, 1999
Published electronically:
November 8, 2000
Additional Notes:
The first author was partially supported by National Science Foundation (USA) grants DMS–9114162, INT–9123481, DMS–9500968, and DMS–9870156.
The second author was partially supported by FONDECYT (Chile) project number 196–0462, a Cátedra Presidencial en Ciencias (Chile), and a John Simon Guggenheim Memorial Foundation fellowship.
Article copyright:
© Copyright 2000
American Mathematical Society