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A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb{S}^n$


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
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Abstract: For a domain $\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\mathbb{S}^n$ we prove the optimal result $\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $\Omega^{\star}$, the symmetric rearrangement of $\Omega$ in $\mathbb{S}^n$, is a geodesic ball in $\mathbb{S}^n$ having the same $n$-volume as $\Omega$. We also show that $\lambda_2/\lambda_1$ for geodesic balls of geodesic radius $\theta_1$ less than or equal to $\pi/2$ is an increasing function of $\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $\theta_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $\theta_1=\pi/2$. Here $j_{\nu,k}$ denotes the $k$th positive zero of the Bessel function $J_{\nu}(t)$. 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 $\mathbb{S}^n$ and having a fixed value of $\lambda_1$ the one with the maximal value of $\lambda_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $\lambda_2/\lambda_1$. Various other results for $\lambda_1$and $\lambda_2$ of geodesic balls in $\mathbb{S}^n$ are proved in the course of our work.


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  • 1. M. Abramowitz and I. A. Stegun, editors, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, D.C., 1964. MR 34:8607
  • 2. M. A. Armstrong, Basic Topology, Springer-Verlag, New York, 1983. MR 84f:55001
  • 3. M. S. Ashbaugh and R. 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), 273-276. MR 89i:81012
  • 4. M. S. Ashbaugh and R. D. 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, P. W. Schaefer, editor, Pitman Research Notes in Mathematics Series, vol. 175, Longman Scientific and Technical, Harlow, Essex, United Kingdom, 1988, pp. 134-145. MR 90c:35157
  • 5. M. S. Ashbaugh and R. D. Benguria, Proof of the Payne-Pólya-Weinberger conjecture, Bull. Amer. Math. Soc. 25 (1991), 19-29. MR 91m:35173
  • 6. M. S. Ashbaugh and R. D. Benguria, A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Annals of Math. 135 (1992), 601-628. MR 93d:35105
  • 7. M. S. Ashbaugh and R. D. Benguria, A second proof of the Payne-Pólya-Weinberger conjecture, Commun. Math. Phys. 147 (1992), 181-190. MR 93k:33002
  • 8. M. S. Ashbaugh and R. D. Benguria, Isoperimetric inequalities for eigenvalue ratios, Partial Differential Equations of Elliptic Type, Cortona, 1992, A. Alvino, E. Fabes, and G. Talenti, editors, Symposia Mathematica, vol. 35, Cambridge University Press, Cambridge, 1994, pp. 1-36. MR 95h:35158
  • 9. M. S. Ashbaugh and R. 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), 402-416. MR 97d:35160
  • 10. M. S. Ashbaugh and R. D. Benguria, On the Payne-Pólya-Weinberger conjecture on the $n$-dimensional sphere, General Inequalities 7 (Oberwolfach, 1995), C. Bandle, W. N. Everitt, L. Losonczi, and W. Walter, editors, International Series of Numerical Mathematics, vol. 123, Birkhäuser, Basel, 1997, pp. 111-128. MR 98k:35139
  • 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. F. E. Baginski, Ordering the zeroes of Legendre functions $P^m_\nu(z_0)$ when considered as a function of $\nu$, J. Math. Anal. Appl. 147 (1990), 296-308. MR 91k:33006
  • 13. F. E. Baginski, Comparison theorems for the $\nu$-zeroes of Legendre functions $P^m_\nu(z_0)$ when $-1<z_0<1$, Proc. Amer. Math. Soc. 111 (1991), 395-402. MR 91i:33003
  • 14. C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in Mathematics, vol. 7, Pitman, Boston, 1980. MR 81e:35095
  • 15. C. Bandle and M. Flucher, Table of inequalities in elliptic boundary value problems, Recent Progress in Inequalities (Nis, 1996), G. V. Milovanovic, editor, Mathematics and its Applications, vol. 430, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998, pp. 97-125. MR 99h:35028
  • 16. B. Baumgartner, Level comparison theorems, Annals of Physics 168 (1986), 484-526. MR 87j:81046
  • 17. B. Baumgartner, Relative concavity of ground state energies as functions of a coupling constant, Phys. Lett. A 170 (1992), 1-4. MR 93i:81022
  • 18. B. Baumgartner, H. Grosse, and A. Martin, The Laplacian of the potential and the order of energy levels, Phys. Lett. B 146 (1984), 363-366. MR 85k:81158
  • 19. B. Baumgartner, H. Grosse, and A. Martin, Order of levels in potential models, Nucl. Phys. B 254 (1985), 528-542. MR 87a:81024
  • 20. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, fourth edition, Wiley, New York, 1989. MR 90h:34001
  • 21. Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Grundlehren der mathematischen Wissenschaften 285, Springer-Verlag, Berlin, 1988. MR 89b:52020
  • 22. I. Chavel, Lowest-eigenvalue inequalities, Proc. Symp. Pure Math., vol. 36, Geometry of the Laplace Operator, R. Osserman and A. Weinstein, editors, Amer. Math. Soc., Providence, Rhode Island, 1980, pp. 79-89. MR 81f:58039
  • 23. I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984. MR 86g:58140
  • 24. G. Chiti, Orlicz norms of the solutions of a class of elliptic equations, Boll. Un. Mat. Ital. (5) 16-A (1979), 178-185. MR 80h:35004
  • 25. 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 83i:35141
  • 26. G. Chiti, An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital. (6) 1-A (1982), 145-151. MR 83i:35140
  • 27. G. Chiti, A bound for the ratio of the first two eigenvalues of a membrane, SIAM J. Math. Anal. 14 (1983), 1163-1167. MR 85f:35158
  • 28. R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I, Interscience Publishers, New York, 1953. MR 16:426a
  • 29. J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. MR 33:1824
  • 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. Helvetici 51 (1976), 133-161. MR 54:568
  • 32. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, second edition, Cambridge University Press, Cambridge, 1952. MR 13:727a
  • 33. E. M. Harrell II and P. L. Michel, Commutator bounds for eigenvalues, with applications to spectral geometry, Commun. Partial Diff. Eqs. 19 (1994), 2037-2055. MR 95i:58182
  • 34. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory, With Applications to Schrödinger Operators, Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996. MR 98h:47003
  • 35. J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, Massachusetts, 1961. MR 23:A2857
  • 36. T. Kato, Perturbation Theory for Linear Operators, 2nd edition, Grundlehren der mathematischen Wissenschaften 132, Springer-Verlag, Berlin, 1976. MR 53:11389
  • 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. J. R. Munkres, Topology, A First Course, Prentice-Hall, Englewood Cliffs, New Jersey, 1975. MR 57:4063
  • 40. J. R. Munkres, Elements of Algebraic Topology, Addison-Wesley, Menlo Park, California, 1984. MR 85m:55001
  • 41. R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238. MR 58:18161
  • 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. M. Reed and B. Simon, Methods of Modern Mathematical Physics, vol. IV: Analysis of Operators, Academic Press, New York, 1978. MR 58:12429c
  • 46. R. D. Richtmyer, Principles of Advanced Mathematical Physics, vol. II, Springer-Verlag, New York, 1981. MR 84h:00024b
  • 47. K. T. Smith, Primer of Modern Analysis, Bogden and Quigley, Tarrytown-on-Hudson, New York, 1971.
  • 48. E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. MR 35:1007
  • 49. E. Sperner, Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317-327. MR 49:5310
  • 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. G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa (4) 3 (1976), 697-718. MR 58:29170
  • 52. H. F. Weinberger, An isoperimetric inequality for the $n$-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 28:4532

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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