Inequalities for Dirichlet and Neumann eingenvalues of the laplacian for domains on spheres

M. Ashbaugh; Howard A. Levine

Journées équations aux dérivées partielles (1997)

  • page 1-15
  • ISSN: 0752-0360

How to cite


Ashbaugh, M., and Levine, Howard A.. "Inequalities for Dirichlet and Neumann eingenvalues of the laplacian for domains on spheres." Journées équations aux dérivées partielles (1997): 1-15. <>.

author = {Ashbaugh, M., Levine, Howard A.},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-15},
publisher = {Ecole polytechnique},
title = {Inequalities for Dirichlet and Neumann eingenvalues of the laplacian for domains on spheres},
url = {},
year = {1997},

AU - Ashbaugh, M.
AU - Levine, Howard A.
TI - Inequalities for Dirichlet and Neumann eingenvalues of the laplacian for domains on spheres
JO - Journées équations aux dérivées partielles
PY - 1997
PB - Ecole polytechnique
SP - 1
EP - 15
LA - eng
UR -
ER -


  1. [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. Zbl0171.38503MR29 #4914
  2. [2] M.S. Ashbaugh and R.D. Benguria, Universal bounds for the low eigenvalues of Neumann Laplacians in n dimensions, SIAM J. Math. Anal. 24 (1993), 557-570. Zbl0796.35122MR94b:35191
  3. [3] 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. Zbl0814.35081MR95h:35158
  4. [4] 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. Zbl0835.58037MR97d:35160
  5. [5] P. Aviles, Symmetry theorems related to Pompeiu's problem, Amer. J. Math. 108 (1986), 1023-1036. Zbl0644.35075MR88d:35070
  6. [6] C. Bandle, Isoperimetric Inequalities and Applications, Pitman Monographs and Studies in Mathematics, vol. 7, Pitman, Boston, 1980. Zbl0436.35063MR81e:35095
  7. [7] 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, R.I., 1980, pp. 79-89. Zbl0467.58025MR81f:58039
  8. [8] I. Chavel, Eigenvalues in Riemannian Geometry, Academic, New York, 1984. Zbl0551.53001MR86g:58140
  9. [9] G. Faber, Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitzungberichte der mathematisch-physikalischen Klasse der Bayerischen Akademie der Wissenschaften zu München Jahrgang, 1923, pp. 169-172. Zbl49.0342.03JFM49.0342.03
  10. [10] 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. Zbl0339.31003MR54 #568
  11. [11] L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 116 (1991), 153-160. Zbl0789.35124MR93h:35146
  12. [12] E. Krahn, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925), 97-100. Zbl51.0356.05JFM51.0356.05
  13. [13] E. Krahn, Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926), 1-44 [English translation in Edgar Krahn 1894-1961: A Centenary Volume, Ü. Lumiste and J. Peetre, editors, IOS Press, Amsterdam, 1994, pp. 139-174]. Zbl52.0510.03JFM52.0510.03
  14. [14] H.A. Levine, Some remarks on inequalities between Dirichlet and Neumann eigenvalues, 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. 121-133. Zbl0669.35091MR89j:35099
  15. [15] H.A. Levine and H.F. Weinberger, Inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal. 94 (1986), 193-208. Zbl0608.35047MR87k:35186
  16. [16] R. Mazzeo, Remarks on a paper of Friedlander concerning inequalities between Neumann and Dirichlet eigenvalues, Int. Math. Research Notices, No. 4 (1991), 41-48. Zbl0752.58035MR93h:35147
  17. [17] R. Molzon, Symmetry and overdetermined boundary value problems, Forum Math. 3 (1991), 143-156. Zbl0789.35118MR92f:58181
  18. [18] L.E. Payne, Inequalities for eigenvalues of membranes and plates, J. Rational Mech. Anal. 4 (1955), 517-529. Zbl0064.34802MR17,42a
  19. [19] L.E. Payne, Some comments on the past fifty years of isoperimetric inequalities, Inequalities: Fifty Years On from Hardy, Littlewood, and Pólya, W.N. Everitt, editor, Marcel Dekker, New York, 1991, pp. 143-161. Zbl0723.52003MR92f:26042
  20. [20] F. Rellich, Darstellung der Eigenwerte Δu + λu = 0 durch ein Randintegral, Math. Z. 46 (1940), 635-636. Zbl0023.04204MR2,56dJFM66.0460.01
  21. [21] G.V. Rozenblum, M.A. Shubin, and M.Z. Solomyak, Spectral Theory of Differential Operators (translated from the Russian by T. Zastawniak), Partial Differential Equations VII, M.A. Shubin, editor, Encyclopedia of Mathematical Sciences, vol. 64, R.V. Gamkrelidze, editor-in-chief, Springer, Berlin, 1994. Zbl0805.35081
  22. [22] E. Sperner, Zur Symmetrisierung von Funktionen auf Sphären, Math. Z. 134 (1973), 317-327. Zbl0283.26015MR49 #5310
  23. [23] G. Szegö, Inequalities for certain eigenvalues of a membrane of given area, J. Rational Mech. Anal. 3 (1954), 343-356. Zbl0055.08802MR15,877c
  24. [24] H.F. Weinberger, An isoperimetric inequality for the n-dimensional free membrane problem, J. Rational Mech. Anal. 5 (1956), 633-636. Zbl0071.09902MR18,63c

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.