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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 25, 2015

Torsional rigidity on compact Riemannian manifolds with lower Ricci curvature bounds

  • Najoua Gamara EMAIL logo , Abdelhalim Hasnaoui and Akrem Makni
From the journal Open Mathematics


In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains


[1] Krahn E., Über eine von Rayleigh formulierte Minmaleigenschaft des Kreises, Math. Ann., 1925, 94, 97-100.10.1007/BF01208645Search in Google Scholar

[2] Faber C., Beweiss, dass unter allen homogenen Membrane von gleicher Fläche und gleicher Spannung die kreisf ˝ ormige die tiefsten Grundton gibt. Sitzungsber.-Bayer. Akad. Wiss., Math.Phys.Munich., 1923, 169-172.Search in Google Scholar

[3] Payne L.E., Rayner M.E., Some isoperimetric norm bounds for solutions of the Helmholtz equation, Z. Angew. Math. Phys., 1973, 24, 105-110.10.1007/BF01594001Search in Google Scholar

[4] Payne L.E., Rayner M.E., An isoperimetric inequality for the first eigenfunction in the fixed membrane problem, Z. Angew. Math. Phys.,1972, 23, 13-15.10.1007/BF01593198Search in Google Scholar

[5] Kohler Jobin M.T., Sur la première fonction propre d’une membrane: une extension à N dimensions de l’inégalité isopérimétrique de Payne-Rayner, Z. Angew. Math. Phys., 1977, 28, 1137-1140.10.1007/BF01601680Search in Google Scholar

[6] Chiti G., A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators, Z. Angew. Math. Phys., 1982, 33, 143-148.10.1007/BF00948319Search in Google Scholar

[7] Hasnaoui A., On the Problem of Queen Dido for a Wedge like membrane and a Compact Riemannian manifold with lower Ricci curvature bound, PhD thesis, University Tunis El Manar, Tunis, Tunisia. 2014.Search in Google Scholar

[8] Cheng S.Y., Eigenvalue comparison theorems and its geometric aplications, Math. Z., 1975, 143, 289-297.10.1007/BF01214381Search in Google Scholar

[9] Bérard P., Meyer D., Inégalités isopérimétriques et applications, Ann. Scient. Ec. Norm. Sup. 1982, 4, 15, 513-542.Search in Google Scholar

[10] Gamara Abdelmoula N., Symétrisation d’inéquations elliptiques et applications géométriques, Math. Z. 1988, 199, 181-190.Search in Google Scholar

[11] Ling J., Lu Z., Bounds of eigenvalues on Riemannian manifolds., Adv. Lect. Math. (ALM), 2010, 10, 241-264.Search in Google Scholar

[12] De Saint-Venant B., Mémoire sur la torsion des prismes. Mémoir. pres. divers. savants, Acad. Sci., 1856, 14, 233-560.Search in Google Scholar

[13] Pólya G., Torsional rigidity, principal frequency, electrostatic capacity, and symmetrization, Quart. of Appl. Maths., 1948, 6, 267-277.10.1090/qam/26817Search in Google Scholar

[14] Makai E., A proof of Saint-Venant’s theorem on torsional rigidity, Acta Math. Acad. Sci. Hungar., 1966, 17, 419-422.10.1007/BF01894885Search in Google Scholar

[15] Bandle C., Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.Search in Google Scholar

[16] Pólya G., Szeg˝o G., Isoperimetric Inequalities in Mathematical Physics. Princeton University Press, 1951.10.1515/9781400882663Search in Google Scholar

[17] Carroll T., Ratzkin J.: Interpolating between torsional rigidity and principal frequency, J. Math. Anal. Appl., 2011, 379, 818-826.10.1016/j.jmaa.2011.02.004Search in Google Scholar

[18] Iversen M., Torsional rigidity of a radially perturbed ball, Oberwolfach Report, 2012, 33, 31-33.Search in Google Scholar

[19] Van den Berg M., Estimates for the torsion function and Sobolev constants, Potential. Anal., 2012, 36, 607-616.10.1007/s11118-011-9246-9Search in Google Scholar

[20] Carroll T., Ratzkin J., A reverse Hölder inequality for extremal Sobolev functions, Potential. Anal, DOI 10.1007/s11118-014-9433-6.Search in Google Scholar

[21] Payne L.E., Some comments on the past fifty years of isoperimetric inequalities, Inequalities (Birmingham, 1987), Lecture Notes in Pure and Appl. Math., 1991, 129, 143-161.Search in Google Scholar

[22] Payne L.E., Some isoperimetric inequalities in the torsion problem for multiply connected regions, Studies in mathematical analysis and related topics: Essays in honor of Georgia Pólya, Stanford Univ. Press, Stanford, Calif., 1962, 270-280.Search in Google Scholar

[23] Payne L.E., Weinberger, H.F., Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl., 1961, 2, 210-216.10.1016/0022-247X(61)90031-2Search in Google Scholar

[24] Ashbaugh M.S., Benguria R. D., A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of Sn., Trans. Amer. Math. Soc., 2001, 353, 1055-1087.10.1090/S0002-9947-00-02605-2Search in Google Scholar

[25] Chavel I., Feldman E. A., Isoperimetric inequalities on curved surfaces., Adv. in Math., 1980, 37, 83-98.10.1016/0001-8708(80)90028-6Search in Google Scholar

[26] Gromov M., Paul Levy’s isoperimetric inequality. Appendix C in Metric structures for Riemannian and non-Riemannian spaces., Progress in Mathematics, 152., Birkh˝auser Boston, Inc., Boston, MA, 1999.Search in Google Scholar

[27] Benguria, R. D.: Isoperimetric inequalities for eigenvalues of the Laplacian. Entropy and the quantum II, 21-60, Contemp. Math., 552, Amer. Math. Soc., Providence, RI (2011).Search in Google Scholar

[28] Chiti G., An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators, Boll. Un. Mat. Ital. A., 1982, 6, 1, 145-151.10.1007/BF00948319Search in Google Scholar

[29] Hardy G.H., Littlewood, J.E., Pólya G., Some simple inequalities satisfied by convex functions, Messenger Math., 1929, 58, 145-152. Reprinted in Collected Papers of G. H. Hardy, Vol.II, London Math. Soc., Clarendon Press: Oxford, 500-508, 1967.Search in Google Scholar

[30] Talenti G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1976, 3, 697-718.Search in Google Scholar

Received: 2015-4-13
Accepted: 2015-9-7
Published Online: 2015-9-25

© 2015

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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