On Isoperimetric Inequalities of Riesz Potentials and Applications

DOI: 10.4236/am.2013.47A001   PDF   HTML   XML   3,656 Downloads   5,338 Views  


In this article, we prove certain isoperimetric inequalities for eigenvalues of Riesz potentials and show some applications of the results to a non-local boundary value problem of the Laplace operator.

Share and Cite:

T. Kalmenov, E. Nysanov and B. Sabitbek, "On Isoperimetric Inequalities of Riesz Potentials and Applications," Applied Mathematics, Vol. 4 No. 7A, 2013, pp. 1-4. doi: 10.4236/am.2013.47A001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. W. Rayleigh, “The Theory of Sound,” Dover Publishing, New York, 1945.
[2] A. Henrot, “Extremum Problems for Eigenvalues of Elliptic Operators,” Birkhauser, Basel, 2006.
[3] D. Daners, “A Faber—Krahn Inequality for Robin Problems in Any Space Dimension,” Mathematische Annalen, Vol. 335, 2006, pp. 767-785. doi:10.1007/s00208-006-0753-8
[4] T. Sh. Kalmenov and D. Suragan, “Boundary Conditions for the Volume Potential for the Polyharmonic Equation,” Differential Equations, Vol. 48, No. 4, 2012, pp. 595-599.
[5] A. Burchard, “A Short Course on Rearrangement Inequalities,” 2009. www.math.toronto.edu/almut/rearrange.pdf
[6] F. Riesz, “Sur Une Inregalitre Intregrale,” Journal of the London Mathematical Society, Vol. 5, No. 3, 1930, pp. 162-168. doi:10.1112/jlms/s1-5.3.162
[7] B. S. Vladimirov, “Equations of Mathematical Physics,” Nauka, Moscow, 1981.
[8] N. S. Landkoff, “Foundations of Modern Potential Theory,” Springer-Verlag, Berlin, 1972. doi:10.1007/978-3-642-65183-0
[9] B. Dittmar, “Sums of Reciprocal Eigenvalues of the Laplacian,” Mathematische Nachrichten, Vol. 237, No. 1, 2002, pp. 45-61. doi:10.1002/1522-2616(200204)237:1<45::AID-MANA45>3.0.CO;2-M
[10] T. Sh. Kalmenov and D. Suragan, “To Spectral Problems for the Volume Potential,” Doklady Mathematics, Vol. 428, No. 1, 2009, pp. 16-19.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.