On Isoperimetric Inequalities of Riesz Potentials and Applications

Tynysbek Sh. Kalmenov, Ernazar Nysanov, Bolys Sabitbek

Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan.

DOI: 10.4236/am.2013.47A001                

Abstract

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.

Keywords

Isoperimetric Inequalities; Eigenvalues of the Laplacian; Riesz Potentials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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.