On some inequalities of spectral geometry for Riesz potentials.
Keywords:
spectral problems, eigenvalues, spectral geometry, Riesz potential, Newton potentialAbstract
In this paper, it is proved that a ball maximizes the first eigenvalue of Riesz potential among domains of given volume. It is also shown that the sum of the squares of the Riesz potential eigenvalues is also maximized in a ball among all domains with the same volume as the ball. In the last section of the paper, authors give an application of the results for a boundary value problem of Laplacian. These results belong to spectral geometry. Spectral geometry is a relatively young and rapidly developing mathematical discipline which combines elements of differential geometry, functional analysis and partial differential equations. Many problems in spectral geometry motivated by issues arising in acoustics, quantum mechanics, and other fields of physics.References
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[2] Henrot A. Extremum problems for eigenvalues of elliptic operators. – Basel: Birkh¨auser,
2006. - 351 p.
[3] Ландкоф Н.С. Основы современной теории потенциала. – Москва: Наука, 1966. – С. 351.
[4] Riesz F. Sur une inґegalitґe intґegrale // J. London Math. Soc. - 1930. - Vol. 5. -P. 162–168.
[5] Burchard A. A Short Course on Rearrangement Inequalities [Электрон. ресурс]. - 2009. -
URL: http://www.math.toronto.edu/almut/rearrange.pdf (дата обращения: 12.12.2012)
[6] Владимиров В.С. Уравнения математической физики. – Москва: Наука, 1981. – С.511.
[7] Daners D. A Faber - Krahn inequality for Robin problems in any space dimension // Math. Ann. – 2006. - Vol. 335. - P. 767–785.
[8] Dittmar B. Sums of reciprocal eigenvalues of the Laplacian // Math. Nachr.– 2002. - Vol. 237. - P. 45-61.
[9] Кальменов Т.Ш., Сураган Д. К спектральным вопросам объемного потенциала // Докл. РАН. – 2009. – №1(428). - С. 16–19.
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