A comparison theorem for eigenvalues of the Newton potential.
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Keywords:
spectral problems, Dirichlet problem, eigenvalues of Laplacian, Neumann problem, Newton potentialAbstract
Dedicated to Prof Mukhtarbay Otelbaev on the occasion of his seventieth birthday There is a wealth of interesting results comparing between Dirichlet and Neumann eigenvalues. In this paper we compare the eigenvalues of the Newton potential with the Dirichlet eigenvalues and the Neumann eigenvalues in a bounded domain in Rd. First we show that the spectral problem of the Newton potential is equivalent to a spectral problem of a non-local boundary value problem of the Laplace operator then it is proved that the nth eigenvalue of the Laplacian with the non-local boundary condition is between the nth eigenvalue of Neumann Laplacian and the nth eigenvalue of Dirichlet Laplacian in a bounded domain of any dimensional euclidian space.References
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[2] Courant R., Hilbert D. Methods of Mathematical Physics. - New York: Interscience., 1953. - 239 p.
[3] Filonov N. On an inequality between Dirichlet and Neumann eigenvalues for the Laplace Operator // St. Petersburg Math. J. - 2005. - Vol. 16, №2. - P. 413–416.
[4] Frank R.L., Laptev A. Inequalities between Dirichlet and Neumann eigenvalues on the Heisenberg group // Internat. Math. Res. Notices - 2010. - Vol. 15. - P. 2889–2902.
[5] Friedlander L. Some inequalities between Dirichlet and Neumann eigenvalues // Arch. Rational Mech. Anal. - 1991. - Vol. 116. - P. 150–160.
[6] Gesztesy F., Mitrea M. Nonlocal Robin Laplacians and some remarks on a paper by Filonov on eigenvalue inequalities // J. Diff. Eq. - 2009. - Vol. 247. - P. 2871–2896.
[7] Kal’menov T.Sh., Suragan D. To Spectral Problems for the Volume Potential // Doklady Mathematics. -2008. - Vol. 80, №2. - P. 646–649.
[8] Laptev A. Spectral Inequalities, Lectures.- London: - Imperial College London, 2010. - 48 p.
[9] Levine H. A. Weinberger H.F. Inequalities between Dirichlet and Neumann eigenvalues // Arch. Rational Mech. Anal. - 1986. - Vol. 94. - P. 193–208.
[10] Mazzeo R. Remarks on a paper of Friedlander concerning inequalities between Dirichlet and Neumann eigenvalues // Internat. Math. Res. Notices - 1991. - Vol. 4. - P. 41–48.
[11] Szego R. Inequalities for certain eigenvalues of a membrane of given area // J. Rational Mech. Anal. - 1954. - Vol. 3. - P. 343–356.
[12] Payne L. E. Inequalities for certain eigenvalues of a membrane of given area // J. Rational Mech. Anal. - 1955. - Vol. 4. - P. 517–529.
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Suragan, D. (2012). A comparison theorem for eigenvalues of the Newton potential. Journal of Mathematics, Mechanics and Computer Science, 75(4), 36–45. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/157
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Differential and Integral Equations
