Comparison of the singular numbers of correct restrictions of elliptic differential operators
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- by V. I. Burenkov and M. Otelbaev
- Trans. Moscow Math. Soc. 2014, 115-131
- DOI: https://doi.org/10.1090/S0077-1554-2014-00229-6
- Published electronically: November 4, 2014
Abstract:
The paper is dedicated to finding the asymptotics of singular numbers of a correct restriction of a uniformly elliptic differential operator of order $2l$ defined on a bounded domain in $\mathbb {R}^n$ with sufficiently smooth boundary, which is in general a non-selfadjoint operator. Conditions are established on a correct restriction, ensuring that its singular numbers $s_k$ are of order $k^{{2l}/n}$ as $k\to \infty$. As an application of this result certain estimates are obtained for the deviation upon domain perturbation of singular numbers of such correct restrictions.References
- Allakhverdiev D. E. On the rate of approximation of completely continuous operators by finite-dimensional operators, Uch. Zap. Azerb. Univ. 1957. Vol. 2. P. 27–35.
- Victor I. Burenkov, Sobolev spaces on domains, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], vol. 137, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1998. MR 1622690, DOI 10.1007/978-3-663-11374-4
- Victor I. Burenkov and Pier Domenico Lamberti, Spectral stability of Dirichlet second order uniformly elliptic operators, J. Differential Equations 244 (2008), no. 7, 1712–1740. MR 2404437, DOI 10.1016/j.jde.2007.12.009
- V. I. Burenkov and M. Otelbaev, On the singular numbers of correct restrictions of non-selfadjoint elliptic differential operators, Eurasian Math. J. 2 (2011), no. 1, 145–148. MR 2910827
- Ky Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 652–655. MR 34519, DOI 10.1073/pnas.35.11.652
- Ky Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. II, Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 31–35. MR 33981, DOI 10.1073/pnas.36.1.31
- Fan K. A minimum property of the eigenvalues of completely continuous operators. Eigenvalues of a sum of hermitian matricies, Amer. Math. Monthly. 1953. Vol. 60, No. 1. P. 48–50.
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
- B. K. Kokebaev, M. Otelbaev, and A. N. Shynybekov, On questions of extension and restriction of operators, Dokl. Akad. Nauk SSSR 271 (1983), no. 6, 1307–1310 (Russian). MR 722342
- J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 2, Travaux et Recherches Mathématiques, No. 18, Dunod, Paris, 1968 (French). MR 0247244
- M. Otelbaev and A. N. Shynybekov, Well-posed problems of Bitsadze-Samarskiĭ type, Dokl. Akad. Nauk SSSR 265 (1982), no. 4, 815–819 (Russian). MR 670839
- Hans Triebel, Approximation numbers in function spaces and the distribution of eigenvalues of some fractal elliptic operators, J. Approx. Theory 129 (2004), no. 1, 1–27. MR 2070178, DOI 10.1016/j.jat.2004.05.003
Bibliographic Information
- V. I. Burenkov
- Affiliation: Faculty of Natural Sciences, People’s Friendship University of Russia, Moscow, Russia
- Email: burenkov@cf.ac.uk
- M. Otelbaev
- Affiliation: Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan
- Email: otelbaevm@mail.ru
- Published electronically: November 4, 2014
- Additional Notes: V. I. Burenkov’s research was supported by a grant from the Russian Scientific Foundation (project 14-11-00443).
- © Copyright 2014 V. I. Burenkov and M. Otelbaev
- Journal: Trans. Moscow Math. Soc. 2014, 115-131
- MSC (2010): Primary 35P15, 35P20, 35J40, 47A75
- DOI: https://doi.org/10.1090/S0077-1554-2014-00229-6
- MathSciNet review: 3308605