D. I. Borisov, “Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients”, Algebra i Analiz, 20:2 (2008), 19–42; St. Petersburg Math. J., 20:2 (2009), 175

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Algebra i Analiz, 2008, Volume 20, Issue 2, Pages 19–42 (Mi aa504)

This article is cited in 26 scientific papers (total in 26 papers)

Research Papers

Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients

D. I. Borisov^ab

^a Bashkir State Pedagogical University
^b Nuclear Physics Institute, Academy of Sciences of the Czech Republic

Full-text PDF (330 kB) Citations (26)

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Abstract: A singularly perturbed second order elliptic system in the entire space is treated. The coefficients of the systems oscillate rapidly and depend both on the slow and fast variables. The homogenized operator is obtained and, in the uniform norm sense, the leading terms of the asymptotic expansion are constructed for the resolvent of the operator described by the system. The convergence of the spectrum is established, and examples are given.

Keywords: Homogenization of differentiable operators, unbounded domain, fast and slow variables.

Received: 30.11.2006

English version:
St. Petersburg Mathematical Journal, 2009, Volume 20, Issue 2, Pages 175–191
DOI: https://doi.org/10.1090/S1061-0022-09-01043-7

Bibliographic databases:

Document Type: Article

MSC: 35B27

Language: Russian

Citation: D. I. Borisov, “Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients”, Algebra i Analiz, 20:2 (2008), 19–42; St. Petersburg Math. J., 20:2 (2009), 175–191

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/aa504

https://www.mathnet.ru/eng/aa/v20/i2/p19

This publication is cited in the following 26 articles:

T. A. Suslina, “Homogenization of elliptic and parabolic equations with periodic coefficients in a bounded domain under the Neumann condition”, Izv. Math., 88:4 (2024), 678–759
T. A. Suslina, “Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients”, Russian Math. Surveys, 78:6 (2023), 1023–1154
V. A. Sloushch, T. A. Suslina, “Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients”, St. Petersburg Math. J., 35:2 (2024), 327–375
Senik N.N., “Homogenization For Locally Periodic Elliptic Operators”, J. Math. Anal. Appl., 505:2 (2022), 125581
M. M. Sirazhudinov, S. P. Dzhamaludinova, “Estimates for the Locally Periodic Homogenization of the Riemann–Hilbert Problem for a Generalized Beltrami Equation”, Diff Equat, 58:6 (2022), 771
Meshkova Yu.M., “On Operator Error Estimates For Homogenization of Hyperbolic Systems With Periodic Coefficients”, J. Spectr. Theory, 11:2 (2021), 587–660
M. M. Sirazhudinov, L. M. Dzhabrailova, “Operatornye otsenki usredneniya zadachi Rimana-Gilberta dlya uravneniya Beltrami s lokalno-periodicheskim koeffitsientom”, Dagestanskie elektronnye matematicheskie izvestiya, 2021, no. 16, 51–61
M. A. Dorodnyi, “Homogenization of periodic Schrödinger-type equations, with lower order terms”, St. Petersburg Math. J., 31:6 (2020), 1001–1054
N. N. Senik, “On homogenization for non-self-adjoint locally periodic elliptic operators”, Funct. Anal. Appl., 51:2 (2017), 152–156
Yu. M. Meshkova, T. A. Suslina, “Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients”, Funct. Anal. Appl., 51:3 (2017), 230–235
Yu. M. Meshkova, T. A. Suslina, “Homogenization of the first initial boundary value problem for parabolic systems: Operator error estimates”, St. Petersburg Math. J., 29:6 (2018), 935–978
Cardone G., “Waveguides With Fast Oscillating Boundary”, Nanosyst.-Phys. Chem. Math., 8:2 (2017), 160–165
Senik N.N., “Homogenization For Non-Self-Adjoint Periodic Elliptic Operators on An Infinite Cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898
S. E. Pastukhova, “Operator Estimates in Homogenization of Elliptic Systems of Equations”, J Math Sci, 226:4 (2017), 445
V. V. Zhikov, S. E. Pastukhova, “Operator estimates in homogenization theory”, Russian Math. Surveys, 71:3 (2016), 417–511
Borisov D. Cardone G. Durante T., “Homogenization and norm-resolvent convergence for elliptic operators in a strip perforated along a curve”, Proc. R. Soc. Edinb. Sect. A-Math., 146:6 (2016), 1115–1158
Meshkova Yu.M. Suslina T.A., “Two-parametric error estimates in homogenization of second-order elliptic systems in ^{ d }”, Appl. Anal., 95:7, SI (2016), 1413–1448
T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition”, Sb. Math., 205:10 (2014), 1492–1527
T. A. Suslina, “Homogenization of elliptic systems with periodic coefficients: operator error estimates in $L_2(\mathbb R^d)$ with corrector taken into account”, St. Petersburg Math. J., 26:4 (2015), 643–693
Borisov D. Cardone G. Faella L. Perugia C., “Uniform Resolvent Convergence for Strip with Fast Oscillating Boundary”, J. Differ. Equ., 255:12 (2013), 4378–4402

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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