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Uspekhi Mat. Nauk, 2016, Volume 71, Issue 3(429), Pages 27–122 (Mi umn9710)  

This article is cited in 16 scientific papers (total in 16 papers)

Operator estimates in homogenization theory

V. V. Zhikova, S. E. Pastukhovab

a Vladimir State University
b Moscow Technological University (MIREA)

Abstract: This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self-adjoint evolution equations, and higher-order elliptic operators.
Bibliography: 62 titles.

Keywords: shift method, integrated estimate, Steklov smoothing, periodicity, problem on the cell, asymptotics of the fundamental solution, spectral method, Bloch representation of an operator, Nash–Aronson estimate.

Funding Agency Grant Number
Russian Science Foundation 14-11-00398
This work was supported by the Russian Science Foundation under grant 14-11-00398.


DOI: https://doi.org/10.4213/rm9710

Full text: PDF file (1244 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2016, 71:3, 417–511

Bibliographic databases:

Document Type: Article
UDC: 517.97
MSC: Primary 35J15, 35K15, 35B27; Secondary 35J30
Received: 21.12.2015

Citation: V. V. Zhikov, S. E. Pastukhova, “Operator estimates in homogenization theory”, Uspekhi Mat. Nauk, 71:3(429) (2016), 27–122; Russian Math. Surveys, 71:3 (2016), 417–511

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. T. A. Suslina, “Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients”, St. Petersburg Math. J., 29:2 (2018), 325–362  mathnet  crossref  isi  elib
    2. 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  mathnet  crossref  crossref  isi  elib
    3. V. V. Zhikov, S. E. Pastukhova, “Asimptotika fundamentalnogo resheniya dlya uravneniya diffuzii v periodicheskoi srede na bolshikh vremenakh i ee primenenie k otsenkam teorii usredneniya”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 223–246  mathnet  crossref  mathscinet
    4. M. Dorodnyi, T. A. Suslina, “Homogenization of a Nonstationary Model Equation of Electrodynamics”, Math. Notes, 102:5 (2017), 645–663  mathnet  crossref  crossref  mathscinet  isi  elib
    5. S. E. Pastukhova, R. N. Tikhomirov, “Operator-type estimates in homogenization of elliptic equations with lower terms”, St. Petersburg Math. J., 29:5 (2018), 841–861  mathnet  crossref  mathscinet  isi  elib
    6. Yu. M. Meshkova, T. A. Suslina, “Usrednenie pervoi nachalno-kraevoi zadachi dlya parabolicheskikh sistem: operatornye otsenki pogreshnosti”, Algebra i analiz, 29:6 (2017), 99–158  mathnet  elib
    7. M. M. Sirazhudinov, “Operatornye otsenki usredneniya obobschennykh uravnenii Beltrami”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 40–46  mathnet  crossref
    8. Pastukhova S.E., “Large-Time Asymptotics of the Fundamental Solution to a Periodic Diffusion Equation and Its Applications”, Proceedings of the International Conference Days on Diffraction (Dd) 2017, eds. Motygin O., Kiselev A., Goray L., Suslina T., Kazakov A., Kirpichnikova A., IEEE, 2017, 258–263  isi
    9. R. Chill, A. F. M. ter Elst, “Weak and strong approximation of semigroups on Hilbert spaces”, Integral Equations Operator Theory, 90:1 (2018), 9, 22 pp.  crossref  mathscinet  isi  scopus
    10. M. A. Dorodnyi, T. A. Suslina, “Spectral approach to homogenization of hyperbolic equations with periodic coefficients”, J. Differential Equations, 264:12 (2018), 7463–7522  crossref  mathscinet  zmath  isi
    11. T. A. Suslina, “Homogenization of the Neumann problem for higher order elliptic equations with periodic coefficients”, Complex Var. Elliptic Equ., 63:7-8 (2018), 1185–1215  crossref  mathscinet  zmath  isi  scopus
    12. T. A. Suslina, “Spectral approach to homogenization of elliptic operators in a perforated space”, Rev. Math. Phys., 30:8 (2018), 1840016, 57 pp.  crossref  mathscinet  isi  scopus
    13. Ch. Chen, A. J. Roberts, J. E. Bunder, “Boundary conditions for macroscale waves in an elastic system with microscale heterogeneity”, IMA J. Appl. Math., 83:3 (2018), 347–379  crossref  mathscinet  isi
    14. T. A. Suslina, “Usrednenie statsionarnoi periodicheskoi sistemy Maksvella v ogranichennoi oblasti v sluchae postoyannoi magnitnoi pronitsaemosti”, Algebra i analiz, 30:3 (2018), 169–209  mathnet  elib
    15. Zecca G., “An Optimal Control Problem For Some Nonlinear Elliptic Equations With Unbounded Coefficients”, Discrete Contin. Dyn. Syst.-Ser. B, 24:3, SI (2019), 1393–1409  crossref  isi
    16. Suslina T.A., “Homogenization of Higher-Order Parabolic Systems in a Bounded Domain”, Appl. Anal., 98:1-2, SI (2019), 3–31  crossref  isi
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