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Algebra i Analiz, 2003, Volume 15, Issue 5, Pages 1–108 (Mi aa817)  

This article is cited in 106 scientific papers (total in 106 papers)

Expository Surveys

Periodic differential operators of second order. Threshold properties and averagings

M. Sh. Birman, T. A. Suslina

St. Petersburg State University, Faculty of Physics

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English version:
St. Petersburg Mathematical Journal, 2004, 15:5, 639–714

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Received: 25.06.2003

Citation: M. Sh. Birman, T. A. Suslina, “Periodic differential operators of second order. Threshold properties and averagings”, Algebra i Analiz, 15:5 (2003), 1–108; St. Petersburg Math. J., 15:5 (2004), 639–714

Citation in format AMSBIB
\Bibitem{BirSus03}
\by M.~Sh.~Birman, T.~A.~Suslina
\paper Periodic differential operators of second order. Threshold properties and averagings
\jour Algebra i Analiz
\yr 2003
\vol 15
\issue 5
\pages 1--108
\mathnet{http://mi.mathnet.ru/aa817}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2068790}
\zmath{https://zbmath.org/?q=an:1072.47042}
\transl
\jour St. Petersburg Math. J.
\yr 2004
\vol 15
\issue 5
\pages 639--714
\crossref{https://doi.org/10.1090/S1061-0022-04-00827-1}


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    This publication is cited in the following articles:
    1. M. Sh. Birman, T. A. Suslina, “Homogenization of a multidimensional periodic elliptic operator in a neighbourhood of the edge of the internal gap”, J. Math. Sci. (N. Y.), 136:2 (2006), 3682–3690  mathnet  crossref  mathscinet  zmath  elib
    2. T. A. Suslina, “On the Homogenization of the Periodic Maxwell System”, Funct. Anal. Appl., 38:3 (2004), 234–237  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. T. A. Suslina, “On Homogenization of Periodic Parabolic Systems”, Funct. Anal. Appl., 38:4 (2004), 309–312  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. V. S. Buslaev, M. Z. Solomyak, D. R. Yafaev, “Mikhail Shlemovich Birman (on the Occasion of his 75th Birthday)”, St. Petersburg Math. J., 16:1 (2005), 1–8  mathnet  crossref  mathscinet  zmath
    5. T. A. Suslina, “On averaging of periodic elliptic operators in a strip domain”, St. Petersburg Math. J., 16:1 (2005), 237–257  mathnet  crossref  mathscinet  zmath
    6. V. V. Zhikov, “On gaps in the spectrum of some divergence elliptic operators with periodic coefficients”, St. Petersburg Math. J., 16:5 (2005), 773–790  mathnet  crossref  mathscinet  zmath
    7. T. A. Suslina, “Averaging of the stationary periodic Maxwell system”, St. Petersburg Math. J., 16:5 (2005), 863–922  mathnet  crossref  mathscinet  zmath
    8. Zhikov V. V., Pastukhova S. E., “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524  mathscinet  zmath  isi  elib
    9. V. V. Zhikov, “On operator estimates in homogenization theory”, Dokl. Math., 72:1 (2005), 535–538  mathnet  mathscinet  zmath  isi  elib  elib
    10. V. V. Zhikov, “Spectral Method in Homogenization Theory”, Proc. Steklov Inst. Math., 250 (2005), 85–94  mathnet  mathscinet  zmath
    11. M. Sh. Birman, T. A. Suslina, “Threshold approximations for the resolvent of a factoried selfadjoint family with corrector”, St. Petersburg Math. J., 17:5 (2006), 745–762  mathnet  crossref  mathscinet  zmath
    12. R. G. Shterenberg, “On the structure of the lower spectral edge for a magnetic Schrödinger operator with small magnetic potential”, St. Petersburg Math. J., 17:5 (2006), 865–873  mathnet  crossref  mathscinet  zmath
    13. M. Sh. Birman, T. A. Suslina, “Averaging of periodic elliptic differential operators with the account of a corrector”, St. Petersburg Math. J., 17:6 (2006), 897–973  mathnet  crossref  mathscinet  zmath  elib
    14. Kamotski V., Matthies K., Smyshlyaev V. P., “Exponential homogenization of linear second order elliptic PDEs with periodic coefficients”, SIAM J. Math. Anal., 38:5 (2006), 1565–1587 (electronic)  crossref  mathscinet  isi  elib  scopus
    15. S. A. Nazarov, “Homogenization of elliptic systems with periodic coefficients: Weighted $L^p$ and $L^\infty$ estimates for asymptotic remainders”, St. Petersburg Math. J., 18:2 (2007), 269–304  mathnet  crossref  mathscinet  zmath  elib
    16. M. Sh. Birman, T. A. Suslina, “Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class $H^1(\mathbb R^d)$”, St. Petersburg Math. J., 18:6 (2007), 857–955  mathnet  crossref  mathscinet  zmath
    17. R. G. Shterenberg, “On homogenization of a periodic magnetic Schrödinger operator with degenerate lower edge of the spectrum”, J. Math. Sci. (N. Y.), 143:2 (2007), 3004–3009  mathnet  crossref  mathscinet  zmath  elib
    18. Pastukhova S.E., “Some estimates from homogenized elasticity problems”, Dokl. Math., 73:1 (2006), 102–106  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  elib  scopus
    19. Borisov D.I., “Asymptotics of the spectrum of the Schrödinger operator perturbed by a rapidly oscillating periodic potential”, Dokl. Math., 73:1 (2006), 19–22  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    20. T. A. Suslina, “Homogenization with corrector for a stationary periodic Maxwell system”, St. Petersburg Math. J., 19:3 (2008), 455–494  mathnet  crossref  mathscinet  zmath  isi
    21. M. Sh. Birman, T. A. Suslina, “Homogenization of the Stationary Periodic Maxwell System in the Case of Constant Permeability”, Funct. Anal. Appl., 41:2 (2007), 81–98  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    22. D. I. Borisov, “Asymptotics of the eigenvalues of elliptic systems with fast oscillating coefficients”, Proc. Steklov Inst. Math. (Suppl.), 259, suppl. 2 (2007), S35–S45  mathnet  crossref  elib
    23. Pastukhova S.E., Tikhomirov R.N., “Operator estimates in reiterated and locally periodic homogenization”, Dokl. Math., 76:1 (2007), 548–553  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    24. Kuchment P., Pinchover Y., “Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds”, Transactions of the American Mathematical Society, 359:12 (2007), 5777–5815  crossref  mathscinet  zmath  isi  scopus
    25. D. I. Borisov, “Asymptotics for the solutions of elliptic systems with rapidly oscillating coefficients”, St. Petersburg Math. J., 20:2 (2009), 175–191  mathnet  crossref  mathscinet  zmath  isi
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    27. V. V. Zhikov, S. E. Pastukhova, “Homogenization of degenerate elliptic equations”, Siberian Math. J., 49:1 (2008), 80–101  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    28. S. E. Pastukhova, “Operator Estimates in Nonlinear Problems of Reiterated Homogenization”, Proc. Steklov Inst. Math., 261 (2008), 214–228  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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    30. E. S. Vasilevskaya, “Homogenization with a corrector for a parabolic Cauchy problem with periodic coefficients”, St. Petersburg Math. J., 21:1 (2010), 1–41  mathnet  crossref  mathscinet  zmath  isi
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    32. Pastukhova S.E., Tikhomirov R.N., “Estimates of locally periodic and reiterated homogenization for parabolic equations”, Dokl. Math., 80:2 (2009), 674–678  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    33. Briet Ph., Kovarik H., Raikov G., Soccorsi E., “Eigenvalue Asymptotics in a Twisted Waveguide”, Communications in Partial Differential Equations, 34:8 (2009), 818–836  crossref  mathscinet  zmath  isi  scopus
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    36. Pastukhova S., “Estimates in homogenization of parabolic equations with locally periodic coefficients”, Asymptotic Analysis, 66:3–4 (2010), 207–228  mathscinet  zmath  isi  elib
    37. N. A. Veniaminov, “Homogenization of periodic differential operators of high order”, St. Petersburg Math. J., 22:5 (2011), 751–775  mathnet  crossref  mathscinet  zmath  isi
    38. V. S. Buslaev, A. A. Pozharskii, “Homogenization in the Scattering Problem”, Funct. Anal. Appl., 44:4 (2010), 243–252  mathnet  crossref  crossref  mathscinet  zmath  isi
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    40. Suslina T., “Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H-1(R-d)”, Mathematical Modelling of Natural Phenomena, 5:4 (2010), 390–447  crossref  mathscinet  zmath  isi  scopus
    41. Andrianov I.V., Awrejcewicz J., Danishevs'kyy V.V., Weichert D., “Wave Propagation in Periodic Composites: Higher-Order Asymptotic Analysis Versus Plane-Wave Expansions Method”, Journal of Computational and Nonlinear Dynamics, 6:1 (2011), 011015  crossref  mathscinet  isi  scopus
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    44. Bunoiu R., Cardone G., Suslina T., “Spectral approach to homogenization of an elliptic operator periodic in some directions”, Math Methods Appl Sci, 34:9 (2011), 1075–1096  crossref  mathscinet  zmath  isi  elib  scopus
    45. M. A. Pakhnin, T. A. Suslina, “Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm”, Funct. Anal. Appl., 46:2 (2012), 155–159  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    49. Kuchment P. Raich A., “Green's Function Asymptotics Near the Internal Edges of Spectra of Periodic Elliptic Operators. Spectral Edge Case”, Math. Nachr., 285:14-15 (2012), 1880–1894  crossref  mathscinet  zmath  isi  elib  scopus
    50. N. N. Senik, “Homogenization for a periodic elliptic operator in a strip with various boundary conditions”, St. Petersburg Math. J., 25:4 (2014), 647–697  mathnet  crossref  mathscinet  zmath  isi  elib
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    53. Cardone G., Pastukhova S.E., Perugia C., “Estimates in Homogenization of Degenerate Elliptic Equations by Spectral Method”, Asymptotic Anal., 81:3-4 (2013), 189–209  crossref  mathscinet  zmath  isi  elib  scopus
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    63. S. E. Pastukhova, “The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization”, Sb. Math., 207:3 (2016), 418–443  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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    68. V. V. Zhikov, S. E. Pastukhova, “Operator estimates in homogenization theory”, Russian Math. Surveys, 71:3 (2016), 417–511  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    69. V. V. Zhikov, S. E. Pastukhova, “On the convergence of bloch eigenfunctions in homogenization problems”, Funct. Anal. Appl., 50:3 (2016), 204–218  mathnet  crossref  crossref  mathscinet  isi  elib
    70. T. A. Suslina, “Homogenization of Schrödinger-Type equations”, Funct. Anal. Appl., 50:3 (2016), 241–246  mathnet  crossref  crossref  mathscinet  isi  elib
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    72. Xu Q., “Convergence Rates for General Elliptic Homogenization Problems in Lipschitz Domains”, SIAM J. Math. Anal., 48:6 (2016), 3742–3788  crossref  mathscinet  zmath  isi  scopus
    73. Kuchment P., “An overview of periodic elliptic operators”, Bull. Amer. Math. Soc., 53:3 (2016), 343–414  crossref  mathscinet  zmath  isi  elib  scopus
    74. Xu Q., “Uniform regularity estimates in homogenization theory of elliptic system with lower order terms”, J. Math. Anal. Appl., 438:2 (2016), 1066–1107  crossref  mathscinet  zmath  isi  elib  scopus
    75. 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  crossref  mathscinet  zmath  isi  elib  scopus
    76. Pastukhova S.E., “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7, SI (2016), 1449–1466  crossref  mathscinet  zmath  isi  elib  scopus
    77. Meshkova Yu.M. Suslina T.A., “Homogenization of initial boundary value problems for parabolic systems with periodic coefficients”, Appl. Anal., 95:8 (2016), 1736–1775  crossref  mathscinet  zmath  isi  elib  scopus
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    83. 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
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    85. 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
    86. 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
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    88. Xu Q., “Convergence Rates and W-1,W-P Estimates in Homogenization Theory of Stokes Systems in Lipschitz Domains”, J. Differ. Equ., 263:1 (2017), 398–450  crossref  mathscinet  zmath  isi  elib  scopus
    89. Senik N.N., “Homogenization For Non-Self-Adjoint Periodic Elliptic Operators on An Infinite Cylinder”, SIAM J. Math. Anal., 49:2 (2017), 874–898  crossref  mathscinet  zmath  isi  scopus
    90. Kha M., Kuchment P., Raich A., “Green'S Function Asymptotics Near the Internal Edges of Spectra of Periodic Elliptic Operators. Spectral Gap Interior”, J. Spectr. Theory, 7:4 (2017), 1171–1233  crossref  mathscinet  zmath  isi  scopus
    91. M. M. Sirazhudinov, “Operatornye otsenki usredneniya obobschennykh uravnenii Beltrami”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 40–46  mathnet  crossref
    92. Cherednichenko K., Waurick M., “Resolvent Estimates in Homogenisation of Periodic Problems of Fractional Elasticity”, J. Differ. Equ., 264:6 (2018), 3811–3835  crossref  mathscinet  zmath  isi  scopus
    93. Chill R., ter Elst A.F.M., “Weak and Strong Approximation of Semigroups on Hilbert Spaces”, Integr. Equ. Oper. Theory, 90:1 (2018), UNSP 9  crossref  mathscinet  isi  scopus
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