Homogenization of the first initial boundary value problem for parabolic systems: Operator error estimates
Algebra i analiz, Tome 29 (2017) no. 6, pp. 99-158.

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}
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Yu. M. Meshkova; T. A. Suslina. Homogenization of the first initial boundary value problem for parabolic systems: Operator error estimates. Algebra i analiz, Tome 29 (2017) no. 6, pp. 99-158. http://geodesic.mathdoc.fr/item/AA_2017_29_6_a2/

[1] “Homogenization of periodic systems with large potentials”, Allaire G., Capdeboscq Y., Piatnitski A., Siess V., Vanninathan M., 174:2 (2004), 179–220 | MR | Zbl

[2] Bakhvalov N. S., Panasenko G. P., Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR

[3] Bensoussan A., Lions J.-L., Papanicolaou G., Asymptotic analysis for periodic structures, Stud. Math. Appl., 5, North-Holland Publ. Co., Amsterdam–New York, 1978 | MR | Zbl

[4] Birman M., Suslina T., “Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics”, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001, 71–107 | MR | Zbl

[5] Birman M. Sh., Suslina T. A., “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i analiz, 15:5 (2003), 1–108 | MR | Zbl

[6] Birman M. Sh., Suslina T. A., “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104 | MR | Zbl

[7] Birman M. Sh., Suslina T. A., “Usrednenie periodicheskikh differentsialnykh operatorov s uchetom korrektora. Priblizhenie reshenii v klasse Soboleva $H^1(\mathbb R^d)$”, Algebra i analiz, 18:6 (2006), 1–130 | MR | Zbl

[8] Borisov D. I., “Asimptotiki reshenii ellipticheskikh sistem s bystro ostsilliruyuschimi koeffitsientami”, Algebra i analiz, 20:2 (2008), 19–42 | MR | Zbl

[9] Choe J. H., Kong K.-B., Lee Ch.-O., “Convergence in $L^p$ space for the homogenization problems of elliptic and parabolic equations in the plane”, J. Math. Anal. Appl., 287:2 (2003), 321–336 | DOI | MR | Zbl

[10] Geng J., Shen Zh., “Convergence rates in parabolic homogenization with time-dependent periodic coefficients”, J. Funct. Anal., 272:5 (2017), 2092–2113 | DOI | MR | Zbl

[11] Griso G., “Error estimate and unfolding for periodic homogenization”, Asymptot. Anal., 40:3/4 (2004), 269–286 | MR | Zbl

[12] Griso G., “Interior error estimate for periodic homogenization”, Anal. Appl., 4:1 (2006), 61–79 | DOI | MR | Zbl

[13] Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR

[14] Zhikov V. V., “Asimptoticheskoe povedenie i stabilizatsiya reshenii parabolicheskogo uravneniya vtorogo poryadka s mladshimi chlenami”, Tr. Mosk. mat. o-va, 46, 1983, 69–98 | MR | Zbl

[15] Zhikov V. V., “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

[16] Zhikov V. V., Pastukhova S. E., “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[17] Zhikov V. V., Pastukhova S. E., “Estimates of homogenization for a parabolic equation with periodic coefficients”, Russ. J. Math. Phys., 13:2 (2006), 224–237 | DOI | MR | Zbl

[18] Zhikov V. V., Pastukhova S. E., “Ob operatornykh otsenkakh v teorii usredneniya”, Uspekhi mat. nauk, 71:3 (2016), 27–122 | DOI | MR | Zbl

[19] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR

[20] Kenig C. E., Lin F., Shen Z., “Convergence rates in $L^2$ for elliptic homogenization problems”, Arch. Rational Mech. Anal., 203:3 (2012), 1009–1036 | DOI | MR | Zbl

[21] Kozlov S. M., “Privodimost kvaziperiodicheskikh differentsialnykh operatorov i usrednenie”, Tr. Mosk. mat. o-va, 46, 1983, 99–123 | MR | Zbl

[22] Kondratev V. A., Eidelman S. D., “Ob usloviyakh na granichnuyu poverkhnost v teorii ellipticheskikh granichnykh zadach”, Dokl. AN SSSR, 246:4 (1979), 812–815 | MR | Zbl

[23] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[24] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964 | MR

[25] Mazya V. G., Shaposhnikova T. O., Multiplikatory v prostranstvakh differentsiruemykh funktsii, Izd-vo LGU, L., 1986 | MR

[26] McLean W., Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[27] Meshkova Yu. M., “Usrednenie zadachi Koshi dlya parabolicheskikh sistem s periodicheskimi koeffitsientami”, Algebra i analiz, 25:6 (2013), 125–177 | MR | Zbl

[28] 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 | DOI | MR | Zbl

[29] Meshkova Yu. M., Suslina T. A., “Two-parametric error estimates in homogenization of second order elliptic systems in $\mathbb R^d$”, Appl. Anal., 95:7 (2016), 1413–1448 | DOI | MR | Zbl

[30] Meshkova Yu. M., Suslina T. A., Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates, 2017, arXiv: 1702.00550v4

[31] Meshkova Yu. M., Suslina T. A., “Usrednenie zadachi Dirikhle dlya ellipticheskikh i parabolicheskikh sistem s periodicheskimi koeffitsientami”, Funkts. anal. i ego pril., 51:3 (2017), 87–93 | DOI

[32] Moskow Sh., Vogelius M., “First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof”, Proc. Roy. Soc. Edinburgh Sect. A, 127:6 (1997), 1263–1299 | DOI | MR | Zbl

[33] Pakhnin M. A., Suslina T. A., “Operatornye otsenki pogreshnosti pri usrednenii ellipticheskoi zadachi Dirikhle v ogranichennoi oblasti”, Algebra i analiz, 24:6 (2012), 139–177 | MR | Zbl

[34] Rychkov V. S., “On restrictions and extensions of the Besov and Triebel–Lizorkin spaces with respect to Lipschitz domains”, J. London Math. Soc., 60 (1999), 237–257 | DOI | MR | Zbl

[35] Sanches-Palensiya E., Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[36] Suslina T. A., “Ob usrednenii periodicheskikh parabolicheskikh sistem”, Funkts. anal. i ego pril., 38:4 (2004), 86–90 | DOI | MR | Zbl

[37] Suslina T. A., “Homogenization of a periodic parabolic Cauchy problem”, Amer. Math. Soc. Transl. (2), 220, Amer. Math. Soc., Providence, RI, 2007, 201–233 | MR | Zbl

[38] Suslina T. A., “Homogenization of a periodic parabolic Cauchy problem in the Sobolev space $H^1(\mathbb R^d)$”, Math. Model. Nat. Phenom., 5:4 (2010), 390–447 | DOI | MR | Zbl

[39] Suslina T. A., “Usrednenie v klasse Soboleva $H^1(\mathbb R^d)$ dlya periodicheskikh ellipticheskikh differentsialnykh operatorov vtorogo poryadka pri vklyuchenii chlenov pervogo poryadka”, Algebra i analiz, 22:1 (2010), 108–222 | MR | Zbl

[40] Suslina T. A., “Homogenization of the Dirichlet problem for elliptic systems: $L_2$-operator error estimates”, Mathematika, 59:2 (2013), 463–476 | DOI | MR | Zbl

[41] Suslina T. A., “Homogenization of the Neumann problem for elliptic systems with periodic coefficients”, SIAM J. Math. Anal., 45:6 (2013), 3453–3493 | DOI | MR | Zbl

[42] Suslina T. A., “Usrednenie ellipticheskikh operatorov s periodicheskimi koeffitsientami v zavisimosti ot spektralnogo parametra”, Algebra i analiz, 27:4 (2015), 87–166 | MR

[43] Xu Q., “Uniform regularity estimates in homogenization theory of elliptic system with lower order terms”, J. Math. Anal. Appl., 438:2 (2016), 1066–1107 | DOI | MR | Zbl

[44] Xu Q., “Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms on the Neumann boundary problem”, J. Differential Equations, 261:8 (2016), 4368–4423 | DOI | MR | Zbl

[45] Xu Q., “Convergence rates for general elliptic homogenization problems in Lipschitz domains”, SIAM J. Math. Anal., 48:6 (2016), 3742–3788 | DOI | MR | Zbl

[46] Xu Q., Zhou Sh., Quantitative estimates in homogenization of parabolic systems of elasticity in Lipschitz cylinders, 2017, arXiv: 1705.01479