Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients

T. A. Suslina

Algebra i analiz, Tome 29 (2017) no. 2, pp. 139-192.

Bibliographie

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

[2] 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

[3] 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

[4] 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

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

[6] 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

[7] Veniaminov N. A., “Usrednenie periodicheskikh differentsialnykh operatorov vysokogo poryadka”, Algebra i analiz, 22:5 (2010), 69–103 | MR | Zbl

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

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

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

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

[12] 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

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

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

[15] 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

[16] Kukushkin A. A., Suslina T. A., “Usrednenie ellipticheskikh operatorov vysokogo poryadka s periodicheskimi koeffitsientami”, Algebra i analiz, 28:1 (2016), 89–149 | MR

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

[18] Meshkova Yu. M., Suslina T. A., Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates, Preprint, 2017, arXiv: 1702.00550v2

[19] Pastukhova S. E., “Operatornye otsenki usredneniya dlya ellipticheskikh uravnenii chetvertogo poryadka”, Algebra i analiz, 28:2 (2016), 204–226 | MR

[20] Pastukhova S. E., “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7 (2016), 1449–1466 | DOI | MR | Zbl

[21] Pakhnin M. A., Suslina T. A., “Usrednenie ellipticheskoi zadachi Dirikhle: otsenki pogreshnosti v $(L_2\to H^1)$-norme”, Funkts. anal. i ego pril., 46:2 (2012), 92–96 | DOI | MR | Zbl

[22] 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

[23] Solonnikov V. A., “Ob obschikh kraevykh zadachakh dlya sistem, ellipticheskikh v smysle A. Daglisa–L. Nirenberga. II”, Tr. Mat. in-ta AN SSSR, 92, 1966, 233–297 | MR | Zbl

[24] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[25] Suslina T. A., “Operatornye otsenki pogreshnosti v $L_2$ pri usrednenii ellipticheskoi zadachi Dirikhle”, Funkts. anal. i ego pril., 46:3 (2012), 91–96 | DOI | MR | Zbl

[26] 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

[27] 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

[28] Suslina T. A., “Usrednenie ellipticheskikh zadach v zavisimosti ot spektralnogo parametra”, Funkts. anal. i ego pril., 48:4 (2014), 88–94 | DOI | MR | Zbl

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