Improved resolvent approximations in homogenization of second order operators with periodic coefficients

S. E. Pastukhova

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 4, pp. 93-104

Résumé

For elliptic divergent self-adjoint second-order operators with $\varepsilon$-periodic measurable coefficients acting on the whole space $\mathbb{R}^d$, resolvent approximations in the operator norm $\|\!\,\boldsymbol\cdot\,\!\|_{H^1\to H^1}$ with remainder of order $\varepsilon^2$ as $\varepsilon\to 0$ are found by the method of two-scale expansions with the use of smoothing.

Détail

Keywords: periodic differential operators, homogenization, correctors, resolvent approximations, operator error estimates.

Bibliographie

[1] N. S. Bakhvalov, “Osrednenie differentsialnykh uravnenii s chastnymi proizvodnymi s bystro ostsilliruyuschimi koeffitsientami”, Dokl. AN SSSR, 221:3 (1975), 516–519 | MR | Zbl

[2] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978 | MR | Zbl

[3] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, M., Nauka, 1984

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

[5] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Kha Ten Ngoan, “Usrednenie i G-skhodimost differentsialnykh operatorov”, UMN, 34:5(209) (1979), 65–133 | MR | Zbl

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

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

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

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

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

[11] D. Kinderlerer, G. Stampakkya, Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Mir, M., 1983 | MR

[12] T. A. Suslina, “Usrednenie statsionarnoi periodicheskoi sistemy Maksvella”, Algebra i analiz, 16:5 (2004), 162–244

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

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

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

[16] S. E. Pastukhova, “Improved approximations of resolvents in homogenization of higher order operators. The selfadjoint case”, J. Math. Sciences, 262:3 (2022), 312–328 | DOI | MR | Zbl

[17] S. E. Pastukhova, “Uluchshennye $L^2$-approksimatsii rezolventy v usrednenii operatorov chetvertogo poryadka”, Algebra i analiz, 34:4 (2022), 74–106

[18] E. S. Vasilevskaya, T. A. Suslina, “Usrednenie parabolicheskikh i ellipticheskikh periodicheskikh operatorov v $L_2(\mathbb{R}^d)$ pri uchete pervogo i vtorogo korrektorov”, Algebra i analiz, 24:2 (2012), 1–103 | MR