Abstract
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.
References
N. S. Bakhvalov, “Averaging of nonlinear partial differential equations with rapidly oscillating coefficients”, Dokl. Akad. Nauk SSSR, 221:3 (1975), 516–519; English transl.:, Sov. Math. Dokl., 16 (1975), 1469–1473.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North Holland, Amsterdam, 1978.
N. S. Bakhvalov and G. P. Panasenko, Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials, Kluwer Acad. Publ., Dordrecht etc., 1989.
V. V. Jikov, S. M. Kozlov, and O. A. Olejnik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1994.
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, and Ha Tien Ngoan, “Averaging and G-convergence of differential operators”, Uspekhi Mat. Nauk, 34:5(209) (1979), 65–133; English transl.:, Russian Math. Surveys, 34:5 (1979), 69–147.
M. Sh. Birman and T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization”, Algebra i Analiz, 15:5 (2003), 1–108; English transl.:, St. Petersbg. Math. J., 15:5 (2004), 639–714.
V. V. Zhikov, “On operator estimates in homogenization theory”, Dokl. Ross. Akad. Nauk, 403:3 (2005), 305–308; English transl.:, Dokl. Math., 72:1 (2005), 534–538.
V. V. Zhikov and S. E. Pastukhova, “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524.
V. V. Zhikov and S. E. Pastukhova, “Operator estimates in homogenization theory”, Uspekhi Mat. Nauk, 71:3 (2016), 27–122; English transl.:, Russ. Math. Surv., 71:3 (2016), 417–511.
O. A. Ladyzhenskaia and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations, A Series of Monographs and Textbooks: Mathematics in Science and Engineering, 46 1968.
D. S. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.
T. A. Suslina, “Homogenization of a stationary periodic Maxwell system”, Algebra i Analiz, 16:5 (2004), 162–244; English transl.:, St. Petersburg Math. J., 16:5 (2005), 863–922.
A. A. Kukushkin and T. A. Suslina, “Homogenization of high order elliptic operators with periodic coefficients”, Algebra i Analiz, 28:1 (2016), 89–149; English transl.:, St. Petersburg Math. J., 28:1 (2017), 65–108.
S. E. Pastukhova, “Estimates in homogenization of higher-order elliptic operators”, Appl. Anal., 95:7 (2016), 1449–1466.
S. E. Pastukhova, “Operator error estimates for homogenization of fourth order elliptic equations”, Algebra i Analiz, 28:2 (2016), 204–226; English transl.:, St. Petersburg Math. J., 28:2 (2017), 273–289.
S. E. Pastukhova, “Improved approximations of resolvents in homogenization of higher order operators. The selfadjoint case”, J. Math. Sciences, 262:3 (2022), 312–328.
S. E. Pastukhova, “Improved resolvent \(L^2\)-approximations in homogenization of fourth order operators”, Algebra i Analiz, 34:4 (2022), 74–106.
E. S. Vasilevskaya and T. A. Suslina, “Homogenization of parabolic and elliptic periodic operators in \(L_2(\mathbb{R}^d)\) with the first and second correctors taken into account”, Algebra i Analiz, 24:2 (2012), 1–103; English transl.:, St. Petersburg Math. J., 24:2 (2013), 185–261.
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 93–104 https://doi.org/10.4213/faa4010.
Translated by S. E. Pastukhova
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Pastukhova, S.E. Improved Resolvent Approximations in Homogenization of Second-Order Operators with Periodic Coefficients. Funct Anal Its Appl 56, 310–319 (2022). https://doi.org/10.1134/S0016266322040086
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DOI: https://doi.org/10.1134/S0016266322040086