Abstract:
We study the homogenization of a fourth-order divergent elliptic operator Aε with rapidly oscillating ε-periodic coefficients, where ε is a small parameter. The homogenized operator A0 is of the same type and has constant coefficients. We apply Zhikov's shift method to obtain an estimate in the (L2→L2)-operator norm of order ε2 for the difference of the resolvents (Aε+1)−1 and (A0+1)−1.
Bibliography: 25 titles.
Keywords:
approximation of resolvents, operator estimate of the homogenization error, corrector, shift method, fourth-order elliptic operator.
\Bibitem{Pas21}
\by S.~E.~Pastukhova
\paper Approximation of resolvents in homogenization of fourth-order elliptic operators
\jour Sb. Math.
\yr 2021
\vol 212
\issue 1
\pages 111--134
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This publication is cited in the following 18 articles:
S. E. Pastukhova, “Improved Homogenization Estimates for Higher-order Elliptic Operators in Energy Norms”, Lobachevskii J Math, 45:7 (2024), 3351
D. I. Borisov, “Operator Estimates in Two-Dimensional Problems with a Frequent Alternation in the Case of Small Parts with the Dirichlet Condition”, Proc. Steklov Inst. Math. (Suppl.), 321, suppl. 1 (2023), S33–S52
S. E. Pastukhova, “On Operator Estimates of the Homogenization of Higher-Order Elliptic Systems”, Math. Notes, 114:3 (2023), 322–338
T. A. Suslina, “Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients”, Russian Math. Surveys, 78:6 (2023), 1023–1154
A. A. Miloslova, T. A. Suslina, “Homogenization of the higher-order parabolic equations with periodic coefficients”, J. Math. Sci., 277:6 (2023), 959
D. I. Borisov, “Homogenization for operators with arbitrary perturbations in coefficients”, Journal of Differential Equations, 369 (2023), 41
D. I. Borisov, D. M. Polyakov, “Resolvent convergence for differential–difference operators with small variable translations”, Mathematics, 11:20 (2023), 4260
V. A. Sloushch, T. A. Suslina, “Operator estimates for homogenization of higher-order elliptic operators with periodic coefficients”, St. Petersburg Math. J., 35:2 (2024), 327–375
A. Piatnitski, V. Sloushch, T. Suslina, E. Zhizhina, “On operator estimates in homogenization of nonlocal operators of convolution type”, Journal of Differential Equations, 352 (2023), 153
S. E. Pastukhova, “On resolvent approximations of elliptic differential operators with periodic coefficients”, Appl. Anal., 101:13 (2022), 4453–4474
D. I. Borisov, “Norm resolvent convergence of elliptic operators in domains with thin spikes”, J. Math. Sci. (N.Y.), 261:3 (2022), 366–392
D. I. Borisov, “Operator estimates for planar domains with irregularly curved boundary. The Dirichlet and Neumann conditions”, J. Math. Sci. (N.Y.), 264:5 (2022), 562–580
S. E. Pastukhova, “Improved approximations of resolvents in homogenization of higher order operators. The selfadjoint case”, J. Math. Sci. (N.Y.), 262:3 (2022), 312–328
D. I. Borisov, M. N. Konyrkulzhaeva, “Operator $L_2$ -estimates for two-dimensional problems with rapidly alternating boundary conditions”, J. Math. Sci. (N.Y.), 267:3 (2022), 319–337
S. E. Pastukhova, “Improved $L^2$-approximation of resolvents in homogenization of fourth order operators”, St. Petersburg Math. J., 34:4 (2023), 611–634
A. A. Miloslova, T. A. Suslina, “Usrednenie parabolicheskikh uravnenii vysokogo poryadka s periodicheskimi koeffitsientami”, Differentsialnye uravneniya s chastnymi proizvodnymi, SMFN, 67, no. 1, Rossiiskii universitet druzhby narodov, M., 2021, 130–191
S. E. Pastukhova, “Improved approximations of resolvent in homogenization of higher order operators”, J. Math. Sci. (N.Y.), 259:2 (2021), 230–243
S. E. Pastukhova, “Improved approximations of resolvents in homogenization of fourth order operators”, J. Math. Sci. (N.Y.), 255:4 (2021), 488–502
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