Abstract:
An elliptic fourth-order differential operator Aε on L2(Rd;Cn) is studied. Here ε>0 is
a small parameter. It is assumed that the operator is given in the factorized form Aε=b(D)∗g(x/ε)b(D), where g(x) is a Hermitian matrix-valued function periodic with respect to some lattice and b(D) is a matrix second-order differential operator. We make assumptions ensuring that the operator Aε is
strongly elliptic. The following approximation for the resolvent (Aε+I)−1 in the operator norm of L2(Rd;Cn) is obtained:
(Aε+I)−1=(A0+I)−1+εK1+ε2K2(ε)+O(ε3).
Here A0 is the effective operator with constant coefficients and K1 and K2(ε) are certain correctors.
Citation:
V. A. Sloushch, T. A. Suslina, “Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account”, Funktsional. Anal. i Prilozhen., 54:3 (2020), 94–99; Funct. Anal. Appl., 54:3 (2020), 224–228
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\paper Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
\jour Funktsional. Anal. i Prilozhen.
\yr 2020
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\pages 94--99
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\jour Funct. Anal. Appl.
\yr 2020
\vol 54
\issue 3
\pages 224--228
\crossref{https://doi.org/10.1134/S0016266320030077}
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Linking options:
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This publication is cited in the following 13 articles:
S. E. Pastukhova, “Improved Homogenization Estimates for Higher-order Elliptic Operators in Energy Norms”, Lobachevskii J Math, 45:7 (2024), 3351
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
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. Raev, V. A. Slousch, T. A. Suslina, “Usrednenie odnomernogo periodicheskogo operatora chetvertogo poryadka s singulyarnym potentsialom”, Matematicheskie voprosy teorii rasprostraneniya voln. 53, Zap. nauchn. sem. POMI, 521, POMI, SPb., 2023, 212–239
A. A. Miloslova, T. A. Suslina, “Homogenization of the higher-order parabolic equations with periodic coefficients”, J. Math. Sci., 277:6 (2023), 959
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, “Improved L2-approximation of resolvents in homogenization of fourth order operators”, St. Petersburg Math. J., 34:4 (2023), 611–634
S. E. Pastukhova, “Approximation of resolvents in homogenization of fourth-order elliptic operators”, Sb. Math., 212:1 (2021), 111–134
V. A. Sloushch, T. A. Suslina, “Threshold approximations for the resolvent of a polynomial nonnegative operator pencil”, St. Petersburg Math. J., 33:2 (2022), 355–385
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
T. A. Suslina, “Homogenization of the Higher-Order Hyperbolic Equations with Periodic Coefficients”, Lobachevskii J Math, 42:14 (2021), 3518
S. E. Pastukhova, “L2- Approximation of Resolvents in Homogenization of Higher Order Elliptic Operators”, J Math Sci, 251:6 (2020), 902