Abstract:
A 4th order elliptic operator AεAε in the diverdence form acting in the entire space Rd and having ε-periodic coefficients is studied (ε is a small parameter).
An approximation for the resolvent (Aε+1)−1 is found with error estimate of order ε3 in the operator (L2→L2)-norm.
The method of double-scale approximation with a generalised shift in the form of smoothing is used.
Keywords:
homogenization, error estimates, approximation of the resolvent, elliptic operator of the 4th order.
Citation:
S. E. Pastukhova, “Improved L2-approximation of resolvents in homogenization of fourth order operators”, Algebra i Analiz, 34:4 (2022), 74–106; St. Petersburg Math. J., 34:4 (2023), 611–634
\Bibitem{Pas22}
\by S.~E.~Pastukhova
\paper Improved $L^2$-approximation of resolvents in homogenization of fourth order operators
\jour Algebra i Analiz
\yr 2022
\vol 34
\issue 4
\pages 74--106
\mathnet{http://mi.mathnet.ru/aa1825}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=548437}
\transl
\jour St. Petersburg Math. J.
\yr 2023
\vol 34
\issue 4
\pages 611--634
\crossref{https://doi.org/10.1090/spmj/1772}
Linking options:
https://www.mathnet.ru/eng/aa1825
https://www.mathnet.ru/eng/aa/v34/i4/p74
This publication is cited in the following 6 articles:
S. E. Pastukhova, “L2-otsenki pogreshnosti usredneniya parabolicheskikh uravnenii s uchetom korrektorov”, SMFN, 69, no. 1, Rossiiskii universitet druzhby narodov, M., 2023, 134–151
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
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, “Improved resolvent approximations in homogenization of second order operators with periodic coefficients”, Funct. Anal. Appl., 56:4 (2022), 310–319