Homogenization of high order elliptic operators with periodic coefficients
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A. A. Kukushkin and T. A. Suslina
Translated by: T. Suslina - St. Petersburg Math. J. 28 (2017), 65-108
- DOI: https://doi.org/10.1090/spmj/1439
- Published electronically: November 30, 2016
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Abstract:
A selfadjoint strongly elliptic operator $A_\varepsilon$ of order $2p$ given by the expression $b(\mathbf {D})^* g(\mathbf {x}/\varepsilon ) b(\mathbf {D})$, $\varepsilon >0$, is studied in $L_2(\mathbb {R}^d;\mathbb {C}^n)$. Here $g(\mathbf {x})$ is a bounded and positive definite $(m\times m)$-matrix-valued function on $\mathbb {R}^d$; it is assumed that $g(\mathbf {x})$ is periodic with respect to some lattice. Next, $b(\mathbf {D})=\sum _{|\alpha |=p} b_\alpha \mathbf {D}^\alpha$ is a differential operator of order $p$ with constant coefficients; the $b_\alpha$ are constant $(m\times n)$-matrices. It is assumed that $m\ge n$ and that the symbol $b({\boldsymbol \xi })$ has maximal rank. For the resolvent $(A_\varepsilon - \zeta I)^{-1}$ with $\zeta \in \mathbb {C} \setminus [0,\infty )$, approximations are obtained in the norm of operators in $L_2(\mathbb {R}^d;\mathbb {C}^n)$ and in the norm of operators acting from $L_2(\mathbb {R}^d;\mathbb {C}^n)$ to the Sobolev space $H^p(\mathbb {R}^d;\mathbb {C}^n)$, with error estimates depending on $\varepsilon$ and $\zeta$.References
- N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788, DOI 10.1007/978-94-009-2247-1
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1
- M. Sh. Birman and T. A. Suslina, Threshold approximations for the resolvent of a factorized selfadjoint family taking a corrector into account, Algebra i Analiz 17 (2005), no. 5, 69–90 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 5, 745–762. MR 2241423, DOI 10.1090/S1061-0022-06-00927-7
- M. Sh. Birman and T. A. Suslina, Averaging of periodic elliptic differential operators taking a corrector into account, Algebra i Analiz 17 (2005), no. 6, 1–104 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897–973. MR 2202045, DOI 10.1090/S1061-0022-06-00935-6
- M. Sh. Birman and T. A. Suslina, Averaging of periodic differential operators taking a corrector into account. Approximation of solutions in the Sobolev class $H^2(\Bbb R^d)$, Algebra i Analiz 18 (2006), no. 6, 1–130 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 18 (2007), no. 6, 857–955. MR 2307356, DOI 10.1090/S1061-0022-07-00977-6
- N. A. Veniaminov, Homogenization of higher-order periodic differential operators, Algebra i Analiz 22 (2010), no. 5, 69–103 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 22 (2011), no. 5, 751–775. MR 2828827, DOI 10.1090/S1061-0022-2011-01166-5
- V. V. Zhikov, On operator estimates in homogenization theory, Dokl. Akad. Nauk 403 (2005), no. 3, 305–308 (Russian). MR 2164541
- V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546, DOI 10.1007/978-3-642-84659-5
- V. V. Zhikov and S. E. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys. 12 (2005), no. 4, 515–524. MR 2201316
- Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452
- V. G. Maz′ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Monographs and Studies in Mathematics, vol. 23, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 785568
- J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939), 78–91.
- S. E. Pastukhova, Operator estimates of homogenization for fourth order elliptic equations, Algebra i Analiz, 28 (2016), no. 2, 204–226.
- Svetlana E. Pastukhova, Estimates in homogenization of higher-order elliptic operators, Appl. Anal. 95 (2016), no. 7, 1449–1466. MR 3499669, DOI 10.1080/00036811.2016.1151495
- T. A. Suslina, Homogenization of elliptic operators with periodic coefficients depending on the spectral parameter, Algebra i Analiz 27 (2015), no. 4, 87–166; English. transl., St. Petersburg Math. J. 27 (2016), no. 4, 651–708.
Bibliographic Information
- A. A. Kukushkin
- Affiliation: St.-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
- Email: beslave@gmail.com
- T. A. Suslina
- Affiliation: St.-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia
- Email: t.suslina@spbu.ru
- Received by editor(s): November 2, 2015
- Published electronically: November 30, 2016
- Additional Notes: Supported by RFBR (grant no. 14-01-00760) and by St. Petersburg State University (project no. 11.38.263.2014).
- © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 28 (2017), 65-108
- MSC (2010): Primary 35B27
- DOI: https://doi.org/10.1090/spmj/1439
- MathSciNet review: 3591067