Skip to main content Accessibility help


  • T. A. Suslina (a1)


Let $\mathcal {O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$ . In the Hilbert space $L_2(\mathcal {O};\mathbb {C}^n)$ , we consider a matrix elliptic second order differential operator $\mathcal {A}_{D,\varepsilon }$ with the Dirichlet boundary condition. Here $\varepsilon \gt 0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf {x}/\varepsilon $ . There are no regularity assumptions on the coefficients. A sharp order operator error estimate $\|\mathcal {A}_{D,\varepsilon }^{-1} - (\mathcal {A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon $ is obtained. Here $\mathcal {A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.



Hide All
[1]Bakhvalov, N. S. and Panasenko, G. P., Homogenization: averaging processes in periodic media. In Mathematical Problems in Mechanics of Composite Materials, Nauka (Moscow, 1984); English transl., Math. Appl. (Soviet Ser.), Vol. 36, Kluwer Academic Publishing Group (Dordrecht, 1989).
[2]Bensoussan, A., Lions, J.-L. and Papanicolaou, G., Asymptotic Analysis for Periodic Structures (Studies in Mathematics and its Applications 5), North-Holland Publishing Co. (Amsterdam–New York, 1978).
[3]Birman, M. Sh. and Suslina, T. A., Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics. In Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000) (Operator Theory: Advances and Applications 129), Birkhäuser (Basel, 2001), 71107.
[4]Birman, M. Sh. and Suslina, T. A., Second order periodic differential operators. Threshold properties and homogenization. Algebra i Analiz 15(5) (2003), 1108; English transl., St. Petersburg Math. J. 15(5) (2004), 639–714.
[5]Birman, M. Sh. and Suslina, T. A., Homogenization with corrector term for periodic elliptic differential operators. Algebra i Analiz 17(6) (2005), 1104; English transl., St. Petersburg Math. J. 17(6) (2006), 897–973.
[6]Birman, M. Sh. and Suslina, T. A., Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class $H^1(\mathbb {R}^d)$. Algebra i Analiz 18(6) (2006), 1130; English transl., St. Petersburg Math. J. 18(6) (2007), 857–955.
[7]Griso, G., Error estimate and unfolding for periodic homogenization. Asymptot. Anal. 40 (2004), 269286.
[8]Griso, G., Interior error estimate for periodic homogenization. Anal. Appl. 4(1) (2006), 6179.
[9]Kenig, C. E., Lin, F. and Shen, Z., Convergence rates in $L^2$ for elliptic homogenization problems. Arch. Ration. Mech. Anal. 203(3) (2012), 10091036.
[10]McLean, W., Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press (Cambridge, 2000).
[11]Pakhnin, M. A. and Suslina, T. A., Homogenization of the elliptic Dirichlet problem: error estimates in the $(L_2 \to H^1)$-norm. Funktsional. Anal. i Prilozhen. 46(2) (2012), 9296; English transl.,Funct. Anal. Appl. 46(2), (2012), 155-159.
[12]Pakhnin, M. A. and Suslina, T. A., Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain. Algebra i Analiz 24(6) (2012), 139177; English transl., St. Petersburg Math. J. 24(6) (2013) (to appear).
[13]Pastukhova, S. E., On some estimates in homogenization problems of elasticity theory. Dokl. Akad. Nauk 406(5) (2006), 604608; English transl., Dokl. Math. 73 (2006), 102–106.
[14]Zhikov, V. V., On the operator estimates in the homogenization theory. Dokl. Akad. Nauk 403(3) (2005), 305308; English transl., Dokl. Math. 72 (2005), 535–538.
[15]Zhikov, V. V., On some estimates of homogenization theory. Dokl. Akad. Nauk 406(5) (2006), 597601; English transl., Dokl. Math. 73 (2006), 96–99.
[16]Zhikov, V. V., Kozlov, S. M. and Olejnik, O. A., Homogenization of Differential Operators, Nauka (Moscow, 1993); English transl., Springer (Berlin, 1994).
[17]Zhikov, V. V. and Pastukhova, S. E., On operator estimates for some problems in homogenization theory. Russ. J. Math. Phys. 12(4) (2005), 515524.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


  • T. A. Suslina (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed