Suslina, T. A. Homogenization of the Dirichlet problem for elliptic systems: \(L_2\)-operator error estimates. (English) Zbl 1272.35021 Mathematika 59, No. 2, 463-476 (2013). Summary: 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 \(\varepsilon > 0\) 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 \(\parallel\mathcal {A}_{D,\varepsilon }^{-1}\parallel_{L_2 \to L_2}\leq C \varepsilon \) is obtained. Here \(\mathcal {A}^0_D\) is the effective operator with constant coefficients and with the Dirichlet boundary condition. Cited in 2 ReviewsCited in 66 Documents MSC: 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35J57 Boundary value problems for second-order elliptic systems 35B45 A priori estimates in context of PDEs Keywords:Dirichlet boundary condition; effective operator PDFBibTeX XMLCite \textit{T. A. Suslina}, Mathematika 59, No. 2, 463--476 (2013; Zbl 1272.35021) Full Text: DOI arXiv References: [1] DOI: 10.1007/s00205-011-0469-0 · Zbl 1258.35086 · doi:10.1007/s00205-011-0469-0 [2] McLean, Strongly Elliptic Systems and Boundary Integral Equations (2000) · Zbl 0948.35001 [3] Griso, Asymptot. Anal. 40 pp 269– (2004) [4] Birman, Algebra i Analiz 18 pp 1– (2006) [5] Birman, Algebra i Analiz 17 pp 1– (2005) [6] Zhikov, Russ. J. Math. Phys. 12 pp 515– (2005) [7] Birman, Algebra i Analiz 15 pp 1– (2003) [8] Zhikov, Homogenization of Differential Operators (1993) [9] DOI: 10.1007/978-3-0348-8362-7_4 · doi:10.1007/978-3-0348-8362-7_4 [10] Zhikov, Dokl. Akad. Nauk 406 pp 597– (2006) [11] Bensoussan, Asymptotic Analysis for Periodic Structures (1978) [12] Zhikov, Dokl. Akad. Nauk 403 pp 305– (2005) [13] Pastukhova, Dokl. Akad. Nauk 406 pp 604– (2006) [14] Pakhnin, Algebra i Analiz 24 pp 139– (2012) [15] DOI: 10.1007/s10688-012-0022-4 · Zbl 1273.47078 · doi:10.1007/s10688-012-0022-4 [16] DOI: 10.1142/S021953050600070X · Zbl 1098.35016 · doi:10.1142/S021953050600070X This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.