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HOMOGENIZATION OF THE DIRICHLET PROBLEM FOR ELLIPTIC SYSTEMS: $L_2$ -OPERATOR ERROR ESTIMATES

  • T. A. Suslina (a1)

Abstract

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.

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