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

Part of: Partial differential equations

Published online by Cambridge University Press:  01 February 2013

T. A. Suslina
T. A. Suslina*
Affiliation:
Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, St. Petersburg, 198504, Russia (email: suslina@list.ru)

Abstract

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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.

MSC classification

Secondary: 35B27: Homogenization; equations in media with periodic structure

Information

Type
Research Article
Information
Mathematika , Volume 59 , Issue 2 , July 2013 , pp. 463 - 476
Copyright
Copyright © 2013 University College London 

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