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Homogenization of the Dirichlet problem for elliptic systems: \(L_2\)-operator error estimates. (English) Zbl 1272.35021

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.

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
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