Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$

Details Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$ Homogenization of the elliptic Dirichlet problem: operator error estimates in $L_2$

2012-01-11

arxiv.org/abs/1201.2286v1

Mathematics

Analysis of PDEs

13 pages

Scientific paper

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. 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 $\mathbf{x}/\varepsilon$. 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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