Operator error estimates for homogenization of the elliptic dirichlet problem in a bounded domain

Abstract

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 ${\mathcal{A}}_{D,\epsilon }$ with the Dirichlet boundary condition. Here $\epsilon >0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\epsilon$. We find approximation of the operator ${\mathcal{A}}_{D,\epsilon }^{-1}$ in the norm of operators acting from $L_2(\mathcal{O};{\mathbb{C}}^n)$ to the Sobolev space $H^1(\mathcal{O};{\mathbb{C}}^n)$ with an error term $O(\sqrt{\epsilon })$. This approximation is given by the sum of the operator $({\mathcal{A}}_D^0{)}^{-1}$ and the first order corrector, where ${\mathcal{A}}_D^0$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.

arXiv comment33 pages

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