Homogenization of the Dirichlet problem for elliptic systems: Two-parametric error estimates

Abstract

Let $\mathcal{O}\subset {\mathbb{R}}^d$ be a bounded domain of class $C^{1,1}$. In $L_2(\mathcal{O};{\mathbb{C}}^n)$, we study a selfadjoint matrix elliptic second order differential operator $B_{D,\epsilon }$, $0<\epsilon ⩽1$, with the Dirichlet boundary condition. The principal part of the operator is given in a factorized form. The operator involves lower order terms with unbounded coefficients. The coefficients of $B_{D,\epsilon }$ are periodic and depend on $\mathbf{x}/\epsilon$. We study the generalized resolvent ${\left(B_{D,\epsilon }-\zeta Q_0(\cdot /\epsilon )\right)}^{-1}$, where $Q_0$ is a periodic bounded and positive definite matrix-valued function, and $\zeta$ is a complex-valued parameter. We obtain approximations for the generalized resolvent in the $L_2(\mathcal{O};{\mathbb{C}}^n)$-operator norm and 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 two-parametric error estimates (depending on $\epsilon$ and $\zeta$).

45 pages, minor revision of the third version. arXiv admin note: text overlap with arXiv:1509.01850

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