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 $C^{1,1}$. In the Hilbert space $L_2(\mathcal {O};\mathbb {C}^n)$, a matrix elliptic second order differential operator $\mathcal {A}_{D,\varepsilon }$ is considered with the Dirichlet boundary condition. Here $\varepsilon >0$ is a small parameter. The coefficients of the operator are periodic and depend on $\mathbf {x}/\varepsilon$. Approximation is found for the operator $\mathcal {A}_{D,\varepsilon }^{-1}$ 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 of $O(\sqrt {\varepsilon })$. This approximation is given by the sum of the operator $(\mathcal {A}^0_D)^{-1}$ and the first order corrector, where $\mathcal {A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.