We consider resolvents $(A_{ϵ}+1{)}^{-1}$ of elliptic second-order differential operators $A_{ϵ}=-\text{div}a\left(x/ϵ\right)\mathrm{\nabla }$ in ${\mathbb{R}}^d$ with ε-periodic measurable matrix $a\left(x/ϵ\right)$ and study the asymptotic behaviour of $(A_{ϵ}+1{)}^{-1}$, as the period ε goes to zero. We provide a construction for the leading terms of the ‘operator asymptotics’ of $(A_{ϵ}+1{)}^{-1}$ in the sense of $L^2$-operator-norm convergence and prove order ${ϵ}^2$ remainder estimates. We apply the modified method of the first approximation with the usage of Steklov's smoothing. The class of operators covered by our analysis includes uniformly elliptic families with bounded coefficients and also with unbounded coefficients from the John–Nirenberg space BMO (bounded mean oscillation).

我们考虑解决方案$({\mathrm{一个}}_{\epsilon }+1{)}^{-1}$椭圆二阶微分算子${\mathrm{一个}}_{\epsilon }=-\text{div}\mathrm{一个}\left(X/\epsilon \right)\mathrm{\nabla }$在${\mathbb{R}}^d$具有ε -周期可测矩阵$\mathrm{一个}\left(X/\epsilon \right)$并研究渐近行为$({\mathrm{一个}}_{\epsilon }+1{)}^{-1}$，随着周期ε变为零。我们提供了“算子渐近线”的主要术语的构造$({\mathrm{一个}}_{\epsilon }+1{)}^{-1}$在某种意义上${\mathrm{大号}}^2$-算子范数收敛和证明顺序${\epsilon }^2$余数估计。我们使用 Steklov 平滑法应用修正后的一阶近似法。我们的分析涵盖的算子类别包括具有有界系数的均匀椭圆族，以及来自 John-Nirenberg 空间BMO（有界平均振荡）的无界系数。