In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was \(L^{\frac{2d}{d-1-\tau }}\)-error estimates \(O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )\) for all \(\tau \in (0,1)\) in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s inequalities (given by Lehrbäck and Vähäkangas in J Funct Anal 271(2):330–364, 2016) and a suboptimal error estimate for the square function of the first-order approximating corrector (earliest investigated by Kenig et al. in Arch Ration Mech Anal 203(3):1009–1036, 2012) under additional regularity assumption on coefficient). The new approach relied on the weighted quenched Calderón–Zygmund estimate (initially appeared in Gloria et al. work Milan J Math 88(1):99–170, 2020 for a quantitative stochastic homogenization theory). The second effort was \(L^2\)-error estimates \(O\big (\varepsilon ^{\frac{5}{6}} \ln ^{\frac{2}{3}}(1/\varepsilon )\big )\) for a Lipschitz domain, followed from a duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem and local-type Sobolev–Poincaré inequalities on perforated domains. Throughout the paper, we do not impose any smoothness assumption on the coefficients.

在目前的工作中，我们为周期性穿孔区域中的线性弹性系统建立了几乎清晰的误差估计。第一个结果是\（L ^ {\ frac {2d} {d-1- \ tau}} \） -误差估计\（O \ big（\ varepsilon ^ {1- \ frac {\ tau} {2}}有界光滑域中所有\（\ tau \ in（0,1）\）的\ big）\），即使对于无孔域上的均质化问题，这也是新的。紧随其后的是加权的Hardy-Sobolev不等式（由Lehrbäck和Vähäkangas在J Funct Anal 271（2）：330-364，2016中给出）和平方函数的次优误差估计一阶近似校正器（最早由Kenig等人在Arch Ration Mech Anal 203（3）：1009-1036，2012中进行的研究）在系数的其他正则性假设下进行。新方法依赖于加权淬灭的Calderón–Zygmund估计（最初出现在Gloria等人的Milan J Math 88（1）：99–170，2020中，用于定量随机均质化理论）。第二项工作是\（L ^ 2 \） -误差估计\（O \ big（\ varepsilon ^ {\ frac {5} {6}} \ ln ^ {\ frac {2} {3}}（1 / \ Lipschitz域的varepsilon）\ big）\），然后是对偶方案和插值不等式。此外，我们开发了新的加权扩展定理和局部类型的Sobolev-Poincaré不等式在穿孔的区域上。在整个论文中，我们不对系数施加任何平滑度假设。