论文标题
穿孔结构域上线性弹性系统均质均质估计的次优误差估计值
suboptimal error estimates for homogenization of linear elasticity systems on perforated domains
论文作者
论文摘要
在目前的工作中，我们确定了定期穿孔域中线性弹性系统的几乎偏差误差估计。第一个结果是$ l^{\ frac {2d} {d-1-τ}} $ - 错误估计$ o \ big（\ varepsilon^{1- \fracτ{2}}} \ big）$，$ 0 <τ<τ<τ<1 $ for一个有界平滑域的$。紧随其后的是加权Hardy-Sobolev的不等式以及对第一阶近似校正器的平方函数的次优误差估计（最早由C. Kenig，F。Lin，Z。Shen \ Shen \ cite {kls}在系数上的其他规律性假设下进行的）。新方法依赖于加权的Calderón-Zygmund估算（最初出现在A. Gloria，S。Neukamm，F。Otto的作品\ Cite {gloria_neukamm_otto_otto_2015}中，以进行定量的随机性同质化理论）。第二次努力是$ l^2 $ -Error估计$ o \ big（\ varepsilon^{\ frac {5} {6}}} \ ln^{\ ln^{\ frac {\ frac {2} {3}}}（1/\ varepsilon）\ big）$ contect confence confence confence confence confence confence confence confence confence confection confece confect confectect confece confecect confection conef。此外，我们为穿孔域开发了一种新的加权扩展定理，并在整个项目中扮演了Z. Shen \ cite {S3}的真正方法。
In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was $L^{\frac{2d}{d-1-τ}}$-error estimates $O\big(\varepsilon^{1-\fracτ{2}}\big)$ with $0<τ<1$ for a bounded smooth domain. It followed from weighted Hardy-Sobolev's inequalities and a suboptimal error estimate for the square function of the first-order approximating corrector (which was earliest investigated by C. Kenig, F. Lin, Z. Shen \cite{KLS} under additional regularity assumption on coefficients). The new approach relied on the weighted quenched Calderón-Zygmund estimate (initially appeared in A. Gloria, S. Neukamm, F. Otto's work \cite{Gloria_Neukamm_Otto_2015} 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 new duality scheme coupled with interpolation inequalities. Also, we developed a new weighted extension theorem for perforated domains, and a real method imposed by Z. Shen \cite{S3} played a fundamental role in the whole project.
