论文标题
在穿孔域中非线性椭圆算子均质化的定量估计值
Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains
论文作者
论文摘要
本文致力于研究穿孔域中非线性椭圆运算符的定量均质化问题。当问题固定在参考域$ \varepsilonΩ$中时，我们获得了急剧错误估计$ O（\ varepsilon）$。如果关于一个有界的穿孔域，则将看到边界层的不良影响，从而导致收敛速率损失$ O（\ varepsilon^{1/2}）$。配备了误差估计，我们在大规模上开发了内部和边界Lipschitz的估计。作为一个应用程序，我们收到了所谓的Quenchedcalderón-Zygumund估计，其真实的论点。为了克服一些困难，我们将扩展理论从（\ cite [theorem 4.3] {osy}）提高到$ l^p $ - 带有$ \ frac {2d} {2D} {d+1}-ε<p <p <\ frac {2d} {2d} {d-1} {d-1} {d-1} {d-1}+ε$和$ 0 <ε$ and $ 0<ε\ llllll1 $。为此，我们在穿孔领域建立了当地类型的庞加莱 - 索伯夫不平等。即使对于相关的线性椭圆模型，本文中的一些结果也是新的。
This paper was devoted to study the quantitative homogenization problems for nonlinear elliptic operators in perforated domains. We obtained a sharp error estimate $O(\varepsilon)$ when the problem was anchored in the reference domain $\varepsilonω$. If concerning a bounded perforated domain, one will see a bad influence from the boundary layers, which leads to the loss of the convergence rate by $O(\varepsilon^{1/2})$. Equipped with the error estimates, we developed both interior and boundary Lipschitz estimates at large-scales. As an application, we received the so-called quenched Calderón-Zygumund estimates by Shen's real arguments. To overcome some difficulties, we improved the extension theory from (\cite[Theorem 4.3]{OSY}) to $L^p$-versions with $\frac{2d}{d+1}-ε<p<\frac{2d}{d-1}+ε$ and $0<ε\ll1$. Appealing to this, we established Poincaré-Sobolev inequalities of local type on perforated domains. Some of results in the present literature are new even for related linear elliptic models.
