论文标题
统一$ w^{1，p} $估计和大规模的规律性，用于穿孔域中的Dirichlet问题
Uniform $W^{1, p}$ Estimates and Large-Scale Regularity for Dirichlet Problems in Perforated Domains
论文作者
论文摘要
在本文中，我们研究了域中拉普拉斯方程式的差异问题$ω_{ε，η} $定期穿孔，并在$ \ mathbb {r}^d $中定期进行小孔，其中$ε$代表了孔之间的最小距离和$η$ spare scale scale和$ $ $ $ $ $ $ $和$ε的比例。我们建立了$ W^{1，P} $估计，该解决方案具有限制常数，根据$ε$和$η$明确。证明依靠大规模的Lipschitz估算到穿孔域中的谐波功能。结果对于$ d \ ge 2 $是最佳的。
In this paper we study the Dirichlet problem for Laplace's equation in a domain $ω_{ε, η}$ perforated periodically with small holes in $\mathbb{R}^d$, where $ε$ represents the scale of the minimal distances between holes and $η$ the ratio between the scale of sizes of holes and $ε$. We establish $W^{1, p}$ estimates for solutions with bounding constants depending explicitly on $ε$ and $η$. The proof relies on a large-scale Lipschitz estimate for harmonic functions in perforated domains. The results are optimal for $d\ge 2$.
