迹部分为零的函数的可拓性
Extendability of functions with partially vanishing trace
摘要：
设$Ω\subseteq \mathbb{R}^d$是开的，而$D\subseteq \partialΩ$是其边界的闭部分。在对$Ω$非常温和的假设下，我们构造了Sobolev空间$\mathrm{W}^{k , p}_D (Ω)$的有界Sobolev扩张算子$1 \leq p < \infty$，它由$\mathrm{W}^{k , p} (Ω)$中所有在$D$上在适当意义下为零的函数组成.与早期的工作相比，这种结构是全球性的，\emph{not}使用本地化参数，这使得工作的边界规律性，这是尖锐的界面划分$D$和$\partial Ω\setminus D$。此外，我们提供齐次和局部估计的扩张算子。另外，我们还讨论了在$D$上具有消失迹条件的Lipschitz函数空间的情形.
英文摘要：
Let $Ω\subseteq \mathbb{R}^d$ be open and $D\subseteq \partialΩ$ be a closed part of its boundary. Under very mild assumptions on $Ω$, we construct a bounded Sobolev extension operator for the Sobolev space $\mathrm{W}^{k , p}_D (Ω)$, $1 \leq p < \infty$, which consists of all functions in $\mathrm{W}^{k , p} (Ω)$ that vanish in a suitable sense on $D$. In contrast to earlier work, this construction is global and \emph{not} using a localization argument, which allows to work with a boundary regularity that is sharp at the interface dividing $D$ and $\partial Ω\setminus D$. Moreover, we provide homogeneous and local estimates for the extension operator. Also, we treat the case of Lipschitz function spaces with a vanishing trace condition on $D$.
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