论文标题
在较高维度的Sobolev扩展域的必要条件
A necessary condition for Sobolev extension domains in higher dimensions
论文作者
论文摘要
我们为域提供了必要条件，即将限制的扩展运算符从$ l^{1，p}（ω）$到$ l^{1，p}（\ Mathbb r^n）$的范围$ 1 <p <2 $。该条件是根据沿着一组局部有限周边及其扩展的度量理论边界的$ω$边界的距离的距离的功率给出的。这概括了平面简单连接域的表征曲线条件，以及$ w^{1,1} $扩展的条件。我们使用必要条件给出曲线条件的定量版本。我们还构建了一个延伸域的示例，该域具有同型球并具有$ n $维边界的示例。
We give a necessary condition for a domain to have a bounded extension operator from $L^{1,p}(Ω)$ to $L^{1,p}(\mathbb R^n)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of $Ω$ integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for $W^{1,1}$-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain that is homeomorphic to a ball and has $n$-dimensional boundary.
