Title: Closed BV-extension and $W^{1,1}$-extension sets Show Chinese title

Title: 闭的BV扩张和$W^{1,1}$-扩张集

Abstract: This paper studies the relations between extendability of different classes of Sobolev $W^{1,1}$ and $BV$ functions from closed sets in general metric measure spaces. Under the assumption that the metric measure space satisfies a weak $(1,1)$-Poincar\'e inequality and measure doubling, we prove further properties for the extension sets. In the case of the Euclidean plane, we show that compact finitely connected $BV$-extension sets are always also $W^{1,1}$-extension sets. This is shown via a local quasiconvexity result for the complement of the extension set. Show Chinese abstract

Abstract: 本文研究了在一般度量测度空间中的闭集上，Sobolev函数类$W^{1,1}$和$BV$的可扩展性之间的关系。 在度量测度空间满足弱$(1,1)$-Poincaré不等式和测度加倍的假设下，我们证明了扩展集的进一步性质。 在欧几里得平面的情况下，我们证明了紧致有限连通的$BV$-扩展集始终也是$W^{1,1}$-扩展集。 这是通过扩展集补集的局部拟凸性结果来证明的。