度量空间中关于非开集的变分容度
The variational capacity with respect to nonopen sets in metric spaces
摘要：
本文系统地讨论了度量空间上的变分容度，并给出了它的基本性质的充分证明。一个新颖之处是，我们研究它的非开集，这是重要的Dirichlet和障碍问题的非开集，在罚款位势论的应用。在度量空间上的标准假设下，证明了变分容量是Choquet容量，并给出了它的几个等价定义，在加权R^n中的开集上，它与文献中通常考虑的变分容量一致.由于某些理想的性质在一般的非开集上不成立，我们引入了一个相关的容度，它在一般的度量空间中是Choquet容度，并且对于许多集，它与变分容度一致.我们提供的例子展示了各种性能的能力和反例时，他们失败了。最后，我们讨论了底层度量空间的变化如何影响变分容量及其极小化函数。
英文摘要：
We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R^n it is shown to coincide with the usual variational capacity considered in the literature. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.
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