局部紧空间中无限多平板冷凝器的内容量
Interior capacities of condensers with infinitely many plates in a locally compact space
摘要：
本文研究了局部紧空间中凝聚子的内容量理论，这里凝聚子被看作是任意集合的可数的局部有限集合，其符号为+1或-1，使得符号相反的集合的闭包互不相交。我们的动机是已知的事实，即在非紧的情况下，理论的主要最小问题通常是不可解的，即使在非常自然的假设下也会发生这种情况（例如，对于欧几里得空间中的牛顿核、格林核或里斯核和许多板的闭凝聚器）。因此，特别有趣的是，发现变分问题的陈述与主极小问题对偶（因此提供了容量的一些新的等价定义），但现在总是可解的（例如，甚至对于无限多个板的非封闭、无界冷凝器）。正定核满足B. Fuglede的强和弱 * 拓扑拓扑之间的一致性条件，提出并解决了具有所需性质的问题。他们的解决方案提供了一个自然的推广，著名的概念，内部容性分布与一组。我们给出了这些解决方案的描述，建立他们的唯一性和连续性的声明，并指出他们的特征属性。
英文摘要：
The study deals with the theory of interior capacities of condensers in a locally compact space, a condenser being treated here as a countable, locally finite collection of arbitrary sets with the sign +1 or -1 prescribed such that the closures of opposite-signed sets are mutually disjoint. We are motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory is in general unsolvable, and this occurs even under very natural assumptions (e.g., for the Newtonian, Green, or Riesz kernels in an Euclidean space and closed condensers of finitely many plates). Therefore it was particularly interesting to find statements of variational problems dual to the main minimum-problem (and hence providing some new equivalent definitions of the capacity), but now always solvable (e.g., even for nonclosed, unbounded condensers of infinitely many plates). For all positive definite kernels satisfying B. Fuglede's condition of consistency between the strong and weak-star topologies, problems with the desired properties are posed and solved. Their solutions provide a natural generalization of the well-known notion of interior capacitary distributions associated with a set. We give a description of those solutions, establish statements on their uniqueness and continuity, and point out their characteristic properties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
