The Variational Capacity with Respect to Nonopen Sets in Metric Spaces

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• 作者：Anders Bj?rn (1)
  Jana Bj?rn (1)
  
• 关键词：Choquet capacity ; Doubling measure ; Metric space ; Newtonian space ; Nonlinear ; Outer capacity ; p ; harmonic ; Poincaré inequality ; Potential theory ; Quasicontinuous ; Sobolev space ; Upper gradient ; Variational capacity ; Primary 31E05 ; Secondary 31C40 ; 31C45 ; 35J20 ; 35J25 ; 35J60 ; 49J40 ; 49J52 ; 49Q20 ; 58J05 ; 58J32
• 刊名：Potential Analysis
• 出版年：2014
• 出版时间：January 2014
• 年：2014
• 卷：40
• 期：1
• 页码：57-80
• 全文大小：464 KB
• 作者单位：Anders Bj?rn (1)
  Jana Bj?rn (1)
  
  1. Department of Mathematics, Link?pings universitet, 581 83, Link?ping, Sweden
  
• ISSN：1572-929X

文摘

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.