2012/10/01 by Anders Björn, Jana Björn
Computer Science · Mathematics · #Convex metric space #Counterexample #Geometric Analysis and Curvature Flows #Injective metric space #Intrinsic metric #Metric (unit) #Metric space #Nonlinear Partial Differential Equations #Optimization and Variational Analysis #Potential theory #Space (punctuation) #Variational analysis #math.AP #msc:31C40 #msc:31C45 #msc:31E05 #msc:35J20 #msc:35J25 #msc:35J60 #msc:49J40 #msc:49J52 #msc:49Q20 #msc:58J05 #msc:58J32
paper · pdf · doi:10.1007/s11118-013-9341-1
published as Potential Anal. 40 (2014), 57-80
arxiv created 2012/10/01 · openalex publication_date 2013/04/27 · arxiv updated 2014/09/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Rn 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.