2015/05/11 by Michael Hinz, Hinz, Michael, Alexander Teplyaev +1
Mathematics · Physics and Astronomy · #26B12 #28A80 #31E05 #47A07 #58A10 #58A14 #60J60 #81Q35 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.CA #math.DG #math.FA #math.MP #math.PR #msc:26B12 #msc:28A80 #msc:31E05 #msc:47A07 #msc:58A10 #msc:58A14 #msc:60J60 #msc:81Q35
paper · pdf · doi:10.48550/arxiv.1505.02819
arXiv admin note: text overlap with arXiv:1201.3548 by other authors
arxiv created 2016/11/16 · arxiv updated 2016/11/17
The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on 1-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial 2-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on 1-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking 1-forms into 2-forms cannot be closable if the martingale dimension is larger than one.