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Dichotomy of global capacity density in metric measure spaces
Hokkaido Univ, Japan.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-9677-8321
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-1238-6751
Univ Cincinnati, OH 45221 USA.
2018 (English)In: Advances in Calculus of Variations, ISSN 1864-8258, E-ISSN 1864-8266, Vol. 11, no 4, p. 387-404Article in journal (Refereed) Published
Abstract [en]

The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

Place, publisher, year, edition, pages
WALTER DE GRUYTER GMBH , 2018. Vol. 11, no 4, p. 387-404
Keywords [en]
Capacitarily stable collection; capacitary potential; capacity density; dichotomy; metric space; Sobolev capacity; variational capacity
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-152075DOI: 10.1515/acv-2016-0066ISI: 000445862300004OAI: oai:DiVA.org:liu-152075DiVA, id: diva2:1258310
Note

Funding Agencies|JSPS KAKENHI [JP25287015, JP25610017, JP17H01092]; Swedish Research Council [621-2011-3139, 621-2014-3974, 2016-03424]; NSF [DMS-1200915, DMS-1500440]

Available from: 2018-10-24 Created: 2018-10-24 Last updated: 2018-10-24

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