Dichotomy of global capacity density in metric measure spaces

Björn, Anders (author): Linköpings universitet,Matematik och tillämpad matematik,Tekniska fakulteten

Björn, Jana (author): Linköpings universitet,Matematik och tillämpad matematik,Tekniska fakulteten

(creator_code:org_t)

2017-05-16
2018
English.
In: Advances in Calculus of Variations. - : WALTER DE GRUYTER GMBH. - 1864-8258 .- 1864-8266. ; 11:4, s. 387-404

Related links:: http://arxiv.org/pdf...; show more...; https://urn.kb.se/re...; https://doi.org/10.1...; show less...

Abstract Subject headings

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.

By the author/editor: Aikawa, Hiroaki; Björn, Anders; Björn, Jana; Shanmugalingam, ...

About the subject

NATURAL SCIENCES: NATURAL SCIENCES; and Mathematics; and Mathematical Ana ...

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