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A new approach to Sobolev spaces in metric measure spaces
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 142, 194-237 p.Article in journal (Refereed) PublishedText
Abstract [en]

Let (X, d(X), mu) be a metric measure space where X is locally compact and separable and mu is a Borel regular measure such that 0 amp;lt; mu(B(x, r)) amp;lt; infinity for every ball B(x, r) with center x is an element of X and radius r amp;gt; 0. We define chi to be the set of all positive, finite non- zero regular Borel measures with compact support in X which are dominated by mu, and M = X boolean OR {0}. By introducing a kind of mass transport metric d(M) on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F : X -amp;gt; R, and then for functions f : X -amp;gt; [-infinity, infinity] by identifying them with the unique element F-f : X -amp;gt; R defined by the mean- value integral: Ff(eta) - 1/vertical bar vertical bar eta vertical bar vertical bar integral f d eta. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space R-n with Lebesgue measure. (C) 2016 Elsevier Ltd. All rights reserved.

Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD , 2016. Vol. 142, 194-237 p.
Keyword [en]
Sobolev space; Metric measure space; Mass transport
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-130256DOI: 10.1016/j.na.2016.04.010ISI: 000378058400009OAI: oai:DiVA.org:liu-130256DiVA: diva2:950674
Available from: 2016-08-01 Created: 2016-07-28 Last updated: 2016-08-22

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Sjödin, Tomas
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Mathematics and Applied MathematicsFaculty of Science & Engineering
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