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Sjödin, Tomas
Publications (6 of 6) Show all publications
Sjödin, T. (2016). A new approach to Sobolev spaces in metric measure spaces. Nonlinear Analysis, 142, 194-237
Open this publication in new window or tab >>A new approach to Sobolev spaces in metric measure spaces
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 142, p. 194-237Article in journal (Refereed) Published
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
Keywords
Sobolev space; Metric measure space; Mass transport
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-130256 (URN)10.1016/j.na.2016.04.010 (DOI)000378058400009 ()
Available from: 2016-08-01 Created: 2016-07-28 Last updated: 2017-11-28
Sjödin, T. & Gardiner, S. J. (2014). Quadrature domains and their two-phase counterparts. In: Vasil'ev A. (Ed.), Harmonic and Complex Analysis and its Applications: (pp. 261-285). Springer
Open this publication in new window or tab >>Quadrature domains and their two-phase counterparts
2014 (English)In: Harmonic and Complex Analysis and its Applications / [ed] Vasil'ev A., Springer, 2014, p. 261-285Chapter in book (Refereed)
Abstract [en]

This survey describes recent advances on quadrature domains that were made in the context of the ESF Network on Harmonic and Complex Analysis and its Applications (2007–2012). These results concern quadrature domains, and their two-phase counterparts, for harmonic, subharmonic and analytic functions.

Place, publisher, year, edition, pages
Springer, 2014
Series
Trends in mathematics, ISSN 2297-0215
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-151992 (URN)10.1007/978-3-319-01806-5_5 (DOI)978-3-319-01805-8 (ISBN)978-3-319-01806-5 (ISBN)
Available from: 2018-10-12 Created: 2018-10-12 Last updated: 2019-01-24
Gardiner, S. J. & Sjödin, T. (2014). Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions. Archive for Rational Mechanics and Analysis, 213(2), 503-526
Open this publication in new window or tab >>Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions
2014 (English)In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 213, no 2, p. 503-526Article in journal (Refereed) Published
Abstract [en]

It is known that corners of interior angle less than pi/2 in the boundary of a plane domain are initially stationary for Hele-Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2014
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-107819 (URN)10.1007/s00205-014-0750-0 (DOI)000336427700005 ()
Available from: 2014-06-23 Created: 2014-06-23 Last updated: 2017-12-05
Shahgholian, H. & Sjödin, T. (2013). Harmonic balls and the two-phase Schwarz function. Complex Variables and Elliptic Equations, 58(6), 837-852
Open this publication in new window or tab >>Harmonic balls and the two-phase Schwarz function
2013 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 6, p. 837-852Article in journal (Refereed) Published
Abstract [en]

In this article, we introduce the concept of harmonic balls in sub-domains of n , through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles / Taylor and Francis, 2013
Keywords
Schwarz function, mean-value property, harmonic functions, two-phase free boundary, quadrature domains, Primary: 35R35, 31A05, 31B05, 31B20
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-94602 (URN)10.1080/17476933.2011.622046 (DOI)000319380200010 ()
Note

Funding Agencies|Swedish Research Council|

Available from: 2013-06-27 Created: 2013-06-27 Last updated: 2017-12-06
Gardiner, S. J. & Sjödin, T. (2012). Two-phase quadrature domains. Journal d'Analyse Mathematique, 116, 335-354
Open this publication in new window or tab >>Two-phase quadrature domains
2012 (English)In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 116, p. 335-354Article in journal (Refereed) Published
Abstract [en]

Recent work on two-phase free boundary problems has led to the investigation of a new type of quadrature domain for harmonic functions. This paper develops a method of constructing such quadrature domains based on the technique of partial balayage, which has proved to be a useful tool in the study of one-phase quadrature domains and Hele-Shaw flows.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2012
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-76547 (URN)10.1007/s11854-012-0009-3 (DOI)000301446000009 ()
Available from: 2012-04-12 Created: 2012-04-11 Last updated: 2017-12-07
Gardiner, S. & Sjödin, T. (2008). Convexity and the exterior inverse problem of potential theory. Proceedings of the American Mathematical Society., 136(5), 1699-1703
Open this publication in new window or tab >>Convexity and the exterior inverse problem of potential theory
2008 (English)In: Proceedings of the American Mathematical Society., Vol. 136, no 5, p. 1699-1703Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-44662 (URN)77250 (Local ID)77250 (Archive number)77250 (OAI)
Available from: 2009-10-10 Created: 2009-10-10 Last updated: 2010-04-22
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