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  • 1.
    Gardiner, S.J.
    et al.
    Natl Univ Ireland Univ Coll Dublin, Sch Math Sci, Dublin 4, Ireland .
    Sjödin, Tomas
    Natl Univ Ireland Univ Coll Dublin, Sch Math Sci, Dublin 4, Ireland .
    Convexity and the exterior inverse problem of potential theory2008In: Proceedings of the American Mathematical Society., Vol. 136, no 5, p. 1699-1703Article in journal (Refereed)
  • 2.
    Gardiner, Stephen J.
    et al.
    University of Coll Dublin, Ireland .
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions2014In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 213, no 2, p. 503-526Article in journal (Refereed)
    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.

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  • 3.
    Gardiner, Stephen J
    et al.
    University of College Dublin.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Two-phase quadrature domains2012In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 116, p. 335-354Article in journal (Refereed)
    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.

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  • 4.
    Shahgholian, Henrik
    et al.
    Royal Institute of Technology, Sweden.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Harmonic balls and the two-phase Schwarz function2013In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 58, no 6, p. 837-852Article in journal (Refereed)
    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.

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  • 5.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A new approach to Sobolev spaces in metric measure spaces2016In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 142, p. 194-237Article in journal (Refereed)
    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.

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  • 6.
    Sjödin, Tomas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Gardiner, Stephen J.
    Natl Univ Ireland Univ Coll Dublin, Sch Math Sci, Dublin 4, Ireland.
    Quadrature domains and their two-phase counterparts2014In: 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.

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