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  • 1.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications2018In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 34, no 3, p. 1323-1360Article in journal (Refereed)
    Abstract [en]

    We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

  • 2.
    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)
  • 3.
    Gardiner, Stephen J.
    et al.
    Univ Coll Dublin, Ireland.
    Ghergu, Marius
    Univ Coll Dublin, Ireland.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Analytic content and the isoperimetric inequality in higher dimensions2018In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 275, no 9, p. 2284-2298Article in journal (Refereed)
    Abstract [en]

    This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in R-N to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory. (C) 2018 Elsevier Inc. All rights reserved.

  • 4.
    Gardiner, Stephen J.
    et al.
    Univ Coll Dublin, Ireland.
    Ghergu, Marius
    Univ Coll Dublin, Ireland.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Isoperimetric Inequalities for Bergman Analytic Content2020In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 69, no 4, p. 1231-1249Article in journal (Refereed)
    Abstract [en]

    The Bergman p-analytic content (1 <= p < infinity) of a planar domain Omega measures the L-p (Omega)-distance between (z) over bar and the Bergman space A(p) (Omega) of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman p-analytic content in terms of the St. Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

  • 5.
    Gardiner, Stephen J.
    et al.
    Univ Coll Dublin, Ireland.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A characterization of annular domains by quadrature identities2019In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 51, no 3, p. 436-442Article in journal (Refereed)
    Abstract [en]

    This note verifies a conjecture of Armitage and Goldstein that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere.

  • 6.
    Gardiner, Stephen J.
    et al.
    University College Dublin, School of Mathematics and Statistics.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Boundary points of angular type form a set of zero harmonic measure2022In: Annales Fennici Mathematici, ISSN 2737-0690, Vol. 47, no 2, p. 641-644Article in journal (Refereed)
    Abstract [en]

    This note addresses a problem of Dvoretzky concerning the harmonic measure of the set of boundary points of a domain in Euclidean space that are of angular type.

  • 7.
    Gardiner, Stephen J.
    et al.
    Univ Coll Dublin, Ireland.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    On a conjecture of Gustafsson and Lin concerning Laplacian growth2022In: Analysis and Mathematical Physics, ISSN 1664-2368, E-ISSN 1664-235X, Vol. 12, no 1, article id 38Article in journal (Refereed)
    Abstract [en]

    Gustafsson and Lin recently published a significant result concerning Laplacian growth problems that start from a simply connected planar domain. However, the validity of their result depends on the verification of a particular conjecture. This paper provides the missing proof.

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  • 8.
    Gardiner, Stephen J.
    et al.
    Univ Coll Dublin, Ireland.
    Sjödin, Tomas
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    On a Conjecture of Král Concerning the Subharmonic Extension of Continously Differentable Functions2020In: Mathematica Bohemica, ISSN 0862-7959, E-ISSN 2464-7136, Vol. 145, no 1, p. 71-73Article in journal (Refereed)
    Abstract [en]

    This note verifies a conjecture of Kral, that a continuously differentiable function, which is subharmonic outside its critical set, is subharmonic everywhere.

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    fulltext
  • 9.
    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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  • 10.
    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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  • 11.
    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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  • 12.
    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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  • 13.
    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.

1 - 13 of 13
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