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  • 1.
    Adamowicz, Tomasz
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Regularity of p(.)-superharmonic functions, the Kellogg property and semiregular boundary points2014In: Annales de l'Institut Henri Poincare. Analyse non linéar, ISSN 0294-1449, E-ISSN 1873-1430, Vol. 31, no 6, p. 1131-1153Article in journal (Refereed)
    Abstract [en]

    We study various boundary and inner regularity questions for p(.)-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for p(.)-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded p(.)-harmonic functions and give some new characterizations of W-0(1,p(.)) spaces. We also show that p(.)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.

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  • 2.
    Adamowicz, Tomasz
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Shanmugalingam, Nageswari
    University of Cincinnati, OH USA .
    Prime ends for domains in metric spaces2013In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 238, p. 459-505Article in journal (Refereed)
    Abstract [en]

    In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.

  • 3.
    Aikawa, Hiroaki
    et al.
    Hokkaido Univ, Japan.
    Björn, Anders
    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.
    Shanmugalingam, Nageswari
    Univ Cincinnati, OH 45221 USA.
    Dichotomy of global capacity density in metric measure spaces2018In: Advances in Calculus of Variations, ISSN 1864-8258, E-ISSN 1864-8266, Vol. 11, no 4, p. 387-404Article in journal (Refereed)
    Abstract [en]

    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.

  • 4.
    Arnlind, Joakim
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Björn, Anders
    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.
    An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces2016In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 134, p. 70-104Article in journal (Refereed)
    Abstract [en]

    We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on ( ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. (C) 2015 Elsevier Ltd. All rights reserved.

  • 5.
    Björn , Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations2009In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, ISSN 0944-2669 , Vol. 35, no 4, p. 481-496Article in journal (Refereed)
    Abstract [en]

    We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincar, inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R (n) .

  • 6.
    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.
    A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets2020In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. 21, p. 293-301Article in journal (Refereed)
    Abstract [en]

    We show that every Sobolev function in W-loc(1, p) (U) on a p-quasiopen set U subset of R-n with a.e.-vanishing p-fine gradient is a.e.-constant if and only if U is p-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality. On unweighted R-n, we also obtain the corresponding result for p-finely open sets in terms of p-fine connectedness, using a deep result by Latvala.

  • 7.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Approximations by regular sets and Wiener solutions in metric spaces2007In: Commentationes mathematicae Universitatis Carolinae, ISSN 0323-0171, Vol. 48, no 2, p. 343-355Article in journal (Refereed)
  • 8.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces2004Report (Other academic)
  • 9.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces2006In: Journal of the Mathematical Society of Japan, ISSN 0025-5645, E-ISSN 1881-1167, Vol. 58, no 4, p. 1211-1232Article in journal (Refereed)
    Abstract [en]

    We study p-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak (1, p)-Poincaré inequality, 1 < p < ∞. We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the one-sided obstacle problem on bounded open sets. Regularity is further characterized in several other ways. Our results apply also to Cheeger p-harmonic functions and in the Euclidean setting to script A sign-harmonic functions, with the usual assumptions on script A sign.

  • 10.
    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.
    Correction: Poincare inequalities and Newtonian Sobolev functions on noncomplete metric spaces (vol 266, pg 44, 2019)2021In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 285, p. 493-495Article in journal (Other academic)
    Abstract [en]

    We correct a mistake in our paper "Poincare inequalities and Newtonian Sobolev functions on noncomplete metric spaces"(Bjorn and Bjorn, 2019 [2]). (C) 2018 Elsevier Inc. All rights reserved.

  • 11.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    First-order Sobolev spaces on metric spaces2009In: Function Spaces, Inequalities and Interpolation (Paseky, 2009) / [ed] Jaroslav Lukes, Lubos Pick, Prague: Matfyzpress , 2009, p. 1-29Conference paper (Refereed)
  • 12.
    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.
    Local and semilocal Poincare inequalities on metric spaces2018In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 119, p. 158-192Article in journal (Refereed)
    Abstract [en]

    We consider several local versions of the doubling condition and Poincare inequalities on metric measure spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincare inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward. We also conclude that many qualitative, as well as quantitative, properties of p-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions. (C) 2018 Elsevier Masson SAS. All rights reserved.

  • 13.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Nonlinear Potential Theory on Metric Spaces2011 (ed. 1)Book (Refereed)
    Abstract [en]

    The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories.

    This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis.

    The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space.

    Each chapter contains historical notes with relevant references and an extensive index is provided at the end of the book.

  • 14.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology2015In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 31, no 1, p. 161-214Article in journal (Refereed)
    Abstract [en]

    We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open E (apart from those which are trivial in this case) and also on R-n.

  • 15.
    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.
    Poincare inequalities and Newtonian Sobolev functions on noncomplete metric spaces2019In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 266, no 1, p. 44-69Article in journal (Refereed)
    Abstract [en]

    Let X be a noncomplete metric measure space satisfying the usual (local) assumptions of a doubling property and a Poincare inequality. We study extensions of Newtonian Sobolev functions to the completion (X) over cap of X and use them to obtain several results on X itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincare inequalities. We also provide a discussion about possible applications of the completions and extension results to p-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations. (C) 2018 Elsevier Inc. All rights reserved.

  • 16.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    POWER-TYPE QUASIMINIMIZERS2011In: ANNALES ACADEMIAE SCIENTIARUM FENNICAE-MATHEMATICA, ISSN 1239-629X, Vol. 36, no 1, p. 301-319Article in journal (Refereed)
    Abstract [en]

    In this paper we examine the quasiminimizing properties of radial power-type functions u(x) = vertical bar x vertical bar(alpha) in R-n. We find the optimal quasiminimizing constant whenever u is a quasiminfinizer of the p-Dirichlet integral, p not equal n, and similar results when u is a quasisub- and quasisuperminimizer. We also obtain similar results for log-powers when p = n.

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  • 17.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Sharp Besov capacity estimates for annuli in metric spaces with doubling measures2023In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 305, no 3, article id 41Article in journal (Refereed)
    Abstract [en]

    We obtain precise estimates, in terms of the measure of balls, for the Besov capacity of annuli and singletons in complete metric spaces. The spaces are only assumed to be uniformly perfect with respect to the centre of the annuli and equipped with a doubling measure.

  • 18.
    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.
    TENSOR PRODUCTS AND SUMS OF p-HARMONIC FUNCTIONS, QUASIMINIMIZERS AND p-ADMISSIBLE WEIGHTS2018In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 146, no 12, p. 5195-5203Article in journal (Refereed)
    Abstract [en]

    The tensor product of two p-harmonic functions is in general not p-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted R-n with p-admissible weights. It is also shown that the tensor product of two p-admissible measures is p-admissible. This last result is generalized to metric spaces.

  • 19.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The Variational Capacity with Respect to Nonopen Sets in Metric Spaces2014In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 40, no 1, p. 57-80Article in journal (Refereed)
    Abstract [en]

    We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R (n) it is shown to coincide with the usual variational capacity considered in the literature. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.

  • 20.
    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.
    Gianazza, Ugo
    University of Pavia, Italy.
    The Petrovskii criterion and barriers for degenerate and singular p-parabolic equations2017In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 368, no 3-4, p. 885-904Article in journal (Refereed)
    Abstract [en]

    In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p amp;gt; 2 and the singular case 1 amp;lt; p amp;lt; 2 We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.

  • 21.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Gianazza, Ugo
    University of Pavia, Italy.
    Parviainen, Mikko
    University of Jyvaskyla, Finland.
    Boundary regularity for degenerate and singular parabolic equations2015In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 52, no 3-4, p. 797-827Article in journal (Refereed)
    Abstract [en]

    We characterise regular boundary points of the parabolic p-Laplacian in terms of a family of barriers, both when p greater than 2 and 1 less than p less than 2. By constructing suitable families of such barriers, we give some simple geometric conditions that ensure the regularity of boundary points.

  • 22.
    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.
    Gianazzac, Ugo
    Univ Pavia, Italy.
    Siljander, Juhana
    Univ Jyvaskyla, Finland.
    Boundary Regularity for the Porous Medium Equation2018In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 230, no 2, p. 493-538Article in journal (Refereed)
    Abstract [en]

    We study the boundary regularity of solutions to the porous medium equation in the degenerate range . In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general-not necessarily cylindrical-domains in . One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/superparabolic functions and weak sub/supersolutions.

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  • 23.
    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.
    Gill, James T.
    St Louis University, MO 63103 USA.
    Shanmugalingam, Nageswari
    University of Cincinnati, OH 45221 USA.
    Geometric analysis on Cantor sets and trees2017In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, Vol. 725, p. 63-114Article in journal (Refereed)
    Abstract [en]

    dUsing uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.

  • 24.
    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.
    Korte, Riikka
    Aalto University, Finland.
    Minima of quasisuperminimizers2017In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 155, p. 264-284Article in journal (Refereed)
    Abstract [en]

    We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q-quasisuperminimizers. More precisely, if u(i) is a Q(i)-quasisuperminimizer, i = 1,2, where 1 amp;lt; Q(1) amp;lt; Q(2), then u = min{u(1), u(2)} is a Q-quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q. We provide the first examples of this phenomenon, i.e. that Q amp;gt; Q(2). In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q(1)Q(2) turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers. (C) 2017 Elsevier Ltd. All rights reserved.

  • 25.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Lahti, Panu
    Chinese Acad Sci, Peoples R China.
    Removable sets for Newtonian Sobolev spaces and a characterization of p-path almost open sets2023In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 39, no 3, p. 1143-1180Article in journal (Refereed)
    Abstract [en]

    We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincare inequality. In particular, when restricted to Euclidean spaces, a closed set E c Rn with zero Lebesgue measure is shown to be removable for W1;p.Rn \ E/ if and only if Rn \ E supports a p-Poincare inequality as a metric space. When p &gt; 1, this recovers Koskelas result (Ark. Mat. 37 (1999), 291-304), but for p = 1, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces L1;p. To be able to include p = 1, we first study extensions of Newtonian Sobolev functions in the case p = 1 from a noncom-plete space X to its completion Xy. In these results, p-path almost open sets play an important role, and we provide a characterization of them by means of p-path open, p-quasiopen and p-finely open sets. We also show that there are nonmeasurable p -path almost open subsets of Rn, n &gt; 2, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with Lp-integrable upper gradients, about p-quasiopen, p-path open and p-finely open sets, and about Lebesgue points for N1;1-functions, to spaces that only satisfy local assumptions.

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  • 26.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    Convergence and local-to-global results for p-superminimizers on quasiopen sets2023In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 365, p. 812-831Article in journal (Refereed)
    Abstract [en]

    In this paper, several convergence results for fine p-(super)minimizers on quasiopen sets in metric spaces are obtained. For this purpose, we deduce a Caccioppoli-type inequality and local-to-global principles for fine p-(super)minimizers on quasiopen sets. A substantial part of these considerations is to show that the functions belong to a suitable local fine Sobolev space. We prove our results for a complete metric space equipped with a doubling measure supporting a p-Poincare inequality with 1 &lt; p &lt; & INFIN;. However, most of & COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

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  • 27.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    Correction: The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces (May, 10.1007/s11118-022-09996-7, 2022)2022In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929XArticle in journal (Other academic)
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  • 28.
    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.
    Latvala, Visa
    University of Eastern Finland, Finland.
    SOBOLEV SPACES, FINE GRADIENTS AND QUASICONTINUITY ON QUASIOPEN SETS2016In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 41, no 2, p. 551-560Article in journal (Refereed)
    Abstract [en]

    We study different definitions of Sobolev spaces on quasiopen sets in a complete metric space X equipped with a doubling measure supporting a p-Poincare inequality with 1 amp;lt; p amp;lt; infinity, and connect them to the Sobolev theory in R-n. In particular, we show that for quasiopen subsets of R-n the Newtonian functions, which are naturally defined in any metric space, coincide with the quasicontinuous representatives of the Sobolev functions studied by Kilpelainen and Maly in 1992.

  • 29.
    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.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces2018In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 135, no 1, p. 59-83Article in journal (Refereed)
    Abstract [en]

    We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincar, inequality, 1 amp;lt; p amp;lt; a. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.

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  • 30.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Latvala, Visa
    Univ Eastern Finland, Finland.
    The Dirichlet Problem for p-minimizers on Finely Open Sets in Metric Spaces2023In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 59, p. 1117-1140Article in journal (Refereed)
    Abstract [en]

    We initiate the study of fine p-(super)minimizers, associated with p-harmonic functions, on finely open sets in metric spaces, where infinity. After having developed their basic theory, we obtain the p-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted . We build this theory in a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality.

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  • 31.
    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.
    Latvala, Visa
    University of Eastern Finland, Finland.
    The Weak Cartan Property for the p-fine Topology on Metric Spaces2015In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 64, no 3, p. 915-941Article in journal (Refereed)
    Abstract [en]

    We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 less than p less than infinity. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all p-superharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients, and do not rely on a vector-valued differentiable structure.

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  • 32.
    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.
    Lehrback, Juha
    Univ Jyvaskyla, Finland.
    Existence and almost uniqueness for p -harmonic Green functions on bounded domains in metric spaces2020In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 269, no 9, p. 6602-6640Article in journal (Refereed)
    Abstract [en]

    We study (p -harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted R n , but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for p -harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a p-Poincar? inequality, or under similar local assumptions.

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  • 33.
    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.
    Lehrback, Juha
    University of Jyvaskyla, Finland.
    Sharp capacity estimates for annuli in weighted R-n and in metric spaces2017In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 286, no 3-4, p. 1173-1215Article in journal (Refereed)
    Abstract [en]

    We obtain estimates for the nonlinear variational capacity of annuli in weighted R-n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R-n. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R-n, which are based on quasiconformality of radial stretchings in R-n.

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  • 34.
    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.
    Lehrback, Juha
    University of Jyväskylä, Finland.
    The annular decay property and capacity estimates for thin annuli2017In: Collectanea Mathematica (Universitat de Barcelona), ISSN 0010-0757, E-ISSN 2038-4815, Vol. 68, no 2, p. 229-241Article in journal (Refereed)
    Abstract [en]

    We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar, inequality. In particular, if the measure has the 1-annular decay property at and the metric space supports a pointwise 1-Poincar, inequality at , then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at . This generalizes the known estimate for the usual variational capacity in unweighted . We also characterize the 1-annular decay property and provide examples which illustrate the sharpness of our results.

  • 35.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Lehrback, Juha
    Univ Jyvaskyla, Finland.
    Volume growth, capacity estimates, p-parabolicity and sharp integrability properties of p-harmonic Green functions2023In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538Article in journal (Refereed)
    Abstract [en]

    In a complete metric space equipped with a doubling measure supporting a p-Poincare inequality, we prove sharp growth and integrability results for p-harmonic Green functions and their minimal p-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R-n and on manifolds.The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

  • 36.
    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.
    Li, Xining
    Sun Yat Sen Univ, Peoples R China.
    Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces2019In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 474, no 2, p. 852-875Article in journal (Refereed)
    Abstract [en]

    We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements. In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space R-+(n), n amp;gt;= 2, is not subadditive on null sets when p not equal 2. Using their result and spherical inversion, we create similar bounded examples in the unit ball B subset of R-n showing that the n-harmonic measure is not subadditive on null sets when n amp;gt;= 3, and neither are the p-harmonic measures in B generated by certain weights depending on p not equal 2 and n amp;gt;= 2. (C) 2019 Elsevier Inc. All rights reserved.

  • 37.
    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.
    Maly, Jan
    University of JE Purkyne, Czech Republic.
    Quasiopen and p-Path Open Sets, and Characterizations of Quasicontinuity2017In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 46, no 1, p. 181-199Article in journal (Refereed)
    Abstract [en]

    In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincar, inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.

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  • 38.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Marola, Niko
    University of Helsinki.
    BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers2010In: ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, ISSN 0294-1449, Vol. 27, no 6, p. 1489-1505Article in journal (Refereed)
    Abstract [en]

    In this paper we use quasiminimizing properties of radial power-type functions to deduce counterexamples to certain Caccioppoli type inequalities and weak Harnack inequalities for quasisuperharmonic functions both of which are well known to hold for p-superharmonic functions We also obtain new bounds on the local integrability for quasisuperharmonic functions Furthermore we show that the logarithm of a positive quasisuperminimizer has bounded mean oscillation and belongs to a Sobolev type space

  • 39.
    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.
    Mirumbe, Ismail
    Makerere Univ, Uganda.
    THE QUASISUPERMINIMIZING CONSTANT FOR THE MINIMUM OF TWO QUASISUPERMINIMIZERS IN R-n2020In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 45, p. 215-225Article in journal (Refereed)
    Abstract [en]

    It was shown in Bjorn-Bjorn-Korte [5] that u := min{u(1), u(2)} is a (Q) over bar -quasisuper-minimizer if u(1) and u(2) are Q-quasisuperminimizers and (Q) over bar = 2Q(2)/(Q+1). Moreover, one-dimensional examples therein show that (Q) over bar is close to optimal. In this paper we give similar examples in higher dimensions. The case when u(1) and u(2) have different quasisuperminimizing constants is considered as well.

  • 40.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Mwasa, Abubakar
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering. Busitema Univ, Uganda.
    Resolutivity and invariance for the Perron method for degenerate equations of divergence type2022In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 509, no 1, article id 125937Article in journal (Refereed)
    Abstract [en]

    We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation in a bounded open set . The vector-valued function A satisfies the standard ellipticity assumptions with and a p-admissible weight w. We show that arbitrary perturbations on sets of (p, w)-capacity zero of continuous (and certain quasicontinuous) boundary data f are resolutive and that the Perron solutions for f and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of (p, w) capacity zero.

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  • 41.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Mäkäläinen, Tero
    University of Jyväskylä.
    Parviainen, Mikko
    Helsinki University of Technology.
    Nonlinear balayage on metric spaces2009In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 71, no 5-6, p. 2153-2171Article in journal (Refereed)
    Abstract [en]

    We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and p-harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.

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  • 42.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Parviainen, Mikko
    Helsinki University of Technology.
    Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces2008Report (Other academic)
  • 43.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Parviainen, Mikko
    Helsinki University Technology.
    Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces2010In: REVISTA MATEMATICA IBEROAMERICANA, ISSN 0213-2230, Vol. 26, no 1, p. 147-174Article in journal (Refereed)
    Abstract [en]

    We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have L-q-Lebesgue points everywhere.

  • 44.
    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.
    Parviainen, Mikko
    Univ Jyvaskyla, Finland.
    The tusk condition and Petrovskii criterion for the normalized p-parabolic equation2019In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 100, no 2, p. 623-643Article in journal (Refereed)
    Abstract [en]

    We study boundary regularity for the normalized p-parabolic equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math. J. 20 (1970) 683-693) showed that the so-called tusk condition guarantees regularity for the heat equation. We generalize this result to the normalized p-parabolic equation, and also obtain Holder continuity. The tusk condition is a parabolic version of the exterior cone condition. We also obtain a sharp Petrovskii criterion for the regularity of the latest moment of a domain. This criterion implies that the regularity of a boundary point is affected if one side of the equation is multiplied by a constant.

  • 45.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Shanmugalingam , Nageswari
    University of Cincinnati.
    A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure2006In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 134, no 3, p. 509-519Article in journal (Refereed)
    Abstract [en]

    A. Baernstein II (Comparison of p-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If G is a union of m open arcs on the boundary of the unit disc D, then is w a,p(G)=w a,p(G), where w a,p denotes the p-harmonic measure? (Strictly speaking he stated this question for the case m=2.) For p=2 the positive answer to this question is well known. Recall that for p≠2 the p-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.

    The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when 1G is the restriction to ∂D of a Sobolev function from W 1,p(C).

    For p≥2 it is no longer true that XG belongs to the trace class. Nevertheless, we are able to show equality for the case m=1 of one arc for all 1, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.

    Finally we show that in a certain sense the equality holds for almost all relatively open sets.

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  • 46.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Björn, Jana
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Shanmugalingam , Nageswari
    University of Cincinnati.
    Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces2008In: Houston Journal of Mathematics, ISSN 0362-1588, Vol. 34, no 4, p. 1197-1211Article in journal (Refereed)
    Abstract [en]

    We show that on complete doubling metric measure spaces X supporting a Poincare inequality, all Newton-Sobolev functions u are quasicontinuous, i.e. that for every epsilon > 0 there is an open set U subset of X such that C-p(U) < epsilon and the restriction of u to X\U is continuous. This implies that the capacity is an outer capacity.

  • 47.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Shanmugalingam, Nageswari
    A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure2003Report (Other academic)
  • 48.
    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.
    Shanmugalingam, Nageswari
    Univ Cincinnati, OH 45221 USA.
    Bounded Geometry andp-Harmonic Functions Under Uniformization and Hyperbolization2021In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 31, p. 5259-5308Article in journal (Refereed)
    Abstract [en]

    The uniformization and hyperbolization transformations formulated by Bonk et al. in"Uniformizing Gromov Hyperbolic Spaces", Asterisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly localp-Poincare inequality, then the transformed measure is globally doubling and supports a globalp-Poincare inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly localp-Poincare inequality, carries nonconstant globally definedp-harmonic functions with finitep-energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain.

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  • 49.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Shanmugalingam, Nageswari
    Department of Mathematical Sciences, University of Cincinnati.
    Classification of metric measure spaces and their ends using p-harmonic functions2022In: Annales Fennici Mathematici, ISSN 2737-0690, Vol. 47, no 2, p. 1025-1052Article in journal (Refereed)
    Abstract [en]

    By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite p-energy p-harmonic and p-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local p-Poincare inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the p-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant p-harmonic functions with finite p-energy as spaces having at least two well-separated p-hyperbolic sequences of sets towards infinity. We also show that every such space X has a function f is an element of/ LP(X) + R with finite p-energy.

  • 50.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Björn, Jana
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Shanmugalingam, Nageswari
    Univ Cincinnati, OH 45221 USA.
    Extension and trace results for doubling metric measure spaces and their hyperbolic fillings2022In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 159, p. 196-249Article in journal (Refereed)
    Abstract [en]

    In this paper we study connections between Besov spaces of functions on a compactmetric space Z, equipped with a doubling measure, and the Newton–Sobolev spaceof functions on a uniform domain Xε. This uniform domain is obtained as auniformization of a (Gromov) hyperbolic filling of Z. To do so, we construct afamily of hyperbolic fillings in the style of Bonk–Kleiner [9] and Bourdon–Pajot [13]. Then for each parameter β > 0 we construct a lift μβ of the doubling measure νon Z to Xε, and show that μβ is doubling and supports a 1-Poincaré inequality.We then show that for each θ with 0 < θ < 1 and p ≥ 1 there is a choice of β = p(1 − θ)ε such that the Besov space is the trace space of the Newton–Sobolev space N1,p(Xε, μβ). Finally, we exploit the tools of potential theory on Xεto obtain fine properties of functions in , such as their quasicontinuity andquasieverywhere existence of Lq-Lebesgue points with q = sνp/(sν − pθ), where sν is a doubling dimension associated with the measure ν on Z. Applying this tocompact subsets of Euclidean spaces improves upon a result of Netrusov [43] in

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