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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.

  • 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, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    A regularity classification of boundary points for p-harmonic functions and quasiminimizers2008In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 338, no 1, p. 39-47Article in journal (Refereed)
    Abstract [en]

    In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to -harmonic functions, with the usual assumptions on .

  • 6.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A regularity classification of boundary points for p-harmonic funktions and quasiminimizers2006Report (Other academic)
  • 7.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A weak Kellogg property for quasiminimizers2004Report (Other academic)
  • 8.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A weak Kellogg property for quasiminimizers2006In: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 81, no 4, p. 809-825Article in journal (Refereed)
    Abstract [en]

    The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set Ω there is a set ⊂ ∂Ω with capacity zero such that for all p-harmonic functions u in Ω with continuous boundary values in Sobolev sense, u attains its boundary values at all boundary points in ∂Ω\ E. In this paper, we show a weak Kellogg property for quasiminimizers: a quasiminimizer with continuous boundary values in Sobolev sense takes its boundary values at quasievery boundary point. The exceptional set may however depend on the quasiminimizer. To obtain this result we use the potential theory of quasisuperminimizers and prove a weak Kellogg property for quasisuperminimizers. This is done in complete doubling metric spaces supporting a Poincaré inequality. © Swiss Mathematical Society.

  • 9.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary behaviour near irregular boundary points for p-harmonic functions2007In: Qualitative Properties of Solutions to Elliptic and Parabolic Equations,2007, 2007Conference paper (Other academic)
  • 10.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Boundary regularity and barriers for elliptic and parabolic problems, and sharp capacity estimates for annuli2013Conference paper (Refereed)
  • 11.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions on metric spaces2004In: Stochastic Calculus with Applications in Geometry and Analysis,2004, 2004Conference paper (Other academic)
  • 12.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Caccioppoli inequalities and Moser iteration2005In: Sobolev inequalites - Different points of view,2005, 2005Conference paper (Other academic)
  • 13.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Characterizations and removability of p-superharmonic functions on metric spaces2004In: Analysis on Metric Measure Spaces,2004, 2004Conference paper (Other academic)
  • 14.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Characterizations of p-superharmonic functions on metric spaces2005In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 169, no 1, p. 45-62Article in journal (Refereed)
    Abstract [en]

    We show the equivalence of some different definitions of p-superharmonic functions given in the literature. We also provide several other characterizations of p-superharmonicity. This is done in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. There are many examples of such spaces. A new one given here is the union of a line (with the one-dimensional Lebesgue measure) and a triangle (with a two-dimensional weighted Lebesgue measure). Our results also apply to Cheeger p-superharmonic functions and in the Euclidean setting to A-superharmonic functions, with the usual assumptions on A.

  • 15.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Characterizations of p-superharmonic functions on metrid spaces2004Report (Other academic)
  • 16.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Cluster sets for p-harmonic functions, quasiminimizers and Sobolev functions2007In: Potential Theory and Applications,2007, 2007Conference paper (Other academic)
  • 17.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Cluster sets for Sobolev functions and quasiminimizers2010In: Journal d'Analyse Mathématique, ISSN 0021-7670, Vol. 112, p. 49-77Article in journal (Refereed)
    Abstract [en]

    In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set has boundary values f in Sobolev sense and f |∂is continuous at x0 ∈ ∂, then the essential cluster set C(u, x0, Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.

     

     

     

  • 18.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Cluster sets for Sobolev functions and quasiminimizers2008Report (Other academic)
    Abstract [en]

       

  • 19.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Correction of The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 10, p. 1756-1757Article in journal (Refereed)
    Abstract [en]

    We fill in a gap in the proofs of Theorems 1.1-1.4 in The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications, to appear in Complex Var. Elliptic Equ., doi:10.1080/17476933.2017.1410799.

  • 20.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    HbA1c according to different standards2016Other (Other (popular science, discussion, etc.))
  • 21.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    HbA1c enligt olika standarder2016Other (Other (popular science, discussion, etc.))
  • 22.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    HbA1c podle různých standardů;2016Other (Other (popular science, discussion, etc.))
  • 23.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Nonlinear potential theory associated with p-harmonic functions on metric spaces2004In: Recent Trends in Potential Theory,2004, 2004Conference paper (Other academic)
  • 24.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    p-harmonic functions on metric spaces2004In: The p-Laplace equation, the infinity-Laplace equation and related topics,2004, 2004Conference paper (Other academic)
  • 25.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    p-harmonic functions with boundary data having jump discontinuities and Baernstein´s problem2009Report (Other academic)
    Abstract [en]

    For p-harmonic functions on unweighted R-2, with 1 andlt; p andlt; infinity, we show that if the boundary values f has a jump at an (asymptotic) corner point zo, then the Perron solution Pf is asymptotically a + b arg(z - z(0)) + o(vertical bar z z(0)vertical bar). We use this to obtain a positive answer to Baernsteins problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p. harmonic measure of (G) over bar. We also obtain various invariance results for functions with jumps and perturbations on small sets. For p andgt; 2 these results are new also for continuous functions. Finally we look at generalizations to R-n and metric spaces.

  • 26.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    p-Harmonic functions with boundary data having jump discontinuities and Baernsteins problem2010In: JOURNAL OF DIFFERENTIAL EQUATIONS, ISSN 0022-0396, Vol. 249, no 1, p. 1-36Article in journal (Refereed)
    Abstract [en]

    For p-harmonic functions on unweighted R-2, with 1 andlt; p andlt; infinity, we show that if the boundary values f has a jump at an (asymptotic) corner point zo, then the Perron solution Pf is asymptotically a + b arg(z - z(0)) + o(vertical bar z z(0)vertical bar). We use this to obtain a positive answer to Baernsteins problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p. harmonic measure of (G) over bar. We also obtain various invariance results for functions with jumps and perturbations on small sets. For p andgt; 2 these results are new also for continuous functions. Finally we look at generalizations to R-n and metric spaces.

  • 27.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    p-harmonic measures and the Perron method for p-harmonic functions2003In: Future Trends in Geometric Function Theory RNC Workshop Jyväskylä 2003,2003, Jyväskylä: University of Jyväskylä, Department of Mathematics and Statistics , 2003, p. 23-Conference paper (Refereed)
  • 28.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Potential theory for quasiminimizers2008In: Nordic-Russian symposium in honour of Vladimir Mazya on the occasion of his 70th birthday,2008, 2008Conference paper (Other academic)
  • 29.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Properties of removable singularities for hardy spaces of analytic functions2002In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 66, no 3, p. 651-670Article in journal (Refereed)
    Abstract [en]

    Removable singularities for Hardy spaces Hp(O) = {f ? Hol(O) :fp = u in O for some harmonic u}, 0 < p < 8 are studied. A set E ? O is a weakly removable singularity for Hp(O\E) if Hp(O\E) ? Hol(O), and a strongly removable singularity for Hp(O\E) if Hp(O\E) = Hp(O). The two types of singularities coincide for compact E, and weak removability is independent of the domain O. The paper looks at differences between weak and strong removability, the domain dependence of strong removability, and when removability is preserved under unions. In particular, a domain O and a set E ? O that is weakly removable for all Hp, but not strongly removable for any Hp(O\E), 0 < p < 8, are found. It is easy to show that if E is weakly removable for Hp(O\E) and q > p, then E is also weakly removable for Hq(O\E). It is shown that the corresponding implication for strong removability holds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, and a comparison is made with the similar situation for weighted Bergman spaces.

  • 30.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Removable singularities for analytic functions in BMO and locally Lipschitz spaces2003In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 244, no 4, p. 805-835Article in journal (Refereed)
    Abstract [en]

    In this paper we study removable singularities for holomorphic functions. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. Dolzhenko (1963), Král (1976) and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on Ω and be holomorphic on Ω \ E, whereas we only assume that the functions belong to the function space on Ω \ E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. Kaufman (1982) obtained some results for s=0. In this paper we obtain a number of examples with certain important properties. Similar examples have earlier been obtained for Hardy Hp classes and weighted Bergman spaces, mainly by the author. Because of the similarities in these three cases, an axiomatic approach is used to obtain some results that hold in all three cases with the same proofs.

  • 31.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Removable singularities for analytic functions in Hardy spaces, BMO and locally Lipschitz spaces2003In: 3rd International Congress of the International Society for Analysis, its Applications and Computation,2001, River Edge, NJ,: World Sci. Publishing , 2003, p. 445-450Conference paper (Other academic)
  • 32.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Removable singularities for bounded p-harmonic and quasi (super) harmonic functions on metric spaces2004Report (Other academic)
  • 33.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Removable singularities for bounded p-harmonic and quasi(super)harmonic functions2006In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 31, p. 71-95Article in journal (Refereed)
  • 34.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Removable singularities for hardy spaces1998In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 35, no 1, p. 1-25Article in journal (Refereed)
    Abstract [en]

    In this paper we study removable singularities for Hardy spaces of analytic funtions on general domains. Two different definitions are given. For compact sets they turn out to be equal and moreover independent of the surrounding domain, as was proved by D. A Hejhal For non-compact sets the difference between the definitions is studied. A survey of the present knowledge is given, except for the special cases of singularities lying on curves and singularities being self-similar Cantor sets, which the author deals with in other papers. Among the results is the non-removability for Hp of sets with dimension greater than ρ. 0 < ρ < 1. Many counterexamples are provided and the Hp capacities are introduced and studied.

  • 35.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Removable singularities for H-p spaces of analytic functions, 0 < p < 12001In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 26, no 1, p. 155-174Article in journal (Refereed)
    Abstract [en]

    In this paper we study removable singularities for Hardy H-p spaces of analytic functions on general domains, mainly for 0 < p < 1. For each p < 1 we prove that there is a self-similar linear Canter set with Hausdorff dimension greater than 0.4p removable for H-p, thereby obtaining the first removable sets with positive Hausdorff dimension for 0 < p < 1. (Cf. the author's older result that a set E removable for H-P, 0 < p < 1, must satisfy dim E p.) We use this to extend some results earlier proved for 1 less than or equal to p < to 0 < p < infinity or 1/2 less than or equal to p < .

  • 36.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Removable singularities for weighted Bergman spaces2006In: Czechoslovak Mathematical Journal, ISSN 0011-4642, E-ISSN 1572-9141, Vol. 56, no 1, p. 179-227Article in journal (Refereed)
    Abstract [en]

    We develop a theory of removable singularities for the weighted Bergman space. The general theory developed is in many ways similar to the theory of removable singularities for Hardy H p spaces, BMO and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable. In the case when weak and strong removability are the same for all sets, in particular if μ is absolutely continuous with respect to the Lebesgue measure m, we are able to say more than in the general case. In this case we obtain a Dolzhenko type result saying that a countable union of compact removable singularities is removable. When dμ = wdm and w is a Muckenhoupt A p weight, 1 < p < ∞, the removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent p′ = p/(p − 1) and the dual weight w′ = w 1/(1 − p).

  • 37.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    The Baernstein problem for p-harmonic functions.2010In: American Mathematical Society Meeting #1057, 2010 Spring Southeastern Sectional Meeting Lexington, KY, March 27-28, 2010, 2010Conference paper (Other academic)
  • 38.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The Dirichlet problem for -harmonic functions on the topologists comb2015In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 279, no 1-2, p. 389-405Article in journal (Refereed)
    Abstract [en]

    In this paper we study the Perron method for solving the -harmonic Dirichlet problem on the topologists comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.

  • 39.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications2019In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 64, no 1, p. 40-63Article in journal (Refereed)
    Abstract [en]

    In this paper, boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincare inequality, but the results are new also in unweighted Euclidean spaces.

  • 40.
    Björn, Anders
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The Perron method for p-harmonic functions: new resolutivity and invariance results2012Conference paper (Refereed)
  • 41.
    Björn, Anders
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    The Perron method for p-harmonic functions: Resolutivity and invariance results2012Conference paper (Refereed)
  • 42.
    Björn, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Weak barriers in nonlinear potential theory2007Report (Other academic)
    Abstract [en]

        

  • 43.
    Björn, Anders
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Weak Barriers in Nonlinear Potential Theory2007In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 27, no 4, p. 381-387Article in journal (Refereed)
    Abstract [en]

    We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for A-harmonic functions under the usual assumptions on A, and for Cheeger p-harmonic functions in metric spaces.

  • 44.
    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)
  • 45.
    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)
  • 46.
    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.

  • 47.
    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)
  • 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.
    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.

  • 49.
    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.

  • 50.
    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.

12 1 - 50 of 90
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