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  • 151.
    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)
  • 152.
    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.

     

     

     

  • 153.
    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]

       

  • 154.
    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)
  • 155.
    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)
  • 156.
    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.

  • 157.
    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.

  • 158.
    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)
  • 159.
    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)
  • 160.
    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.

  • 161.
    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.

  • 162.
    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)
  • 163.
    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)
  • 164.
    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)
  • 165.
    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.

  • 166.
    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 < .

  • 167.
    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).

  • 168.
    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)
  • 169.
    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]

        

  • 170.
    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.

  • 171.
    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)
  • 172.
    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)
  • 173.
    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.

  • 174.
    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)
  • 175.
    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.

  • 176.
    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.

  • 177.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Applied Mathematics.
    Power-type quasiminimizers2009Report (Other academic)
  • 178.
    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

  • 179.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Applied Mathematics.
    Björn, Jana
    Linköping University, Department of Mathematics. Linköping University, Department of Mathematics, Applied Mathematics.
    Marola, Niko
    Department of Mathematics and Systems Analysis, Helsinki University of Technology.
    BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers2009Report (Other academic)
  • 180.
    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.

  • 181.
    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)
  • 182.
    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.

  • 183.
    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.

  • 184.
    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.

  • 185.
    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)
  • 186.
    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
    Sobolev extensions of Hölder continuous and characteristic functions on metric spaces2005Report (Other academic)
  • 187.
    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
    Department of Mathematical Sciences University of Cincinnati.
    Sobolev extensions of Hölder continuous and characteristic functions on metric spaces2007In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-2479, Vol. 59, no 6, p. 1135-1153Article in journal (Refereed)
    Abstract [en]

    We study when characteristic and Holder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions. ©Canadian Mathematical Society 2007.

  • 188.
    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
    The Dirichlet problem for p-harmonic functions on metric spaces2003In: Journal für die Reine und Angewandte Mathematik, ISSN 0075-4102, E-ISSN 1435-5345, no 556, p. 173-203Article in journal (Refereed)
    Abstract [en]

    We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a PoincarΘ inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces.

  • 189.
    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.
    The Perron method for p-harmonic functions in metric spaces2003In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 195, no 2, p. 398-429Article in journal (Refereed)
    Abstract [en]

    We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.

  • 190.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Marola, Niko
    Institute of Mathematics Helsinki University of Technology.
    Moser iteration for (quasi)minimizers on metric spaces2005Report (Other academic)
  • 191.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Marola, Niko
    Helsinki University of Technology.
    Moser iteration for (quasi)minimizers on metric spaces2006In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 121, no 3, p. 339-366Article in journal (Refereed)
    Abstract [en]

    We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.

  • 192.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Martio, Olli
    Department of Mathematics and Statistics University of Helsinki.
    Pasting lemmas and characterizations of boundary regularity for quasiminimizers2008Report (Other academic)
    Abstract [en]

    Quasiharmonic functions correspond to p-harmonic functions when minimizers of the p-Dirichlet integral are replaced by quasiminimizers. In this paper, boundary regularity for quasiharmonic functions is characterized in several ways; in particular it is shown that regularity is a local property of the boundary. For these characterizations we employ a version of the so called pasting lemma; this is a useful tool in the theory of superharmonic functions and our version extends the classical pasting lemma to quasiharmonic functions and quasiminimizers. The results are obtained for metric measure spaces, but they are new also in the Euclidean spaces.

  • 193.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Martio, Olli
    University Helsinki, Department Math, FI-00014 Helsinki, Finland .
    Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers2009In: RESULTS IN MATHEMATICS, ISSN 1422-6383, Vol. 55, no 3-4, p. 265-279Article in journal (Refereed)
    Abstract [en]

    Quasiharmonic functions correspond to p-harmonic functions when minimizers of the p-Dirichlet integral are replaced by quasiminimizers. In this paper, boundary regularity for quasiminimizers is characterized in several ways; in particular it is shown that regularity is a local property of the boundary. For these characterizations we employ a version of the so called pasting lemma; this is a useful tool in the theory of superharmonic functions and our version extends the classical pasting lemma to quasisuperharmonic functions and quasisuperminimizers. The results are obtained for metric measure spaces, but they are new also in the Euclidean spaces.

  • 194.
    Björn, Anders
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Riesel, Hans
    Royal Institute of Technology.
    Correction: FACTORS OF GENERALIZED FERMAT NUMBERS (vol 67, No. 221, pg 441-446, 1998)2011In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 80, no 275, p. 1865-1866Article in journal (Refereed)
    Abstract [en]

    We note that one more factor is missing from Table 1 in Bjorn-Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441 446, in addition to the three already reported upon in Bjorn-Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), p. 2099.

  • 195.
    Björn, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Riesel, Hans
    NADA Kungliga tekniska högskolan.
    Table errata: Factors of generalized Fermat numbers [Math. Comp. 67 (1998), 441--446]2005Other (Other academic)
  • 196.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Approximations by regular sets in metric spaces2004Report (Other academic)
  • 197.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions on metric spaces2004In: International Workshop on Potential Theory in Matsue 2004,2004, 2004Conference paper (Other academic)
    Abstract [en]

       

  • 198.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions on metric spaces2004In: Analysis on Metric Measure Spaces,2004, 2004Conference paper (Other academic)
    Abstract [en]

      

  • 199.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Boundary regularity for p-harmonic functions on metric spaces2004In: The p-Laplace equation, the infinity-Laplace equation and related topics,2004, 2004Conference paper (Other academic)
    Abstract [en]

      

  • 200.
    Björn, Jana
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Dirichlet problem for $p$-harmonic functions in metric spaces2003In: Future Trends in Geometric Function Theory RNC Workshop Jyväskylä 2003,2003, Jyväskylä: University of Jyväskylä, Department of Mathematics and Statistics , 2003, p. 31-Conference paper (Refereed)
1234567 151 - 200 of 721
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