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  • 1.
    Bodin, Mats
    et al.
    Umeå University, Faculty of Science and Technology, Department of Ecology and Environmental Sciences.
    Pietrusica-Paluba, Katarzyna
    Harmonic functions representation of Besov-Lipschitz functions on nested fractals2012In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 37, no 2, p. 509-523Article in journal (Refereed)
    Abstract [en]

    R. S. Strichartz proposes a discrete definition of Besov spaces on self-similar fractals having a regular harmonic structure. In this paper, we characterize some of these Holder-Zygmund and Besov-Lipschitz functions on nested fractals by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.

  • 2.
    Lundström, Niklas L.P.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Nyström, Kaj
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    The Boundary Harnack Inequality for Solutions to Equations of Aronsson type in the Plane2011In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 36, p. 261-278Article in journal (Refereed)
    Abstract [en]

    In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain \Omega \subset R2 and which are solutions to a general equation of p-Laplace type, 1 < p < \infty. We also establish the same type of result for solutions to the Aronsson type equation \nabla (F(x,\nabla u)) \cdot F\eta(x,\nabla u) = 0. Concerning \Omega we only assume that \partial\Omega is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

  • 3.
    Lundström, Niklas L.P.
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Vasilis, Jonatan
    Decay of a p-harmonic measure in the plane2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, p. 351-366Article in journal (Refereed)
    Abstract [en]

    We study the asymptotic behaviour of a p-harmonic measure w(p), p is an element of (1, infinity], in a domain Omega subset of R-2, subject to certain regularity constraints. Our main result is that w(p) (B (w, delta) boolean AND partial derivative Omega, w(0)) approximate to delta(q) as delta -> 0(+), where q = q(v,p) is given explicitly as a function of v and p. Here, v is related to properties of Omega near w. If p = infinity, this extends to some domains in R-n. By a result due to Hirata, our result implies that the p-Green function for p is an element of (1, 2) is not quasi-symmetric in plane C-1,C-1-domains.

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  • 4.
    Sjödin, Tord
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Polar sets and capacitary potentials in homogeneous spaces2013In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, p. 771-783Article in journal (Refereed)
    Abstract [en]

    A set E in a space X is called a polar set in X, relative to a kernel k(x; y), if thereis a nonnegative measure in X such that the potential Uk(x) = ∞ precisely when x ∈ E. Polarsets have been characterized in various classical cases as G-sets (countable intersections of opensets) with capacity zero. We characterize polar sets in a homogeneous space (X; d; ) for severalclasses of kernels k(x; y), among them the Riesz -kernels and logarithmic Riesz kernels. The latercase seems to be new even in Rn.

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