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  • 1.
    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.

  • 2.
    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.

  • 3.
    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.

  • 4.
    Björn, Jana
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Maz´ya, Vladimir G.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form2000In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 12, no 1, p. 81-113Article in journal (Refereed)
    Abstract [en]

    We consider the Dirichlet problem for A-harmonic functions, i.e. the solutions of the uniformly elliptic equation div(A(x)del u(x)) = 0 in an n-dimensional domain Omega, n greater than or equal to 3. The matrix A is assumed to have bounded measurable entries. We obtain pointwise estimates for the A-harmonic functions near a boundary point. The estimates are in terms of the Wiener capacity and the so called capacitary interior diameter. They imply pointwise estimates for the A-harmonic measure of the domain Omega, which in turn lead to a sufficient condition for the Holder continuity of A-harmonic functions at a boundary point. The behaviour of A-harmonic functions at infinity and near a singular point is also studied and theorems of Phragmen-Lindelof type, in which the geometry of the boundary is taken into account, are proved. We also obtain pointwise estimates for the Green function for the operator -div(A(.)del u(.)) in a domain Omega and for the solutions of the nonhomogeneous equation -div(A(x)del u(x)) = mu with measure on the right-hand side.

  • 5.
    Kozlov, Vladimir
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    van den Berg, M.
    Gilkey, P.
    Kirsten, K.
    Heat content asymptotics for Riemannian manifolds with Zaremba boundary conditions2007In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 26, no 3, p. 225-254Article in journal (Refereed)
    Abstract [en]

    The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion are determined in terms of geometric invariants, partial information is obtained about the fourth coefficient.

  • 6.
    Luo, Gou
    et al.
    Department of Mathematics, The Ohio State University, Columbus OH, USA.
    Maz´ya, Vladimir G.
    Department of Mathematics, The Ohio State University, Columbus OH, USA.
    Weighted positivity of second order elliptic systems2007In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 27, no 3, p. 251-270Article in journal (Refereed)
    Abstract [en]

    Integral inequalities that concern the weighted positivity of a differential operator have important applications in qualitative theory of elliptic boundary value problems. Despite the power of these inequalities, however, it is far from clear which operators have this property. In this paper, we study weighted integral inequalities for general second order elliptic systems in ℝ n (n ≥ 3) and prove that, with a weight, smooth and positive homogeneous of order 2–n, the system is weighted positive only if the weight is the fundamental matrix of the system, possibly multiplied by a semi-positive definite constant matrix.

  • 7.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Mosco Type Convergence of Bilinear Forms and Weak Convergence of n-Particle Systems2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 2, p. 241-267Article in journal (Refereed)
    Abstract [en]

    It is well known that Mosco (type) convergence is a tool in order to verify weak convergence of finite dimensional distributions of sequences of stochastic processes. In the present paper we are concerned with the concept of Mosco type convergence for non-symmetric stochastic processes and, in particular, n-particle systems in order to establish relative compactness.

  • 8.
    Thim, Johan
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Two Weight Estimates for the Single Layer Potential on Lipschitz Surfaces with Small Lipschitz Constant2015In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 1, p. 79-95Article in journal (Refereed)
    Abstract [en]

    This article considers two weight estimates for the single layer potential - corresponding to the Laplace operator in R (N+1) - on Lipschitz surfaces with small Lipschitz constant. We present conditions on the weights to obtain solvability and uniqueness results in weighted Lebesgue spaces and weighted homogeneous Sobolev spaces, where the weights are assumed to be radial and doubling. In the case when the weights are additionally assumed to be differentiable almost everywhere, simplified conditions in terms of the logarithmic derivative are presented, and as an application, we prove that the operator corresponding to the single layer potential in question is an isomorphism between certain weighted spaces of the type mentioned above. Furthermore, we consider several explicit weight functions. In particular, we present results for power exponential weights which generalize known results for the case when the single layer potential is reduced to a Riesz potential, which is the case when the Lipschitz surface is given by a hyperplane.

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