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  • 1.
    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.

  • 2.
    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
    University of Cincinnati, OH 45221 USA.
    The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities2015In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 7, p. 3078-3114Article in journal (Refereed)
    Abstract [en]

    In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples. (C) 2015 Elsevier Inc. All rights reserved.

  • 3.
    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.

  • 4.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On bounded solutions of the Emden-Fowler equation in a semi-cylinder2002In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 179, no 2, p. 456-478Article in journal (Refereed)
    Abstract [en]

    Bounded solutions of the Emden-Fowler equation in a semi-cylinder are considered. For small solutions the asymptotic representations at infinity are derived. It is shown that there are large solutions whose behavior at infinity is different. These solutions are constructed when some inequalities between the dimension of the cylinder and the homogeneity of the nonlinear term are fulfilled. If these inequalities are not satisfied then it is proved, for the Dirichlet problem, that all bounded solutions tend to zero and have the same asymptotics as small solutions. © 2002 Elsevier Science (USA).

  • 5.
    Kozlov, Vladimir
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On the Hadamard formula for nonsmooth domains2006In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 230, no 2, p. 532-555Article in journal (Refereed)
    Abstract [en]

    We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1, 1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications. © 2006 Elsevier Inc. All rights reserved.

  • 6.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Kuznetsov, Nikolay
    Russian Academy of Science, St. Petersburg.
    Bounds for steady water waves with vorticity2012In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 252, no 1, p. 663-691Article in journal (Refereed)
    Abstract [en]

    The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Bounds on the free-surface profiles and on the total head are obtained under minimal assumptions about properties of solutions to the problem and the vorticity distribution.

  • 7.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Lokharu, E.
    Lund Univ, Sweden.
    Small-amplitude steady water waves with critical layers: Non-symmetric waves2019In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 267, no 7, p. 4170-4191Article in journal (Refereed)
    Abstract [en]

    The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. The reduced system describes all small-amplitude solutions of the problem and, as an application, we give a proof of the existence of non-symmetric steady water waves whenever the number of roots of the dispersion equation is greater than one. (C) 2019 Published by Elsevier Inc.

    The full text will be freely available from 2021-05-08 07:58
  • 8.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Rossmann, Juergen
    University of Rostock, Germany.
    On the nonstationary Stokes system in a cone2016In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 260, no 12, p. 8277-8315Article in journal (Refereed)
    Abstract [en]

    The authors consider the Dirichlet problem for the nonstationary Stokes system in a threedimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions. (C) 2016 Elsevier Inc. All rights reserved.

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