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  • 1.
    Johansson, Tomas
    Linköping University, The Institute of Technology. Linköping University, Department of Science and Technology.
    An iterative procedure for solving a Cauchy problem for second order elliptic equations2004In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 272, p. 46-54Article in journal (Refereed)
    Abstract [en]

    An iterative method for reconstruction of solutions to second order elliptic equations by Cauchy data given on a part of the boundary, is presented. At each iteration step, a series of mixed well-posed boundary value problems are solved for the elliptic operator and its adjoint. The convergence proof of this method in a weighted L2 space is included. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  • 2.
    Karlsson, John
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Löbus, Jörg-Uwe
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
    Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula2016In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, no 17-18, p. 2192-2222Article in journal (Refereed)
    Abstract [en]

    The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron- Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

  • 3.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Nazarov, Alexander
    St.-Petersburg Department of Steklov Institute, St.-Petersburg, Russia; St Petersburg State University, Russia .
    The Dirichlet problem for non-divergence parabolic equations with discontinuous in time coefficients in a wedge2014In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 287, no 10, p. 1142-1165Article in journal (Refereed)
    Abstract [en]

    We consider the Dirichlet problem in a wedge for parabolic equation whose coefficients are measurable function of t. We obtain coercive estimates in weighted L-p,L-q-spaces. The concept of "critical exponent" introduced in the paper plays here the crucial role. Various important properties of the critical exponent are proved. We give applications to the Dirichlet problem for linear and quasi-linear non-divergence parabolic equations with discontinuous in time coefficients in cylinders Omega x (0, T), where Omega is a bounded domain with an edge or with a conical point.

  • 4.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Rossmann, J
    University of Rostock, Germany .
    Asymptotics of solutions of the heat equation in cones and dihedra2012In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 285, no 11-12, p. 1422-1449Article in journal (Refereed)
    Abstract [en]

    The authors deal with the asymptotics of solutions of the first boundary value problem for the heat equation near vertices of cones or edges. They obtain estimates for the remainder in weighted Lp, q Sobolev spaces. The paper generalizes the main result of Kozlov and Mazya 6 to the case \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$p,q\not=2$\end{document} and to edges.

  • 5.
    Kozlov, Vladimir
    et al.
    Linköping University, Department of Mathematics, Analysis and Mathematics Education. Linköping University, Faculty of Science & Engineering.
    Rossmann, Jurgen
    Univ Rostock, Germany.
    On the Neumann problem for the nonstationary Stokes system in angles and cones2023In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 296, no 4, p. 1504-1533Article in journal (Refereed)
    Abstract [en]

    The authors consider the Neumann problem for the nonstationary Stokes system in a two-dimensional angle or a three-dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.

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  • 6.
    Lifshits, Mikhail
    et al.
    Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. St. Petersburg State University, Russia.
    Linde, Werner
    Friedrich-Schiller-Universität Jena, Germany.
    Fractional integration operators of variable order: continuity and compactness properties2014In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 287, no 8-9, p. 980-1000Article in journal (Refereed)
    Abstract [en]

    Let alpha : [0, 1] -greater than R be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator of variable order defined by (R-alpha(.) f) (t) := 1/Gamma(alpha(t)) integral(t)(0)(t - s)(alpha(t)-1) f(s) ds, t is an element of [0, 1], as an operator from L-p[0, 1] to L-q[0, 1]. Our first aim is to study its continuity properties. For example, we show that R-alpha(.) is always bounded (continuous) in L-p[0, 1] provided that 1 less than p less than= infinity. Surprisingly, this becomes false for p = 1. In order R-alpha(.) to be bounded in L-1 [0, 1], the function alpha(.) has to satisfy some additional assumptions. In the second, central part of this paper we investigate compactness properties of R-alpha(.). We characterize functions alpha(.) for which R-alpha(.) is a compact operator and for certain classes of functions alpha(.) we provide order-optimal bounds for the dyadic entropy numbers e(n)(R-alpha(.)).

  • 7.
    Malý, Lukáš
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Calderón-type theorems for operators with non-standard endpoint behavior on Lorentz spaces2012In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 285, no 11-12, p. 1450-1465Article in journal (Refereed)
    Abstract [en]

    The Calderón theorem states that every quasilinear operator, which is bounded both from to , and from  to  for properly ordered values of , is bounded on some rearrangement-invariant space if and only if the so-called Calderón operator is bounded on the corresponding representation space. We will establish Calderón-type theorems for non-standard endpoint behavior, where Lorentz Λ and M spaces will be the endpoints of the interpolation segment. Two distinctive types of non-standard behavior are to be discussed; we’ll explore the operators bounded both from Λ(X1) to Λ(Y1), and from Λ(X2) to M(Y2) using duality arguments, thus, we need to study the operators bounded both from Λ(X1) to M(Y1), and from M(X2) to M(Y2) first. For that purpose, we evaluate Peetre's K-functional for varied pairs of Lorentz spaces.

  • 8.
    Maz´ya, Vladimir G.
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rossman, Jurgen
    Rostock University.
    Pointwise estimates for Green's kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone2005In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 278, no 15, p. 1766-1810Article in journal (Refereed)
    Abstract [en]

    The paper deals with a mixed boundary value problem for the Stokes system in a polyhedral cone. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the sides of the polyhedron. The authors obtain regularity results for weak solutions in weighted L2 Sobolev spaces and prove point estimates of Green's matrix.

  • 9.
    Maz´ya, Vladimir G.
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Rossmann, Jurgen
    University of Rostock.
    Lp estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains2007In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 280, no 7, p. 751-793Article in journal (Refereed)
    Abstract [en]

    A mixed boundary value problem for the Stokes system in a polyhedral domain is considered. Here different boundary conditions (in particular, Dirichlet, Neumann, free surface conditions) are prescribed on the faces of the polyhedron. The authors prove the existence of solutions in (weighted and non-weighted) Lp Sobolev spaces and obtain regularity assertions for weak solutions. The results are based on point estimates of Green's matrix.

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