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  • 1.
    Pankratova, Iryna
    et al.
    Narvik University College, Narvik, Norway; Ecole Polytechnique, Palaiseau Cedex, France.
    Piatnitski, A.
    Lebedev Physical Institute RAS, Moscow, Russia.
    Homogenization of spectral problem for locally periodic elliptic operators with sign-changing density function2011In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 7, p. 3088-3134Article in journal (Refereed)
    Abstract [en]

    The paper deals with homogenization of a spectral problem for a second order self-adjoint elliptic operator stated in a thin cylinder with homogeneous Neumann boundary condition on the lateral boundary and Dirichlet condition on the bases of the cylinder. We assume that the operator coefficients and the spectral density function are locally periodic in the axial direction of the cylinder, and that the spectral density function changes sign. We show that the behavior of the spectrum depends essentially on whether the average of the density function is zero or not. In both cases we construct the effective 1-dimensional spectral problem and prove the convergence of spectra.

  • 2.
    Pettersson, Irina
    et al.
    The Arctic University of Norway, UiT, Norway.
    Piatnitski, Andrey
    The Arctic University of Norway, UiT, Norway; Institute for Information Transmission Problems of Russian Academy of Sciences, Russian Federation.
    Stationary convection-diffusion equation in an infinite cylinder2018In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 264, no 7, p. 4456-4487Article in journal (Refereed)
    Abstract [en]

    We study the existence and uniqueness of a solution to a linear stationary convection–diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.

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