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Networks and Heterogeneous Media

2008, Volume 3, Issue 3: 675-689. doi: 10.3934/nhm.2008.3.675
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Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain

  • 1.
    Department of Differential Equations, Faculty of Mechanics and Mathematics, Moscow State University, Moscow
  • 2.
    Department of Pure and Applied Mathematics, Russian Open State Technical University of Railway Transport, Moscow
  • Received: 01 April 2008

  • Primary: 35B27, 35K85.

  • This paper is aimed at a homogenization problem for a parabolic variational inequality with unilateral constraints. The constraints on solutions are imposed on disk-shaped subsets belonging to the boundary of the domain and forming a periodic structure, so that one has a problem with rapidly oscillating boundary conditions on a part of the boundary. Under certain conditions on the relation between the period of the structure and the radius of the disks, the homogenized problem is obtained. With the help of special auxiliary functions, the solutions of the original variational inequalities are shown to converge to the solution of the homogenized problem in Sobolev space as the period of the structure tends to zero.

    Citation: T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain[J]. Networks and Heterogeneous Media, 2008, 3(3): 675-689. doi: 10.3934/nhm.2008.3.675

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  • This paper is aimed at a homogenization problem for a parabolic variational inequality with unilateral constraints. The constraints on solutions are imposed on disk-shaped subsets belonging to the boundary of the domain and forming a periodic structure, so that one has a problem with rapidly oscillating boundary conditions on a part of the boundary. Under certain conditions on the relation between the period of the structure and the radius of the disks, the homogenized problem is obtained. With the help of special auxiliary functions, the solutions of the original variational inequalities are shown to converge to the solution of the homogenized problem in Sobolev space as the period of the structure tends to zero.


  • This article has been cited by:

    1. T. A. Mel’nik, O. A. Sivak, Asymptotic approximations for solutions to quasilinear and linear parabolic problems with different perturbed boundary conditions in perforated domains, 2011, 177, 1072-3374, 50, 10.1007/s10958-011-0447-y
    2. T. A. Mel’nik, O. A. Sivak, Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain, 2010, 164, 1072-3374, 427, 10.1007/s10958-009-9756-9
    3. D. Gómez, M. Lobo, M.E. Pérez, T.A. Shaposhnikova, Averaging of variational inequalities for the Laplacian with nonlinear restrictions along manifolds, 2013, 92, 0003-6811, 218, 10.1080/00036811.2011.602635
    4. Mariya Ptashnyk, Homogenization of some degenerate pseudoparabolic variational inequalities, 2019, 469, 0022247X, 44, 10.1016/j.jmaa.2018.08.047
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  • © 2008 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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