Abstract:
We consider the homogenization of non-linear diffusion problems with various boundary conditions in periodically perforated domains. These problems are stated as variational inequalities defined by non-linear strictly monotone operators of second order with periodic rapidly oscillating coefficients. We establish the relevant convergence of solutions of the problems to solutions of two-scale and macroscale limiting variational inequalities. We give methods for deriving such limiting variational inequalities. In the case of potential operators, we establish relations between the limiting variational inequalities obtained and the two-scale and macroscale constrained minimization problems.
Citation:
G. V. Sandrakov, “Homogenization of variational inequalities for non-linear diffusion problems in perforated domains”, Izv. Math., 69:5 (2005), 1035–1059
This publication is cited in the following 14 articles:
Jesús Ildefonso Díaz, Alexander Vadimovich Podolskiy, Tatiana Ardolionovna Shaposhnikova, “Unexpected regionally negative solutions of the homogenization of Poisson equation with dynamic unilateral boundary conditions: critical symmetric particles”, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 118:1 (2024)
Jake Avila, “Homogenization and corrector results of elliptic problems with Signorini boundary conditions in perforated domains”, Annali di Matematica, 2024
G. V. Sandrakov, S. I. Lyashko, V. V. Semenov, “Simulation of Filtration Processes for Inhomogeneous Media and Homogenization*”, Cybern Syst Anal, 59:2 (2023), 212
Alexander A. Kovalevsky, “Nonlinear variational inequalities with variable regular bilateral constraints in variable domains”, Nonlinear Differ. Equ. Appl., 29:6 (2022)
V. V. Semenov, S. V. Denisov, G. V. Sandrakov, O. S. Kharkov, “Convergence of the Operator Extrapolation Method for Variational Inequalities in Banach Spaces*”, Cybern Syst Anal, 58:5 (2022), 740
Vedel Ya.I., Sandrakov G.V., Semenov V.V., Chabak L.M., “Convergence of a Two-Stage Proximal Algorithm For the Equilibrium Problem in Hadamard Spaces”, Cybern. Syst. Anal., 56:5 (2020), 784–792
G. V. Sandrakov, A. L. Hulianytskyi, “SOLVABILITY OF HOMOGENIZED PROBLEMS WITH CONVOLUTIONS FOR WEAKLY POROUS MEDIA”, JNAM, 2020, no. 2 (134), 59
G. V. Sandrakov, “HOMOGENIZED MODELS FOR MULTIPHASE DIFFUSION IN POROUS MEDIA”, JNAM, 2019, no. 3 (132), 43
Kovalevsky A.A., “On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains”, Nonlinear Anal.-Theory Methods Appl., 147 (2016), 63–79
Jaeger W., Neuss-Radu M., Shaposhnikova T.A., “Homogenization of a Variational Inequality for the Laplace Operator with Nonlinear Restriction for the Flux on the Interior Boundary of a Perforated Domain”, Nonlinear Anal.-Real World Appl., 15 (2014), 367–380
T. A. Mel’nyk, Iu. A. Nakvasiuk, “Homogenization of a parabolic signorini boundary value problem in a thick plane junction”, J Math Sci, 2012
Mel'nyk T.A., Nakvasiuk Iu.A., Wendland W.L., “Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem”, Math. Meth. Appl. Sci, 34:7 (2011), 758–775
Kazmerchuk Yu.A., Mel'nyk T.A., “Homogenization of the signorini boundary-value problem in a thick plane junction”, Nonlinear Oscill., 12:1 (2009), 45–59
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