A note on H-convergence

O.P. Kogut (Institute of Applied System Analysis, National Technical University KPI)
P.I. Kogut (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0003-1593-0510
T.N. Rudyanova (Dnipropetrovsk State Finance Academy), https://orcid.org/0000-0002-8685-4132


In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $$$\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$$$. We derive the sufficient conditions, which guarantee the coincidence of $$$H$$$-limit with the weak-* limit of such sequences in $$$L^{\infty} (D; \mathbb{R}^{n \times n})$$$.

Full Text:



Bucur D., Buttazzo G. "Variational Methods in Shape Optimization Problems", Progr. in Nonlinear Diff. Equations and their Appl., Birkhauser, 2005; 65.

Bucur D., Trebeschi P. "Shape optimization problem governed by nonlinear state equations", Proc. Roy. Soc. Edinburgh. Ser. A, 1998; 128: pp. 943-963.

Calvo-Jurado C., Casado-Daz J. "Optimization by the homogenization method for nonlinear elliptic Dirichlet problems", Mediterr. J. Math., 2007; 4: pp. 53-63. doi:10.1007/s00009-007-0102-5

Dal Maso G., Murat F. "Asymptotic behaviour and correctors for Dirichlet problem in perforated domains with homogeneous monotone operators", Ann. Scoula Norm. Sup. Pisa Cl.Sci., 1997; 24(4): pp. 239-290.

Gaevskii H., Greger K., Zaharias K. Nonlinear Operator Equations and Operator Differential Equations, Mir, 1978. (in Russian)

Lions J.-L. Some methods of Solving Non-Linear Boundary Value Problems, Dunod-Gauthier-Villars, 1969.

Lurie K.A. Applied Optimal Control Theory of Distributed Systems, Plenum Press, 1993.

Murat F. "Un contre-exemple pour le probleme du controle dans les coefficients", C.R.A.S. Paris, Ser. A, 1971; 273: pp. 708-711. (in French)

Murat F. "Teoremes de non-existence pour des problemes de controle dans le coefficients", C.R.A.S. Paris, Ser. A, 1972; 274: pp. 395-398. (in French)

Murat F., Tartar L. "H-convergence, Topics in the mathematical modelling of composite materials", Progr. Nonlinear Diff. Eq. Appl., 1997; 31: pp. 21-43.

Tartar L. "Estimations fines de coefficients homogeneises", Ennio de Giorgi colloq., Paris, 1983; Res. Notes in Math., Pitman, 1985; 125: pp. 168-187. (in French)

Zgurovsky M.Z., Mel'nik V.S. Nonlinear Analysis and Control of Physical Processes and Fields, Springer-Verlag, 2004. doi:10.1007/978-3-642-18770-4

Zhikov V.V., Kozlov S.M., Oleinik O.A. Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, 1994. doi:10.1007/978-3-642-84659-5


  • There are currently no refbacks.

Copyright (c) 2009 O.P. Kogut, P.I. Kogut, T.N. Rudyanova

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Registered in

ISSN (Online): 2664-5009
ISSN (Print): 2664-4991