Almost-periodic functions and solutions of differential equations, quality properties of equations in Banach spaces, spectral theory of differential operators, stabilization of solutions of parabolic equations. Homogenization of differential operators and variational functionals. Lavrentiev phenomenon. Variational problem for lagrangians with non-standard grows conditions. The concept of two-scale convergence associated with a fixed periodic Borel measure is introduced. In the case when our measure is Lebegue measure on the torus convergence in the sense of Nguetseng-Allaire is obtained. An application of two-scale convergence to the homogenization of some problems in the theory of porous media (the double-porosity model) is presented. A mathamatical notion of "softly or weakly coupled parallel flows" is worked out. A homogenized operator is constructed, and the convergence result itself is interpreted as a "strong two-scale resolvent convergence". Problems concerning the behaviour of the spectrum under homogenization are touched upon in this connection. We presented a Homogenization Theory on periodic networks, junctions and, more generally, Multi-dimensional Structures. We has shown that the Homogenized Problem has a non-classical character in most cases. This important fact is a distinctive feature of Elasticity Problems, in contrast to scalar Problems. A weighted Sobolev space is constructed in which smooth functions are not dense, and their closure is of codimension one. With the help of this weighted space, counterexamples to natural hypotheses on the passage to the limit in non-uniformly-elliptic equations and on the structure of the limit equation are constructed. We introduced a new class of weight functions (partially Muckenhoupt weights"). For the corresponding elliptic equation we proved the Holder continuity. At the same time the Harnack inequality, weight Sobolev inequality and double-condition fail. In particulary, the old problem by Fabes, Birolli, Serapioni is solved.
Main publications:
Levitan B. M., Zhikov V. V., Pochti-periodicheskie funktsii i differentsialnye uravneniya, Izd. MGU, M., 1978
Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993
Jikov V. V., Kozlov S. M., Oleinik O. A., Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994
Zhikov V. V., “Svyaznost i usrednenie. Primery fraktalnoi provodimosti”, Matem. sb., 187:8 (1996), 3–40
Zhikov V. V., “Usrednenie zadach teorii uprugosti na singulyarnykh strukturakh”, Izvestiya RAN, ser. matem., 66:2 (2002), 81–148
