Zhikov Vasilii Vasil'evich

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Total publications: 97
Scientific articles: 91
Presentations: 4

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This page:9733
Abstract pages:34474
Full texts:10229
Doctor of physico-mathematical sciences (1975)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 14.08.1940
Keywords: almost-periodic functions, elliptic and parabolic equations, homogenization, Lavrentiev phenomenon, Sobolev spaces, singular structures, nonstandard growth conditions, relaxation functional.


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.


Graduated from Faculty of Mathematics and Mechanics of M. V. Lomonosov Moscow State University (MSU) in 1963 (department of theory of functions and functional analysis). Ph. D. thesis was defended in 1970. D. Sci. thesis was defended in 1975. Since 1978 I have led the reseach seminar at VGPU on Partial differential equations. Since 2000 I have led the reseach seminar at MSU on Homogenization.

Main publications:
  1. Levitan B. M., Zhikov V. V., Pochti-periodicheskie funktsii i differentsialnye uravneniya, Izd. MGU, M., 1978  mathscinet  zmath
  2. Zhikov V. V., Kozlov S. M., Oleinik O. A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993  mathscinet  zmath
  3. Jikov V. V., Kozlov S. M., Oleinik O. A., Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994  mathscinet
  4. Zhikov V. V., “Svyaznost i usrednenie. Primery fraktalnoi provodimosti”, Matem. sb., 187:8 (1996), 3–40  mathnet  mathscinet  zmath
  5. Zhikov V. V., “Usrednenie zadach teorii uprugosti na singulyarnykh strukturakh”, Izvestiya RAN, ser. matem., 66:2 (2002), 81–148  mathnet  mathscinet  zmath
List of publications on Google Scholar
List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. Homogenization and two-scale convergence in Sobolev space with oscillating exponent
V. V. Zhikov, S. E. Pastukhova
Algebra i Analiz, 30:2 (2018),  114–144
2. Large time asymptotics of fundamental solution for the diffusion equation in periodic medium and its application to estimates in the theory of averaging
V. V. Zhikov, S. E. Pastukhova
CMFD, 63:2 (2017),  223–246
3. On the convergence of bloch eigenfunctions in homogenization problems
V. V. Zhikov, S. E. Pastukhova
Funktsional. Anal. i Prilozhen., 50:3 (2016),  47–65
4. Operator estimates in homogenization theory
V. V. Zhikov, S. E. Pastukhova
Uspekhi Mat. Nauk, 71:3(429) (2016),  27–122
5. О плотности гладких функций в весовых соболевских пространствах с переменным показателем
V. V. Zhikov, M. D. Surnachev
Algebra i Analiz, 27:3 (2015),  95–124
6. On integral representation of $\Gamma$-limit functionals
V. V. Zhikov, S. E. Pastukhova
Fundam. Prikl. Mat., 19:4 (2014),  101–120
7. Uniform convexity and variational convergence
V. V. Zhikov, S. E. Pastukhova
Tr. Mosk. Mat. Obs., 75:2 (2014),  245–276
8. The $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions
V. V. Zhikov, S. E. Pastukhova
Mat. Sb., 205:4 (2014),  33–68
9. Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent
Yu. A. Alkhutov, V. V. Zhikov
Mat. Sb., 205:3 (2014),  3–14
10. Introduction to the theory of two-scale convergence
V. V. Zhikov, G. A. Yosifian
Tr. Semim. im. I. G. Petrovskogo, 29 (2013),  281–332
11. On the Navier–Stokes equations: Existence theorems and energy equalities
V. V. Zhikov, S. E. Pastukhova
Tr. Mat. Inst. Steklova, 278 (2012),  75–95
12. Estimates of the Nash–Aronson type for degenerating parabolic equations
V. V. Zhikov
CMFD, 39 (2011),  66–78
13. Homogenization of Monotone Operators Under Conditions of Coercitivity and Growth of Variable Order
V. V. Zhikov, S. E. Pastukhova
Mat. Zametki, 90:1 (2011),  53–69
14. Hölder continuity of solutions of parabolic equations with variable nonlinearity exponent
Yu. A. Alkhutov, V. V. Zhikov
Tr. Semim. im. I. G. Petrovskogo, 28 (2011),  8–74
15. On the Property of Higher Integrability for Parabolic Systems of Variable Order of Nonlinearity
V. V. Zhikov, S. E. Pastukhova
Mat. Zametki, 87:2 (2010),  179–200
16. Lemmas on compensated compactness in elliptic and parabolic equations
V. V. Zhikov, S. E. Pastukhova
Tr. Mat. Inst. Steklova, 270 (2010),  110–137
17. Existence theorems for solutions of parabolic equations with variable order of nonlinearity
Yu. A. Alkhutov, V. V. Zhikov
Tr. Mat. Inst. Steklova, 270 (2010),  21–32
18. New Approach to the Solvability of Generalized Navier–Stokes Equations
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 43:3 (2009),  33–53
19. On the Technique for Passing to the Limit in Nonlinear Elliptic Equations
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 43:2 (2009),  19–38
20. On passage to the limit in nonlinear elliptic equations
V. V. Zhikov
Dokl. Akad. Nauk, 420:3 (2008),  300–305
21. Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent
V. V. Zhikov, S. E. Pastukhova
Mat. Sb., 199:12 (2008),  19–52
22. Homogenization of degenerate elliptic equations
V. V. Zhikov, S. E. Pastukhova
Sibirsk. Mat. Zh., 49:1 (2008),  101–124
23. Solvability of the Three-Dimensional Thermistor Problem
V. V. Zhikov
Tr. Mat. Inst. Steklova, 261 (2008),  101–114
24. On the Trotter–Kato Theorem in a Variable Space
V. V. Zhikov, S. E. Pastukhova
Funktsional. Anal. i Prilozhen., 41:4 (2007),  22–29
25. Diffusion in an incompressible random flow. Estimates of Nash-Aronson type for transition probabilities, and the central limit theorem
V. V. Zhikov
Dokl. Akad. Nauk, 407:4 (2006),  439–442
26. Homogenization of random singular structures and random measures
V. V. Zhikov, A. L. Piatnitski
Izv. RAN. Ser. Mat., 70:1 (2006),  23–74
27. Estimates of the Nash–Aronson type for the diffusion equation with non-symmetric matrix and their application to homogenization
V. V. Zhikov
Mat. Sb., 197:12 (2006),  65–94
28. Derivation of the limit equations of elasticity theory on thin nets
V. V. Zhikov, S. E. Pastukhova
Tr. Semim. im. I. G. Petrovskogo, 25 (2006),  55–97
29. Homogenized Tensor on Networks
V. V. Zhikov, S. E. Pastukhova
Tr. Mat. Inst. Steklova, 250 (2005),  105–111
30. Spectral Method in Homogenization Theory
V. V. Zhikov
Tr. Mat. Inst. Steklova, 250 (2005),  95–104
31. On gaps in the spectrum of some divergence elliptic operators with periodic coefficients
V. V. Zhikov
Algebra i Analiz, 16:5 (2004),  34–58
32. Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 38:3 (2004),  15–28
33. On density of smooth functions in Sobolev–Orlich spaces
V. V. Zhikov
Zap. Nauchn. Sem. POMI, 310 (2004),  67–81
34. Homogenization for elasticity problems on periodic networks of critical thickness
V. V. Zhikov, S. E. Pastukhova
Mat. Sb., 194:5 (2003),  61–96
35. Homogenization of elasticity problems on singular structures
V. V. Zhikov
Izv. RAN. Ser. Mat., 66:2 (2002),  81–148
36. To the Problem of Passage to the Limit in Divergent Nonuniformly Elliptic Equations
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 35:1 (2001),  23–39
37. On an extension of the method of two-scale convergence and its applications
V. V. Zhikov
Mat. Sb., 191:7 (2000),  31–72
38. On the Homogenization Technique for Variational Problems
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 33:1 (1999),  14–29
39. The leading term of the spectral asymptotics for the Kohn–Laplace operator in a bounded domain
Yu. A. Alkhutov, V. V. Zhikov
Mat. Zametki, 64:4 (1998),  493–505
40. Weighted Sobolev spaces
V. V. Zhikov
Mat. Sb., 189:8 (1998),  27–58
41. Meyer-type estimates for solving the nonlinear Stokes system
V. V. Zhikov
Differ. Uravn., 33:1 (1997),  107–114
42. Diffusion in an Incompressible Random Flow
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 31:3 (1997),  10–22
43. Homogenization of non-linear second-order elliptic equations in perforated domains
V. V. Zhikov, M. E. Rychago
Izv. RAN. Ser. Mat., 61:1 (1997),  69–88
44. Connectedness and homogenization. Examples of fractal conductivity
V. V. Zhikov
Mat. Sb., 187:8 (1996),  3–40
45. Asymptotic problems related to a second-order parabolic equation in nondivergence form with randomly homogeneous coefficients
V. V. Zhikov
Differ. Uravn., 29:5 (1993),  859–869
46. The limit load and homogenization
O. O. Barabanov, V. V. Zhikov
Izv. RAN. Ser. Mat., 57:5 (1993),  15–43
47. Averaging in perforated random domains of general type
V. V. Zhikov
Mat. Zametki, 53:1 (1993),  41–58
48. Threshold of conductivity for a random cubic structure
V. V. Zhikov
Mat. Zametki, 52:6 (1992),  15–24
49. On passage to the limit in nonlinear variational problems
V. V. Zhikov
Mat. Sb., 183:8 (1992),  47–84
50. The Lavrent'ev effect and averaging of nonlinear variational problems
V. V. Zhikov
Differ. Uravn., 27:1 (1991),  42–50
51. Estimates for the averaged matrix and the averaged tensor
V. V. Zhikov
Uspekhi Mat. Nauk, 46:3(279) (1991),  49–109
52. Problems of the continuation of functions in connection with averaging theory
V. V. Zhikov
Differ. Uravn., 26:1 (1990),  39–50
53. Asymptotic problems connected with the heat equation in perforated domains
V. V. Zhikov
Mat. Sb., 181:10 (1990),  1283–1305
54. Spectral approach to asymptotic diffusion problems
V. V. Zhikov
Differ. Uravn., 25:1 (1989),  44–50
55. Effective conductivity of random homogeneous sets
V. V. Zhikov
Mat. Zametki, 45:4 (1989),  34–45
56. Remarks on the problem of residual diffusion
V. V. Zhikov
Uspekhi Mat. Nauk, 44:6(270) (1989),  155–156
57. Averaging of a system of Beltrami equations
V. V. Zhikov, M. M. Sirazhudinov
Differ. Uravn., 24:1 (1988),  64–73
58. Averaging of functionals of the calculus of variations and elasticity theory
V. V. Zhikov
Izv. Akad. Nauk SSSR Ser. Mat., 50:4 (1986),  675–710
59. Estimates for the trace of the averaged matrix
V. V. Zhikov
Mat. Zametki, 40:2 (1986),  226–237
60. Stabilization of the solution of the Cauchy problem for parabolic equations
V. N. Denisov, V. V. Zhikov
Mat. Zametki, 37:6 (1985),  834–850
61. Questions of convergence, duality, and averaging for functionals of the calculus of variations
V. V. Zhikov
Izv. Akad. Nauk SSSR Ser. Mat., 47:5 (1983),  961–998
62. Asymptotic behavior and stabilization of solutions of a second-order parabolic equation with lowest terms
V. V. Zhikov
Tr. Mosk. Mat. Obs., 46 (1983),  69–98
63. Averaging of singularly perturbed elliptic operators
V. V. Zhikov, E. V. Krivenko
Mat. Zametki, 33:4 (1983),  571–582
64. $G$-convergence of elliptic operators
V. V. Zhikov
Mat. Zametki, 33:3 (1983),  345–356
65. Averaging of parabolic operators
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik
Tr. Mosk. Mat. Obs., 45 (1982),  182–236
66. Averaging of parabolic operators with almost periodic coefficients
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik
Mat. Sb. (N.S.), 117(159):1 (1982),  69–85
67. On $G$-compactness of a class of nondivergence elliptic operators of second order
V. V. Zhikov, M. M. Sirazhudinov
Izv. Akad. Nauk SSSR Ser. Mat., 45:4 (1981),  718–733
68. $G$-convergence of parabolic operators
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik
Uspekhi Mat. Nauk, 36:1(217) (1981),  11–58
69. The averaging of nondivergence second order elliptic and parabolic operators and the stabilization of solutions of the Cauchy problem
V. V. Zhikov, M. M. Sirazhudinov
Mat. Sb. (N.S.), 116(158):2(10) (1981),  166–186
70. Averaging and $G$-convergence of differential operators
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Hà Tiên Ngoan
Uspekhi Mat. Nauk, 34:5(209) (1979),  65–133
71. A point stabilization criterion for second order parabolic equations with almost periodic coefficients
V. V. Zhikov
Mat. Sb. (N.S.), 110(152):2(10) (1979),  304–318
72. Proof of the Favard theorem on the existence of almost-periodic solution for an arbitrary Banach space
V. V. Zhikov
Mat. Zametki, 23:1 (1978),  121–126
73. Favard theory
V. V. Zhikov, B. M. Levitan
Uspekhi Mat. Nauk, 32:2(194) (1977),  123–171
74. On the stabilization of solutions of parabolic equations
V. V. Zhikov
Mat. Sb. (N.S.), 104(146):4(12) (1977),  597–616
75. Some admissibility and dichotomy questions. The averaging principle
V. V. Zhikov
Izv. Akad. Nauk SSSR Ser. Mat., 40:6 (1976),  1380–1408
76. The invertibility of the operator $d/dt+A(t)$ in the space of bounded functions
V. V. Zhikov, V. M. Tyurin
Mat. Zametki, 19:1 (1976),  99–104
77. Solvability of linear equations in the Besicovitch and Bohr classes of almost periodic functions
V. V. Zhikov
Mat. Zametki, 18:4 (1975),  553–560
78. Some new results in abstract Favar theory
V. V. Zhikov
Mat. Zametki, 17:1 (1975),  33–40
79. The averaging principle for parabolic equations with variable principal term
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 208:1 (1973),  32–35
80. Monotonicity in the theory of almost periodic solutions of nonlinear operator equations
V. V. Zhikov
Mat. Sb. (N.S.), 90(132):2 (1973),  214–228
81. On the theory of admissibility of pairs of function spaces
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 205:6 (1972),  1281–1283
82. The stability and instability of Levinson's center
V. V. Zhikov
Differ. Uravn., 8:12 (1972),  2167–2170
83. Certain functional methods in the theory of almost periodic solutions. I
V. V. Zhikov
Differ. Uravn., 7:2 (1971),  215–225
84. Remarks on compactness conditions related to the work of M. I. Kadets on integration of abstract almost-periodic functions
V. V. Zhikov
Funktsional. Anal. i Prilozhen., 5:1 (1971),  30–36
85. The existence of Levitan almost-periodic solutions of linear systems (second complement to Favard's classical theory)
V. V. Zhikov
Mat. Zametki, 9:4 (1971),  409–414
86. Almost periodic solutions of linear and nonlinear equations in a Banach space
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 195:2 (1970),  278–281
87. Addition to the classical theory of Favard
V. V. Zhikov
Mat. Zametki, 7:2 (1970),  239–246
88. On a problem of Bochner and von Neumann
V. V. Zhikov
Mat. Zametki, 3:5 (1968),  529–538
89. On inverse Sturm–Liouville problems on a finite segment
V. V. Zhikov
Izv. Akad. Nauk SSSR Ser. Mat., 31:5 (1967),  965–976
90. Conditions for discreteness and finiteness of the negative spectrum of Schrödinger's operator equation
M. G. Gasymov, V. V. Zhikov, B. M. Levitan
Mat. Zametki, 2:5 (1967),  531–538
91. On the question of harmonic analysis of bounded solutions of operator equations
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 169:6 (1966),  1254–1257
92. Almost periodic solutions of differential equations in Hilbert space
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 165:6 (1965),  1227–1230
93. Abstract equations with almost-periodic coefficients
V. V. Zhikov
Dokl. Akad. Nauk SSSR, 163:3 (1965),  555–558

94. To the memory of Boris Moiseevich Levitan (on the 100th anniversary of his birth)
V. V. Zhikov, A. A. Shkalikov
Uspekhi Mat. Nauk, 71:3(429) (2016),  207–209
95. In memory of Boris Moiseevich Levitan (1914–2004)
V. V. Zhikov, A. A. Shkalikov
Tr. Mosk. Mat. Obs., 75:2 (2014),  105–106
96. Vladimir Alexandrovich Kondratiev. July 2, 1935 – March 11, 2010
M. S. Agranovich, I. V. Astashova, L. A. Bagirov, V. V. Vlasov, V. V. Zhikov, Yu. S. Ilyashenko, V. V. Kozlov, A. A. Kon'kov, S. I. Pokhozhaev, E. V. Radkevich, N. Kh. Rozov, I. N. Sergeev, A. L. Skubachevskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova
CMFD, 39 (2011),  5–10
97. Olga Arsenjevna Oleinik
I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. Yu. Goritskii, N. V. Denisova, V. V. Zhikov, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, E. V. Radkevich, O. S. Rozanova, È. R. Rozendorn, N. Kh. Rozov, M. S. Romanov, I. N. Sergeev, I. V. Filimonova, A. V. Filinovskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova
Tr. Semim. im. I. G. Petrovskogo, 28 (2011),  5–7
98. Ol'ga Arsen'evna Oleinik (obituary)
T. D. Venttsel', V. S. Vladimirov, V. V. Zhikov, A. M. Il'in, V. A. Il'in, V. A. Kondrat'ev, L. D. Kudryavtsev, E. F. Mishchenko, S. M. Nikol'skii, Yu. S. Osipov, E. V. Radkevich, N. Kh. Rozov, V. A. Sadovnichii, L. D. Faddeev, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, A. A. Shkalikov
Uspekhi Mat. Nauk, 58:1(349) (2003),  165–174
99. Sergei Mikhailovich Kozlov (obituary)
N. S. Bakhvalov, A. Yu. Belyaev, M. I. Vishik, V. V. Zhikov, V. P. Maslov, O. A. Oleinik, G. P. Panasenko, A. L. Piatnitski
Uspekhi Mat. Nauk, 51:4(310) (1996),  145–146

Presentations in Math-Net.Ru
1. Spectral problems in some models of photonic crystals
V. V. Zhikov
International Conference on Differential Equations and Dynamical Systems
July 9, 2014 10:30
2. Соболевские пространства с переменным показателем, эффект Лаврентьева, неклассические эллиптические задачи
V. V. Zhikov
Seminar on Theory of Functions of Several Real Variables and Its Applications to Problems of Mathematical Physics
November 2, 2011 16:00
3. On an approach to solvability of generalized Navier–Stokes equations
V. V. Zhikov
Meetings of the Moscow Mathematical Society
December 9, 2008
4. Вариационные задачи и эллиптические уравнения с нестандартными условиями роста
V. V. Zhikov
Meetings of the Moscow Mathematical Society
April 18, 2006

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