Vassily Vassilievich Zhikov, an outstanding mathematician passed away on 12 February 2017. The present special issue of Applicable Analysis is dedicated to him. V.V. Zhikov was born on 14 August 1940 in Novocherkassk (USSR). Graduated from Moscow State University M.V. Lomonosov in 1963, V.V. Zhikov defended his PhD thesis in 1966 and in 1975 the Doctor es Science thesis. Since 1966 he worked as an associate professor and since 1978 as a professor of Vladimir State University. Since 2000 he became a professor of Moscow State University M.V. Lomonosov, keeping as main position his professorship at Vladimir State University. His scientific interests were focused at partial differential equations, convex analysis, homogenization, functional analysis. In particular, he contributed to the theory of almost periodic functions, proved a non-stationary compensated compactness lemma, studied the G-convergence of solutions for elliptic and parabolic equations of arbitrary order, contributed to percolation theory and theory of equations with random coefficients, generalized the homogenization theory for the partial differential equations with Lavrentiev's effect, invented an original method of detection of lacunae in the spectrum of elliptic operators with periodic coefficients and with important applications to photonic crystals. He supervised 15 PhD students and some of his former students contribute in this special issue.
The special issue is a collection of papers dealing with the homogenization and qualitative theory of differential equations, close to the functional analysis, the fields where V.V. Zhikov obtained fundamental results. The first part of the special issue studies some new problems of the homogenization theory. The paper by T.A. Suslina is devoted to the homogenization of higher order parabolic problems via the resolvent analysis, developed earlier by M.Sh. Birman and T.A. Suslina. The spectral analysis for the elasticity equations with strongly oscillating boundary is considered in the paper by A.G. Chechkina, C. d’Apice and U. De Maio. The paper by A. Braides and V. Chiado-Piat considers the homogenization of one-dimensional networks of thin subdomains as a part of a three-dimensional domain. In the article by I.V. Kamotski and V.P. Smyshlyaev the two scale homogenization is applied to partial differential equations modeling high contrast heterogeneous media. The stochastic homogenization of high contrast heterogeneous medium is developed in the paper by M. Cherdantsev, K. Cherednichenko and I. Velcic. In the paper by G. Cardone and J.L. Wouekeng the homogenization corrector to a nonlinear elliptic equation is constructed and justified. An asymptotic analysis of a thin poroelastic membrane is provided in the paper by A. Mikelic and J. Tambaca. A shell limit model is derived. In the paper by G. Panasenko and R. Stavre the limit boundary conditions for a thin stiff cylindrical heterogeneous elastic tube filled with a fluid are derived by means of asymptotic expansions with respect to the small parameter which is the ratio of the thickness of the wall and the diameter of the tube. In the paper by A. Piatnitski, S. Pirogov and E. Zhizhina the long time behavior of a high contrast media is studied via Markov semigroups analysis. The paper by J.I. Diaz, D. Gomez-Castro, T.A. Shaposhnikova and M.N. Zubova is devoted to the homogenization of diffusion equation with the Robin-type condition at the boundary of small particles of critical size. The convergence of attractors for diffusion-reaction equations with random coefficients is studied in the paper by K.A. Bekmaganbetov, G.A. Chechkin and V.V. Chepyzhov.
The second part of the issue is devoted to the general qualitative theory of PDE. In the paper by P. Aceveda, C. Amrouche and C. Conca new results on the existence and uniqueness of a solution to the Boussinesq equations (the Navier–Stokes equations coupled with a diffusion equation) are obtained. The solvability and properties of Navier–Stokes equations with the slip boundary condition are studied in the paper by J.E.C. Lope, R. Sato and B. Vernescu. The regularity of solutions to nonlinear parabolic problems with variable nonlinearity is considered in the paper by S. Antontsev and S. Shmarev. The Harnac inequality for a transmission problem for p(x)-Laplacian is the topic of the paper by Yu.A. Alkhutov and M.D. Surnachev. Galerkin method for anisotropic p(x)-Laplacian is analyzed in the paper by S.E. Pastukhova and D.A. Yakubovich. Article by S.V. Bankevich and A.I. Nazarov studies monotonicity of functionals with variable exponent. Sharp A-harmonic approximation of Sobolev functions is studied in the paper by D. Cruz-Uribe and L. Diening. The Bohr-Neugebauer property for almost automorphic partial functional differential equations is the topic of the paper by B. Es-Sebbara, Kh. Ezzinbia and G.M. N’Guerekata. A new effective method for solution of unsteady Schrodinger equation in the case of big dimension is presented in the article by F. Lanzara, V.G. Maz’ya and G. Schmidt. Singular weighted Sobolev spaces in relation to the diffusion equation are considered in the paper by A. Chiarini and P. Mathieu. The paper by V. Barrera-Figueroa, VS. Rabinovich and M. Maldonado-Rosas is devoted to numerical estimates of spectrum of quantum graphs with delta-interaction at vertices.