Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
7.06.1966
Phone: +7 (499) 613 45 35
Fax: +7 (495) 939 20 90
E-mail: Website: https://abris.tv/grisha Keywords: homogenization of differential operators; spectral theory of differential operators; asymptotic methods; integral estimates of solutions of PDE; behavior of thin plates, rod structures and junctions; microinhomogenious media.
UDC: 517.9, 517.946, 517.95, 517.956.2, 517.956.225, 517.956.226, 517.956.6, 517.956.8, 517.98, 517.984.4, 517.984.6, 519.632.4, 517.955.8
MSC: 35b20, 35b27, 35b40, 35b45, 35c20, 35j05, 35j25, 35m10, 35p15, 74b99, 74k10, 74k15, 74k20, 74k30, 74q99, 35J25, 35B25, 39A10, 39A11, 39A70, 39B62, 41A44, 45A05
Subject:
Boundary homogenization for PDE in domains with microinhomogeneous structure. There were formulated and proved homogenization Theorems for boundary value problems with rapidly changing type of boundary conditions with periodic and nonperiodic boundary conditions. Also there were obtained the estimates of deviation of solutions of initial problem from solutions of homogenizaed problem. Vibration of thin plates, rods and junctions. Vibration of thin nonsymmetric plates with rough surface and other singular constructions were described. Weighted Korn type inequality for such domains were proved. Homogenization of random structures. We gave new definition of domains with random structure and proved homogenization theroems. Domains with oscillating boundary. We studied the behavior of bodies with the oscillation of external boundary and internal perforation. Also there were proved homogenization theorems and were constructed leading terms of the asymptotic expansion of solutions with respect to the small parameter characterized the microinhomogenity.Singular Measures and homogenization. We introduced new approach to homogenization problems in domains with thin and infinitisimally thin elements. We define the Sobolev function-spaces, proved the embedding theorems and Weyl-type theorems. Also we proved homogenization theorems for such structures.
Main publications:
The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary (jointly with A. Friedman and A. L. Piatnitski) // Journal of Mathematical Analysis and Applications (JMAA), 1999, v. 231, no. 1, p. 213–234.
Effective Membrane Permeability: Estimates and Low Concentration Asymptotics (Jointly with A. G. Belyaev and R. R. Gadylshin) // SIAM J. Appl. Math., 2000, v. 60, no. 1, p. 84–108.
