Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail: Keywords: nonlinear elliptic equations and variational inequalities; homogenization of boundary value problems in variable domains; G-convergence of nonlinear operators; Г-convergence of integral functionals; existence and properties of solutions of nonlinear equations with L1-data; regularity of solutions of degenerate nonlinear high-order equations.
Subject:
Necessary and sufficient conditions for Г-convergence of integral functionals with varying domain of definition were established and theorems on Г-compactness for these functionals were proved. New results on $G$-compactness of sequences of nonlinear elliptic operators (including high-order operators) corresponding to Dirichlet and Neumann problems in variable domains were obtained. G-convergence of nonlinear operators of Neumann problems in domains of framework-type periodic structure with thin channels was studied and representations for coefficients of the G-limit operator were obtained. The asymptotic behaviour of solutions of Neumann problems for nonlinear elliptic equations in three-dimensional domains with periodically allocated simple and double accumulators was investigated. It was shown that these solutions converge in a certain sense to a solution of a problem for a system of some functional equations and one differential equation. An effect of double homogenization was first established in regard to Dirichlet problems for nonlinear elliptic second-order equations with coefficients depended on a parameter in variable domains of general structure. A notion of entropy solution of Dirichlet problem for some classes of nonlinear elliptic high-order equations with L1-data was introduced and results on existence, uniqueness and summability of such a kind of solutions were proved. New results on summability of solutions of nonlinear elliptic second-order equations with right-hand sides in logarithmic classes of functions were established.
Main publications:
Kovalevskii A.A. G-convergence and homogenization of nonlinear elliptic operators in divergence form with variable domain // Russ. Acad.Sci. Izv. Math., 1995, 44(3), 431–460.
Kovalevsky A. An effect of double homogenization for Dirichlet problems in variable domains of general structure // Comptes Rendus Acad. Sci. Paris, Ser. I, 1999, 328(12), 1151–1156.
Kovalevskii A.A. Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in L1 // Izv. Math., 2001, 65(2), 231–283.
Kovalevsky A.A. Integrability and boundedness of solutions to some anisotropic problems // J. Math. Anal. Appl. 2015, 432(2), 820–843.
Kovalevsky A.A., Skrypnik I.I., Shishkov A.E. Singular Solutions of Nonlinear Elliptic and Parabolic Equations. Berlin: De Gruyter, 2016. 436 p.
Kovalevsky A.A. On the convergence of solutions to bilateral problems with the zero lower constraint and an arbitrary upper constraint in variable domains // Nonlinear Anal., 2016, 147, 63–79.
Kovalevsky A.A. Variational problems with variable regular bilateral constraints in variable domains // Rev. Mat. Complut., 2019, 32(2), 327–351.
Kovalevsky A.A. On the convergence of solutions of variational problems with variable implicit pointwise constraints in variable domains // Ann. Mat. Pura Appl., 2019, 198(4), 1087–1119.
