All research projects
Functional-analytical methods for the study of boundary value problems for differential and functional differential equations with partial derivatives
The main idea of the project: research of new classes of differential and functional-differential equations, inequalities and systems and the application of the obtained results to interdisciplinary research in mathematical models of physical and biological processes.
Many problems that are studied in mathematical analysis can be reformulated in terms of the action of various operators in function spaces.
Construction of spherically symmetric stationary solutions of the Vlasov-Poisson system of equations describing the stationary distribution of particles in a gravitational field. Obtaining sufficient conditions for confining high-temperature plasma in a “mirror-trap” fusion reactor.
The project analyzes boundary value problems for elliptic functional-differential equations in bounded domains and half-space, as well as elliptic functional-differential equations in the entire space R^n.
The project is devoted to the development of new qualitative and geometric methods for the study of boundary value problems for differential and functional differential equations, their application to the Vlasov equations (kinetics of high-temperature plasma), the Kato problem of the square root of the operator, mathematical biology and mathematical medicine.
To study linear elliptic differential-difference equations, symmetrized matrices corresponding to difference operators are used, while the skew-symmetric component does not violate the strong ellipticity of the linear operator and the smoothness properties of generalized solutions. Previously, the solvability criteria for nonlinear elliptic differential-difference equations were proposed, in which the difference operators are described by symmetric matrices. It was shown that, unlike the linear case for nonlinear problems, the skew-symmetric part affects ellipticity. In this project, we propose to use previously developed methods to study nonlinear elliptic problems with difference operators, which correspond to triangular matrices.