Noncommutative elliptic theory
Construction of a noncommutative elliptic theory for a new class of operators associated with the representation of a group by quantized canonical transformations on various manifolds. New operators are presented in three forms: operators on closed manifolds, relative (non-commutative) elliptic theory, and non-commutative elliptic theory on manifolds with singularities. In all cases, the Fredholmness property of the studied operators is studied and the index formulas for them are presented. The results of the project have applications in the theory of inverse problems for hyperbolic equations, the theory of dynamical systems, and also in noncommutative geometry.
Linear and nonlinear differential and functional differential equations and their applications
Development of new qualitative and geometric methods for studying boundary value problems for differential and functional differential equations, their application to the Vlasov equations (kinetics of high-temperature plasma), Kato's square root problem of the operator, mathematical biology and mathematical medicine. As part of the study, algebraic conditions will be obtained for the unique solvability of the Dirichlet problem for a functional-differential equation with incommensurable contractions, as well as a continuous dependence of the solution on the compression parameters. It is planned to obtain formulas for the volume of a hyperbolic 5-simplex through the coordinates of vertices and edge lengths, as well as new classes of stationary solutions of the Vlasov-Poisson equations.
Elliptic functional differential equations in bounded and unbounded domains
Boundary value problems for elliptic functional differential equations in bounded domains and half-space, as well as elliptic functional differential equations in the whole space R ^ n are investigated. The Dirichlet problem in a half-space for strongly elliptic differential-difference equations is investigated, both necessary and sufficient conditions for the coercivity of differential-difference equations with incommensurable shifts of arguments in bounded domains are obtained, and applications of these problems to the Kato root square problem of the operator are considered. It is planned to prove the Kato’s conjecture for elliptic differential-difference operators with degeneration and to investigate the solvability of parabolic problems for differential-difference equations in weighted spaces. Field of study: the theory of boundary value problems for elliptic differential-difference equations with argument shifts in terms with higher exponent has important applications in the theory of multilayer plates and shells, the theory of multidimensional laser systems, and control theory.
Non-classical variational and boundary value problems and their applications
To study linear elliptic differential-difference equations, symmetrized matrices corresponding to difference operators are used, and the skew-symmetric component does not violate the strong ellipticity of the linear operator and the smoothness property of generalized solutions. Previously, criteria for the solvability of nonlinear elliptic differential-difference equations were proposed, in which the difference operators are described by symmetric matrices. It was shown that, in contrast to the linear case for nonlinear problems, the skew-symmetric part affects the ellipticity. In this project, it is proposed to use the previously developed methods to study nonlinear elliptic problems with difference operators, which correspond to triangular matrices. Linear elliptic differential-difference equations with degeneration with variable coefficients are also considered. For these equations, a connection with nonlocal elliptic boundary value problems of type, and the question of the existence of traces of generalized solutions on manifolds generated by shifts of the boundary inside the domain will be established and investigated. In addition, the question of the spectral stability of the Schrödinger operator is considered. Variational-Hamiltonian methods for studying the qualitative properties of the motion of infinite-dimensional dynamical systems will be developed and applied. It is planned to construct Hamilton action functionals for the equations of motion of nonpotential systems, including differential and functional differential equations, and also to find the first integrals of these equations using approaches based on the application of variable transformation theory to study the invariance of the equations of motion and their corresponding functionals.