Non-Classical variational and boundary value problems and their applications
Non-Classical variational and boundary value problems and their applications

Area of science:

  • Mathematics, mechanics and informatics.

Scientific direction:

  • Partial differential equations,
  • Mathematical physics, 
  • Computational mathematics, 
  • Theoretical mechanics.

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.

Linear elliptic differential-difference equations with degeneracy with variable coefficients will be also reviewed. For these equations, a connection with nonlocal elliptic boundary value problems of the types A. V. Bitsadze, A. A. Samara will be established, and the question of the existence of traces of generalized solutions on manifolds generated by shifts of the boundary inside the domain will be researched.

The spectral stability of the Schrodinger operator will also be reviewed. Variational-Hamiltonian methods of researching of qualitative properties of motion of infinite-dimensional dynamical systems will be developed and applied. It is planned to construct Hamiltonian action functionals for the equations of motion of non-potential systems, including differential and functional-differential equations, as well as to find the first integrals of these equations, using approaches based on the application of the theory of transformation of variables to study the invariance of the equations of motion and the corresponding functional. It is proposed to develop a theory of regularization and numerical methods for estimating the error of solving inverse problems for differential and functional differential equations with different a priori information about the desired solution.

List of key publications on the project:

  1. Solonukha O.V. On Nonlinear and Quasilinear Functional Differential Equations //Discrete and Continuous Dynamical Systems - Series S, V.9, N 3, 2016, pp. 869-893 ISSN 1937-1632 (print) 1937-1179 (online)
  2. Skubachevskii A.L. Nonlocal Elliptic Problems in Infinite Cylinder and Applications//Discrete and Continuous Dynamical Systems, Ser. S, 2016, vol. 9, № 3, pp. 847-868.
  3. Budochkina S.A., Savchin V.M. An operator equation with the second time derivative and Hamiltonian-admissible equations // Doklady Mathematics, 2016, Vol. 94, No. 2, pp. 487-489.
  4. Burenkov V. I., Goldshtein V., Ukhlov A. Conformal spectral stability estimates for the Neumann Laplacian. Mathematische Nachrichten 289 (2016), p. 1-17. DOI:10.1002/mana.201500439
  5. G.M. Kuramshina, A.G. Yagola. Applications of regularizing algorithms in structural chemistry. – Eurasian Journal of Mathematical and Computer Applications, 2017, Vol. 5, Issue 3, pp. 53-72.
  6.  A.S. Leonov, A.N. Sharov, A.G. Yagola. A posteriori error estimates for numerical solutions to inverse problems of elastography. - Inverse Problems in Science and Engineering, 2017, v. 25, issue 1, pp. 114-1287, DOI: 10.1080/17415977.2016.1138949.
  7. A.S. Leonov, A.N. Sharov and A.G. Yagola. Solution of the inverse elastography problem for parametric classes of inclusions with a posteriori error estimate. – Journal of Inverse and Ill-Posed Problems, 2017, v. 25, pp. 1-7, DOI: 10.1515/jiip-2017-0043.
  8. G.G. Onanov and A.L. Skubachevskii. Nonlocal problems in the mechanics of three-layer shells// Math. Model. Nat. Phenom. 12 (2017) 192-207.
  9. V.M. Savchin, S.A. Budochkina, Yake Gondo, A.V. Slavko, ON THE CONNECTION BETWEEN FIRST INTEGRALS, INTEGRAL INVARIANTS AND POTENTIALITY OF EVOLUTIONARY EQUATIONS, EURASIAN MATHEMATICAL JOURNAL, 2018, 9(4),  с. 82-90.
  10. Burenkov V. I., Chigambaeva D., Nursultanov E. D., Marcinkiewicz-type interpolation theorem and estimates for convolutions for Morrey-type spaces, Eurasian Mathematical Journal, 2018, 2, с. 82–88.
  11.  Savchin Vladimir Mikhailovich, Budochkina Svetlana Aleksandrovna, Bi-variational evolutionary systems and approximate solutions, International Journal of Advanced and Applied Sciences, 2019.
Project goals
  • To research the solvability of the Dirichlet problem for essentially nonlinear elliptic differential-difference equations in a bounded domain with a shift operator corresponding to triangular matrices. To determine solvability criteria for essentially nonlinear and quasi-linear problems.
  • To research the properties of generalized solutions of boundary value problems for elliptic differential-difference equations with degeneracy and variable coefficients by the method of establishing the interaction with nonlocal elliptic problems relating the values of the desired function on different compacts. To obtain necessary and sufficient terms for the existence of traces of generalized solutions on manifolds generated by boundary shifts.
  • Review of obtaining accurate evaluations of the deviation of the eigenvalues and eigenfunctions of the Schrodinger operator in perturbation of the domain of definition through various geometric characteristics of the deviation of the domains.
  • Development of methods for studying the equations of motion of infinite-dimensional potential and non-potential systems. Using Euler and non-Euler classes of functionals, the construction of corresponding Hamiltonian actions for various classes of equations of motion of infinite-dimensional systems and the questions of their representability in the form of Euler-Lagrange equations with non-potential force densities, including differential and functional-differential equations are investigated.
  • To research the invariant properties of the constructed Hamilton actions and the equations of motion themselves with respect to different groups of transformations. In this regard, we can expect to find the laws of conservation in a non-classical form (for example, in the form of integrals with convolutions, etc.).
  • The solution of the fundamental problem of computational mathematics is the development and application of the theory of extraoptimal regularizing algorithms for solving linear and nonlinear multidimensional ill-posed problems for differential equations.
Project leader All participants
Vladimir Filippov

Vladimir Filippov

Rector of the RUDN University, doctor of physics-mathematics, professor, academician of the Russian Academy of Education
Project results
Criteria for the solvability of elliptic functional differential equations in the form of terms on triangular matrices corresponding to the difference operator were formulated, provided that the differential operator is either quasilinear and satisfying algebraic conditions of strong ellipticity, or has a special form, for example, R-Laplacian form. The materiality of the obtained criteria is illustrated by counterexamples.
The interaction between the first boundary value problem for an elliptic differential-difference equation with degeneration and an elliptic differential equation with nonlocal boundary conditions relating the values of the desired function on different compacts obtained from each other by shifts to the elements of the group generated by difference operators was established.
For the class of domains described by a fixed atlas, estimates of the eigenvalue difference module of the Schrodinger operator were obtained for the first boundary value problem considered on two different domains through the atlas distance between these domains. In this case, areas with arbitrarily strong degeneration are allowed. By modifying the transition operators, estimates are obtained through the Lebesgue measure of the symmetric difference of the regions. Similar questions will be reviewed for the eigenvalues of the Schrodinger operator for the second boundary value problem, which will require a different technique compared to the first boundary value problem. The dependence of the constant in these estimates on the eigenvalue number will be also researched.
It was proved that the symmetry generators (variational and symmetry equations) form a Lie algebra with respect to the commutator of two generators. The terms under which the symmetry generators (variational and symmetry equations) form a Lie algebra with respect to the G-switch of two generators are obtained. The terms under which the symmetry generators (variational and symmetry equations) form a Lie-admissible algebra with respect to (S, T) - the product of two generators were obtained. Different formulation of the inverse problem of elastography. A software package for solving the inverse problem of elastography in a finite-parametric formulation.
Application area
  • The study of inverse problems for differential equations is associated with important applications to the problems of Geophysics, Computer tomography, Financial mathematics, etc.