Year 2018

Department S.M. Nikol’skii Mathematical Institute

**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:**

- 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)
- 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.
- 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.
- 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
- 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.
- 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.
- 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.
- G.G. Onanov and A.L. Skubachevskii. Nonlocal problems in the mechanics of three-layer shells// Math. Model. Nat. Phenom. 12 (2017) 192-207.
- 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.
- 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.
- Savchin Vladimir Mikhailovich, Budochkina Svetlana Aleksandrovna, Bi-variational evolutionary systems and approximate solutions, International Journal of Advanced and Applied Sciences, 2019.

- 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.

Vladimir Filippov

- The study of inverse problems for differential equations is associated with important applications to the problems of Geophysics, Computer tomography, Financial mathematics, etc.