Year 2019

**Area of science:**

Mathematics, mechanics and informatics.

**Scientific direction: **

Partial differential equations.

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 Dirichlet problem in the half-space for strongly elliptic differential-difference equations will be studied. Both necessary and sufficient conditions for coercivity of differential-difference equations with incommensurable shifts of arguments in bounded domains were obtained, and applications of these problems to the Kato problem of the square root of the operator were analyzed. The Kato hypothesis for elliptic degenerate differential-difference operators will be proved and the solvability of parabolic problems for differential-difference equations in weight spaces will be studied. Elliptic functional-differential equations with different compressions whose logarithms are incommensurable will be analyzed, and the terms for the unique and Fredholm solvability of the boundary value problem and the continuous dependence of the solutions on the compression coefficients will be found. The solvability of functional-differential equations in the weight spaces containing one-variable compression and stretching in the higher part of the compression (such transformations are called orthotropic compressions) will be also studied.

**List of key publications on the project:**

- Skubachevskii A.L., Elliptic differential-difference operators with degeneration and the Kato square root problem, Mathematische Nachrichten, 2018, т. 291, 2660–2692.
- Selitsky A., On the solvability of parabolic functional differential equations in Banach spaces (II), Eurasian Mathematical Journal, 2018, 9(4), принято в печать.
- Muravnik A. B., On absence of global positive solutions of elliptic inequalities with KPZ-nonlinearities, Complex Variables and Elliptic Equations, 2018
- Muravnik A. B., On the half-plane Dirichlet problem for differential-difference elliptic equations with several nonlocal terms, Math. Model. Nat. Phenom., Vol. 12, No.6, 2017, С. 130-143.
- Rossovskii L., Elliptic Functional Differential Equations with Incommensurable Contractions, Math. Model. Nat. Phenom., Vol. 12, No.6, 2017, С. 226-239.
- Россовский Л. Е., К фильтрации изображений с использованием анизотропной диффузии, ЖВМиМФ, т.57, №3, 2017, С. 396–403.
- Tasevich A., Analysis of Functional-Differential Equation with Orthotropic Contractions// Math. Model. Nat. Phenom., Vol. 12, No.6, 2017, С. 240-248.
- Skubachevskii A.L., The Kato Square Root Problem for Some Class of Regular Accretive Operators// The 43th Сonference on Evolution Equations and Applications. AbstractBook. Tokyo, Japan, December 25-27, 2017. С. 92-93.

- Proof of solvability, determination of uniqueness classes, obtaining an integral representation of solutions to the Dirichlet problem in half-space for elliptic differential-difference equations and studying their qualitative properties. Obtaining explicitly necessary and sufficient terms for the implementation of Garding inequality for differential-difference equations with incommensurable shifts in a bounded domain.
- Proof of the Kato hypothesis for elliptic differential-difference operators of the second order with degeneracy in some bounded domains, as well as a study of the solvability of parabolic problems for differential-difference equations in weight spaces. Obtaining conditions for solvability of boundary value problems for different types of functional differential equations containing compression and stretching of arguments of higher derivatives of an unknown function, including equations with incommensurable compression and orthotropic compression.

Alexander Skubachevskii

- The theory of boundary value problems for elliptic differential-difference equations with argument shifts in the senior terms has important applications in the theory of multilayer plates and shells, the theory of multidimensional laser systems, and control theory.