Alexander Skubachevskii
Doctor of Physics and Mathematics
Professor, Scientific director,
s.M. Nikol’skii Mathematical Institute
To do science under any conditions.
1970
Entered Moscow Aviation Institute (MAI). He was a Lenin scholar. From the 3rd year of study he was actively engaged in the qualitative theory of differential equations.
1974
Solved the problem of existence of unbounded oscillating solutions of the second order functional-differential equation, which was previously formulated as an unsolved problem. Being a student, he published 3 scientific works.
1976
Graduated from Moscow Aviation Institute with honours and entered the postgraduate school of the Faculty of “Applied Mathematics”.
1979
Graduated from the postgraduate course and worked at the same Faculty as an assistant, senior lecturer, associate professor.
1980
PhD thesis on “Boundary value problems for elliptic equations with deviating arguments in senior terms” was defended.
1987
Doctoral thesis on “Nonlocal elliptic boundary value problems” was defended at the Steklov Mathematical Institute of the Academy of Sciences of the USSR, specialty “Differential Equations”.
1990
Academic title Professor was awarded.
1997
The medal “In the 850-th anniversary of Moscow” was awarded.
1988-2005
Head of the Department of Differential Equations at Moscow Aviation Institute.
2005-2015
Head of the Department of Differential Equations and Mathematical Physics of Peoples’ Friendship University of Russia (RUDN).
2010
The breastplate “For merits in development of science of the Republic of Kazakhstan” was awarded.
2012
Winner of the Second all-Russian competition of the scientific and methodological Mathematics Board of the Ministry of Education and Science of the Russian Federation “Best educational publication in Mathematics” (textbook “Nonlocal Boundary Value Problems and their Applications to the Study of Multidimensional Diffusion Processes and Thermoregulation of Living Cells”).
2013
Diploma of the Ministry of Education and Science of the Russian Federation for achievements in education and training of highly qualified personnel was awarded.
2015
A. L. Skubachevskii is an honorary worker of higher professional education of the Russian Federation.
2016
The I.G. Petrovsky prize of the Russian Academy of Sciences for a series of works “Non-classical boundary value problems” was received.
2015 - 2018
Head of the Department of Applied Mathematics, Peoples’ Friendship University of Russia (RUDN).
2018 - present time
Director of the S.M. Nikol’skii Mathematical Institute (RUDN University).
Teaching
- A. L. Skubachevskii developed the training courses - the basis for the following tutorials in the English language:
- Nonlocal Boundary Value Problems and Functional Differential Equations Part I: Boundary Value Problems for Differential-Difference Equations. Moscow, RUDN, 2013, p. 1-199.
- Nonlocal Boundary Value Problems and Functional Differential Equations Part III: Nonlocal Elliptic Boundary Value Problems. Moscow, RUDN, 2014, p. 1−241.
- From 1979 to 2005 A. L. Skubachevskii taught at the Department of Differential Equations, Moscow Aviation Institute (MAI). Read the following courses:
- “Ordinary Differential Equations” (specialty “Automatic Control Systems”),
- “Equations of Mathematical Physics” (specialty “Applied Mathematics and Informatics”),
- “Functional Analysis” (specialty “Automatic Control Systems”),
- “Linear Algebra” (specialty “Automatic Control Systems”),
- “Mathematical Analysis” (specialty “Applied Mathematics and Informatics”),
- “Theory of Functions of Complex Variable” (specialty “Automatic Control Systems”).
- The following special courses were developed in Moscow Aviation Institute MAI:
- “Nonlocal Elliptic Boundary Value Problems” (specialty “Applied Mathematics and Informatics”),
- “Boundary Value Problems for Functional Differential Equations” (specialty “Applied Mathematics and Informatics”).
- A special authorial course for bachelors was developed and is taught in RUDN:
- “Equations of Mathematical Physics” (directions “Mathematics”, “Applied Mathematics and Informatics”).
- 15 Candidates of Science and 3 Doctors of Science were trained.
- A. L. Skubachevskii gave lectures to German professors and postgraduate students of Justus Liebig University Giessen as a visiting Professor under the Mercator programme of the German Science Foundation (DFG), (Giessen, Germany, Justus Liebig University Giessen):
- “Elliptic Functional Differential Equations” (1999-2000).
- “Nonlocal Elliptic Problems” (2002-2003).
Science
- The problem of the existence of unbounded oscillating solutions of a functional-differential equation of the 2nd order, which was previously formulated as an unsolved problem was solved. The obtained results can help to develop a general theory of oscillation of functional differential equations.
- The theory of boundary value problems for elliptic and parabolic functional differential equations was put forward. The theory allows us to investigate elastic deformations of three-layer plates and shells with goffered filler used in aviation and cosmonautics. The fact that the regular difference operator performs an isomorphism of the first order Sobolev subspace with homogeneous Dirichlet conditions on the first order Sobolev subspace with nonlocal boundary conditions was proved. This result made it possible to apply the theory of elliptic differential-difference equations to the study of spectral properties of nonlocal elliptic boundary value problems. Through the use of the results devoted to quasi-linear parabolic functional differential equations new conditions for the emergence of self-oscillations in nonlinear optical systems with two-dimensional feedback were obtained. The results obtained for elliptic functional-differential operators allowed us to prove that strongly elliptic functional-differential operators and elliptic differential-difference operators with degeneracy satisfy the Kato conjecture on the square root of an operator.
- A general theory of elliptic problems with nonlocal conditions was put forward for the first time. The question of solvability of such problems was formulated in the literature as an unsolved problem. A method for investigating the solvability of nonlocal elliptic problems in Sobolev spaces and in weight spaces, applicable to various cases of the structure of nonlocal terms was developed, an asymptotic solution near the singularity points was obtained. The theory of nonlocal elliptic problems was applied to the well-known problem of the existence of Feller semigroups arising in the theory of multidimensional diffusion processes was applied.
- The solvability and smoothness of generalized solutions of boundary value problems for functional differential equations of one variable in the non-self-adjoint case was investigated. The obtained results allowed to generalize N. N. Krasovsky theorem on the damping of the control system with aftereffect in the case of equations of neutral type.
- Along with a well-known German Professor Hans-Otto Walter, sufficient conditions for hyperbolicity of periodic solutions of nonlinear functional differential equations were obtained. These results have an important application in the study of stability of periodic solutions of nonlinear control systems with after-effect.
- Sufficient conditions for the existence of classical solutions of mixed problems for the system of Vlasov-Poisson equations describing the kinetics of high-temperature plasma in a thermonuclear reactor were obtained for the first time. An estimate of plasma confinement time was obtained.
Monographs:
- A.L. Skubachevskii Elliptic Functional Differential Equations and Applications// Birkhäuser: Basel-Boston-Berlin: Birkhäuser Verlag, 1997. 293 p.
- A.L. Skubachevskii Non-classical boundary value problems I. The journal “Contemporary Mathematics. Fundamental Directions” M.: publishing house RUDN, 2007. 26. p. 3-132; Non-classical boundary value problems II. The journal “Contemporary Mathematics. Fundamental Directions.” M.: publishing house RUDN, 2009. 33. p. 3-179.
Eng. trans.: Nonclassical boundary value problems I. J. of Math. Sciences, Springer, 2008, v.155, № 2, p. 199-334; Nonclassical boundary value problems II. J. of Math. Sciences, Springer, 2010, v.166, №4, p. 377-561.
Scientific interests
- Oscillation of solutions of functional differential equations.
- Boundary value problems for functional differential equations.
- Theory of control of systems with after-effect.
- Boundary value problems for elliptic and parabolic functional differential equations.
- Multilayer plates and shells.
- Self-oscillation of nonlinear feedback laser systems.
- Automatic thermocontrol problem with hysteresis.
- Nonlocal elliptic boundary value problems.
- Feller Semigroups.
- Kato problem on the square root of the operator.
- Mixed problems for Vlasov-Poisson equations.
The paper deals with the second boundary-value problem for a second-order differential-difference equation with variable coefficients on the interval (0,d) as well as with the question of conditions on the right-hand side of the equation that ensure the smoothness of the generalized solutions of the boundary-value problem on the whole interval (0,d) for d∉N.
The stellar dynamic models considered here deal with triples (f,ρ,U) of three functions: the distribution function f, the local density ρ, and the Newtonian potential U, where f is a function q of the local energy E=U(r)+u22. Our first result is an answer to the following question: Given a (positive) function p=p(r) on a bounded interval [0,R], how can one recognize p as the local density of a stellar dynamic model of the given type (“inverse problem”)? If this is the case, we say that p is “extendable” (to a complete stellar dynamic model). Assuming that p is strictly decreasing we reveal the connection between p and F, which appears in the nonlinear integral equation p=FU[p] and the solvability of Eddington’s equation between F and q (Theorem 4.1). Second, we investigate the following question (“direct problem”): Which q induce distribution functions f of the form f=q(−E(r,u)−E0) of a stellar dynamic model? This leads to the investigation of the nonlinear equation p=FU[p] in an approximative and constructive way by mainly numerical methods.
We consider the existence of stationary solutions to the Vlasov-Poisson system on a domain Ω⊂R^3 describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of Ω. In a first part we prove an existence result for a generalized system of Vlasov-Poisson type and investigate the relation between the strength of the external magnetic field and the amount of plasma that is confined measured in terms of the total charges. The key tools here are the method of sub-/supersolutions and the use of first integrals in combination with cutoff functions. In a second part we apply these general results to the usual Vlasov-Poisson equation in three different settings: the infinite and finite cylinder, as well as domains with toroidal symmetry. This way we prove the existence of stationary solutions corresponding to a two-component plasma confined in a Mirror trap, as well as a Tokamak.
We consider the second boundary-value problem for a second-order differential-difference equation with variable coefficients on the interval (0,d). We investigate the existence of a generalized solution and obtain conditions on the right-hand side of the equation which ensure the smoothness of generalized solutions on the entire interval (0,d).
We consider the second boundary value problem for a second-order differential-difference equation with variable coefficients on the interval (0,d). It was obtained the necessary and sufficient condition for the existence of a generalized solution. It was proved that, if the right-hand side of the equation is orthogonal to some functions, then a generalized solution from the Sobolev space W^1_2(0,d) belongs to the space W^2_2(0,d).
A linear control system described by a system of differential-difference equations of neutral type with several delays and variable matrix coefficients is considered. The relationship between the variational problem for a nonlocal functional describing the multidimensional control system with delays and the corresponding boundary value problem for the system of differential-difference equations is established. The existence and uniqueness of a generalized solution to this boundary value problem are proved.
We consider the three-dimensional stationary Vlasov-Poisson system of equations with respect to the distribution function of the gravitating matter f, the local density ρ, and the Newtonian potential U. For a given function p, we obtain sufficient conditions for p to be “extendable.” This means that there exists a stationary spherically symmetric solution (f,ρ,U) of the Vlasov-Poisson system depending on the local energy E.
Mixed boundary value problems for arbitrary strongly elliptic differential-difference equations of the second order and non-local mixed problems for second order elliptic differential equations in a cylinder are considered. The relationship of these problems, as well as their unique solvability, is established.
Strongly elliptic differential-difference equations having the form of product for the Laplacian and difference operator with mixed boundary conditions in a cylindrical domain are considered. The relationship of such problems with non-local mixed problems for strongly elliptic differential equations, as well as their unique solvability, is shown.
A control system described by a system of neutral-type differential equations with variable matrix coefficients and several delays is considered. The relation between the variational problem for a non-local functional describing a multidimensional control system with delays and the corresponding boundary value problem for a system of differential-difference equations is shown. The existence and uniqueness of the generalized solution of the boundary value problem are proved.
We consider elliptic differential-difference operators of the second order with degeneration in a cylinder. It is proved that these operators satisfy the hypothesis of T. Kato about the square root of an operator.
In this paper, we consider second-order elliptic differential-difference operators with degeneration in a cylinder associated with closed, densely defined, sectorial semilinear forms in L_2(Q). It is proved that these operators satisfy Kato's square root conjecture.
We consider elliptic differential-difference operators with degeneration in a bounded domain with piecewise smooth boundary. It is proved that these operators are regular accretive and satisfy the Kato square root conjecture.
The first mixed problem for a system of Vlasov–Poisson equations in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a high-temperature plasma. It is shown that the characteristics of the Vlasov equations do not cross the cylinder boundary if the external magnetic field is sufficiently large. New sufficient conditions for the existence and uniqueness of a classical solution to the system of Vlasov–Poisson equations with supports of the ion and electron distribution densities lying at a certain distance from the cylinder boundary are obtained.
The first mixed problem for the Vlasov-Poisson system in an infinite cylinder is considered. This problem describes the kinetics of charged particles in a high-temperature two-component plasma under an external magnetic field. For an arbitrary electric field potential and a sufficiently strong external magnetic field, it is shown that the characteristics of the Vlasov equations do not reach the boundary of the cylinder. It is proved that the Vlasov-Poisson system with ion and electron distribution density functions supported at some distance from the cylinder boundary has a unique classical solution.
We consider a regular difference operator with variable coefficients in a bounded domain. It will be proved that this operator maps continuously and bijectively the Sobolev space of order k with the homogeneous Dirichlet boundary conditions to the subspace of Sobolev space of order k with nonlocal boundary conditions on the shifts of the boundary. This allows to apply the results obtained for nonlocal elliptic problems to the investigation of elliptic differential-difference equations.
Skubachevskii A. L. Elliptic differential-difference operators with degeneration and the Kato square root problem // Math. Nachr., Vol. 291, No. 17-18, 2018, pp. 2660-2692 DOI:10.1002/mana.201700475
We prove correctness of the Kato square root conjecture for elliptic differential-difference operators with degeneration.
We consider an elastic system consisting of two coaxial cylindrical shells connected by goffered filler. Using a variational principle, this model is reduced to a boundary value problem for strongly elliptic system of differential-difference equations. It is proved the existence and uniquenesss of generalized solution of the above problem, smoothness of solution, and convergence of the Ritz method.
The first mixed problem for the Vlasov–Poisson equations with an external magnetic field in half-space is considered. This problem describes the evolution of the density distribution functions of ions and electrons in a high-temperature plasma with a given electric field potential at the boundary. It is shown that for an arbitrary electric field potential and a sufficiently large external magnetic field induction, the characteristics of the Vlasov equations do not reach the half-space boundary. For sufficiently small initial density distribution functions of charged particles, the existence and uniqueness of the classical solution with supports of the density distribution functions of charged particles lying at some distance from the boundary is proved.
We consider a damping problem for control system with delay described by the system of differential-difference equations of neutral type, and establish the relationship of the variational problem for the nonlocal functionals and the corresponding boundary value problem for differential-difference equations. We prove the existence and uniqueness of generalized solution to the boundary value problem for this system of differential-difference equations.
Skubachevskii A. L., Tsuzuki Y., Vlasov-Poisson equations for a two-component plasma in a half-space // Dokl. Math., Vol. 94, No. 3, 2016, pp. 681-683.
The Vlasov-Poisson equations for a two-component high-temperature plasma with an external magnetic field in a half-space are considered. The electric field potential satisfies the Dirichlet condition on the boundary, and the initial density distribution functions of charged particles satisfy the Cauchy conditions. Sufficient conditions for the induction of the external magnetic field and the initial charged-particle density distribution functions are obtained that guarantee the existence of a classical solution for which the supports of the charged-particle density distributions are located at some distance from the boundary.
We consider a unique solvability of nonlocal elliptic problems in an infinite cylinder in weighted spaces and in Hölder spaces. These results allow us to prove the existence and uniqueness of classical solutions for the Vlasov-Poisson equations with nonlocal conditions in an infinite cylinder for sufficiently small initial data.
Rossovskii L.E., Skubachevskii A. L. Introduction to the theory of partial differential equations // MTsNMO, 2021.
This textbook on partial differential equations is intended for the first acquaintance with the subject. It is distinguished by the combination of modern language and rigor with accessibility and a unified approach to the presentation of the material, based on the use of Sobolev spaces and the concept of a generalized solution. The book can become an indispensable assistant to third-year students and the main textbook on partial differential equations for students studying in the areas of "Mathematics" and "Applied Mathematics and Computer Science".
In this paper, we study the smoothness of generalized solutions to boundary value problems for strongly elliptic differential-difference equations on a boundary of neighboring subdomains of smoothness.
The Vlasov–Poisson equations with an external magnetic field in an infinite cylinder for a two-component high-temperature plasma with initial conditions for the distribution densities of charged particles and a non-local boundary condition for the electric field potential are considered. For sufficiently small initial distribution densities, the existence and uniqueness of the classical solution with supports of the distribution densities of charged particles lying in some inner cylinder are proved.
This paper deals with the solvability of nonlocal elliptic problems in an infinite cylinder in weighted and Hölder spaces. Interest in this formulation is motivated by applications to controlled thermonuclear fusion in magnetic traps of the type of magnetic mirrors, which have the shape of a long cylinder.