Alexander Skubachevskii
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

  1. 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.
  2. 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”).
  3. 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”).
  4. A special authorial course for bachelors was developed and is taught in RUDN:
    • “Equations of Mathematical Physics” (directions “Mathematics”, “Applied Mathematics and Informatics”).
  5. 15 Candidates of Science and 3 Doctors of Science were trained. 
  6. 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.
Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. There were presented applications of these results to the theory of non-local elliptic problems, to the Kato square root problem of an operator, to elasticity theory, and to problems of non-linear optics.
An elliptic equation of order 2m with nonlocal conditions near the boundary is considered. The presence of nonlocal terms in the boundary conditions leads to the appearance of singularities in the solution. The Fredholm property is proved for this problem in weighted spaces and asymptotics of the solutions are obtained. These results are applied to the study of the solvability and smoothness of solutions of elliptic differential-difference equations.
Skubachevskii A.L. On some problems for multidimensional diffusion processes // Soviet Math. Dokl. 1990. 40. № 1, pp. 75-79.
We consider an elliptic operator of second order with nonlocal boundary conditions connecting the values of a function and its derivatives on the boundary Г of an arbitrary domain Q in Rn with values on manifolds Гs in Q. It is proved that the closure of the operator in question in the space C(Q ̅) is the infinitesimal generator of an M-semigroup. It is shown that in the case where the manifold on whish the nonlocal conditions are given intersect Г this assertion may fail to hold.
Skubachevskii A.L. Elliptic differential-difference equations with degeneration // Trans. Moscow Math. Soc. 1998. 59, pp. 217-256.
We consider a differential-difference equation of order 2m in a bounded domain Q in Rn. We assume that the differential operator is strongly elliptic in Q ̅, and that the Hermitian part of the difference operator is a nonnegative degenerate operator. We obtain energy inequalities and construct the Friedrichs extension L_R of the differential-difference operator under consideration. We prove that spectrum of the operator L_R consists of eigenvalues λ_0=0 of infinite multiplicity and λ_s (s=1, 2, …) of finite multiplicity. An elliptic differential-difference equation with degeneration can have a solution that does not even belong to the Sobolev space W1(Q). However, in this article we show that the orthogonal projection of a solution on the image of the difference operator has definite smoothness, not on the whole space Q, but in some domains Q_r in Q.
A field rotation in the two-dimensional feedback loop leads to generation of multi-petal waves. A mathematical model of such a system is described by bifurcation of periodic solutions for nonlinear parabolic functional differential equation with transformation of space variables. In this paper we obtain sufficient conditions for the emergence of the Andronov-Hopf bifurcation for the equations under consideration.
For periodic solutions to the autonomous delay differential equation x′(t)=−μx(t)+f(x(t−1)) with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a criterion for hyperbolicity of certain periodic solutions, which are rapidly oscillating in the sense that the delay 1 is larger than the distance between consecutive zeros. The criterion is used to find periodic orbits which are unstable and hyperbolic. An example of a non-autonomous periodic linear delay differential equation with a monodromy operator which is not hyperbolic shows how subtle the conditions of the hyperbolicity criteria in the present paper and in its predecessor are. We also derive first results on Floquet multipliers in case of irrational periods. These are based on approximations by periodic solutions with rational periods.
This paper is concerned with the first mixed problem for the Vlasov-Poisson equations in an infinite cylinder, a problem describing the evolution of the density distribution of ions and electrons in a high temperature plasma under an external magnetic field. A stationary solution is constructed for which the charged-particle density distributions are supported in a strictly interior cylinder. A classical solution for which the supports of the charged-particle density distributions are at a distance from the cylindrical boundary is shown to exist and to be unique in some neighbourhood of the stationary solution.