Leonid Rossovskii
Doctor of Physics and Mathematics
Associate Professor,
s.M. Nikol’skii Mathematical Institute
Mathematics creates beauty
1993
Graduated from Moscow Aviation Institute (MAI), the Faculty of Applied Mathematics.
1996
Candidate thesis on “Boundary value problems for functional differential equations with linearly transformed argument” was presented, Lomonosov Moscow State University (MSU), scientific adviser - Professor A. L. Skubachevskii.
2003
Academic title Associate Professor at the Department of Differential Equations was awarded.
2012
The winner of the Second all-Russian competition of Scientific and Methodical Council on Mathematics of the Ministry of education and science of the Russian Federation “THE BEST EDUCATIONAL MANUAL OF MATHEMATICS”, the nomination “Mathematics in classical universities and technical universities (with enhanced mathematical training)” for the manual: “Qualitative theory of differential and functional differential equations”, M.: RUDN, 2008, 190 p.
2013
Doctoral thesis on “Elliptic functional-differential equation with contractions and stretchings of arguments of unknown functions”, Peoples’ Friendship University of Russia.
1997-2005
associate Professor of the Department of Differential Equations at Moscow Aviation Institute.
2005-2015
Associate professor of the Department of Applied Mathematics (till 2014 - the Department of Differential Equations and Mathematical Physics) RUDN.
2015 – 2018
Professor of the Department of Applied Mathematics, RUDN University.
2011-2016
Associate professor of the Department of Mathematical Analysis of Chechen State University (ChSU).
2016 - present time
Professor of the Department of Applied Mathematics and Computer Technology (till 2018 - the Department of Computational Mathematics and Computer Technology) ChSU.
Teaching
- Rossovskii L. E. prepared a number of new training courses, the most significant are:
- Controlled systems with aftereffect
- Functional differential equations
- Nonlocal boundary value problems
- The following training manuals on their basis were created:
- L.E. Rossovskii. Qualitative theory of differential and functional-differential equations, RUDN, Moscow, 2008, 190 p.
- E.M. Varfolomeev, L.E. Rossovskii. Functional-differential equations and their applications to the study of neural networks and information transfer by nonlinear laser systems with feedback, RUDN, Moscow, 2008, 263 p.
- L.Е. Rossovskii. Nonlocal Boundary Value Problems and Functional Differential Equations, RUDN, Moscow, 2014, 155 p.
- L.E. Rossovskii, A.L. Skubachevskii. Partial Differential Equations. Part I. Function Spaces. Elliptic Problems, RUDN, Moscow, 2016, 136 p.
- L.E. Rossovskii, A.L. Skubachevskii. Partial Differential Equations. Part II. Evolution Equations, RUDN, Moscow, 2016, 105 p.
- Courses for bachelor’s students at RUDN:
- “Functional Analysis” (direction “Applied Mathematics and Informatics”)
- “Basics of Functional Analysis” (Academy of Engineering RUDN)
- “Controlled Systems with Aftereffect” (direction “Applied Mathematics and Informatics”)
- Courses for master’s students at RUDN:
- “Functional-differential Equations” (direction “Applied Mathematics and Informatics”)
- “Non-local boundary value problems” (direction “Applied Mathematics and Informatics”)
- Has been teaching at the Faculty of Mathematics and Computer Technology of Chechen State University (ChSU) since 2011, currently a course for bachelor’s students:
- “Numerical Methods” (direction “Applied Mathematics and Informatics”)
Science
- The theory of boundary value problems for elliptic functional differential equations with stretching and compression of independent variables in senior terms was developed:
- * necessary and sufficient coerciveness conditions: (fulfillment of the Garding-type inequality);
- * unique solvability and smoothness of solutions of Dirichlet and Neumann problems were studied;
- * the Fredholm property of general boundary value problem with contraction in the Sobolev space was proved;
- * conditions of unique solvability of the equation in weight spaces were obtained;
- * spectral stability of the Neumann problem with respect to small deformations of the domain was studied;
- * dependence of solutions on compression coefficients was studied, the influence of multiplicative incommensurable compressions on the solvability and properties of solutions of the boundary value problem was considered.
- Currently, there are many papers devoted to functional differential equations with extensions and compressions for functions of one variable, which is caused by extensive applications (for example, the classical pantograph equation was derived in several areas simultaneously: astrophysics, engineering, biology). In addition, they are model in a class of equations with unlimited delay. The partial differential equations containing in the senior terms the stretching and compression of the arguments of the desired function are a relatively new object in the theory of differential and functional-differential equations, and their study is essential in the construction of the general theory of elliptic boundary value problems for equations with infinite non-isometric group of shifts. From a mathematical point of view, an important feature of such equations, which has a decisive influence on the methods of research and properties of solutions, is that the transformation of the arguments generates inside the infinite orbit, condensing near the origin or coordinate axes. Modern theory of functional spaces, Gelfand theory of commutative Banach algebras, theory of pseudo-differential operators are widely used in the research. At the same time, the general approaches known for elliptic equations and systems required significant modification. For example, in the transition from constant coefficients in the equation to variable coefficients, it is not possible to apply the known localization method associated with the “freezing” of coefficients, due to the lack of a suitable partition of one. A new approach based on the construction of a special decomposition of the functional-differential operator in the class of considered functional operators and pseudo-differential operators was developed.
- The obtained results on elliptic equations with extensions and compressions are related to a number of fundamentally new moments. The possible presence of an infinite-dimensional kernel or a cokernel of the operator of the boundary value problem was demonstrated, as well as the presence of non-smooth solutions. It was also found that the Fredholm property of operator boundary problems under the assumption of ellipticity of its local part is influenced by the values of the coefficients of the equation in the members with the grips only at the origin. The dependence of solutions of elliptic functional differential equations on the shear or compression coefficients was not considered in the literature before. In addition, the study of the conditions of uniform coercivity of differential-difference equations and equations with stretching and compression of arguments shows that this property is not stable in the following sense: the coercivity of the problem at a certain value of the parameter does not provide coercivity of the problem in an arbitrarily small vicinity of this value. Against the background of the general development of the theory of elliptic functional-differential equations, there are very few meaningful results about equations with incommensurable shifts or compressions; expressed in algebraic form, the results of solvability for such equations are among the first in this direction.
- Sphere of application:
- * functional-differential equations with affine transformations of the argument (i.e. with combinations of compressions/strains and shifts), generalizing the well-known pantograph equation, are used in a variety of fields: astrophysics, nonlinear oscillations, biology, number theory, probability theory. They describe the absorption of light in the interstellar medium, are included in the mathematical model of the dynamics of the contact wire of the electric power supply of the rolling stock, arise in the study of the process of cell growth and division. Elliptic functional-differential equations are also widely used. A strongly elliptic system of differential-difference equations describes an elastic model of a three-layer plate with a corrugated filler (such plates are widely used in aircraft and rocket science). Nonlinear laser systems with two-dimensional feedback containing non-local light field transformations, as well as a number of problems arising in the theory of plasma and the theory of multidimensional diffusion processes (Feller semigroups) lead to the need to study elliptic functional differential equations.
- * the obtained results and the developed methods are a significant step in the construction of the general theory of elliptic boundary value problems for functional differential equations with infinite non-isometric group of shifts;
- * a new class of operators satisfying the conjecture of T. Kato on the square root of the operator was distinguished;
- * the results obtained for elliptic functional differential equations are the basis for the study of nonlocal parabolic problems, see e.g.,
- L.E Rossovskii, A.R. Hanalyev. Coercive solvability of nonlocal boundary value problems for parabolic equations // Contemporary Mathematics. Fundamental Directions. 2016, V. 62, p. 140–151.
- Elliptic functional differential equations
- Nonlocal boundary value problems
Scientific interests
- Elliptic functional differential equations
- Nonlocal boundary value problems
A system of nonlinear parabolic equations describing the evolution of a color image is considered. The existence and uniqueness of a global solution to the mixed problem for this system is proved.
We obtain a number of necessary and sufficient conditions of strong ellipticity for a functional-differential equation containing orthotropic contractions of the argument of the unknown function in the principal part. We establish the unique solvability of the first boundary-value problem and the discreteness, semiboundedness, and sectorial structure of its spectrum.
A boundary value problem for an elliptic functional-differential equation with contraction and dilatation of the arguments of the desired function in the leading part is considered in a starshaped bounded domain. Estimates for the modification of eigenvalues of the operator of the problem under internal deformations of the domain are obtained.
L.E. Rossovskii. Boundary value problems for elliptic functional-differential equations with dilatations and compressions of the arguments//Transactions of the Moscow Mathematical Society. – 2001. – V. 62. – С.185-212.
In a starshaped domain, a 2mth order functional differential equation with the compressed arguments in the principal part and variable coefficients is considered. The Fredholm solvability of a general boundary value problem is established.
The “commensurability” of transformations has been a crucial assumption in the study of solvability and regularity of solutions for elliptic functional differential equations in domains, while equations with incommensurable transformations are much less studied. In the paper, we consider an equation containing multiplicatively incommensurable contractions of the arguments of the unknown function in the principal part. Algebraic conditions for unique solvability of the Dirichlet problem are obtained as well as conditions ensuring the existence of an infinite-dimensional null-space. As a complementary conclusion, we observe that the spectral properties of functional operators with contractions are unstable with respect to small perturbations of scaling parameters.
The conditions for the unique solvability of the boundary value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. In the paper, we study this dependence.
We study the Dirichlet problem for a functional differential equation containing shifted and contracted argument under the Laplacian sign. We establish conditions for the unique solvability and demonstrate also that the problem may have an infinite dimensional solution manifold.
In an arbitrary Banach space, the nonlocal problem is considered for an abstract parabolic equation with a linear unbounded strongly positive operator whose domain is independent of time and is everywhere dense in this space. This operator generates an analytic semigroup. We prove the coercive solvability of the problem in the weighted Hölder space. Earlier, this result was known only for constant operators. We consider applications in the class of parabolic functional differential equations with transformation of spatial variables and in the class of parabolic equations with nonlocal conditions on the boundary of the domain. Thus, this describes parabolic equations with nonlocal conditions both with respect to time and with respect to spatial variables.
In this monograph, the theory of boundary value problems for elliptic functional differential equations containing the compressed and the dilated arguments of the higher derivatives is constructed, including the prototype boundary-value problem, the strongly elliptic equations, general boundary-value problems for high-order equations in Sobolev spaces, solvability in weighted spaces, and the spectral stability of functional differential operators.
Rossovskii L. E., Tasevich A. L. Unique solvability of a functional-differential equation with orthotropic contractions in weighted spaces // Differential Equations, Vol. 53, No. 12, 2017, pp. 1631–1644
The conditions for the unique solvability of the boundary-value problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing “chaotic” dependence on the compression ratio. For example, it turns out that the spectral radius of the operator
L_2(R^n)∋u(x)↦u(p^{−1}x+h)−u(p^{−1}x−h)∈L_2(R^n),p>1,h∈R^n,
is equal to 2p^{n/2} for transcendental values of p, and depends on the coefficients of the minimal polynomial for p in the case where p is an algebraic number. In this paper, we study this dependence. The starting point is the well-known statement that, given a velocity vector with rationally independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for p. The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of p.
We examine the Dirichlet problem in a bounded plane domain for a strongly elliptic functional-differential equation of the second order containing the argument transformations x↦px (p>0) and x↦−x in higher-order derivatives. The study of solvability of the problem relies on a Gårding-type inequality for which some necessary and sufficient conditions are obtained in algebraic form.