Maria Korovina

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• Professor
• Professor at Lomonosov Moscow State University

The aim of the article is to create a method for studying the asymptotics of solutions to second-order differential equations with irregular singularities. The method allows us to prove the convergence of formal series included in the asymptotics of solutions for a wide class of second-order differential equations in the neighborhoods of their irre...

The purpose of the textbook is to present the theory of generalized functions, its methods and its application to solving problems of mathematical physics in various spaces. The book covers the main spaces of generalized functions, including the space of generalized functions of slow growth and Sobolev spaces. Much attention is paid to methods asso...

We construct the uniform asymptotics of solutions to a third order equation whose holomorphic coefficients have an arbitrary irregular singularity in the space of functions of exponential growth. In general, the problem of constructing the asymptotics of solutions of differential equations in a neighborhood of an irregular singular point was formul...

This paper is devoted to describing the asymptotic behavior of solutions to linear differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. In this paper, to construct a uniform asymptotics, we use resurgent analysis methods based on the Laplace–Borel integral transformation.

We study the problem of wave propagation in the medium whose velocity characteristics change under an external impact. We will study a three-dimensional case. The aim of our study is construct the asymptotic of the solution for the wave operator with a variable time depended coefficient of the Laplacian at infinity. The case of meromorphic and holo...

A new Special Issue edited by Professor Hovik Matevossian, Professor Maria Korovina, and Professor Sergey Lurie, is now open for submission! Submission Deadline: 31 October 2023

We consider the problem of obtaining asymptotics for solutions of dierential operators in a neighborhood of an irregular singular point; more precisely, the construction of uniform asymptotics for solutions of linear dierential equations with second-order meromorphic coe-cients in a neighborhood of a singular point. Examples are also given that con...

In present paper, we consider the problem of obtaining the asymptotics of solutions of a hyperbolic equation with holomorphic coefficients depending on the time parameter in the space of functions of exponential growth. Sufficient conditions for the convergence of asymptotic series presented in the asymptotics of solutions to a boundary value probl...

In this paper, we apply methods of resurgent analysis (including the requantization method) to the construction of asymptotics for solutions of linear ordinary differential equations with holomorphic coefficients. We provide a classification of various types of asymptotics depending on the principal symbol of the differential operator. Using the re...

The paper is devoted to studying the behavior of solutions of the Cauchy problem for large values of time—more precisely, obtaining an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients for large values of the time parameter t. To o...

The main goal of this article is to study the behavior of solutions of non-stationary problems at large timescales, namely, to obtain an asymptotic expansion characterizing the behavior of the solution of the Cauchy problem for a one-dimensional second-order hyperbolic equation with periodic coefficients at large values of the time parameter t. To...

In this paper, we consider the problem of obtaining the asymptotics of solutions of differential operators in a neighborhood of an irregular singular point. More precisely, we construct uniform asymptotics for solutions of linear differential equations with second-order meromorphic coefficients in a neighborhood of a singular point and apply the re...

We study the problem of constructing the asymptotics of solutions to a boundary value problem for the hyperbolic equation with holomorphic coefficients depending on the time parameter t in the space of functions of exponential growth. In addition, sufficient conditions are established for the convergence of asymptotic series contained in the asympt...

The problem of wave propagation in the medium whose velocity characteristics change under an external impact is considered. We investigate a three-dimensional case. The asymptotics of the solution for the Klein–Gordon–Fock operator with a variable coefficient of the Laplacian at infinity have been constructed. The case of holomorphic coefficients h...

In this paper we investigate the construction of uniform asymptotics of solutions for the Klein-Gordon-Fock boundary value problem with meromorphic coefficients as t → ∞. The difficulty in solving these problems is due to the fact that infinity, as known, is an irregular singular point of such equations. In the case of holomorphic coefficients, the...

This paper is a review of results concerning the construction of asymptotics of solutions to degenerate linear differential equations with holomorphic coefficients. We consider both cases of ordinary and partial differential equations.

We study the problem of wave propagation in the medium whose velocity characteristics change under an external impact. We will study a three-dimensional case. The aim of our study is constructing the asymptotic of the solution for the wave operator with a variable time depended coefficient of the Laplacian at infinity. The case of meromorphic and h...

This study is devoted to the description of the asymptotic expansions of solutions of linear ordinary differential equations with holomorphic coefficients in the neighborhood of an infinitely distant singular point. This is a classical problem of analytical theory of differential equations and an important particular case of the general Poincare pr...

We study the problem of wave propagation in the medium whose velocity characteristics change under an external impact. The aim of our study is constructing the asymptotics of the solution of a wave operator with a variable coefficient for the Laplacian at infinity. This study allows us to study the elements of transport infrastructure for the prese...

In the paper we study the Poincare problem for second-order linear differential equations and also classification of asymptotic expansions of solutions in vicinities of irregular singular points for linear differential equations with holomorphic coefficients

The aim of this article is constructing asymptotics of solution of ordinary differential equations with holomorphic coefficients in neighborhood of infinity. Since infinity in general is irregular singular point then problem of representing asymptotics of solution of an equation is a special case of Poincare problem.

Целью работы является описание методов ресургентного анализа в применении к построению асимптотик решений линейных обыкновенных дифференциальных уравнений с голоморфными коэффициентами, в том числе метода повторного квантования. В статье приводится классификация различных типов асимптотик в зависимости от основного символа дифференциального операто...

In this work, we derive asymptotics of solutions of ordinary differential equations with holomorphic coefficients in the neighborhood of infinity. This problem represents a particular case of the general problem of constructing asymptotics of linear differential equations with irregular singularities, namely the Poincare problem. The case of infini...

The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse Laplace–Borel transform is solved for a particular typ...

The asymptotics of linear differential equations with cusp-type degeneration are studied. The problem of constructing asymptotics at infinity for equations with holomorphic coefficients can be reduced to that problem. The main result is the construction of asymptotics of solutions of such equations in the case of multiple roots of the highest-order...

In this paper, we construct the asymptotics for second order linear differential equations with higher-order singularity for the case where the principle symbol has multiple roots. In addition, we solve the problem of constructing asymptotic solutions of Laplace's equation on a manifold with a second order cuspidal singularity. Abstract-In this pap...

We study the asymptotics of solutions of homogeneous nth-order differential equations with a cusp degeneration for the case in which the principal symbol has multiple roots. We describe a new method for constructing the asymptotics, which we call the repeated quantization method. Examples of application of the method are given.

We study the asymptotics of solutions of partial differential equations with higher degenerations. Such equations arise, for example, when studying solutions of elliptic equations on manifolds with cuspidal singular points. We construct the asymptotics of a solution of the Laplace equation defined on a manifold with a cuspidal singularity of order...

We study inhomogeneous differential equations with higher-order degeneration in the coefficients in the resonance-free case and construct the asymptotics of their solutions.

Asymptotic decompositions of solutions to degenerate equations that include a differential operator with smooth coefficients are studied. Laplace-Borel transform is applied to homogeneous equations in the case where the roots of the operator symbol are simple. The space of holomorphic functions of exponential growth in the domain is considered and...

Degenerate elliptic equations that include an operator-valued function on Banach spaces, which depends on p polynomially and is an analytic function with respect to r in some neighborhood of the point r = 0, are studied. A function is said to be infinitely extendable if, for any number R, there exists a discrete set such that the function admits an...

We study the existence of resurgent solutions of differential equations with higher-order degeneration. The proof of the existence of a resurgent solution for the case in which the right-hand side of the equation is a resurgent function is the main result of the present paper.

In the present paper, we consider differential equations with cusp-type degeneration in the principal symbol. We prove the existence of an endlessly continuable solution in the case of Fredholm equations with endlessly continuable right-hand side. For the case in which the principal symbol of the operator has simple singularities, we obtain asympto...

The present paper deals with differential equations with edge degeneration in spaces with asymptotics. We give the definition of an edge space with asymptotics, prove the continuity of operators with edge degeneration in the scale of these spaces, present statements of problems with edge operators, and state conditions providing their Fredholm prop...

We define a space with asymptotics isomorphic to the direct sum of the corresponding Sobolev spaces for any values of their indices. In spaces with asymptotics, we consider problems for differential operators with conical degeneration and state conditions providing their Fredholm property.

This paper studies relative elliptic morphisms. The necessity of considering such morphisms arises in solving many problems, such as Sobolev’s problem, the problem of constructing self-adjoint extensions of the operator Laplace in the many-particle problem, etc. A relative morphism is defined as an operator morphism associated with a pair ( M , X )...

We study Sobolev problems for the case in which the boundary condition is posed on a submanifold X that is a stratified manifold, more precisely, a union of several transversally intersecting smooth manifolds. Using the theory of elliptic morphisms, we state conditions for a Sobolev problem to be Fredholm and construct the solution.

The theory of relative elliptic morphisms is generalized to a stratified submanifold that is union of transversally intersecting manifolds. The operators are classified in the the corresponding operator morphism, and the operator morphisms form an algebra involving an admissible composition of operators in a morphism that is defined on the pseudodi...

This paper is a continuation of [the author, Differ. Equ. 40, No. 2, 227–240 (2004); translation from Differ. Uravn. 40, No. 2, 216–228 (2004; Zbl 1084.58006)], where the theory of Sobolev problems was constructed in spaces with asymptotics corresponding to a pair (M,X), where M is a smooth compact manifold without boundary and X is the union of tw...

The notion of a function space with asymptotics was introduced for the rst time in (1) in the analysis of elliptic problems on manifolds with singularities. Later, the denition of a Sobolev problem in a space with asymptotics was given in (2) for the case in which the manifold near which the asymptotic type is dened is smooth. In the present paper,...

We continue the research [M. V. Korovina, Differ. Equ. 38, No. 6, 816–829 (2002); translation from Differ. Uravn. 38, No. 6, 775–786 (2002; Zbl 1030.35029)] and consider Schrödinger equations with a potential concentrated on a pencil of planes whose intersection has a nonzero dimension. In other words, the problem is to construct self-adjoint exten...

oand coinciding with on this domain. Here x 2 R 3 and y 2 R 3 are coordinates along P1 and P2, respectively, transversal to each other. In the subspaces P1 and P2, we introduce spherical coordinate systems (rx;!x )a nd (ry;!y), respectively, where rx and ry are the radii and !x and !y are points of the corresponding two-dimensional spheres. By [2],...

Ag(x,t) = g*(x,t) + Obviously, 5(t)c(x) E H -2 (R'~). Denoting the domain of the adjoint operator by Da~, we find that functions belonging to DA~ C L2 (R '~) in some e-neighborhood of X have the form g(x, t) = go(x,t) + gl(x)/r, where go(x,t) e H -1/2 (U~) and gl(x) e H-1/2(X) are functions such that g(x,t) E H 2 (G~). Moreover, the operator A~: Da...