Matematicheskii Sbornik
Peer-review mathematical journal
Editor-in-chief
- Boris S. Kashin, Member of the Russian Academy of Sciences, Doctor of physico-mathematical sciences, Professor
Founders
- Russian Academy of Sciences
- Steklov Mathematical Institute of RAS
Main webpage: https://www.mathnet.ru/eng/sm
About
Frequency
The journal is published monthly.
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- Russian Science Citation Index (elibrary.ru)
- Math-Net.Ru
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Scope
The journal publishes original scientific research containing full results in the author's field of study in the field of mathematical analysis, ordinary differential equations, partial differential equations, mathematical physics, geometry and topology, algebra and number theory, and functional analysis.
Main webpage: https://www.mathnet.ru/sm
English version, Sbornik: Mathematics 1064-5616 (print), 1468-4802 (online)
Access to the English version journal dating from the first translation volume is available at https://www.mathnet.ru/eng/sm.
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卷 216, 编号 8 (2025)
- 年: 2025
- 文章: 9
- URL: https://bakhtiniada.ru/0368-8666/issue/view/20351
Tribute to Arlen Mikhailovich Il'in
3-4
Boyarsky-Meyers estimate of solution to the Zaremba problem for Poisson's equation with drift
摘要
An estimate for the increased integrability is obtained for the gradient of the solution to the Zaremba problem for Poisson's equation with lower terms in a bounded domain with Lipschitz boundary and fast alternation of Dirichlet and Neumann conditions.
5-21
On operator estimates for elliptic equations in two-dimensional domains with fast oscillating boundary and frequent alternation of boundary conditions
摘要
A second-order semilinear elliptic equation is considered in an arbitrary two-dimensional domain with boundary that is rapidly oscillating with small amplitude. The oscillations are arbitrary, with no assumption of periodicity or local periodicity. Fast alternating Dirichlet/Neumann boundary conditions are imposed on this boundary. In the case under consideration a Dirichlet problem with the same differential equation arises in the homogenization limit. The main results obtained are $W^1_2$ and $L_2$-operator estimates.
22-40
The rigidity theorem for the equation of characteristics of a second-order linear equation of mixed type on a plane at a point where the coefficients are zero
摘要
Binary differential equations (that is, equations of the form $a(x,y) dy^2+2b(x,y) dx dy+c(x,y) dx^2=0$, where the coefficients $a$, $b$ and $c$ are analytic functions in a neighbourhood of the point $(0,0)$) are considered. A rigidity theorem is proved for degenerate singular points of such equations (that is, for $a(0,0)=b(0,0)=c(0,0)=0$): if generic binary differential equations of this form are formally equivalent, then they are analytically equivalent.
41-81
Asymptotics of a solution to a terminal control problem with two small parameters
摘要
An optimal control problem is considered in the class of piecewise continuous controls with smooth geometric constraints on a fixed interval of time for a linear autonomous system with two small positive independent parameters, one of which, $\varepsilon$, multiplies some derivatives in the equations of the system, while the other, $\mu$, is involved in the initial conditions. The quality functional is convex and terminal, and depends only on the values of the slow variables at the terminal instant. A limit relation as the small parameters tend independently to zero is verified for the vector describing the optimal control. Two cases are considered: the regular case, when the optimal control in the limiting problem if continuous, and the singular case, when this control has a singularity. In the regular case the solution is shown to expand in a power series in $\varepsilon$ and $]\mu$, while in the singular case the solution is asymptotically represented by an Erdelyi series — in either case the asymptotics is with respect to the standard gauge sequence $\varepsilon^k+\mu^k$, as $\varepsilon+\mu\to0$.
82-111
Global uniform asymptotics in the form of Airy functions for the problem of scattering on a repulsive Coulomb potential and Keplerian trajectories
摘要
Results announced in the authors' note [1] are presented in detail. For the scattering problem on a Coulomb potential we present an approach allowing us to construct an appropriate Lagrangian manifold formed by Keplerian orbits and to find an asymptotic representation for the solution by means of the Maslov canonical operator. Using resent results on an effective representation of the Maslov canonical operator in extended neighbourhoods of Lagrangian singularities (caustics) we can represent the asymptotic behaviour of the solution globally and uniformly as an Airy function of a complicated argument.
112-128
Meromorphy of solutions for a system of $N$ equations of Painleve 34 type related to negative symmetries of the Korteweg–de Vries equation
摘要
We prove the property of meromorphic extendability for every local holomorphic solution of a system of nonlinear nonautonomous ordinary differential equations. This system is a vector generalization of Painleve's 34 equation (which is in its turn equivalent to the second Painleve equation) and coincides with the stationary part of a symmetry of the Korteweg–de Vries equation obtained as the sum of the stationary parts of the classical Galilean symmetry and $N$ negative symmetries of this integrable evolutionary equation.
129-154
Distribution of poles of real-valued solutions of the third Painleve equation $P_{\mathrm{III}}^{(6)}$
摘要
We study a two-parameter family of real solutions of a special Painleve equation of the third kind,which is used in many models of mathematical physics. Using the method of isomonodromic deformations we construct asymptotic formulae as $x\to\infty$ on the real semi-axis, including the distribution of poles of the singular solution. For $n\gg1$ we show that there are no real poles with $x
155-170
A priori estimate of solutions to the first mixed problem for Vlasov–Poisson system and plasma confinement
摘要
171-186