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Том 241, № 3 (2019)
- Год: 2019
- Статей: 11
- URL: https://bakhtiniada.ru/1072-3374/issue/view/15022
Article
N. V. Stepanov and His Geometric Theory of Ordinary Differential Equations
Аннотация
We review the main results of the geometric theory of ordinary differential equations obtained by the prominent Russian geometer N. V. Stepanov (1926–1991). Some results obtained by Stepanov are illustrated by examples of third- and five-order equations.
245-250
Description of Functionals that are Minimized by Φ-Triangulations
Аннотация
We obtain condition for a function f defined on the set of simplexes S under which the values \( F(T)=\sum \limits_{S\in T}f(S)\;\mathrm{or}\;{F}_f^m(T){=}_{S\in T}^{\mathrm{max}}f(S) \) are minimal for Φ-triangulations of T . As consequences, we also obtain certain extremal properties of the classical Delone triangulation.
251-257
Existence of Entropic Solutions of an Elliptic Problem in Anisotropic Sobolev–Orlicz Spaces
Аннотация
We consider the Dirichlet problem in an arbitrary unbounded domain with inhomogeneous boundary conditions for a certain class of anisotropic elliptic equations whose right-hand sides belong to the class L1 and prove the existence of entropic solutions in anisotropic Sobolev–Orlicz spaces.
258-284
285-289
Existence of Weak Solutions to an Elliptic-Parabolic Equation with Variable Order of Nonlinearity
Аннотация
We consider an equation with variable nonlinearity of the form |u|p(x), in which the parabolic term can vanish, i.e., in the corresponding domain the parabolic equation becomes “elliptic.” Under the weak monotonicity conditions (nonstrict inequality) we prove the existence of a solution to the first mixed problem in a cylinder with a bounded base.
290-305
Solution of Periodic Boundary-Value Problems of the Spatial Theory of Elasticity in the Vector Form
Аннотация
We discuss boundary-value problems for the system of equations of the spatial theory of elasticity in the class of double-periodic functions and obtain a general solution of the system. We distinguish six types of elementary Floquet waves and examine their energy characteristics. We consider fundamental boundary-value problems in the half-space in the vector form. The diffraction problem for an elastic wave on a periodic system of defects in the vector form is reduced to the paired summator functional equation.
306-317
On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity
Аннотация
We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture <2π by the method of dominant balance). We compare intermediate results with results obtained by methods of three-dimensional power geometry. We find possible asymptotics in terms of elliptic functions, construct a power series, which represents an asymptotic expansion of the solution to the third Painlevé equation in a certain sector, estimate the aperture of this sector, and obtain a recurrent relation for the coefficients of the series.
318-326
327-339
Stochastic Perturbations of Stable Dynamical Systems: Trajectory-Wise Approach
Аннотация
We study stochastic perturbations of a dynamical system with a locally stable fixed point. The perturbed system has the form of Ito stochastic differential equations. We assume that perturbations do not vanish at the equilibrium of the deterministic system. Using the approach based on consideration of trajectories to the analysis of stochastic differential equations, we find restrictions for perturbations under which the stability of the equilibrium is preserved with probability 1.
340-353
On Optimal Approximations of the Norm of the Fourier Operator by a Family of Logarithmic Functions
Аннотация
The Lebesgue constant corresponding to the classical Fourier operator is approximated by a family of logarithmic functions depending on two parameters. We find optimal values of parameters for which the best uniform approximation of the Lebesgue constant by a specific function of this family is achieved. The case where the corresponding remainder strictly increases is also considered.
354-363
Basic Bifurcation Scenarios in Neighborhoods of Boundaries of Stability Regions of Libration Points in the Three-Body Problem
Аннотация
In this paper, we construct stability regions (in the linear approximation) of triangular libration points for the planar, restricted, elliptic three-body problem and examine bifurcations that occur when parameters of the system pass through the boundaries of these regions. A new scheme for the construction of stability regions is presented, which leads to approximation formulas describing these boundaries. We prove that on one part of the boundary, the main scenario of bifurcation is the appearance of nonstationary 4π-periodic solutions that are close to a triangular libration point, whereas on the other part, the main scenario is the appearance of quasiperiodic solutions.
364-378