Vol 63, No 2 (2017): Proceedings of the Crimean Autumn Mathematical School-Symposium

New Results

Construction of Energetic Functions for Ω-Stable Diffeomorphisms on 2and 3-Manifolds

Grines V.Z., Pochinka O.V.


In this paper, we review the results connected to existence of the energetic function for discrete dynamical systems. Also we consider technique of construction of such functions for some classes of Ωstable and structurally stable diffeomorphisms on manifolds of dimension 2 and 3.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):191-222
pages 191-222 views

Large Time Asymptotics of Fundamental Solution for the Diffusion Equation in Periodic Medium and Its Application to Estimates in the Theory of Averaging

Zhikov V.V., Pastukhova S.E.


The diffusion equation is considered in an infinite 1-periodic medium. For its fundamental solution we find approximations at large values of time t. Precision of approximations has pointwise and integral estimates of orders O(t(-d+j+1)/2) and O(t(-j+1)/2), j=0,1,…, respectively. Approximations are constructed based on the known fundamental solution of the averaged equation with constant coefficients, its derivatives, and solutions of a family of auxiliary problems on the periodicity cell. The family of problems on the cell is generated recurrently. These results are used for construction of approximations of the operator exponential of the diffusion equation with precision estimates in operator norms in Lp-spaces, 1≤p≤∞. For the analogous equation in an ε-periodic medium (here ε is a small parameter) we obtain approximations of the operator exponential in Lp-operator norms for a fixed time with precision of order O(εn), n=1,2,….
Contemporary Mathematics. Fundamental Directions. 2017;63(2):223-246
pages 223-246 views

Model of the Maxwell Compressible Fluid

Zakora D.A.


A model of viscoelastic barotropic Maxwell fluid is investigated. The unique solvability theorem is proved for the corresponding initial-boundary value problem. The associated spectral problem is studied. We prove statements on localization of the spectrum, on the essential and discrete spectra, and on asymptotics of the spectrum.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):247-265
pages 247-265 views

Removal of Isolated Singularities of Generalized Quasiisometries on Riemannian Manifolds

Ilyutko D.P., Sevostyanov E.A.


For mappings with unbounded characteristics we prove theorems on removal of isolated singularities on Riemannian manifolds. We prove that if a mapping satisfies certain inequality of absolute values and its quasiconformity characteristic has a majorant of finite average oscillation at a fixed singular point, then it has a limit at this point.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):266-277
pages 266-277 views

On Some Problems Generated by a Sesquilinear Form

Kopachevskii N.D., Yakubova A.R.


Based on the generalized Green formula for a sesquilinear nonsymmetric form for the Laplace operator, we consider spectral nonself-adjoint problems. Some of them are similar to classical problems while the other arise in problems of hydrodynamics, diffraction, and problems with surface dissipation of energy. Properties of solutions of such problems are considered. Also we study initial-boundary value problems generating considered spectral problems and prove theorems on correct solvability of such problems on any interval of time.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):278-315
pages 278-315 views

Matching Spectral and Initial-Boundary Value Problems

Radomirskaya K.A.


Based on the approach to abstract matching boundary-value problems introduced in [18], we consider matching spectral problems for one and two domains. We study in detail the arising operator pencil with self-adjoint operator coefficients. This pencil acts in a Hilbert space and depends on two parameters. Both possible cases are considered, where one parameter is spectral and the other is fixed, and properties of solutions are obtained depending on this. Also we study initial-boundary value problems of mathematical physics generating matching problems. We prove theorems on unique solvability of a strong solution ranging in the corresponding Hilbert space.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):316-339
pages 316-339 views

On Multiple Completeness of the Root Functions of Ordinary Differential Polynomial Pencil with Constant Coefficients

Rykhlov V.S.


In the space of square integrable functions on a finite segment we consider a class of polynomial pencils of nth-order ordinary differential operators with constant coefficients and two-point boundary-value conditions (at the edges of the segment). We suppose that roots of the characteristic equation of pencils of this class are simple and nonzero. We establish sufficient conditions for m-multiple completeness (1≤m≤n) of the system of root functions of pencils from this class in the space of square integrable functions on this segment.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):340-361
pages 340-361 views

Spectral Analysis of Higher-Order Differential Operators with Discontinuity Conditions at an Interior Point

Yurko V.A.


Higher-order differential operators on a finite interval with jump conditions inside the interval are studied. Properties of spectral characteristics are obtained, and completeness and expansion theorems are proved for this class of operators.
Contemporary Mathematics. Fundamental Directions. 2017;63(2):362-372
pages 362-372 views

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