Transactions of A Razmadze Mathematical Institute最新文献

In the current paper we consider an integral representation of functions and embedding theorems of multianisotropic Sobolev spaces in the three-dimensional case when the completely regular polyhedron has an arbitrary number of anisotropic vertices. This work generalizes results obtained in Karapetyan (in press) and Karapetyan (2016). Particularly, in Karapetyan (in press) the two-dimensional case was fully solved and in Karapetyan (2016) analogous results were obtained for the case of one anisotropic vertex. The problem takes root from various works of Sobolev, particularly, Sobolev (1938) and Sobolev (0000) [4], [5]. Related results were obtained by many authors and can be found in Besov et al. (1967), Reshetnyak (1971), Smith (1961), Nikolsky (0000) and Il’in (1967) [6], [7], [8], [9], [10]. The monograph (Besov, 1978) contains an overview of the problem. The results obtained in this paper are based on a generalization of regularization by a quasi-homogeneous polynomial (see Uspenskii (1972) and Karapetyan (1990) [11], [12]).

In this paper there is considered the Elastoplastic problem for infinite plate, that is weakened by two identical square holes. The boundaries of the holes are partially unknown contours. The plate is in a stressed state, a region of plasticity contains only unknown parts of holes contours and does not spread inside of the plate. Applying the theory of functions of a complex variable and the conformal mapping theory the problem is reduced to a boundary value problem of the analytic function theory and the solution of this problem is obtained, the unknown parts of the holes contours are defined.

In this paper, we prove a Hölder’s type inequality for fully measurable grand Lebesgue spaces, which involves the notion of fully measurable small Lebesgue spaces. It is proved that these spaces are non-reflexive rearrangement invariant Banach function spaces. Moreover, under certain continuity assumptions, along with several properties of fully measurable small Lebesgue spaces, we establish Levi’s theorem for monotone convergence and that grand and small spaces are associated to each other.

Our aim is to establish sharp weighted bounds for the Hilbert transform of odd and even functions in terms of the mixed type characteristics of weights. These bounds involve $A_p$ and $A_{\infty }$ type characteristics. As a consequence, we obtain weighted bounds in terms of so-called Andersen–Muckenhoupt type characteristics.

Generalized stochastic integral from predictable operator-valued random process with respect to a cylindrical Wiener process in an arbitrary Banach space is defined. The question of existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The sufficient condition of existence of the stochastic integral in terms of $p$-absolutely summing operators is given. The stochastic differential equation for generalized random processes is considered and existence and uniqueness of the solution is developed. As a consequence, the corresponding results of the stochastic differential equations in an arbitrary Banach space are given.

The boundary contact problem for a micropolar homogeneous elastic hemitropic medium with a friction is investigated. Here, on a part of the elastic medium surface with a friction, instead of a normal component of force stress there is prescribed the normal component of the displacement vector. We give their mathematical formulation of the Problem in the form of spatial variational inequalities. We consider two cases, the so-called coercive case (when elastic medium is fixed along some part of the boundary) and semi-coercive case (the boundary is not fixed). Based on our variational inequality approach, we prove the existence and uniqueness theorems and show that solutions continuously depend on the data of the original problem. In the semi-coercive case, the necessary condition of solvability of the corresponding contact problem is written out explicitly. This condition under certain restrictions is sufficient, as well.

We consider Dirichlet boundary value problem for Laplace–Beltrami Equation On Hypersurface $S$, when the Laplace–Beltrami operator on the surface is described explicitly in terms of Günter’s differential operators. Using the calculus of Günter’s tangential differential operators on hypersurfaces we establish Finite Element Method for the considered boundary value problem and obtain approximate solution in explicit form.

We investigate the mixed boundary value problems of the generalized thermo-electro-magneto-elasticity theory for homogeneous anisotropic solids with interior cracks. Using the potential methods and theory of pseudodifferential equations on manifolds with boundary we prove the existence and uniqueness of solutions. We analyse the asymptotic behaviour and singularities of the mechanical, electric, magnetic, and thermal fields near the crack edges and near the curves, where different types of boundary conditions collide. In particular, for some important classes of anisotropic media we derive explicit expressions for the corresponding stress singularity exponents and demonstrate their dependence on the material parameters. The questions related to the so called oscillating singularities are treated in detail as well.

For a one-dimensional wave equation with integral nonlinearity, the second Darboux problem is considered for which the questions on the existence and uniqueness of a global solution are investigated.

The present paper investigates natural oscillations and stability of shells of revolution which are close by their form to cylindrical ones, with elastic filler and under the action of meridional forces, external pressure and temperature. The shell is assumed to be thin and elastic. A filler is simulated by an elastic base. The shells of positive and negative Gaussian curvature are considered. Formulas for finding the least frequencies and a form of wave formation are written out. The questions dealing with the higher frequencies and stability of shells of revolution are studied, and formulas for critical loadings are also written out.