# Vol 19, No 1 (2015)

**Year:**2015**Articles:**11**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1215

### On the completeness of a pair of biorthogonally conjugated systems of functions

#### Abstract

In this paper we studied the spectral problem for an ordinary second order differential equation on a ﬁnite interval with a discontinuous coefficient of the highest derivative. At the ends of the segment the boundary conditions of the ﬁrst kind are given. We found eigenvalues with their asymptotic behavior as the roots of the transcendental equation. The system of eigenfunctions is the trigonometric sine on one half of the segment, and the hyperbolic sine on the other. The system of eigenfunctions is not orthogonal in the space of square integrable functions. The corresponding biorthogonal system of functions was built as a solution to the dual problem. In the proof of the completeness of the biorthogonal system we used well known Keldysh theorem about the completeness of the eigenfunctions system of a nonselfadjoint operator.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):7-18

7-18

### On the Dirichlet problem for an elliptic equation

#### Abstract

It is well known that the concept of a generalized solution from the Sobolev space $W_2^1$ of the Dirichlet problem for a second order elliptic equation is not a generalization of the classical solution sensu stricto: not every continuous function on the domain boundary is a trace of some function from $W_2^1$. The present work is dedicated to the memory of Valentin Petrovich Mikhailov, who proposed a generalization of both these concepts. In the Mikhailov's deﬁnition the boundary values of the solution are taken from the $L_2$; this deﬁnition extends naturally to the case of boundary functions from $L_p$, $p > 1$. Subsequently, the author of this work has shown that solutions have the property $(n - 1)$-dimensional continuity; n is a dimension of the space in which we consider the problem. This property is similar to the classical deﬁnition of uniform continuity, but traces of this function on the measures from a special class should be considered instead of values of the function at points. This class is a little more narrow than the class of Carleson measures. The trace of function on the measure is an element of $L_p$ with respect to this measure. The property $(n - 1)$-dimensional continuity makes it possible to give another deﬁnition of the solution of the Dirichlet problem (a deﬁnition of $(n - 1)$-dimensionally continuous solution), which is in the form close to the classical one. This deﬁnition does not require smoothness of the boundary. The Dirichlet problem in the Mikhailov's formulation and especially for the $(n - 1)$-dimensionally continuous solution was studied insufficiently (in contrast to the cases of classical and generalized solutions). First of all, it refers to conditions on the right side of the equation, in which the Dirichlet problem is solvable. In this article the new results in this direction are presented. In addition, we discuss the conditions on the coefficients of the equation and the conditions on the boundary of a domain in which the problem is considered. The results about the solvability and about the boundary behavior of solutions are compared with the analogous theorems for classical and generalized solutions. Some unsolved problems arising from such comparison are discussed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):19-43

19-43

### On solutions of elliptic equations with nonpower nonlinearities in unbounded domains

#### Abstract

The paper highlighted some class of anisotropic elliptic equations of second order in divergence form with younger members with nonpower nonlinearities $$ \sum\limits_{\alpha=1}^{n}(a_{\alpha}({\bf x},u,\nabla u))_{x_{\alpha}}-a_0({\bf x},u,\nabla u)=0. $$ The condition of total monotony is imposed on the Caratheodory functions included in the equation. Restrictions on the growth of the functions are formulated in terms of a special class of convex functions. These requirements provide limited, coercive, monotone and semicontinuous corresponding elliptic operator. For the considered equations with nonpower nonlinearities the qualitative properties of solutions of the Dirichlet problem in unbounded domains $ \Omega \subset \mathbb {R} _n, \; n \geq 2$ are studied. The existence and uniqueness of generalized solutions in anisotropic Sobolev-Orlicz spaces are proved. Moreover, for arbitrary unbounded domains, the Embedding theorems for anisotropic Sobolev-Orlicz spaces are generalized. It makes possible to prove the global boundedness of solutions of the Dirichlet problem. The original geometric characteristic for unbounded domains along the selected axis is used. In terms of the characteristic the exponential estimate for the rate of decrease at infinity of solutions of the problem with finite data is set.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):44-62

44-62

### The multiple interpolation de la Vallée Poussin problem

#### Abstract

This article is concerned with the solving of multiple interpolation de La Vallée Poussin problem for generalized convolution operator. Particular ate tention is paid to the proving of the sequential sufficiency of the set of solutions of the generalized convolution operator characteristic equation. In the generalized Bargmann-Fock space the adjoint operator of multiplication by the variable $z$ is the generalized differential operator. Using this operator we introduce the generalized shift and generalized convolution operators. Applying the chain of equivalent assertions we obtain the fact that the multiple interpolation de La Vallée Poussin problem is solvable if and only if the composition of generalized convolution operator with multiplication by the ﬁxed entire function $\psi(z)$ is surjective. Zeros of the function $\psi(z)$ are the nodes of interpolation. The surjectivity of composition of the generalized convolution operator with the multiplication comes down to the proof of the sequential sufficiency of the set of zeros of a generalized convolution operator characteristic function in the set of solutions of the generalized convolution operator with the characteristic function $\psi(z)$. In the proof of the sequential sufficiency it became necessary to consider the relation of eigenfunctions for different values of $\mu_i$. The eigenfunction with great value of µi tends to inﬁnity faster than eigenfunction with a lower value for $z$ tends to inﬁnity.The derivative of the eigenfunction of higher order tends to inﬁnity faster than lower-order derivatives with the same values of $\mu_i$. A signiﬁcant role is played by the fact that the kernel of the generalized convolution operator with characteristic function $\psi(z)$ is a ﬁnite sum of its eigenfunction and its derivatives. Using the Fischer representation, Dieudonne-Schwartz theorem and Michael's theorem on the existence of a continuous right inverse we obtain that if the zeros of the characteristic function of a generalized convolution operator are located on the positive real axis in order of increasing then multiple interpolation de La Vallée Poussin problem is solvable in the interpolation nodes.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):63-77

63-77

### Nonlocal problem for partial differential equations of fractional order

#### Abstract

A nonlocal problem is investigated for the partial differential equation (diffusion equation of fractional order) in a finite domain. The boundary condition contains a linear combination of generalized operators of fractional integro-differentiation used on the solution in the characteristics and the solution and its derivative in the degenerating line. The uniqueness of the solution is proved by a modified Tricomi method. The existence of the solution is equivalently reduced to the question of the solvability of Fredholm integral equations of the second kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):78-86

78-86

### Ultrametric diffusion in a strong centrally symmetric field

#### Abstract

A random process at the boundary of a ﬁnite regularly branching tree encapsulated in the central-symmetric external ﬁeld is considered with respect to introduced ultrametricity. We demonstrate an explicit procedure of reduction of dimensionality of the problem. In addition, we consider the strongﬁeld-limit and show that in this case the problem can be solved exactly. The exact solution of the strong-ﬁeld-limit problem related to the case of linearly growing hierarchy of barriers is exempliﬁed and supplemented by estimations of the transition kinetics into the ground state.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):87-104

87-104

### Bigravity in Hamiltonian formalism

#### Abstract

Theory of bigravity is one of approaches proposed to solve the dark energy problem of the Universe. It deals with two metric tensors, each one is minimally coupled to the corresponding set of matter ﬁelds. The bigravity Lagrangian equals to a sum of two General Relativity Lagrangians with the different gravitational coupling constants and different ﬁelds of matter accompanied by the ultralocal potential. As a rule, such a theory has 8 gravitational degrees of freedom: the massless graviton, the massive graviton and the ghost. A special choice of the potential, suggested by de Rham, Gabadadze and Toley (dRGT), allows to avoid of the ghost. But the dRGT potential is constructed by means of the matrix square root, and so it is not an explicit function of the metrics components. One way to do with this difficulty is to apply tetrads. Here we consider an alternative approach. The potential as a differentiable function of metrics components is supposed to exist, but we never appeal to the explicit form of this function. Only properties of this function necessary and sufficient to exclude the ghost are studied. The ﬁnal results are obtained from the constraint analysis and the Poisson brackets calculations. The gravitational variables are the two induced metrics and their conjugated momenta. Also lapse and shift variables for both metrics are involved. After the exclusion of 3 auxiliary variables we stay with 4 ﬁrst class constraints and 2 second class ones responsible for the ghost exclusion. The requirements for the potential are as follows: 1) the potential should satisfy a system of the ﬁrst order linear differential equations; 2) the potential should satisfy the homogeneous Monge-Ampere equation in 4 auxiliary variables; 3) the Hessian of the potential in 3 auxiliary variables is non-degenerate.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):105-116

105-116

### Contractions on ranks and quaternion types in clifford algebras

#### Abstract

In this paper we consider expressions in real and complex Clifford algebras, which we call contractions or averaging. We consider contractions of arbitrary Clifford algebra element. Each contraction is a sum of several summands with different basis elements of Clifford algebra. We consider even and odd contractions, contractions on ranks and contractions on quaternion types. We present relation between these contractions and projection operations onto ﬁxed subspaces of Cliﬀord algebras - even and odd subspaces, subspaces of ﬁxed ranks and subspaces of ﬁxed quaternion types. Using method of contractions we present solutions of system of commutator equations in Clifford algebras. The cases of commutator and anticommutator are the most important. These results can be used in the study of different ﬁeld theory equations, for example, Yang-Mills equations, primitive ﬁeld equation and others.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):117-135

117-135

### Inverse problem for a nonlinear partial differential equation of the eighth order

#### Abstract

We study the questions of solvability of the inverse problem for a nonlinear partial differential equation of the eighth order, left-hand side of which is the superposition of pseudoparabolic and pseudohyperbolic operators of the fourth order. The applicability of the Fourier method of separation of variables is proved in study of mixed and inverse problems for a nonlinear partial differential equation of the eighth order. Using the method of separation of variables, the mixed problem is reduced to the study of the countable system of nonlinear Volterra integral equations of the second kind. Use the given additional conditions led us to study of nonlinear Volterra integral equation of the ﬁrst kind with respect to the second unknown function (with respect to restore function). With the help of nonclassical integral transform the one-value restore of the second unknown function is reduced to study of the unique solvability of nonlinear Volterra integral equation of the second kind. As a result is obtained a system of two nonlinear Volterra integral equations of the second kind with respect to two unknown functions. This system is one-value solved by the method of successive approximations. Further the stability of solutions of the mixed and inverse problems is studied with respect to initial value and additional given functions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):136-154

136-154

### On the inner turbulence paradigm

#### Abstract

In the paper we study the reproducing of the initial phase of the inner turbulence (without regard for the boundary effects). The atypical regularization of multiple-component Euler system is made by the viscosity and diffuse layering introduction. The analogue of Hugoniot condition and the analogue of Lax stability condition are constructed for it. The problem of local accessibility of the phase space points is investigated. The bifurcations of one-front solutions of the abridged Euler system to the two-front solutions are obtained. The supersonic behaviour of bifurcations appearance is shown. The reconstruction of the initial phase of the inner turbulence (without regard for the boundary effects) is made including the mathematical description of the birth of two-speed ﬂow (the Riemann-Hugoniot catastrophe) and alternation.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):155-185

155-185

### Hyperbolic theories and problems of continuum mechanics

#### Abstract

Theories and problems of that part of continuum thermomechanics which can not be properly formulated without partial diﬀerential equations of hyperbolic analytical type are considered. Special attention is paid to comparatively new hyperbolic continuum theories: the theory of three-dimensional perfect plasticity and the theory of micropolar thermoelasticity. The latter is accepted as type-II thermoelasticity. Three-dimensional statical and kinematical equations of the perfect plasticity theory by Ishlinskii and Ivlev are studied in order to elucidate their analytical type and opportunity to obtain integrable equations along some special lines. A new approach to hyperbolic formulations of thermoelasticity presumes consideration of referential gradients of thermodynamic state variables and extra ﬁeld variables (rapid variables) as independent functional arguments in the action density. New hyperbolic thermomechanics of micropolar thermoelastic media is developed within the framework of classical ﬁeld theory by the variational action integral and the least action principle.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(1):186-202

186-202