Vol 19, No 1 (2015)

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.

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.

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.

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.

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.