# Volume 45, № 11, 1993

### Approximation of harmonic functions on compact sets in ℂ

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1467–1475

The direct theorem of the theory of approximation of harmonic functions is established in the case of functions defined on a compact set, the complement of which with respect to ℂ is a John domain.

### Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1476–1483

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.

### On conformal mapping of polygonal regions

Gutlyanskii V. Ya., Zaidan A. O.

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1484–1494

We develop Kufarev's method for determining unknown parameters in the Schwarz-Christoffel integral in the case of conformal mapping of polygonal regions with boundary normalization.

### Partial regularity of traces of solutions of higher-order nonlinear elliptic systems

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1495–1502

For traces of generalized solutions of elliptic systems on smooth manifolds, we study the dependence of the Hausdorff dimension of the set of points at which a solution is not smooth on the modulus of ellipticity of a system.

### Averaging of Neumann problems for nonlinear elliptic equations in regions of framework type with thin channels

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1503–1513

The $G$-convergence of operators of the Neumann problem is established in regions with framework-type periodic structure with thin channels. A representation of the coefficients of a $G$-limiting operator is obtained.

### Large deviations in the problem of distinguishing the counting processes

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1514–1521

We prove the general limit theorem on probability of large deviations of the logarithm of the likelihood ratio with the null hypothesis and alternative. Weaker versions of the principle of large deviations are obtained in predictable terms for the problem of distinguishing the counting processes. The case of counting processes with deterministic compensators is studied.

### On the rate of rational approximation of functions on tangent continua

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1522–1533

Upper and lower bounds are established for the rate of rational approximation of functions piecewise analytic on tangent continua. In some special cases, these bounds are coordinated depending on the mutual location of the continua.

### $G$-convergence of parabolic operators and weak convergence of solutions of diffusion equations

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1534–1541

The relationship between the $G$-convergence of parabolic operators and the weak convergence of solutions of diffusion equations is established. Constructive conditions are obtained for determining coefficients of a $G$-limiting operator.

### Asymptotic expansion of solutions of quasilinear parabolic problems in perforated domains

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1542–1566

The asymptotic expansion of solutions to quasilinear parabolic problems with the Dirichlet boundary condilions is constructed in the regions with a fine-grain boundary. It is shown that the sequence of the remainders of the expansion strongly converges to zero in the space $W^{1,1/2}_2$.

### On the existence of initial values of solutions of weakly nonlinear parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1567–1570

We study the properties of solutions of weakly nonlinear parabolic equations in cylindrical domains. The existence conditions are established for local nontangential limits as $t \rightarrow 0$.

### Qualitative properties of solutions of the Neumann problem for a higher-order quasilinear parabolic equation

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1571–1579

The property of localization of perturbations is proved for a solution of an initial boundary-value Neumann problem in a region $D = \Omega x,\quad t > 0$, where $\Omega$ is a region in $R^n$ with a noncompact boundary.

### Asymptotic behavior of a class of stochastic semigroups in the Bernoulli scheme

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1580–1584

The family of subalgebras that describe the space of complex-valued $2 \times 2$ matrices is selected. In this space, the stochastic semigroup $Y_n = X_n X_{n-1} ... X_1, \; n = \overline{1, \infty}$, is considered, where $\{X_ , і = \overline{1, \infty}\}$ are independent equally distributed random matrices taking two values. For the stochastic semigroup $Y_n$, whose phase space belongs to one of the subalgebras, the index of exponential growth is calculated explicitly.