# Volume 44, № 2, 1992

### On the 80th Aniversary of Birthday of V. V. Gnedenko, Member of the Academy of Sciences pf the Ukraine

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 147-148

### Uniqueness of solutions of mixed problems and a Cauchy problem for parabolic equations of high order with unbounded coefficients

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 149–155

New uniqueness classes of generalized solutions (generalized Tacklind classes) of initial-boundary-value problems are found for linear and quasilinear divergent parabolic equations of high order with coefficients increasing at infinity.

### Stefan problem with a kinetic and the classical conditions at the free boundary

Bazalii B. V., Degtyarev S. P.

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 155–166

The Stefan problem is considered with the kinetic condition u^{+}=u^{−}=ɛk(y, τ)-ɛv at the phase interface, where k(y, τ) is the half-sum of the principal curvatures of the free boundary and v is the speed of its shifting in the direction of a normal. The solvability of a modified Stefan problem in spaces of smooth functions and the convergence of its solutions as ɛ → 0 to a solution of the classical Stefan problem are proved.

### Behavior of solutions of the Dirichlet problem for a second-order quasilinear elliptic equation of general form close to a corner point

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 167–173

The Dirichlet problem for the uniformly elliptic equation $$a_{ij} (x,u,u_x )u_{x_i x_j } + a(x,u,u_x ) = 0$$ is considered in a bounded plane region. It is assumed that there is a corner point on the boundary of the region (the origin), and that the coefficients of the equation satisfy minimal smoothness conditions and appropriate conditions of growth on the gradient (not greater than quadratic). For a smooth solution, it is shown that, in a neighborhood of the corner point,$$u(x) = O(|x|^{\pi /\omega } ),\nabla u(x) = O(|x|^{\pi /\omega - 1} ),$$ where $ω$ is the angle in which the two arcs of the boundary of the region intersect at the origin.

### Solutions with singularities of a certain equation of mathematical physics

Gutlyanskii V. Ya., Ryazanov V. I.

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 174-177

A theorem for the representation of solutions with singularities of logarithmic type of a certain equation of mathematial physics in terms of quasiconformal mappings is obtained.

### Families of functions with equiabsolutely continuous integrals

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 178–184

The approximation of functions from Hardy classes by bounded analytic functions is investigated. A theorem is proved, characterizing the sets of functions with equiabsolutely continuous integrals as limit points of the family of bounded subsets of the space H^{∞}.

### Properties of the solutions of multilinear elliptical systems of nondivergence type

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 184–191

Results on the existence of finite and infinite singular points are obtained for multilinear elliptical systems that satisfy an analogue of the Cordes condition.

### Convergence of solutions of variational inequalities with two-sided obstructions in perforated regions

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 191–197

Conditions and the type of convergence of the solutions of elliptical variational inequalities with two-sided obstructions in perforated regions are established.

### Averaging evolutionary equations perturbed by random processes with jumps

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 197–207

Weak convergence of measures generated by solutions of an evolutionary equation dependent on a small parameter to the unique solution of the martingale problem corresponding to the stochastic evolutionary equation is proved. The coefficients of the initial equation depend on random Markov processes with jumps.

### Approximation of analytic functions by polynomials “close” to polynomials of best approximation

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 208–214

The rate of approximation of analytic functions at interior points of compact sets with connected complement by polynomials “close” to polynomials of best approximation is investigated.

### Certain classes of extension of a lacunary Hermitian operator

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 215-232

Certain classes of extensions of a lacunary Hermitian operator are described in terms of abstract boundary conditions. The connection between the asymptotic behavior of eigenvalues of an extension near the boundary of lacuna and the asymptotic of the negative spectrum of the corresponding boundary operator is found.

### Remarks on the $L_p — L_q$ estimates of solutions of the Klein-Gordon equation

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 233–245

We clarify some difficulties in the proofs of estimates, which were established earlier, in the paper by Marshall, Straus and Wainger devoted to the L_{p} —L_{q} estimates of solutions to the Klein-Gordon equation.

### Asymptotic behavior of integral functionals of diffusion processes with periodic coefficients

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 245–252

A central limit theorem is proved for functionals of integral form and with its help is established the local asymptotic normality of the logarithm of the likelihood ratio for diffusion processes with periodic coefficients.

### Searching for partially ordered structures

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 253–260

General methods are developed to search for minimal and dead-end payoff operators for a class of sources endowed with a partial order structure. It is shown that the results can be used in the design of experiments with automata and in the minimization of Boolean functions.

### Multiplicative inequalities in domains with noncompact boundary

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 260–268

Exact embedding theorems of the multiplicative type are established for functions of Sobolev spaces defined in a domain Ω ⊂*R* ^{n},*n*⩾2, whose boundary is not compact. The main condition on the domain is of the isoperimetric type.

### The Stefan thermodiffusion problem in the presence of convection

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 269–274

The Stefan thermodiffusion problem is considered, taking into account convective motions in the liquid phase. The solvability of this problem is proved in spaces of smooth functions.

### On strong Carleman means of multiple trigonometric Fourier series

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 275–279

We prove the strong Carleman summability of the Fourier series of continuous functions on the $m$-dimensional torus, with partial sums constructed over polyhedra of a certain class.

### On the behavior of solutions of quasilinear elliptic equations of second order in unbounded domains

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 279–283

Analogues of the well known in the theory of analytic functions Phragmén-Lindelöff theorem are formulated for the solutions of a wide class of quasilinear equations of elliptic type. Examples are given which illustrate the sharpness of the obtained results for solutions of equations of the form div(|∇u|^{α−2}∇u)=f(x, u), where the function f(x, u) is locally bounded in IR^{n+1},*f*(*x*, 0)=0,*uf*(*x, u*)⩾a¦*u*¦^{1+q},*a*>0,*α*>*1*,*α*-1>q⩾0, n⩾2.

### Convergence of diffusion processes

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 284–289

For stochastic diffusion equations with coefficients depending on a parameter, necessary and sufficient conditions of the weak convergence of solutions to the solution of a stochastic diffusion equation are obtained.

### A criterion for differentiability in the sense of Belinskii and its consequences

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 289–294

There are obtained necessary and sufficient conditions for differentiability in the sense of Belinskii of quasiconformal mappings at a point.

### Conferences on nonlinear problems of mathematical physics and problems with free boundaries

Bazalii B. V., Skrypnik I. V., Tedeev A. F.

Ukr. Mat. Zh. - 1992. - 44, № 2. - pp. 295-297