# Volume 52, № 11, 2000

### Igor Volodymyrovych Skrypnik (On His 60th Birthday)

Berezansky Yu. M., Kharlamov P. V., Khruslov E. Ya., Kit G. S., Korneichuk N. P., Korolyuk V. S., Kovalev A. M., Kovalevskii A. A., Lukovsky I. O., Mitropolskiy Yu. A., Samoilenko A. M., Savchenko O. Ya.

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1443-1445

### On the Solvability of the Hele–Shaw Model Problem in Weighted Hölder Spaces in a Plane Angle

Bazalii B. V., Vasil'eva N. V.

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1446-1457

We study a nonstationary boundary-value problem for the Laplace equation in a plane angle with time derivative in a boundary condition. We obtain coercive estimates in weighted Hölder spaces.

### On the Theory of Generalized Toeplitz Kernels

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1458-1472

A new proof of the integral representation of the generalized Toeplitz kernels is given. This proof is based on the spectral theory of the corresponding differential operator that acts in the Hilbert space constructed from a kernel of this sort. A theorem on conditions that should be imposed on the kernel to guarantee the self-adjointness of the operator considered (i.e., the uniqueness of the measure in the representation) is proved.

### On the Spectrum of an Equivariant Extension of the Laplace Operator in a Ball

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1473-1483

We study the relationship between the well-posedness of an equivariant problem for the Poisson equation in a ball and the spectrum of the operator generated by it.

### On the Cauchy Problem for $\mathop {2b}\limits^ \to$ -Parabolic Systems with Growing Coefficients

Ivasyshen S. D., Pasichnyk H. S.

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1484-1496

For $\mathop {2b}\limits^ \to$ -parabolic dissipative systems and systems with growing coefficients as $| x | → ∞$ in the presence of degeneracies in the initial hyperplane, we investigate the fundamental matrix of solutions and the solvability of the Cauchy problem.

### On the Convergence of Certain Numerical Characteristics of Variational Dirichlet Problems in Variable Domains

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1497-1512

We prove two theorems that enable one to reduce the problem of convergence of general characteristics of variational Dirichlet problems in variable domains to the problem of convergence of simpler characteristics of these problems. We describe the case where the convergence of simpler characteristics takes place.

### Multivariational Inequalities and Operator Inclusions in Banach Spaces with Mappings of the Class $(S)_{+}$

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1513-1523

We prove theorems on the existence of solutions of variational inequalities and operator inclusions in Banach spaces with multivalued mappings of the class (*S*)_{+}. We justify the method of penalty operators for variational inequalities.

### Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1524-1533

We prove a convergence theorem and obtain asymptotic (as ε → 0) estimates for a solution of a parabolic initial boundary-value problem in a junction Ω_{ε} that consists of a domain Ω_{0} and a large number *N* ^{2} of ε-periodically located thin cylinders whose thickness is of order ε = *O*(*N* ^{−1}).

### On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1534-1549

We consider a sequence of Dirichlet problems for a nonlinear divergent operator *A*: *W* _{ m } ^{1}(Ω_{ s }) → [*W* _{ m } ^{1}(Ω_{ s })]^{*} in a sequence of perforated domains Ω_{ s } ⊂ Ω. Under a certain condition imposed on the local capacity of the set Ω \ Ω_{ s }, we prove the following principle of compensated compactness: \({\mathop {\lim }\limits_{s \to \infty }} \left\langle {Ar_s ,z_s } \right\rangle = 0\) , where *r* _{s}(*x*) and *z* _{s}(*x*) are sequences weakly convergent in *W* _{ m } ^{1}(Ω) and such that *r* _{s}(*x*) is an analog of a corrector for a homogenization problem and *z* _{s}(*x*) is an arbitrary sequence from \({\mathop {W_m^1 }\limits^ \circ} (\Omega _s)\) whose weak limit is equal to zero.

### Regularity of a Boundary Point for Degenerate Parabolic Equations with Measurable Coefficients

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1550-1565

We investigate the continuity of solutions of quasilinear parabolic equations in the neighborhood of the nonsmooth boundary of a cylindrical domain. As a special case, one can consider the equation \(\frac{{\partial u}}{{\partial t}} - \Delta _p u = 0\) with the *p*-Laplace operator Δ*p*. We prove a sufficient condition for the regularity of a boundary point in terms of *C* _{p}-capacity.

### Dynamic Game Problems of Approach for Fractional-Order Equations

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1566-1583

We propose a general method for the solution of game problems of approach for dynamic systems with Volterra evolution. This method is based on the method of decision functions and uses the apparatus of the theory of set-valued mappings. Game problems for systems with Riemann–Liouville fractional derivatives and regularized Dzhrbashyan–Nersesyan derivatives (fractal games) are studied in more detail on the basis of matrix Mittag-Leffler functions introduced in this paper.