2019
Том 71
№ 11

Skrypnik I. V.

Articles: 23
Article (Russian)

Uniform approximation of solutions of nonlinear parabolic problems in perforated domains

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1244-1258

We investigate the behavior of a remainder of an asymptotic expansion for solutions of a quasi-linear parabolic Cauchy-Dirichlet problem in a sequence of domains with fine-grained boundary. By using a modification of an asymptotic expansion and new pointwise estimates for a solution of a model problem, we prove the uniform convergence of the remainder to zero.

Anniversaries (Ukrainian)

Dmytro Yakovych Petryna (on his 70 th birthday)

Ukr. Mat. Zh. - 2004. - 56, № 3. - pp. 291-292

Article (Russian)

Convergence of Eigenvalues and Eigenfunctions of Nonlinear Dirichlet Problems in Domains with Fine-Grain Boundary

Ukr. Mat. Zh. - 2003. - 55, № 6. - pp. 824-839

We study the behavior of eigenvalues and eigenfunctions of the Dirichlet problem for nonlinear elliptic second-order equations in domains with fine-grain boundary.

Anniversaries (Ukrainian)

Mykola Ivanovych Shkil' (On His 70th Birthday)

Ukr. Mat. Zh. - 2002. - 54, № 12. - pp. 1589-1591

Anniversaries (Russian)

Naum Il'ich Akhiezer (on his 100-th birthday)

Ukr. Mat. Zh. - 2001. - 53, № 3. - pp. 291-293

Article (English)

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 s ) → [W m 1 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.

Anniversaries (Ukrainian)

Yurii Makarovich Berezanskii

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 579-581

Article (Ukrainian)

A priori estimates of solutions of linear parabolic problems with coefficients from Sobolev spaces

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1534–1548

We consider the general initial-boundary-value problem for a linear parabolic equation of arbitrary even order in anisotropic Sobolev spaces. We prove the existence and uniqueness of a solution and establish ana priori estimate for it.

Chronicles (Ukrainian)

The International Conference “Nonlinear Partial Differential Equations”

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 1007–1008

Anniversaries (Ukrainian)

Anatolii Mikhailovich Samoilenko (on his 60th birthday)

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4

Article (Russian)

Principle of additivity in averaging of degenerate nonlinear Dirichlet problems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 118–135

We study the problem of averaging of Dirichlet problems for degenerate nonlinear elliptic equations of the second order in domains with fine-grained boundary under the condition that the weight function belongs to a certain Muckenhoupt class. We prove a pointwise estimate for solutions of the model degenerate nonlinear problem. The averaged boundary-value problem is constructed under new structural conditions for a perforated domain. In particular, we do not assume that the diameters of cavities are small as compared with the distances between them.

Article (Russian)

On asymptotic decompositions of o-solutions in the theory of quasilinear systems of difference equations

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 672–677

We consider a quasilinear system of difference equations with certain conditions. We prove that there exists a formal partial o-solution of this system in the form of functional series of special type. We also prove a theorem on the asymptotic behavior of this solution.

Article (Russian)

Pointwise estimates of potentials for higher-order capacities

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 149–163

In a domain D=Ω\ER n , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W p m (D)W q 1 (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)−kf(x)W p m (D)W q 1 (D), where kR 1, f(x)C 0 (Ω), and f(x)=1 for xF. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.

Article (Russian)

New conditions for averaging of nonlinear dirichlet problems in perforated domains

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 675-694

We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.

Brief Communications (Ukrainian)

To the memory of Valentin Anatol'evich Zmorovich

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1110–1111

Article (Russian)

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 $W^{1,1/2}_2$.

Article (Russian)

On the Hölder property for functions from the class $B_{q, s}$

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1020–1028

A class of functions $B_{q, s}$ containing generalized solutions of some higher-order quasilinear parabolic equations is defined. We prove that $B_{q, s}$ is imbedded in the space of Hölder functions.

Chronicles (Russian)

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

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

Anniversaries (Russian)

Samuil Davidovich Eidelman (On his sixtieth birthday)

Ukr. Mat. Zh. - 1991. - 43, № 5. - pp. 578

Article (Ukrainian)

Regular points of generalized solutions of nonlinear parabolic systems of higher order

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 429–436

Article (Ukrainian)

A nonlinear periodic optimal control problem for a system with a small parameter in part of the derivatives

Ukr. Mat. Zh. - 1987. - 39, № 3. - pp. 289–295

Article (Ukrainian)

Application of topological methods to equations with monotonic operators

Ukr. Mat. Zh. - 1972. - 24, № 1. - pp. 69–79

Brief Communications (Russian)

$A$ -harmonic fields with peculiarities

Ukr. Mat. Zh. - 1965. - 17, № 4. - pp. 130-133