# Kovalevskii A. A.

### On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity

Kovalevskii A. A., Voitovich M. V.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1511–1524

We consider the Dirichlet problem for a class of nonlinear divergent equations of the fourth order characterized by the condition of strengthened ellipticity imposed on their coefficients. The main result of the present paper shows how the summability of generalized solutions of the given problem improves, depending on the variation in the exponent of summability of the right-hand side of the equation beginning with a certain critical value. The exponent of summability that guarantees the boundedness of solutions is determined more exactly.

### On the convergence of functions from a Sobolev space satisfying special integral estimates

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 168–183

For sequences of functions from a Sobolev space satisfying special integral estimates, we, in one case, establish a lemma on the choice of pointwise convergent subsequences and, in a different case, prove a theorem on convergence of the corresponding sequences of generalized derivatives in measure. These results are applied to the problem of existence of the entropy solutions of nonlinear equations with degenerate coercivity and *L* ^{1}-data.

### 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 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.

### Averaging of integral functionals related to domains of frame-type periodic structure with thin channels

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 616-625

We establish the Γ-convergence of a sequence of integral functionals related to domains of frame-type periodic structure with thin channels. We obtain a representation for the integrand of a Γ-limit functional.

### γ-Convergence of integral functionals and the variational dirichlet problem in variable domains

Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1236–1254

By using special local characteristics of domains Ω_{ s }⊂Ω,*s*=12,..., we establish necessary and sufficient conditions for the γ-convergence of sequences of integral functionals*I* _{λs }:*W* ^{ k,m }(Ω_{ s })→ℝ, λ⊂Ω to interal functionals defined on W^{ k,m }(Ω).

### On the $Γ$-Convergence of integral functionals defined on sobolev weakly connected spaces

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 614-628

We introduce and study the concept of Γ-convergence of functionate*I* _{ s }:*W* ^{ k,m }(Ω)→ℝ,*s*=1,2,..., to a functional defined on (*W* ^{ k,m }(Ω))^{2} and describe the relationship between this type of convergence and the convergence of solutions of Neumann variational problems. For a sequence of integral functionate*I* _{ s }:*W* ^{ k,m }(Ω)→ℝ, we prove a theorem on the selection of a subsequence Γ-convergent to an integral functional defined on (*W* ^{ k,m }(Ω))^{2}.

### Averaging of the Neumann problems for nonlinear elliptic equations in domains with accumulators

Ukr. Mat. Zh. - 1995. - 47, № 2. - pp. 194–212

We study the asymptotic behavior of solutions of the Neumann problems for nonlinear elliptic equations in domains with accumulators, which simulate porous media. An effective description is given for an averaged problem, which, in the case of simple accumulators, is a problem for the system of a functional equation and a differential equation; in the case of double accumulators, it is a problem for the system of two functional equations and a differential equation.

### 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.

### On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 948–962

A notion of $G$-convergence of operators $A_s :\; W_s \rightarrow W_s^*$ to the operator $A:\; W \rightarrow W^*$ is introduced and studied under certain connection conditions for the Banach spaces $W_s,\; s = 1, 2, ... ,$ and the Banach space $W$. It has been established that the connection conditions for abstract space are satisfied by the Sobolev spaces $\overset{\circ}{W}^{k, m}(\Omega_s),\quad \overset{\circ}{W}^{k, m}(\Omega)$ ($\{\Omega_s\}$ is a sequence of perforated domains contained in a bounded domain $\Omega \subset \mathbb{R}^n$). Hence, the results obtained for abstract operators can be applied to the operators of Dirichlet problems in the domains $\Omega_s$.

### 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.