Skrypnik I. V.

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.

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

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.

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.

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.

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.

In a domain D=Ω\E∈ R 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 k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. 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.

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.

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

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.