The Steklov–Zaremba problem for the Laplace operator in a bounded domain with a strictly Lipschitz boundary is considered. A homogeneous Dirichlet condition is specified on the closed part of the boundary of the domain, and the Steklov boundary condition with a spectral parameter is assumed to be satisfied on the complement to the closed part. This problem is a natural generalization of the classical Steklov problem. With respect to a closed set on the boundary of the domain, where the homogeneous Dirichlet boundary condition is specified, its Wiener capacity is assumed to be positive. It follows from this condition on the capacity that it is natural to consider the problem in the Sobolev space of functions that are square-integrable together with all generalized (weak) first-order derivatives. The aim of the paper is to find an estimate for the maximum modulus of normalized eigenfunctions to the problem under consideration. The proofs of the main results make substantial use of the iterative technique of Jurgen Moser.

We study homogenization of random reaction-diffusion systems with rapidly oscillating terms in domains with rapidly oscillating boundary and containing a small parameter epsilon that characterizes the oscillation rate of coefficients in equations and in the boundary conditions. We consider reactiondiffusion systems that obey some general dissipative conditions. We study the asymptotic behavior of trajectory attractors of the considered systems as epsilon -> 0+. We combine homogenization methods and the theory of trajectory attractors. We construct trajectory attractors 2t epsilon for the considered random reactiondiffusion systems and the trajectory attractor 2t for the corresponding limit (homogenized) deterministic reaction-diffusion system including the limit boundary conditions. After that, we prove that, with probability one, the random trajectory attractor 2t epsilon converges to the (non-random) trajectory attractor 2t as epsilon -> 0 in the weak topology of some natural functional space that contains the trajectory spaces of the reaction-diffusion systems.

In the present paper, reaction–diffusion systems (RD-systems) with rapidly oscillating coefficients and righthand sides in equations and in boundary conditions were considered in domains with locally periodic oscillating (wavering) boundary. We proved a weak convergence of the trajectory attractors of the given systems to the trajectory attractors of the limit (homogenized) RD-systems in domain independent of the small parameter, characterizing the oscillation rate. We consider the critical case in which the type of boundary condition was preserved. For this aim, we used the approach of Chepyzhov and Vishik concerning trajectory attractors of evolutionary equations. Also, we applied the homogenization (averaging) method and asymptotic analysis to derive the limit (averaged) system and to prove the convergence. Defining the appropriate axillary functional spaces with weak topology, we proved the existence of trajectory attractors for these systems. Then, we formulated the main theorem and proved it with the help of auxiliary lemmata.

We establish the unique solvability of the Zaremba problem for linear second order elliptic equations in divergence form with measurable coefficients and lower order terms.

A reaction -diffusion system in a domain with randomly located obstacles was considered. When studying the problem, we sat the homogeneous Dirichlet condition on the outer boundary of the domain and the Neumann condition on the boundary of the cavities. Under such assumptions, it was proven that random trajectory attractors of this system with random coefficients converge in some weak topology to the deterministic trajectory attractor of a homogenized reaction -diffusion system with deterministic coefficients in a homogeneous domain without obstacles. In the case of uniqueness, we obtained weak convergence of random global attractors to a deterministic global attractor.

In the paper, we consider a linear second order elliptic problem with drift in a domain perforated along the boundary. Setting homogeneous Dirichlet condition on the boundary of the cavities and homogeneous Neumann condition on the outer boundary of the domain, we prove the higher integrability of the gradient of the solution to the problem (the Boyarsky–Meyers estimate). DOI 10.1134/S1061920824030051

We consider the Ginzburg-Landau equation in the perforated domain, with rapidly oscillating coefficients. We derive the homogenized Ginzburg-Landau equation with a 'strange term' (potential) and prove that the trajectory attractors of the given equation tend in a weak sense to the trajectory attractors of the homogenized one. Assuming additional conditions to be satisfied for the coefficients, we provide also a convergence of the global attractor.

In this paper the Ginzburg-Landau equation is considered in locally periodic porous medium, with rapidly oscillating terms in the equation and boundary conditions. It is proved that the trajectory attractors of this equation converge in a weak sense to the trajectory attractors of the limit Ginzburg-Landau equation with an additional potential term. For this aim we use an approach from the papers and monographs of V.V. Chepyzhov and M.I. Vishik concerning trajectory attractors of evolution equations. Also we apply homogenization methods appeared at the end of the XX-th century. First, we apply the asymptotic methods for formal construction of asymptotics, then, we verify the leading terms of asymptotic series by means of the methods of functional analysis and integral estimates. Defining the appropriate axillary functional spaces with weak topology, we derive the limit (homogenized) equation and prove the existence of trajectory attractors for this equation. Then we formulate the main theorem and prove it with the help of axillary lemmas.

Nonclassical problems in mathematical hydrodynamics arise when studying the motion of rheologically complex media, as well as under boundary conditions different from classical ones. In this paper, existence and uniqueness theorems are established for the classical solution to the problem of a stationary boundary layer of a liquid with the rheological law of Ladyzhenskaya near a solid wall with given conditions characterizing the force of surface tension and the phenomenon of slipping near this wall.

We establish the increased integrability of the gradient of the solution to the Dirichlet problem for the Laplace operator with lower terms and prove the unique solvability of this problem.

We prove the higher integrability of the gradient of solutions of the Zaremba problem in a bounded strongly Lipschitz domain for an inhomogeneous p( · ) -Laplace equation with a variable exponent p having a logarithmic continuity modulus.

In the paper we consider a problem for complex Ginzburg–Landau equations in a medium with locally periodic small obstacles. It is assumed that the obstacle surface can have different conductivity coefficients. We prove that the trajectory attractors of this system converge in a certain weak topology to the trajectory attractors of the homogenized Ginzburg–Landau equations with an additional potential (in the critical case), without an additional potential (in the subcritical case) in the medium without obstacles, or disappear (in the supercritical case).

The two-dimensional system of Navier–Stokes equations in a medium with anisotropic variable viscosity and periodic small obstacles is considered. It is proved that the trajectory attractors of the system tend in a certain weak topology to the trajectory attractors of the homogenized system of Navier–Stokes equations with an additional potential in a medium without obstacles.

An Erratum to this paper has been published: https://doi.org/10.1134/S1064562422340026

We study the higher integrability of solutions to the Zaremba problem for the p(∙)-Laplacian in a plane domain with Lipschitz boundary. We prove that the Boyarsky–Meyers estimates for solutions are valid under a special ratio between the parts of the Dirichlet and Neumann conditions on the boundary.

A higher integrability of the gradient of a solution to the Zaremba problem in a bounded Lipschitz plane domain is proved for the inhomogeneous p(·) -Laplace equation.

We study reaction–diffusion systems with rapidly oscillating terms in the coefficients of equations and in the boundary conditions, in media with periodic obstacles. The non-linear terms of the equations only satisfy general dissipation conditions. We construct trajectory attractors for such systems in the strong topology of the corresponding trajectory dynamical systems. By means of generalized Fatou’s lemma we prove the strong convergence of the trajectory attractors of considered systems to the trajectory attractors of the corresponding homogenized reaction–diffusion systems which contain an additional potential.

The variational solution to the Zaremba problem for a divergent linear second order elliptic equation with measurable coefficients is considered. The problem is set in a local Lipschitz graph domain. An estimate in $$L_{2+\delta }$$ , $$\delta >0$$ , for the gradient of a solution, is proved. An example of the problem with the Dirichlet data supported by a fractal set of zero $$(n-1)$$ -dimensional measure and non-zero p-capacity, $$p>1$$ is constructed.

An existence and uniqueness theorem for a classical solution to the system of equations describing thermal boundary layers in viscous media with the Ladyzhenskaya rheological law is generalized.

In a perforated domain, we consider the two-dimensional system of Navier–Stokes equations with rapidly oscillating terms in the equations and boundary conditions. We prove that the trajectory attractors of this system converge in some weak topology to trajectory attractors of the homogenized Navier–Stokes equations with an additional potential.