Homogenization of the elliptic Dirichlet problem: operator error estimates in L2L_2

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain of class $C^2$. In the Hilbert space $L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal{A}_{D,\varepsilon}$ with the Dirichlet boundary condition. Here $\varepsilon>0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf{x}/\varepsilon$. A sharp order operator error estimate $\|\mathcal{A}_{D,\varepsilon}^{-1} - (\mathcal{A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon$ is obtained. Here $\mathcal{A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.Comment: 13 page

Homogenization of random Navier-Stokes-type system for electrorheological fluid

The paper deals with homogenization of Navier-Stokes-type system describing electrorheologial fluid with random characteristics. Under non-standard growth conditions we construct the homogenized model and prove the convergence result. The structure of the limit equations is also studie

Uniform resolvent convergence for strip with fast oscillating boundary

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change

Spectral approach to homogenization of an elliptic operator periodic in some directions

The operator $$A_{\varepsilon}= D_{1} g_{1}(x_{1}/\varepsilon, x_{2}) D_{1} + D_{2} g_{2}(x_{1}/\varepsilon, x_{2}) D_{2}$$ is considered in $L_{2}({\mathbb{R}}^{2})$, where $g_{j}(x_{1},x_{2})$, $j=1,2,$ are periodic in $x_{1}$ with period 1, bounded and positive definite. Let function $Q(x_{1},x_{2})$ be bounded, positive definite and periodic in $x_{1}$ with period 1. Let $Q^{\varepsilon}(x_{1},x_{2})= Q(x_{1}/\varepsilon, x_{2})$. The behavior of the operator $(A_{\varepsilon}+ Q^{\varepsilon}$ as $\varepsilon\to0$ is studied. It is proved that the operator $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}$ tends to $(A^{0} + Q^{0})^{-1}$ in the operator norm in $L_{2}(\mathbb{R}^{2})$. Here $A^{0}$ is the effective operator whose coefficients depend only on $x_{2}$, $Q^{0}$ is the mean value of $Q$ in $x_{1}$. A sharp order estimate for the norm of the difference $(A_{\varepsilon}+ Q^{\varepsilon})^{-1}- (A^{0} + Q^{0})^{-1}$ is obtained. The result is applied to homogenization of the Schr\"odinger operator with a singular potential periodic in one direction.Comment: 3

Smoluchowski-Kramers approximation in the case of variable friction

We consider the small mass asymptotics (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski-Kramers approximation. Some applications of the Smoluchowski-Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered.Comment: already publishe

Effective macroscopic dynamics of stochastic partial differential equations in perforated domains

An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domain without holes. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of \emph{convergence in probability distribution}, and the effectivity of the macroscopic system in the sense of \emph{convergence in energy} are also proved

Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solutio

The Stokes and Poisson problem in variable exponent spaces

We study the Stokes and Poisson problem in the context of variable exponent spaces. We prove the existence of strong and weak solutions for bounded domains with C^{1,1} boundary with inhomogenous boundary values. The result is based on generalizations of the classical theories of Calderon-Zygmund and Agmon-Douglis-Nirenberg to variable exponent spaces.Comment: 20 pages, 1 figur

A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface

It is proved that small periodic singular perturbation of a cylindrical waveguide surface may open a gap in the continuous spectrum of the Dirichlet problem for the Laplace operator. If the perturbation period is long and the caverns in the cylinder are small, the gap certainly opens.Comment: 24 pages, 9 figure