Convergence Rates in L^2 for Elliptic Homogenization Problems

We study rates of convergence of solutions in L^2 and H^{1/2} for a family of elliptic systems {L_\epsilon} with rapidly oscillating oscillating coefficients in Lipschitz domains with Dirichlet or Neumann boundary conditions. As a consequence, we obtain convergence rates for Dirichlet, Neumann, and Steklov eigenvalues of {L_\epsilon}. Most of our results, which rely on the recently established uniform estimates for the L^2 Dirichlet and Neumann problems in \cite{12,13}, are new even for smooth domains.Comment: 25 page

The Navier wall law at a boundary with random roughness

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size \eps \ll 1. In a parent paper, we derived a homogenized boundary condition of Navier type as \eps \to 0. We show here that for a large class of boundaries, this Navier condition provides a O(\eps^{3/2} |\ln \eps|^{1/2}) approximation in $L^2$, instead of O(\eps^{3/2}) for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demon- strates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.Comment: 18 pages, 5 figure

Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough ($L^\infty(\Omega)$, $\Omega \subset \R^d$) coefficients $a(x)$ that, in particular, model media with non-separated scales and high contrast in material properties. We define the flux norm as the $L^2$ norm of the potential part of the fluxes of solutions, which is equivalent to the usual $H^1$-norm. We show that in the flux norm, the error associated with approximating, in a properly defined finite-dimensional space, the set of solutions of the aforementioned PDEs with rough coefficients is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard space (e.g., piecewise polynomial). We refer to this property as the {\it transfer property}. A simple application of this property is the construction of finite dimensional approximation spaces with errors independent of the regularity and contrast of the coefficients and with optimal and explicit convergence rates. This transfer property also provides an alternative to the global harmonic change of coordinates for the homogenization of elliptic operators that can be extended to elasticity equations. The proofs of these homogenization results are based on a new class of elliptic inequalities which play the same role in our approach as the div-curl lemma in classical homogenization.Comment: Accepted for publication in Archives for Rational Mechanics and Analysi