Homogenization

On the Precise Asymptotics of the Constant in Friedrich's Inequality for Functions Vanishing on the Part of the Boundary with Microinhomogeneous Structure

“…Friedrich's inequality, when the functions vanishes on a part of the boundary is sometimes called "Poincaré's inequality," but we prefer to say "Friedrich's" or "Friedrich's type inequality" keeping the name "Poincaré's inequality" for the following (see, e.g., [6]):…”

Nematodynamics and random homogenization

“…In further analysis, we use the Birkhoff theorem in the following form (see, for instance, [5,7]): Theorem 3.1 (Birkhoff ergodic theorem) Let T y satisfy the Definition 3.1 and assume that ψ ∈ L q ( , μ), q 1. Then for almost all ω ∈ , the realization ψ(T y ω) has the spatial meanvalue M(ψ (T y ω)).…”

“…for example, [1][2][3][4][5][6][7]) are basic methods for studying mathematical problems appearing in theory of media with nontrivial microstructure. These methods allow to simplify modeling of composites, skeletons, reinforced structures, perforated materials (porous media), cell-structures, bodies with concentrated masses, stratified flow (of newtonian and nonnewtonian fluids), mixture of fluids with different viscosity (density) or mixture of fluids and gas (fluids and solid particles), and many other.…”

Billiards, scattering by rough obstacles, and optimal mass transportation

“…The projection of the imparted momentum on the direction of the body's motion equals (v −v + )·v. Summing up over all particles incident on the body over a long period of time (and the summation amounts to integration over ν B,C ), one gets R(B) (1), that is, the mean value of the resistance force along the direction of the body's motion.…”

“…Note that rough surfaces have been intensively studied in natural sciences for more than 100 years in various contexts. For instance, much work has been done on heat transfer through a rough interface between two media (see, e.g., [1]). A large part of contact mechanics is intended to describe the contact between two rough surfaces [6].…”