On some problems of homogenization

Abstract

This article concerns a type of partial differential equations in which coefficients occur that are periodic functions of the basic independent variables or coordinates x 1 {x_1} , x 2 {x_2} (we restrict ourselves to two-dimensional problems). The period length H H , here assumed to be the same in both directions, is supposed to be small in comparison with the scale of the general field, and interest is directed to what happens when H → 0 H \to 0 . An example is the equation \[ Σ ∂ ∂ x j ( a i j ∂ u ∂ x i ) = f ( x 1 , x 2 ) \Sigma \frac {\partial }{{\partial {x_j}}}\left ( {{a_{ij}}\frac {{\partial u}}{{\partial {x_i}}}} \right ) = f\left ( {{x_1},{x_2}} \right ) \] with the boundary condition u = 0 u = 0 on a given closed curve C C in the X 1 {X_1} , x 2 {x_2} -plane. Size and shape of C C are independent of H H , while the coefficients a 11 {a_{11}} , a 12 {a_{12}} , a 21 {a_{21}} , a 22 {a_{22}} are periodic functions of auxiliary variables ξ i = x 1 / H {\xi _i} = {x_1}/H with period 1 in ξ 1 {\xi _1} , ξ 2 {\xi _2} . The problem of interest is whether the unknown function u u , which must react upon all the fluctuations of the a i j {a_{ij}} , can be related to a function U ( x 1 , x 2 ) U\left ( {{x_1}, {x_2}} \right ) which is determined by a partial differential equation with constant coefficients, and how these coefficients can be obtained from the a i j {a_{ij}} . This is called the problem of "homogenization".