THE NEEDLE PROBLEM APPROACH TO NON-PERIODIC HOMOGENIZATION

We introduce a new method to homogenization of non-periodic problems and illustrate the approach with the elliptic equation -del.(a(epsilon)u(epsilon)) = f. On the coefficients a(epsilon) assume that solutions u(epsilon) of homogeneous epsilon-problems on simplices with average slope xi is an element of R-n have the property that flux-averages f a(epsilon)del(epsilon) is an element of R-n converge, for epsilon -> 0, to some limit a*(xi), independent of the simplex. Under this assumption, which is comparable to H-convergence, we show the homogenization result for general domains and arbitrary right hand side. The proof uses a new auxiliary problem, the needle problem. Solutions of the needle problem depend on a triangulation of the domain, they solve an epsilon-problem in each simplex and are affine on faces.