Nonlocal Effects Induced by Homogenization

It is indeed surprising that a weak limit of inverses of second order elliptic operators is also the inverse of a second order elliptic operator. Of course in order to obtain this result one has to consider a class larger than the class of operators of the form $$ - \sum\limits_{i = 1}^N {{\partial \over {\partial {x_i}}}\left[ {a(x){\partial \over {\partial {x_i}}}} \right]}$$ (1) with a positive function a, and include the operators of the form $$ - \sum\limits_{i,j = 1}^N {{\partial \over {\partial {x_i}}}\left[ {{a_{ij}}(x){\partial \over {\partial {x_j}}}} \right]}$$ (2) where the functions a ij are the entries of a symmetric positive matrix A(x).