Non-local interactions resulting from the homogenization of a linear diffusive medium

It is well known that the effective properties of a heterogeneous diffusive medium may contain a non-local part. We are interested in the set of all n on-local interactions which can arise from homogenization. We consider a bo unded domain Omega of R-3 and we show that any non-local energy of the kind integral (Omega) alpha (x)\ \ delu \ \ (2) dx + integral (Omega) u(x)(2) v (dx) + integral (Omegax Omega) (u(x) - u(y))(2) mu (dx, dy) can be obtained , provided that the Radon measures v and CL make this energy functional low er semi-continuous. (C) 2001 Academie des sciences/Editions scientifiques e t medicales Elsevier SAS.