Non‐local interactions in the homogenization closure of thermoelectric functionals

It is a very well‐known fact that the effective properties of a heterogeneous electrical medium may contain a non‐local term. The same phenomenon can occur within a heterogeneous thermally conducting medium. We are interested in the set of all non‐local interactions which may arise from the homogenization of a thermoelectric medium where there are couplings between the temperature and the electric field. We consider a bounded open subset Ω⊂ $\mathbb{R}^{3}$ and we show that any non‐local energy of the kind

F(u,v)=∫_Ω×Ω $\pmatrix{u(x)-u(y)\cr v(x)-v(y)\cr}$ ·μ(dx,dy) $\pmatrix{u(x)-u(y)\cr v(x)-v(y)\cr}$ ,

belongs to the closure of the set of thermoelectric functionals, provided that the positive definite symmetric matrix‐valued measure μ(dx,dy) makes this energy functional continuous in the strong topology of L²(Ω, $\mathbb{R}^{2}$ ).