Non‐local interactions in the homogenization closure of thermoelectric functionals

Article type: Research Article

Authors: Camar‐Eddine, M. | Milton, G.W.

Affiliations: Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112‐0090, USA E‐mails: {camar, milton}@math.utah.edu

Note: [] Corresponding author.

Abstract: 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,$\mathbb{R}^{2}$).

Keywords: homogenization, gamma‐convergence, Mosco‐convergence, composite materials, thermoelectricity, non‐local phenomena

Journal: Asymptotic Analysis, vol. 41, no. 3-4, pp. 259-276, 2005