Asymptotics for the voltage potential in a periodic network with localized defects

We compare the potential uh of a 2D regular lattice of conductors of size h, to the potential uh,d of a defective lattice, where some conductive links have different conductivities. We show that to first order in h, each defect contributes to the difference uh,d-uh as a product of three terms: A polarization matrix, the gradient of the potential u of the limiting continuous medium obtained as h (right arrow)0, and the gradient of Greenʹs function of the limiting medium. Establishing the asymptotics of uh,d-uh involves uniform W1, (infinity) estimates on the potentials uh.