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

摘要

We compare the potential u(h) of a 2D regular lattice of conductors of size h, to the potential u(h,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 u(h,d)-u(h) as a product of three terms: A polarization matrix, the gradient of the potential a of the limiting continuous medium obtained ash h -> 0, and the gradient of Green's function of the limiting medium. Establishing the asymptotics of u(h,d)-u(h) involves uniform W-1,W-infinity estimates on the potentials u(h). Copyright (C) 2007 John Wiley & Sons, Ltd.