Asymptotics for the voltage potential in a periodic network with localized defects
Résumé
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 u of the limiting continuous medium obtained as $h\to0$, and the gradient of Green's function of the limiting medium. Establishing the asymptotics of $u_{h,d}−u_h$ involves uniform $W^{1,\infty}$ estimates on the potentials $u_h$.