The conductivity of a sheet containing inclusions with sharp corners

We have written down an integral equation for the surface charge on the interface between an n-sided regular polygonal inclusion with conductivity o1 and a host with conductivity <00. The conductivity of a sheet with few inclusions can be determined from the induced dipole associated with this charge distribution. The integral equation is solved numerically and detailed results are given for n= 3, 4 and 6. The circle limit is approached for large n. We show that our results are consistent with the reciprocity theorem for two-dimensional conducting media, and we reproduce the known analytic results in the limits when the inclusion is either a hole or a superconducting region. We also find the conductivity of a sheet containing a general n-pointed star. Analytic fits to our numerical results are given which are used as input for effective medium theories. The results given in this paper for the electrical conductivity, apply also to the thermal conductivity and to dielectric media.