Blow-up of Electric Fields between Closely Spaced Spherical Perfect Conductors∗

In the present work, we establish an optimal estimate for the electric potential difference between closely adjacent spherical perfect conductors in n dimensional space (n ≥ 2). This result indicates that electric fields blow up as a pair of spherical perfect conductors approach each other, and provides the lower bound with the optimal blowup rate which was recently established by Bao, Li and Yin. There are two notable advantages in our work: an explicit derivation on the blow-up phenomena and a blowup rate expressed in terms of the radii of conductors. The work is originally motivated by estimate for high stress occurring in stiff fiber-reinforced composites. The practical significance have stimulated several related works by mathematical analysis. Much recently, Bao, Li and Yin accomplished a valuable result in the both fields of theoretical and applied mathematics. Their method employs innovative analytical techniques, and the result is valid for a large class of shapes containing spherical cases and provides the quite surprising blow-up rate. Compared with their way, our work will aid our practical understanding of the blow-up phenomenon. In this respect, our method to employ the point charge technique is familiar enough to easily get the optimal lower bound under the assumption of a pair of spherical conductors. With the simple derivation we are able to understand the blow-up phenomenon easily. Especially in R, our estimate also provides a blow-up rate in terms of the radii of conductors, which is unable to be described with Bao, Li and Yin’s result. Our blow-up rate in terms of radii plays an important role in an understanding of the relation between the blow-up and geometric properties, and surely has a significant meaning in the study on the blow-up. MSC-class: 15A15, 15A09, 15A23