An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions

We consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with an extreme conductivity (insulators or perfect conductors) are closely located, the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero. The gradient estimate in between insulators in three dimensions has been regarded as a challenging problem, while the optimal blow-up rates in terms of the distance were successfully obtained for the other extreme conductivity problems in two and three dimensions, and are attained on the shortest line segment between inclusions. In this paper, we establish upper and lower bounds of gradients on the shortest line segment between two insulating unit spheres in three dimensions. These bounds present the optimal blow-up rate of gradient on the line segment which is substantially different from the rates in the other problems.