Estimates for Electric Fields Blown Up between Closely Adjacent Conductors with Arbitrary Shape

It may be well known in practice that high stress concentrations occur in fiberreinforced composites. There have been several works by analysis to estimate for the stresses between closed spaced fibers. However, the known results on stiff fibers have until now been restricted to the particular case of circular cross-sections. Thus, we extend the blow-up results on the stresses specialized only for disks to the general case of arbitrary shapes. Moreover, we prove that the blowup rate of the general case is exactly the same as that of disks. Nevertheless, from the viewpoint of methodology, the technique we use is significantly different from the previous one restricted to the case of disks. Referring to antiplane shear problems, these works are reduced to the gradient estimates for the solution to the conductivity problem containing two closely spaced conductors which represent the cross-sections of fibers. We establish a novel representation for the solution on conductors by a probability function. Based on this, the general blow-up results are derived by a simpler method.