Quantitative estimates of the field excited by an emitter in a narrow region between two circular inclusions

A field excited by an emitter can be enhanced due to presence of closely located inclusions. In this paper we consider such field enhancement when inclusions are disks of the same radii, and the emitter is of dipole type and located in the narrow region between two inclusions. We derive quantitatively precise estimates of the field enhancement in the narrow region. The estimates reveal that the field is enhanced by a factor of $\epsilon^{-1/2}$ in most area, where $\epsilon$ is the distance between two inclusions. This factor is the same as that of gradient blow-up when there is a smooth back-ground field, not a field excited by an emitter. The method of deriving estimates shows clearly that enhancement is due to potential gap between two inclusions.