Optimal estimate of field concentration between multiscale nearly-touching inclusions for 3-D Helmholtz system

We are concerned with the field concentration between two nearly-touching inclusions with high-contrast material parameters, which is a central topic in the theory of composite materials. The degree of concentration is characterised by the blowup rate of the gradient of the underlying field. In this paper, we derive optimal gradient estimates for the wave filed of the 3-D Helmholtz system in the quasi-static regime. There are two salient features of our results that are new to the literature. First, we cover all the possible scenarios that the size of the inclusions are in different scales in terms of the asymptotic distance parameter $\epsilon$, which can be used to characterise the curvature effects of the shape of the inclusions on the field concentration. Second, our estimates can not only recover the known results in the literature for the static case, but can also reveal the interesting frequency effect on the field concentration. In fact, a novel phenomena is shown that even if the static part vanishes, field blowup can still occur due to the (low) frequency effect.