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

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