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 ǫ, 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.
We are concerned with the field concentration between two nearlytouching 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 field 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 , 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 phenomenon is shown that even if the static part vanishes, field blowup can still occur due to the (low) frequency effect.
In this paper we demonstrate the surface plasmon resonance of nanoparticles in a two-dimensional elastic system. We use the layer potential technique related to the elastic equations and make an asymptotic expansion of the disturbed elastic wave field about the size of nanoparticles. The principal term in the asymptotic expansion mentioned earlier is related to the Neumann-Poincáre operator in the elastic system. Finally, the surface plasmon resonance is investigated by the spectral properties of the Neumann-Poincáre operator.
We consider the recovery of Lame constants and an unknown inner core in elastic system. In this paper, we use layer potential technique to represent the solution of the equation and analyze the obtained solution using transmission conditions across the boundary. Firstly, in a single-layer structure, using the same boundary measurements, we utilize the obtained solution to uniquely recover the Lame constant. Then, in a two-layer structure, we also prove a Calderon-type identity and use this identity to uniquely recover the piecewise Lame constant through the same boundary measurements. Finally, we prove that in a two-layer structure, the unique recovery of piecewise Lame constant in the quasi-static regime.
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