Mathematical Study of Fields Concentration and Gradient Estimates with Applications

Description

This project is concerned with the mathematical study of fields concentration due to closely-spaced irregular material inclusions. We aim to establish sharp gradient estimates to quantitatively characterise the fields concentration in acoustic, electromagnetic and elastic scattering that arise in novel applications. Fields concentration widely occurs in practical applications including inverse problems and composite materials. The degree of concentration is characterised by the blowup rate of the underlying gradient field, which is a central topic in mathematical physics and applied mathematics. Motivated by cutting-edge applications, we shall consider the gradient estimates of the wave fields in acoustic, electromagnetic and elastic scattering in new scenarios. There are several novel and salient features of the proposed study. First, we shall consider our study in the quasi-static regime. In comparison, most of the existing studies are concerned with the static case. Though the static model corresponds to the limiting case when the frequency goes to zero, there are rich and subtle structures in the quasi-static regime that deserve a careful and thorough investigation. In fact, we shall not simply include the frequency effect as a higher-order term with the leading term recovering the known results in the static case. Instead, we shall sharply quantify the frequency effect, and show that if coupled with the asymptotic distance parameter as well as the material parameters, new gradient blowup phenomena can be induced. Second, the sizes of the material inclusions may be of different scales. That is, though both the material inclusions are small in size compared to the operating wavelength, the degree of smallness may be multi-scale (compared to the asymptotic distance of the inclusions). Third, some of the proposed study involves coupled-physics process. In fact, we shall study the gradient field in bubbly elastic mediums with closely-spaced air bubbles. The proposed study on this problem is new to the literature and arises in the effective construction of elastic metamaterials via bubbly elastic mediums. We shall develop techniques that combine layer-potential techniques, a-priori estimates, asymptotic analysis and singularity decompositions to derive the sharp quantitative estimates of the gradient fields in various scenarios. Our estimates shall provide a comprehensive understanding of the frequency effect, material effect, multi-scale effect, geometric effect as well as the multi-physical effect of the field-concentration phenomena with practical implications. Finally, we shall consider interesting applications/implications of the gradient estimate results to related inverse problems and the theory of composite materials.