Mathematical Studies on Field Concentration Problem

Abstract

This thesis is concerned with the mathematical studies of field concentration problem for the wave scattering system. Field concentration refers to the intensification of an electric, magnetic, or acoustic wave field within a specific area or volume. We examine field concentration problem from two perspectives: the concentration occurring due to material and geometric irregularities, and the generation of field concentration for practical applications.

Material irregularity is typically a key factor for field concentration. We initially investigate the asymptotic behaviors of time-harmonic scalar waves generated by an incident wave interacting with high-contrast inhomogeneous medium inclusions. We derive classical sound-hard and sound-soft obstacle scattering models and introduce two novel models when material parameters reach extreme values. These results offer a rigorous mathematical characterization of the connection between inhomogeneous medium scattering and obstacle scattering. We then consider field concentration problem between two nearly touching inclusions with high-contrast material parameters, where the concentration degree is characterized by the gradient blowup rate of the underlying field. We derive optimal gradient estimates for the wave field of three dimensional Helmholtz system in the quasi-static regime. The gradient estimate not only recovers known results for the static case but also reveals interesting frequency effects on field concentration.

Subsequently, we explore the generation of field concentration for scalar waves. We develop a mathematical framework for generating customized field concentration through surface transmission resonance. The generation critically depends on specific geometric properties of transmission eigenfunctions. We first demonstrate the existence of a sequence of transmission eigenfunctions for a particular wavenumber that exhibit distinct surface resonant behaviors, including strong localization and oscillation properties. As surface transmission resonant modes, these eigenfunctions fulfill requirements for generating field concentration. Then we show that for a given inclusion within a uniformly homogeneous background space, one can design an incident field to generate strong localized field concentration at specified locations around the inclusion. Our study has several salient features. The proposed method imposes no restrictions on the geometry of inclusions or the frequency regime for scalar waves. While most researches focus on multiple nearly touching inclusions, or resonance in subwavelength regimes. The field concentration can occur at any specified location even multiple locations, and extend beyond subwavelength regime. Our study opens new avenues for the effective utilization of field concentration in wave scattering system.

Date of Award	18 Aug 2025
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Hongyu LIU (Supervisor)