Detectability of Internal Structures in Coupled Physical Systems with Boundary Measurements

Abstract

This thesis investigates how internal structures become detectable in several classes of coupled physical systems governed by partial differential equations (PDEs). In such systems, boundary or exterior measurements may encode information about interior physical processes and structural features through the analytic properties of the governing equations. The configurations considered in this thesis include geometric defects, embedded impenetrable obstacles, and variations in material parameters. The main objective is to clarify the mathematical mechanisms through which such hidden interior features influence observable data.

Rather than formulating these problems within a single unified model, the thesis develops a common detectability perspective across several coupled PDE settings. Under this perspective, the thesis studies three different mechanisms through which internal structures become externally observable. For coupled elliptic systems with transparent boundary conditions, a quantitative relationship is established between discrepancies in internal physical processes and boundary measurements. In particular, localized geometric configurations, especially product-type ends with small geometric scales and high extrinsic curvature, are shown to impose structural constraints on solution behavior, thereby linking localized interior structures to observable boundary effects.

The thesis also investigates detectability in transmission eigenvalue problems and electromagnetic scattering. For coupled-physics transmission eigenvalue problems, vanishing properties of transmission eigenfunctions associated with geometric singularities and material contrasts are established, leading to visibility and uniqueness results in inverse scattering. For electromagnetic scattering with embedded impenetrable obstacles, an effective medium approximation is developed and rigorously justified, providing an equivalent penetrable representation of the corresponding scattering behavior at the level of far-field response.

These results show that interior structures are encoded in boundary observables through different mechanisms involving geometry, material parameters, and coupling effects. This encoding bears a conceptual resemblance to holographic-type principles in physics, in the sense that boundary observables reflect structural information from the interior. The thesis thus provides a mathematical perspective on how hidden bulk features become detectable from accessible boundary or exterior measurements, and contributes to the analysis of inverse problems in complex inhomogeneous media.

Date of Award	11 Aug 2026
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Hongyu LIU (Supervisor)