Yueran Geng | Searcharxiv

arXiv2026

Spectral structures of elastic-electromagnetic transmission eigenvalue problems

The time-harmonic elastic-electromagnetic interior transmission eigenvalue problem (EEITEP) arises when an elastic body becomes invisible to an incident electromagnetic wave. This spectral problem is typically non-elliptic and non-self-adjoint, making its analysis delicate. In this paper, we study the discreteness of transmission eigenvalues and the boundary localization of the associated eigenfunctions. For a general bounded Lipschitz domain, we prove that the set of positive transmission eigenvalues, if non-empty, is discrete with $\infty$ as its only possible accumulation point. For a radially symmetric domain, we demonstrate the existence of a sequence of transmission eigenvalues and derive their asymptotic behavior. We rigorously show that the corresponding transmission eigenfunctions exhibit boundary localization in their electromagnetic components, whereas the elastic displacement field remains globally distributed throughout the domain. Finally, we derive lower bounds for the $L^{\infty}$-norms of the electromagnetic gradients normalized by their $L^2$-norms, quantifying their blow-up behavior near the boundary along this sequence. These findings reveal a potential spectral mechanism for developing super-resolution imaging methods in elastic-electromagnetic scattering.

math.AP↗

arXiv2025

Non-radiating elastic sources in inhomogeneous elastic media at corners with applications

This paper is concerned with non-radiating elastic sources in inhomogeneous elastic media. We demonstrate that the value of non-radiating elastic sources must vanish at convex corners of their support, provided the sources exhibit Hölder continuous regularity near the corner. Additionally, their gradient must satisfy intricate algebraic relationships with the angles defining the underlying corners, assuming the sources have $C^{1,α}$ regularity with $α\in (0,1)$ in the neighborhood of the corners. Our analysis employs complex geometrical optics (CGO) solutions as test functions within a partial differential system to conduct asymptotic analysis near the corners. These characterizations enable us to establish unique identifiability results for determining the position and shape of radiating elastic sources from a single far-field measurement, both locally and globally. The uniqueness of such identification is a longstanding challenge in inverse scattering with a rich history. Specifically, when the support of a radiating elastic source is a convex polygon and the source is Hölder continuous at the corners, we can simultaneously determine the source's shape and its values at the corners. Furthermore, when the source function exhibits $C^{1,α}$ regularity in the neighborhood of a corner, the gradient at that corner can typically be determined. Additionally, when the support includes a convex sectorial corner and the elastic source satisfies certain generic conditions, we demonstrate that such a source must radiate at any frequency.

math.AP↗