The analysis of resonant frequencies and blow-up estimates of close-to-touching subwavelength resonators in the two-dimensional Helmholtz system
Hongjie Dong, Hongjie Li, Longjuan Xu
TL;DR
This work analyzes two closely spaced high-contrast inclusions in the 2D Helmholtz system, revealing two subwavelength resonances with distinct leading-order behavior and providing rigorous gradient blow-up estimates for the wave field in the gap. The authors develop a boundary-integral framework, derive small-$k$ and NP-operator expansions, and use symmetry to obtain explicit resonance frequencies: one governed by a logarithmic balance $\omega^2\ln\omega=O(\delta)$ and another by a spectral term $\omega_2 \sim \sqrt{\delta}$ scaled by $|D_1|$ and $\alpha_{12}-\alpha_{11}$. They further quantify the singular behavior of the fields in the narrow region, showing a sharp $O(1/\varepsilon)$-type blow-up for the gradient in the gap and clarifying the dependence on boundary data. The results highlight how geometry and resonance tilt the blow-up mechanism in 2D, offering design principles for multi-frequency metamaterials and underscoring essential differences from the 3D setting.
Abstract
In this paper, we investigate wave scattering by a pair of closely spaced inclusions embedded in a homogeneous medium, characterized by a high contrast physical parameters. The system is modeled by the two-dimensional Helmholtz equation. We show that this configuration exhibits two sub-wavelength resonant modes, whose frequencies display distinct leading-order asymptotic behaviors. These findings differ significantly from those in the three-dimensional Helmholtz setting. Furthermore, we provide a quantitative analysis of the gradient blow-up rates for the wave field localized between the two resonators.