The analysis of resonant frequencies and blow-up estimates of close-to-touching subwavelength resonators in the two-dimensional Helmholtz system

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.