Mathematical Analysis of Subwavelength Resonances and Gradient Blow-up for Two Close-to-Touching Inclusions within the Two-Dimensional Elasticity
Subwavelength elastic resonators can concentrate wave energy at length scales far below the incident wavelength, but their behavior becomes especially delicate when two resonators almost touch. In this paper, we give a rigorous analysis of a two-dimensional dimer made of two high-contrast hard inclusions embedded in a soft elastic matrix. The analysis confronts two features that are absent from the corresponding three-dimensional theory: the logarithmic low-frequency singularity of the two-dimensional elastic fundamental solution and the possible non-invertibility of the static single-layer potential. We overcome these difficulties by proving the invertibility of the correct frequency-dependent leading-order operator and then using it to reduce the resonance problem to a finite-dimensional system. For generally convex resonators satisfying natural symmetry assumptions, we derive six subwavelength resonant frequencies and identify their dependence on the material contrast $\delta$ and the inter-inclusion distance $\varepsilon$. We further quantify the resonant field concentration in the narrow gap. In the regime $\varepsilon=\Ocal(\delta^\beta)$, $0<\beta<2$, the gradients of the eigenmodes display sharply classified blow-up behavior: some modes attain the stronger rate $\Ocal(1/\varepsilon)$ at the closest point of the gap, while others blow up at the rate $\Ocal(1/\sqrt{\varepsilon})$ away from the centerline; the remaining mode is governed by a boundary mismatch mechanism. These results uncover resonance-induced singularities that are markedly stronger and more structured than those in static or non-resonant elasticity, and they provide a framework for analyzing larger clusters of closely spaced elastic subwavelength resonators.