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 talk, 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.

徐龙娟，首都师范大学研究人员，主要从事弹性波传播、亚波长共振及奇异场集中等领域的数学理论研究。