Subwavelength Phononic Bandgaps in High-Contrast Elastic Media

Inspired by [25], this paper investigates subwavelength bandgaps in phononic crystals consisting of periodically arranged hard elastic materials embedded in a soft elastic background medium. Our contributions are threefold. First, we introduce the quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form to characterize the subwavelength resonant frequencies, which are identified through the condition that the determinant of a certain matrix vanishes. Second, we derive asymptotic expansions for these resonant frequencies and the corresponding non-trivial solutions, thereby establishing the existence of subwavelength phononic bandgaps in elastic media. Finally, we analyze dilute structures in three dimensions, where the spacing between adjacent resonators is significantly larger than the characteristic size of an individual resonator, allowing the inter-resonator interactions to be neglected. In particular, an illustrative example is presented in which the resonator is modeled as a ball.