Eigenvalue Asymptotics in High-Contrast Media

Abstract

We investigate the eigenfrequency asymptotics of a bounded acoustic cavity containing a shrinking high-contrast inclusion. This configuration is motivated by contrast-enhanced ultrasound imaging, in which microbubbles are employed as acoustic contrast agents. We identify dimension-dependent material scalings that keep the inclusion-induced resonance in a fixed order-one frequency regime as the inclusion shrinks. Our main result gives a complete asymptotic description of the spectrum in any prescribed bounded frequency window. Two distinct spectral mechanisms arise: the background Dirichlet eigenfrequencies persist as perturbed eigenvalue clusters, while the singular material contrast creates an additional Minnaert eigenvalue branch. The limiting Minnaert frequency is explicit in both dimensions, with a capacitance-based expression in three dimensions and an area-based expression in two dimensions. Two-dimensional numerical experiments confirm both the perturbation of the background Dirichlet eigenfrequencies and the emergence of the additional Minnaert branch.