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.
Related
- Learning Coarse Spaces for Two-Grid Methods from Smoothed Errors with Applications to High-Contrast Porous Media Flow
- Asymptotics for the k-Hessian Eigenvalue on the Unit Ball
- Dirichlet eigenvalue asymptotics on conformally compact Einstein manifolds
- Sharp gradient bounds for uniformly elliptic PDE, eigenvalue asymptotics, and the Landis conjecture
- Regularity Effects on the Small-Diffusion Asymptotics of the Principal Eigenvalue: Non-convergence and Convergence
BibTeX
@misc{diao2026eigenvalue,
title = {{Eigenvalue Asymptotics in High-Contrast Media}},
author = {Huaian Diao and Long Li and Mourad Sini and Qilong Zhai},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30721},
eprint = {2608.30721},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30721},
}
RIS
TY - UNPB
TI - Eigenvalue Asymptotics in High-Contrast Media
AU - Huaian Diao
AU - Long Li
AU - Mourad Sini
AU - Qilong Zhai
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30721
UR - https://arxiv.org/abs/2608.30721
AB - 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.
ER -