高对比度介质中的本征值渐近行为
Eigenvalue Asymptotics in High-Contrast Media
- School of Mathematics, Jilin University（吉林大学数学学院）
- RICAM, Austrian Academy of Sciences（奥地利科学院计算与应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对含收缩高对比度内含物的有界声腔，确定维度相关材料缩放关系，给出任意有界频率窗口内频谱的完整渐近描述，发现两类频谱机制，经二维数值实验验证。
AI中文摘要：
我们研究包含不断缩小的高对比度内含物的有界声腔的本征频率渐近行为，该构型的研究动机是造影增强超声成像，其中微泡被用作声造影剂。我们确定了与维度相关的材料缩放关系，使得在内含物缩小时，内含物诱导的共振保持在固定的量级为1的频率范围内。我们的主要结果给出了任意给定有界频率窗口内频谱的完整渐近描述，存在两种不同的频谱机制：背景Dirichlet本征频率作为受扰动的本征值簇持续存在，而奇异材料对比度则产生了额外的Minnaert本征值分支。极限Minnaert频率在两个维度上均有明确表达式，三维中采用基于电容的表达式，二维中采用基于面积的表达式。二维数值实验证实了背景Dirichlet本征频率的扰动以及额外Minnaert分支的出现。
英文摘要：
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.
