环形域和球壳上的显式Robin格林函数、共振谱和阻抗恢复

中文摘要 AI 辅助

本文构造了环形域和球壳域上Helmholtz方程的显式闭式格林函数，其内外边界上具有两个独立的Robin阻抗。Graf加法定理和超球面加法定理将每个角模态简化为一个显式的$2\times2$线性系统，共振谱由特征行列式控制，该行列式关于两个阻抗是双线性的。这种双线性具有直接的逆问题后果：一个非径向角模态的两个共振频率通过一个显式二次方程最多生成两个候选阻抗对，第三个共振选择物理对，一个精确的反射对称性障碍确定了径向球模态无法恢复阻抗的情况。在谱方面，我们证明了分支在两个阻抗中均为正、单重且严格递增；推导了阻抗平面四个角处的一阶渐近，系数由边界质量和极限本征函数的法向导数给出，以及在Dirichlet-Dirichlet角处的显式混合二阶系数，对于径向模态，该系数以闭式形式求值为$2\pi/(R_2-R_1)^3$；建立了低频无共振带，其阈值展开在三维中显式到二阶，并在整个阻抗范围内具有精确的有理逼近；证明了具有壳曲率修正的普适高频间距定律。还包括基于雅可比矩阵的灵敏度和条件数准则。核和谱可计算到机器精度，所有渐近区域均通过数值确认，核为有限元和边界元验证提供了参考解。

英文摘要

This paper constructs explicit closed-form Green's functions for the Helmholtz equation on annular and spherical-shell domains with two independent Robin impedances on the inner and outer boundaries. Graf's and the hyperspherical addition theorems reduce each angular mode to an explicit $2\times2$ linear system, and the resonance spectrum is governed by a characteristic determinant bilinear in the two impedances. This bilinearity has a direct inverse-problem consequence: two resonant frequencies of one non-radial angular mode generate at most two candidate impedance pairs via an explicit quadratic equation, a third resonance selects the physical pair, and an exact reflection-symmetry obstruction identifies where the radial spherical mode cannot recover the impedances. Spectrally, we prove the branches positive, simple and strictly increasing in both impedances; derive first-order asymptotics at the four corners of the impedance plane, with coefficients given by boundary masses and normal derivatives of the limiting eigenfunctions, and an explicit mixed second-order coefficient at the Dirichlet--Dirichlet corner, which for the radial mode evaluates in closed form to $2π/(R_2-R_1)^3$; establish a low-frequency resonance-free band, with a second-order threshold expansion explicit in dimension three and a rational approximation accurate over the whole impedance range; and prove the universal high-frequency spacing law with a shell-curvature correction. Jacobian-based sensitivity and conditioning criteria are included. The kernels and spectra are computable to machine precision, all asymptotic regimes are confirmed numerically, and the kernels provide reference solutions for finite-element and boundary-element validation.

发表机构

• Southeast University（东南大学）

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