标题： 辐射和非辐射弹性源及介质的几何表征及其应用 显示英文标题

标题： Geometrical characterizations of radiating and non-radiating elastic sources and mediums with applications

摘要： 在本文中，我们研究了两种类型的时谐弹性波散射问题。 第一种涉及由具有紧支集的主动弹性源产生的散射波。 第二种涉及由具有紧支集的非均匀介质引起的弹性波散射。 我们推导出了一些关于散射体几何性质、相关源或入射波场以及物理参数的新定量结果。 特别是，我们证明了具有小支撑或高曲率边界点的散射体必须在任何频率下辐射。 这些定性特征使我们能够建立几个局部和全局唯一性结果，以从单个远场测量中确定源或介质散射体的支撑。 此外，我们揭示了弹性传输本征函数的新几何特性。 为了推导辐射或非辐射源的强度与其支撑直径之间的定量关系，我们利用了Helmholtz分解、Lamé算子的平移不变$L^2$-范数估计和全局能量估计。 另一种关键的技术方法结合了复几何光学（CGO）解与局部正则性估计，有助于在可接受的$K$-曲率边界点附近进行微局部分析。 显示英文摘要

摘要： In this paper, we investigate two types of time-harmonic elastic wave scattering problems. The first one involves the scattered wave generated by an active elastic source with compact support. The second one concerns elastic wave scattering caused by an inhomogeneous medium, also with compact support. We derive several novel quantitative results concerning the geometrical properties of the underlying scatterer, the associated source or incident wave field, and the physical parameters. In particular, we show that a scatterer with either a small support or high-curvature boundary points must radiate at any frequency. These qualitative characterizations allow us to establish several local and global uniqueness results for determining the support of the source or medium scatterer from a single far-field measurement. Furthermore, we reveal new geometric properties of elastic transmission eigenfunctions. To derive a quantitative relationship between the intensity of a radiating or non-radiating source and the diameter of its support, we utilize the Helmholtz decomposition, the translation-invariant $L^2$-norm estimate for the Lam\'e operator, and global energy estimates. Another pivotal technical approach combines complex geometric optics (CGO) solutions with local regularity estimates, facilitating microlocal analysis near admissible $K$-curvature boundary points.