On a local geometric property of the generalized elastic transmission eigenfunctions and application
On a local geometric property of the generalized elastic transmission eigenfunctions and application
复制标题
DOI：
10.1088/1361-6420/ac23c2
复制
发表时间：
2021-02
期刊：
影响因子：
2.1
通讯作者：
H. Diao;Hongyu Liu;B. Sun
中科院分区：
文献类型：
--
作者：
H. Diao;Hongyu Liu;B. Sun
考虑非线性且完全连续的散射图 S(Ω;λ,μ,V),ui=ut∞(x^),x^∈Sn−1 ，由于入射波场 u i 通过 Lamé 系统，该图将非均匀弹性散射体 (Ω; λ, μ, V) 发送到其远场模式 ut∞。这里，(λ，μ，V)表示以Ω紧密支撑的弹性散射体的介质构型。在本文中，我们关注 S 核空间的内在几何结构，这对于弹性波的逆散射和隐形隐身理论具有根本重要性，并且最近受到了相当多的关注。事实证明，该研究是分析某一非自共轭非椭圆传输特征值问题的几何性质。我们提出了广义弹性传输特征值问题，并证明了在通用正则性准则下，传输特征函数在 ∂Ω 角点附近局部消失。正则性准则的特征在于传输本征函数的霍尔德连续性或特定的傅里叶扩展性质（根据赫戈尔兹波近似）。作为一个有趣且重要的应用，我们应用局部几何特性通过单个远场测量得出长期存在的逆弹性问题的几个新颖的独特可识别性结果。
Consider the nonlinear and completely continuous scattering map S(Ω;λ,μ,V),ui=ut∞(x^),x^∈Sn−1 , which sends an inhomogeneous elastic scatterer (Ω; λ, μ, V) to its far-field pattern ut∞ due to an incident wave field u i via the Lamé system. Here, (λ, μ, V) signifies the medium configuration of an elastic scatterer that is compactly supported in Ω. In this paper, we are concerned with the intrinsic geometric structure of the kernel space of S , which is of fundamental importance to the theory of inverse scattering and invisibility cloaking for elastic waves and has received considerable attention recently. It turns out that the study is contained in analysing the geometric properties of a certain non-selfadjoint and non-elliptic transmission eigenvalue problem. We propose a generalized elastic transmission eigenvalue problem and prove that the transmission eigenfunctions vanish locally around a corner of ∂Ω under generic regularity criteria. The regularity criteria are characterized by the Hölder continuity or a certain Fourier extension property (in terms of the Hergoltz wave approximation) of the transmission eigenfunctions. As an interesting and significant application, we apply the local geometric property to derive several novel unique identifiability results for a longstanding inverse elastic problem by a single far-field measurement.