Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell's Equations
Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell's Equations
复制标题
时谐麦克斯韦方程各向异性电磁参数的边界恢复
DOI：
10.1007/s12220-023-01499-0
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发表时间：
2024
期刊：
影响因子：
--
通讯作者：
Holman S
中科院分区：
文献类型：
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作者：
Holman S
这项工作涉及微分 1 型时谐麦克斯韦方程组的逆边值问题。我们在 3 维紧凑且简单连接的黎曼流形 M 上制定边值问题，边界赋予黎曼度量 g。假设介电常数和磁导率是实值各向异性（即 (1, 1)-张量），我们的目标是确定由这些参数引起的某些度量，表示为 byandat。我们表明，阻抗和导纳图的知识决定了在其边界法线坐标中的切向条目，尽管由于固定边界的微分同胚产生的非唯一性，无法在此类坐标中确定背景体积形式。然后，我们证明在某些情况下，我们也可以将法线分量恢复为共形多重边界法线坐标。最后，我们建立一个归纳证明来证明，如果在边界法线坐标 处确定，那么对于所有阶的法线导数在 处也是如此。
This work concerns inverse boundary value problems for the time-harmonic Maxwell’s equations on differential 1-forms. We formulate the boundary value problem on a 3-dimensional compact and simply connected Riemannian manifoldMwith boundaryendowed with a Riemannian metricg. Assuming that the electric permittivityand magnetic permeabilityare real-valued anisotropic (i.e (1, 1)-tensors), we aim to determine certain metrics induced by these parameters, denoted byandat. We show that the knowledge of the impedance and admittance maps determines the tangential entries ofandatin their boundary normal coordinates, although the background volume form cannot be determined in such coordinates due to a non-uniqueness occuring from diffeomorphisms that fix the boundary. Then, we prove that in some cases, we can also recover the normal components ofup to a conformal multiple atin boundary normal coordinates for. Last, we build an inductive proof to show that ifandare determined atin boundary normal coordinates for, then the same follows for their normal derivatives of all orders at.