تقرير
Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell's Equations
| العنوان: | Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell's Equations |
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| المؤلفون: | Holman, Sean, Torega, Vasiliki |
| سنة النشر: | 2023 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs, Mathematics - Differential Geometry, 35R30, 35S15, 35Q61, 53B50 |
| الوصف: | 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 manifold $M$ with boundary $\partial M$ endowed with a Riemannian metric $g$. Assuming that the electric permittivity $\varepsilon$ and magnetic permeability $\mu$ are real-valued anisotropic (i.e $(1,1)-$ tensors), we aim to determine certain metrics induced by these parameters, denoted by $\hat{\varepsilon}$ and $\hat{\mu}$ at $\partial M$. We show that the knowledge of the impedance and admittance maps determines the tangential entries of $\hat{\varepsilon}$ and $\hat{\mu}$ at $\partial M$ in 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 of $\hat{\mu}$ up to a conformal multiple at $\partial M$ in boundary normal coordinates for $\hat{\varepsilon}$. Last, we build an inductive proof to show that if $\hat{\varepsilon}$ and $\hat{\mu}$ are determined at $\partial M$ in boundary normal coordinates for $\hat{\varepsilon}$, then the same follows for their normal derivatives of all orders at $\partial M$. Comment: 40 pages, 1 figure |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2303.06688 |
| رقم الأكسشن: | edsarx.2303.06688 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |