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- Title
Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell's Equations.
- Authors
Holman, Sean; Torega, Vasiliki
- Abstract
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 ∂ M endowed with a Riemannian metric g. Assuming that the electric permittivity ε and magnetic permeability μ are real-valued anisotropic (i.e (1, 1)-tensors), we aim to determine certain metrics induced by these parameters, denoted by ε ^ and μ ^ at ∂ M . We show that the knowledge of the impedance and admittance maps determines the tangential entries of ε ^ and μ ^ at ∂ 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 μ ^ up to a conformal multiple at ∂ M in boundary normal coordinates for ε ^ . Last, we build an inductive proof to show that if ε ^ and μ ^ are determined at ∂ M in boundary normal coordinates for ε ^ , then the same follows for their normal derivatives of all orders at ∂ M .
- Subjects
MAXWELL equations; BOUNDARY value problems; RIEMANNIAN metric; DIFFERENTIAL equations; PERMITTIVITY; MAGNETIC permeability; DIFFEOMORPHISMS
- Publication
Journal of Geometric Analysis, 2024, Vol 34, Issue 3, p1
- ISSN
1050-6926
- Publication type
Article
- DOI
10.1007/s12220-023-01499-0