Boundary Recovery of Anisotropic Electromagnetic Parameters for the Time Harmonic Maxwell’s Equations

Sean Holman, Vasiliki Torega

Research output: Contribution to journalArticlepeer-review

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 εˆ and µˆ at ∂M. We show that the knowledge of the impedance and admittance maps determines the tangential entries of εˆ and µˆ 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 µˆ up to a conformal multiple at ∂M in boundary normal coordinates for εˆ. Last, we build an inductive proof to show that if εˆ and µˆ are determined at ∂M in boundary normal coordinates for εˆ,
then the same follows for their normal derivatives of all orders at ∂M.
Original languageEnglish
Article number68
JournalJournal of Geometric Analysis
Volume34
DOIs
Publication statusPublished - 9 Jan 2024

Keywords

  • Maxwell’s equations
  • Inverse Problems
  • Anisotropic
  • Electromagnetic Parameters
  • Boundary Normal Coordinates
  • Pseudodifferential Operators

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