Inverse Boundary Value Problems of Time Harmonic Maxwell's Equations

  • Vasiliki Torega

Student thesis: Phd

Abstract

This thesis studies the inverse problem of time harmonic Maxwell's equations with anisotropic electromagnetic parameters. We formulate the time harmonic Maxwell's equations on differential forms on a 3-d Riemannian manifold with smooth boundary. The electric permittivity and magnetic permeability denoted by e and m, respectively, are assumed real valued smooth (1,1)- tensors and not multiples of each other at any point. Our aim is given the impedance and admittance maps, to recover the metrics induced by e and m at the boundary. Thus, we use the calculus of pseudodifferential operators to calculate the principal symbols of the boundary mappings. Using these principal symbols, we prove the boundary recovery of the tangential entries of the metrics induced by the electromagnetic parameters in their boundary normal coordinates up to a conformal factor. The non-uniqueness in the recovery arises from diffeomorphisms that fix the boundary. Next, we additionally investigate the boundary recovery of the normal components of the metric induced by m in boundary normal coordinates for the metric induced by e. Last, we work under the aforementioned choice of coordinates to show inductively that if the metrics induced by the electromagnetic parameters are determined at the boundary, then the same holds for their normal derivatives of all orders.
Date of Award31 Dec 2022
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorWilliam Lionheart (Supervisor) & Sean Holman (Supervisor)

Keywords

  • boundary normal coordinates
  • pseudodifferential operators
  • impedance map
  • admittance map
  • Maxwell's equations
  • inverse problems
  • anisotropic

Cite this

'