This book presents an in-depth treatment of various mathematical aspects of electromagnetism
and Maxwell's equations: from modeling issues to well-posedness results and the coupled models
of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD).
These equations and boundary conditions are discussed including a brief review of absorbing
boundary conditions. The focus then moves to well-posedness results. The relevant function
spaces are introduced with an emphasis on boundary and topological conditions. General
variational frameworks are defined for static and quasi-static problems time-harmonic problems
(including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue
problems) and time-dependent problems with or without constraints. They are then applied to
prove the well-posedness of Maxwell's equations and their simplified models in the various
settings described above. The book is completed with a discussion of dimensionally reduced
models in prismatic and axisymmetric geometries and a survey of existence and uniqueness
results for the Vlasov-Poisson Vlasov-Maxwell and MHD equations. The book addresses mainly
researchers in applied mathematics who work on Maxwell's equations. However it can be used for
master or doctorate-level courses on mathematical electromagnetism as it requires only a
bachelor-level knowledge of analysis.
