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The Mathematical Theory of Maxwell’s Equations

Document "The Mathematical Theory of Maxwell’s Equations" give you the knowledge: The Variational Expansion into Wave Functions, Scattering From a Perfect Conductor, Approach to the Cavity Problem, Boundary Integral Equation Methods for Lipschitz Domains,. | The Mathematical Theory of Maxwell s Equations Andreas Kirsch and Frank Hettlich Department of Mathematics Karlsruhe Institute of Technology KIT Karlsruhe Germany May 24 2014 2 Preface This book arose from lectures on Maxwell s equations given by the authors between 2007 and 2013. Graduate students from pure and applied mathematics physics - including geophysics - and engineering attended these courses. We observed that the expectations of these groups of students were quite different In geophysics expansions of the electromagnetic fields into spherical vector- harmonics inside and outside of balls are of particular interest. Graduate students from numerical analysis wanted to learn about the variational treatments of interior boundary value problems including an introduction to Sobolev spaces. A classical approach in scattering theory - which can be considered as a boundary value problem in the unbounded exterior of a domain - uses boundary integral equation methods which are particularly helpful for deriving properties of the far field behaviour of the solution. This approach is for polygonal domains or more generally Lipschitz domains also of increasing relevance from the numerical point of view because the dimension of the region to be discretized is reduced by one. In our courses we wanted to satisfy all of these wishes and designed an introduction to Maxwell s equations which coveres all of these concepts - but restricted ourselves almost completely except Section 4.3 to the time-harmonic case or in other words to the frequency domain and to a number of model problems. The Helmholtz equation is closely related to the Maxwell system for time-harmonic fields . As we will see solutions of the scalar Helmholtz equation are used to generate solutions of the Maxwell system Hertz potentials and every component of the electric and magnetic field satisfies an equation of Helmholtz type. Therefore and also for didactical reasons we will consider in each of our .