ABSTRACT
In this chapter we will study several problems related to Maxwell’s equations, phrased as equations where the curl operator is used. The natural space for these equations H(curl, Ω) is considerably more complicated than the classical Sobolev spaces, and this will be particularly visible in two aspects: (a) the natural trace in this space is defined in a weak form (like the normal component in H(div, Ω)), but its range will be quite hard to identify; (b) the lack of compactness of the injection of H(curl, Ω) into L 2(Ω) will cause trouble when dealing with time-harmonic equations and eigenvalues. The entire chapter will deal with vector fields in domains of three-dimensional space or defined on their boundaries. To simplify notation, we will write D ( Ω ) := D ( Ω ) 3 , C ∞ ( R 3 ) := C ∞ ( R 3 ) 3 , S ( R 3 ) := S ( R 3 ) 3 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath16_1.jpg"/> and so on.
