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158 3. Representation Formulae

bookkeeping of the respective orders of differentiation, all of the calculations
presented in this chapter can be carried out under weaker assumptions for
C k,κ boundaries according to the particular formulae and derivations. In or-
der not to be overwhelmingly descriptive we leave these details to the reader.
As far as generalizations to mixed and screen boundary value problems
are concerned, as indicated in Section 2.5, one has to modify the represen-
tation formulae as e.g. (3.9.18) by splitting the generalized Cauchy data
ϕ jD = ϕ j + ϕ j and λ jN = λ j + λ j in order to obtain systems of bound-

ary integral equations instead of (3.9.23) and (3.9.26). Also, singularities of
the charges at the collision points need to be incorporated which is crucial
for the corresponding analysis.
In this Chapter we also did not include one of the very important appli-
cations of integral equations, namely those for the general Maxwell equations
in electromagnetic theory, which have drawn much attention in recent years.
To this end, we refer the reader to the monographs by Cessenat [38] and
Nedelec [234] and also to Costabel and Stephan [54] and other more recent
results by Buffa et al [27, 28, 29] and Hiptmair [126].