The investigation deals with the, so-called, electric boundary value problems (BVP) for the Helmholtz vector equation in domains with interior cuts (craks) in the form of open surfaces. The study is carried out in the Bessel potential and Besov spaces with the help of the theory of integral (pseudodifferential) equations on the manifolds with boundary. The uniqueness and existence theorems are proved and C~α -smoothness (with α < 1/2) of solutions is established in the neighbourhood of the boundaries of crack surfaces.
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