Some properties of the operators of potential theory and their application to the investigation of the basic equation of electrostatics and magnetostatics

Some new properties are proved for the operatorB ^* of the direct value of the potential of a double layer on a closed surfaceS=?ω, in particular the existence inH ^1/2(S) of a basis of eigenfunctions. On the basis of these properties it is proved that the vector integral equation $$\alpha {\rm M}(x) + \nabla \int\limits_\Omega {M(y)} \nabla _y |x - y|dy = H(x), \alpha \geqslant 0,\Omega \subset R^3 ,$$ which is encountered in classical problems of electro- and magnetostatics, is equivalent to the well-known scalar equation with operatorB ^*. The properties of the operator on the left-hand side and of the solutions of the vector equation are investigated.