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

Abstract

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.