Solution of Searched Function Normal Derivative Jump Problem with Internal or External Neumann Condition for the Laplacian in R<sup>3</sup> by Means of Simple and Double Layer Potentials

Abstract

Modeling of electrostatic fields at the environments with different characters leads to necessity of solution of the jump boundary value problems for the Laplacian in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> . The normal derivative jump problem at the Hilbert space the normal derivative elements of which have the jump through boundary surface was considered in (Nedelec, 1973). Solution of this problem was searched as simple layer potential. At the Hilbert space elements of which as their normal derivatives have the jump through boundary surface only normal derivative jump condition is not sufficient for obtaining of searched function. We have to add to this condition additional internal or external Neumann condition and suggest to look for the solution of this problem the sum of simple and double layer potentials (Polishchuk, 2003).