Double- and Simple-Layer Potentials for a Three-Dimensional Elliptic Equation with a Singular Coefficient and Their Applications

The double- and simple-layer potentials play an important role in solving boundary value problems for elliptic equations. The desired solution represents a potential of a certain layer with unknown density. One finds it with the help of the theory of Fredholm integral equations of the second kind. The potential, in turn, is expressed via the fundamental solution to the given elliptic equation. At present, fundamental solutions to Helmholtz multidimensional singular equations are known, nevertheless, the potential theory is constructed only for two-dimensional degenerate equations. In this paper, we study the mentioned potentials for a three-dimensional elliptic equation with one singular coefficient and apply the obtained results to solving the Dirichlet problem.

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