Double- and simple-layer potentials for generalized singular elliptic equations and their applications to the solving the Dirichlet problem

Abstract: Potentials play an important role in solving boundary value problems for elliptic equations. In the middle of the last century, a potential theory was constructed for a two-dimensional elliptic equation with one singular coefficient. In the study of potentials, the properties of the fundamental solutions of the given equation are essentially and fruitfully used. At the present time, fundamental solutions of a multidimensional elliptic equation with several singular coefficients are already known. In this paper, we investigate the double- and simple-layer potentials for this kind of elliptic equations. Results from potential theory allow us to represent the solution of the boundary value problems in integral equation form. By using a decomposition formula and other identities for the Lauricella's hypergeometric function in many variables, we prove limiting theorems and derive integral equations concerning a densities of the double- and simple-layer potentials. The obtained results are applied to find an explicit solution of the Dirichlet problem for the generalized singular elliptic equation in the some part of the multidimensional ball.