Nonlocal Problem for an Elliptic Equation with Singular Coefficients in a Semi-infinite Parallelepiped

Lauricella hypergeometric function and its application to the solution of the Neumann problem for a multidimensional elliptic equation with several singular coefficients in an infinite domain

“…We will denote via u = u 1 − u 2 . Then it is clear that the function u satisfies the equation ( 2) and homogeneous boundary conditions (8) and condition (9).…”

“…We note the joint work of M.Salakhitdinov and A.Hasanov [7], in which solutions of the Dirichlet and Holmgren problems for a multidimensional elliptic equation with one singular coefficient in the half-space were found in an explicit forms. For three-and four-dimensional elliptic equations with two [8], three [9] and four [10] singular coefficients boundary value problems in infinite domains are investigated. Recently [11], a solution to the external Dirichlet problem for an equation (2) in the hyperoctant Ω is found explicitly.…”

Dirichlet-Neumann Spectral Problems for Three-Dimensional Elliptic Equations With Singular Coefficients

“…Many studies have been devoted to finding eigenvalues and eigenfunctions of boundary value problems for various equations of elliptic and mixed types in the plane, among which the works [13], [14], and others should be mentioned. Problems of this type for three-dimensional elliptic and mixed equations are also well studied, for example, in works [15], [16], [17], [18], [19], [20], [21].…”

“…Based on formulas (9), let us find a general solution of equation ( 31) and, taking into account the introduced notations, we obtain a general solution of equation (20) in the form…”

“…Assuming in (32) v = v l , l ∈ N, we obtain the eigenfunctions of the problem { (20), (21)} corresponding to eigenvalues χ l :…”