Nonlocal boundary value problems for a three-dimensional elliptic equation with singular coefficients in a semi-infinite parallelepiped

Abstract

The investigated two nonlocal problems for an elliptic equation with two singular coefficients in a semi-infinite parallelepiped. The proof of the uniqueness of the solution and its construction is carried out by the method of spectral analysis. The solution to the problem is constructed as the sum of the biorthogonal series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

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].…”

“…where r ∈ (0, 1), b 3 and b 4 are arbitrary constants. It follows from (29) that a solution of equation ( 18) satisfying the first condition (19) exists at Rev ≥ 1/2 + β + γ and it is defined by equality…”

“…Now taking into account that v l , l ∈ N − known numbers defined by equations (33), we find the values of the parameter λ from (30). To this end, substituting (30) into the second condition (19), we obtain…”