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

Abstract

We consider a three-dimensional elliptic equation with two singular coefficients, for which a nonlocal problem is studied in a semi-infinite parallelepiped. The study of the problem is carried out using the method of separation of Fourier variables and spectral analysis. For the problem posed, using the Fourier method, two one-dimensional spectral problems are obtained. Based on the completeness property of the systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution to the problem is constructed in the form of the sum of a double Fourier series with respect to trigonometric and Bessel functions. 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.