The Dirichlet Problem for an Elliptic Equation with Singular Coefficients in a Semi-Cylindrical Domain

Abstract

We consider a three-dimensional equation of elliptic type with two singular coefficients, for which the Dirichlet problem is studied in a half-cylinder. The study of the problem is carried out using the method of spectral analysis. For the problem posed, using the Fourier method, one-dimensional spectral problems are obtained. Based on the completeness property of systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution of the problem under study is constructed in the form of the sum of the Fourier–Bessel 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.