The Unique Solvability of Boundary Value Problems for a 3D Elliptic Equation With Three Singular Coefficients

We state and study several boundary value problems for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of solutions of the stated problems. For proving the existence of solutions, we use the spectral Fourier method based on separation of variables. We construct solutions to the stated problems in the form of the sum of the double Fourier—Bessel series. We justify the uniform convergence of the constructed series with the help of asymptotic bounds for Bessel functions of real and imaginary arguments. Using these bounds, we estimate each term of the series, which allows us to prove the convergence of the series and its derivatives up to the second order inclusive and to justify the existence theorem in the class of regular solutions.

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