Dirichlet Problem for the Laplace Equation in the Hyperoctant of a Multidimensional Ball

The present work is devoted to study the boundary value problems for a multidimensional Laplace equation with the Dirichlet conditions on the lateral faces of the ball’s hyperoctant and on the sphere’s hyperoctant. The main result is establishment of necessary and sufficient conditions for unique solvability of the boundary value problems. An energy integral method and Green’s function method are used in the proof of the main result. All solutions are found in an explicit forms, which contain singular solution of Laplace equation.