In this work we show that the eigenvalues of the Laplacian with homogeneous Neumann B.C. cannot be positive and we solve the Dirichlet
BVP for the Laplace equation ∆u = 0 in the region between two concentric spheres (r1 = 1 and r2 = 2). Finally, we show the simplifications
in the solution if Dirichlet boundary data are independent of the Azimuthal angle.