Two functionals connected to the Laplacian in a class of doubly connected domains on rank one symmetric spaces of non-compact type

Let B (1) be a ball in a non-compact rank-one symmetric space and let B (0) be a smaller ball inside it. It is shown that if y is the solution of the problem -Delta u = 1 in vanishing on the boundary, then the Dirichlet-energy of y is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on is maximal if and only if the two balls are concentric. The formalism of Damek-Ricci harmonic spaces is used.