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

Abstract

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 −Δu = 1 in B1∖B0¯ 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 B1∖B0¯ is maximal if and only if the two balls are concentric. The formalism of Damek-Ricci harmonic spaces is used.