We study the Landau damping of Bogoliubov excitations in two- and three-dimensional optical lattices at finite temperatures, extending our recent work on one-dimensional (1D) optical lattices. We use a Bose-Hubbard tight-binding model and the Popov approximation to calculate the temperature dependence of the number of condensate atoms
nc0(T) in each lattice well. As with 1D optical lattices, damping only occurs if the Bogoliubov excitations exhibit anomalous dispersion (i.e., the excitation energy bends upward at low momentum), analogous to the case of phonons in superfluid
4He. This leads to the disappearance of all damping processes in a
D-dimensional simple cubic optical lattice when $U{n}^{\mathrm{c}0}\ensuremath{\geqslant}6DJ$, where
U is the on-site interaction, and
J is the hopping matrix element.