  We investigate the steady state heat current in two and three dimensional
disordered harmonic crystals in a slab geometry, connected at the boundaries to
stochastic white noise heat baths at different temperatures.The disorder causes
short wavelength phonon modes to be localized so the heat current in this
system is carried by the extended phonon modes which can be either diffusive or
ballistic. Using ideas both from localization theory and from kinetic theory we
estimate the contribution of various modes to the heat current and from this we
obtain the asymptotic system size dependence of the current. These estimates
are compared with results obtained from a numerical evaluation of an exact
formula for the current, given in terms of a frequency transmission function,
as well as from direct nonequilibrium simulations. These yield a strong
dependence of the heat flux on boundary conditions. Our analytical arguments
show that for realistic boundary conditions the conductivity is finite in three
dimensions but we are not able to verify this numerically, except in the case
where the system is subjected to an external pinning potential. This case is
closely related to the problem of localization of electrons in a random
potential and here we numerically verify that the pinned three dimensional
system satisfies Fourier's law while the two dimensional system is a heat
insulator. We also investigate the inverse participation ratio of different
normal modes.
