  Harmonically trapped lattice bosons with strong repulsive interactions
exhibit a superfluid-Mott-insulator heterostructure in the form of a "wedding
cake". We discuss the mesoscopic aspects of such a system within a
one-dimensional scattering matrix approach and calculate the scattering
properties of quasi-particles at a superfluid-Mott-insulator interface as an
elementary building block to describe transport phenomena across such a
boundary. We apply the formalism to determine the heat conductivity through a
Mott layer, a quantity relevant to describe thermalization processes in the
optical lattice setup. We identify a critical hopping below which the heat
conductivity is strongly suppressed.
