  Recent experiments on strongly-interacting bosons in optical lattices have
revealed the co-existence of spatially-separated Mott-insulating and
number-fluctuating phases. The description of this inhomogeneous situation is
the topic of this Letter. We establish that the number-fluctuating phase forms
a superfluid trapped between the Mott-insulating regions and derive the
associated collective mode structure. We discuss the interlayer's crossover
between two- and three-dimensional behavior as a function of the lattice
parameters and estimate the critical temperatures for the transition of the
superfluid phase to a normal phase.
