  We study the effect of spatial inhomogeneity on the physics of a strongly
correlated electron system exhibiting a metallic phase and a Mott insulating
phase, represented by the simple Hubbard model. In three dimensions, we
consider various geometries, including vacuum-metal-vacuum, a junction between
a weakly and a strongly correlated metal, and finally the double junctions
metal-Mott insulator-metal and metal-strongly correlated metal- metal. We
applied to these problems the self-consistent Gutzwiller technique recently
developed in our group, whose approximate nature is compensated by an extreme
flexibility,ability to treat very large systems, and physical transparency. The
main general result is a clear characterization of the position dependent
metallic quasiparticle spectral weight. Its behavior at interfaces reveals the
ubiquitous presence of exponential decays and crossovers, with decay lengths of
clear physical significance. The decay length of metallic strength in a
weakly-strongly correlated metal interface is due to poor screening in the
strongly correlated side. The decay length of metallic strength from a metal
into a Mott insulator (or into vacuum) is due to tunneling. In both cases, the
decay length is a bulk property, and diverges with a critical exponent ($\sim
1/2$ in the present approximation, mean field in character) as the (continuous,
paramagnetic) Mott transition is approached.
