  A novel Thomas-Fermi (TF) approach to inhomogeneous superfluid Fermi-systems
is presented and shown that it works well also in cases where the Local Density
Approximation (LDA) breaks down. The novelty lies in the fact that the
semiclassical approximation is applied to the pairing matrix elements not
implying a local version of the chemical potential as with LDA. Applications
will be given to the generic fact that if a fermionic superfluid in the BCS
regime overflows from a narrow container into a much wider one, pairing is
substantially reduced at the overflow point. Two examples pertinent to the
physics of the outer crust of neutron stars and superfluid fermionic atoms in
traps will be presented. The TF results will be compared to quantal and LDA
ones.
