  We consider a many-body system of fermionic atoms interacting via a local
pair potential and subject to an external potential within the framework of BCS
theory. We measure the free energy of the whole sample with respect to the free
energy of a reference state which allows us to define a BCS functional with
boundary conditions at infinity. Our main result is a lower bound for this
energy functional in terms of expressions that typically appear in
Ginzburg-Landau functionals.
