We consider the $N$-component Ginzburg-Landau model in the large $N$ limit,
the system being embedded in an external constant magnetic field and confined
between two parallel planes a distance $L$ apart from one another. On physical
grounds, this corresponds to a material in the form of a film in the presence
of an external magnetic field. Using techniques from dimensional and
$zeta$-function regularization, modified by the external field and the
confinement conditions, we investigate the behavior of the system as a function
of the film thickness $L$. This behavior suggests the existence of a minimal
critical thickness below which superconductivity is suppressed.