  We consider the Euclidean $D$-dimensional $-\lambda |\phi |^4+\eta |\phi |^6$
($\lambda ,\eta >0 $) model with $d$ ($d\leq D$) compactified dimensions.
Introducing temperature by means of the Ginzburg--Landau prescription in the
mass term of the Hamiltonian, this model can be interpreted as describing a
first-order phase transition for a system in a region of the $D$-dimensional
space, limited by $d$ pairs of parallel planes, orthogonal to the coordinates
axis $x_1, x_2, ..., x_d$. The planes in each pair are separated by distances
$L_1, L_2, ..., L_d$. We obtain an expression for the transition temperature as
a function of the size of the system, $% T_c(\{L_i\})$, $i=1, 2, ..., d$. For
D=3 we particularize this formula, taking $L_1=L_2=... =L_d=L$ for the
physically interesting cases $d=1$ (a film), $d=2$ (an infinitely long wire
having a square cross-section), and for $d=3$ (a cube). For completeness, the
corresponding formulas for second-order transitions are also presented.
Comparison with experimental data for superconducting films and wires shows
qualitative agreement with our theoretical expressions
