We investigate the critical behavior of the N-component Euclidean lf4 model, in
the large N limit, in three situations: confined between two parallel planes a distance
L apart from one another; confined to an infinitely long cylinder having a
square transversal section of area L2; and to a cubic box of volume L3. Taking the
mass term in the form m0 2 =asT−T0d, we retrieve Ginzburg–Landau models which
are supposed to describe samples of a material undergoing a phase transition,
respectively, in the form of a film, a wire and of a grain, whose bulk transition
temperature sT0d is known. We obtain equations for the critical temperature as
functions of L and of T0, and determine the limiting sizes sustaining the
transition.