  We study the structure of the constrained minimizers of the
Gates-Lebowitz-Penrose free-energy functional ${\mathcal F}_{\rm GLP}(m)$,
non-local functional of a density field $m(x)$, $x\in {\mathcal T}_L$, a
$d$-dimensional torus of side length $L$. At low temperatures, ${\mathcal
F}_{\rm GLP}$ is not convex, and has two distinct global minimizers,
corresponding to two equilibrium states. Here we constrain the average density
$L^{-d}\int_{{\cal T}_L}m(x)\dd x$ to be a fixed value $n$ between the
densities in the two equilibrium states, but close to the low density
equilibrium value. In this case, a "droplet" of the high density phase may or
may not form in a background of the low density phase, depending on the values
$n$ and $L$. We determine the critical density for droplet formation, and the
nature of the droplet, as a function of $n$ and $L$. The relation between the
free energy and the large deviations functional for a particle model with
long-range Kac potentials, proven in some cases, and expected to be true in
general, then provides information on the structure of typical microscopic
configurations of the Gibbs measure when the range of the Kac potential is
large enough.
