  The aim of this paper is to investigate the minimization problem related to a
Ginzburg-Landau energy functional, where in particular a nonlinear diffusion of
mean curvature-type is considered, together with a classical double well
potential. A careful analysis of the corresponding Euler-Lagrange equation,
equipped with natural boundary conditions and mass constraint, leads to the
existence of an unique Maxwell solution, namely a monotone increasing solution
obtained for small diffusion and close to the so-called Maxwell point. Then, it
is shown that this particular solution (and its reversal) has least energy
among all the stationary points satisfying the given mass constraint. Moreover,
as the viscosity parameter tends to zero, it converges to the increasing
(decreasing for the reversal) single interface solution, namely the constrained
minimizer of the corresponding energy without diffusion. Connections with
Cahn-Hilliard models, obtained in terms of variational derivatives of the total
free energy considered here, are also presented.
