Minimization of a Ginzburg–Landau functional with mean curvature operator in 1-D

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.