Boundary Effects in the Gradient Theory of Phase Transitions

We consider the van der Waals’ free energy functional, with scaling parameter ε, in the plane domain R+ × R+, with inhomogeneous Dirichlet boundary conditions. We impose the two stable phases on the horizontal boundaries R+×{0} and R+×{∞}, and free boundary conditions on {∞}×R+. Finally, the datum on {0} ×R+ is chosen in such a way that the interface between the pure phases is pinned at some point (0, y). We show that there exists a critical scaling, y = yε, such that, as ε → 0, the competing effects of repulsion from the boundary and penalization of gradients play a role in determining the optimal shape of the (properly rescaled) interface. This result is achieved by means of an asymptotic development of the free energy functional. As a consequence, such analysis is not restricted to minimizers but also encodes the asymptotic probability of fluctuations.