We consider the two dimensional van der Waals' free energy
functional, with scaling parameter epsilon, in the positive quadrant with
inhomogeneous Dirichlet boundary conditions. We impose the two stable phases
on the horizontal boundaries and free boundary conditions on the vertical right
boundary. Finally, the datum on the left vertical side is chosen in such a
way that the interface between the pure phases is pinned at some point (0,y).
We prove the existence of a critical scaling of the pin location y where
the competing effects of repulsion from the boundary and penalization of
gradients both play a role in determining the optimal shape of the interface.
This result develops the study of the boundary layer in the variational theory
of phase separation, which goes back to the 1987 paper by L. Modica.
This is a joint work with L. Bertini and A. Garroni.