Boundary effects in the gradient theory of phase transitions (Q2904731)

In the paper it is considered the van der Waals' free energy functional, with scaling parameter \(\varepsilon\): \(E_\varepsilon(u)=\int[\varepsilon|\nabla u|^2+\varepsilon^{-1}V(u)]dr\), in the plane domain \(\mathbb{R}_+ \times \mathbb{R}_+\), with inhomogeneous Dirichlet boundary conditions. In the paper it is taken \(V(u)=(u^2-1)^2\). It is imposed the two stable phases on the horizontal boundaries \(\mathbb{R}_+\times\{0\}\) and \(\mathbb{R}_+\times\{+\infty\}\), and free boundary conditions on \(\{+\infty\}\times \mathbb{R}_+\). The datum on \(\{0\}\times \mathbb{R}_+\) is chosen in such a way that the interface between the pure phases is pinned at some point \((0, y)\). It is established the existence of a critical scaling, \(y = y_\varepsilon\), such that, as \(\varepsilon\to\;0\), 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. One of the key points in this result is an asymptotic development of the free energy functional.