• DOI:10.1137/110821469
• Corpus ID: 17618779

Boundary Effects in the Gradient Theory of Phase Transitions

@article{Bertini2011BoundaryEI,
  title={Boundary Effects in the Gradient Theory of Phase Transitions},
  author={Lorenzo Bertini and Paolo Butt{\`a} and Adriana Garroni},
  journal={SIAM J. Math. Anal.},
  year={2011},
  volume={44},
  pages={926-945}
}

• L. Bertini, P. Buttà, A. Garroni
• Published 3 January 2011
• Mathematics
• SIAM J. Math. Anal.

We consider the van der Waals' free energy functional, with a scaling small parameter epsilon, in the plane domain given by the first quadrant, and inhomogeneous Dirichlet boundary conditions. The boundary data are chosen in such a way that the interface between the pure phases tends to be horizontal and is pinned at some point on the y-axis which approaches zero as epsilon converges to zero. We show that there exists a critical scaling for the pinning point, such that, as the small parameter epsilon tends to zero, 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.