Boundary effects in the gradient theory of phase transitions

TL;DR

This paper analyzes how boundary effects influence phase transition interfaces in a scaled van der Waals model, revealing a critical pinning point that balances boundary repulsion and gradient penalization, affecting interface shape and fluctuations.

Contribution

It introduces a critical scaling analysis for boundary pinning in phase transitions, combining asymptotic free energy development with fluctuation probability insights.

Findings

Existence of a critical pinning point scaling.

Balance between boundary repulsion and gradient penalization.

Asymptotic description of interface shape and fluctuations.

Abstract

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.