Calculus of Variations and Geometric Measure Theory
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L. Bertini - P. Buttà - A. Garroni

Boundary effects in the gradient theory of phase transition

created by garroni on 31 Dec 2010

[BibTeX]

Preprint

Inserted: 31 dec 2010

Year: 2010

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.

Keywords: Gradient theory of phase transitions, Development by Gamma-convergence, Boundary layer


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