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Higher-order phase transitions with line-tension effect

Published online by Cambridge University Press:  23 April 2010

Bernardo Galvão-Sousa
Bernardo Galvão-Sousa*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton ON L8S 4K1, Canada. beni@mcmaster.ca
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Abstract

The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal.144 (1998) 1–46] for a first-order perturbation model. This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies

$\[{\cal F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu),\]$

where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon\lambda_{\varepsilon}^{\frac{2}{3}} \sim 1$.

Keywords

Gamma limitfunctions of bounded variationsfunctions of bounded variations on manifoldsphase transitions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 603 - 647
Copyright
© EDP Sciences, SMAI, 2010

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