Higher-order phase transitions with line-tension effect
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 603-647

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 linéaire 4 (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 ${ℱ}_{\epsilon }\left(u\right):={\epsilon }^{3}{\int }_{\Omega }{|{D}^{2}u|}^{2}+\frac{1}{\epsilon }{\int }_{\Omega }W\left(u\right)+{\lambda }_{\epsilon }{\int }_{\partial \Omega }V\left(Tu\right),$ 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 $\epsilon {\lambda }_{\epsilon }^{\frac{2}{3}}\sim 1$.

DOI : https://doi.org/10.1051/cocv/2010018
Classification:  49Q20,  49J45,  58E50,  76M30
Keywords: gamma limit, functions of bounded variations, functions of bounded variations on manifolds, phase transitions
@article{COCV_2011__17_3_603_0,
author = {Galv\~ao-Sousa, Bernardo},
title = {Higher-order phase transitions with line-tension effect},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {603-647},
doi = {10.1051/cocv/2010018},
zbl = {1228.49048},
mrnumber = {2826972},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_603_0}
}

Galvão-Sousa, Bernardo. Higher-order phase transitions with line-tension effect. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 603-647. doi : 10.1051/cocv/2010018. http://www.numdam.org/item/COCV_2011__17_3_603_0/

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