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 ε (u):=ε 3 Ω |D 2 u| 2 +1 ε Ω W(u)+λ ε Ω 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 ελ ε 2 3 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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