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Phase transition with the line-tension effect. (English) Zbl 0915.76093

Summary: We make the connection between the geometric model for capillarity with line tension and the Cahn-Hilliard model of two-phase fluids. To this aim, we consider the energy \(F_\varepsilon(u):= \varepsilon \int_\Omega| Du|^2+ {1\over \varepsilon} \int_\Omega W(u)+ \lambda \int_{\partial\Omega} V(u)\), where \(u\) is a scalar density function and \(W\) and \(V\) are double-well potentials. We show that the behaviour of \(F_\varepsilon\) in the limit \(\varepsilon\to 0\) and \(\lambda\to\infty\) depends on the limit of \(\varepsilon\log\lambda\). If this limit is finite and strictly positive, then the singular limit of the energy \(F_\varepsilon\) leads to a coupled problem of bulk and surface phase transitions, and under certain assumptions agrees with the relaxation of the capillary energy with line tension.

MSC:

76T99 Multiphase and multicomponent flows
76B45 Capillarity (surface tension) for incompressible inviscid fluids
82B26 Phase transitions (general) in equilibrium statistical mechanics
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