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The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. I: Mathematical analysis. (English) Zbl 0797.35172
The paper studies the Cahn-Hilliard equation for phase separation, that models quenching of binary alloys. The problem is modeled by a parabolic variational inequality, that is the limit problem for the Cahn-Hilliard equation with a double well free energy, when the temperature decreases below the critical temperature. The corresponding limit free energy is a double well with infinite walls.
The global existence, regularity and asymptotic behavior for a weak formulation that possesses a Lyapunov functional is proven. The existence of stationary solutions and the limit as the phenomenological parameter converges to zero is studied. The existence of a sequence of minimizers is proven, minimizers that converge to a piecewise \(\pm 1\) limit, that has a minimal interface between the two sets.
The problem is similar to a Stefan problem with surface tension.

MSC:
35R35 Free boundary problems for PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
80A22 Stefan problems, phase changes, etc.
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