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The gradient theory of the phase transitions in Cahn-Hilliard fluids with Dirichlet boundary conditions. (English) Zbl 0847.49018

Summary: We are interested in the asymptotic behavior of minimizers (as \(\varepsilon\to 0\)) of the variational problems under the Dirichlet condition \[ \inf\Biggl\{ \int_\Omega \Biggl[\varepsilon |\nabla u|^2+ {1\over \varepsilon} W(x, u)\Biggr] dx \Biggl|u\in W^{1,2} (\Omega; \mathbb{R}^n), u= g\text{ on } \partial\Omega\Biggr\}, \] where \(W(x,\cdot)\) is a nonnegative function with only two zeros \(\alpha\) and \(\beta\). Here \(\alpha\) and \(\beta\) are independent of the space variable \(x\). In this paper, we show that the limit of a sequence of minimizers \(\{u_\varepsilon\}_{\varepsilon> 0}\) (as \(\varepsilon\to 0\)) is a solution of another variational problem without boundary condition. However, the limit variational problem has a boundary integral corresponding to transition layers near \(\partial\Omega\). Our analysis relies mainly on the theory of gamma convergence. In order to overcome the difficulty of inhomogeneity of the boundary condition, we approximate \(g(x)\) by suitable piecewise smooth functions near the boundary \(\partial\Omega\).

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
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