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Connectivity of phase boundaries in strictly convex domains. (English) Zbl 0911.49025
Let \(\Omega\) be a bounded domain in \({\mathbb R}^n\) and let \(W:{\mathbb R}\rightarrow\) \({\mathbb R}\) be a non-negative “double-well” potential vanishing at two points \(a\) and \(b\). The authors study a model for phase transitions which correspond to stable critical points of the constrained minimization problem \[ \inf_{u\in {\mathcal A}_m}E_\varepsilon (u), \] where \[ E_\varepsilon (u)=\int_\Omega \left(\frac{1}{\varepsilon} W(u)+\frac{\varepsilon}{2} | \nabla u| ^2\right)dx \] and \({\mathcal A}_m\) denotes the class of those \(H^1\)-functions satisfying the additional condition \(\int_\Omega u=m\), for some prescribed real number \(m\). The above problem arises in the study of the van der Waals-Cahn-Hilliard model of phase transitions. The main result of the paper shows that if \(\Omega\) is strictly convex then stable critical points of \(E_\varepsilon\) have a connected, thin transition layer separating the two phases. More exactly, if \(u_\varepsilon\) are critical points of \(E_\varepsilon\) within \({\mathcal A}_m\) having non-negative second variation and satisfying the condition \(\varepsilon E_\varepsilon (u_\varepsilon)\rightarrow 0\) as \(\varepsilon\rightarrow 0\) then this family of minimizers has a connected transition layer bridging the two constant phases \(a\) and \(b\) provided that \(\varepsilon\) is small enough.

49Q20 Variational problems in a geometric measure-theoretic setting
82B26 Phase transitions (general) in equilibrium statistical mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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