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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.

##### MSC:
 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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