Connectivity of Phase Boundaries in Strictly Convex Domains

Abstract

We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem \(\inf_{\int_{\Omega} u\,dx=m} \int_{\Omega} \frac{1}{\varepsilon}W(u)+\frac{\varepsilon}{2} \big|\nabla u\big|^2 dx.\) Here W is a double‐well potential and \(\Omega\subset\R^n\) is a strictly convex domain. For e small, this is closely related to the problem of partitioning Ω into two subdomains of fixed volume, where the subdomain boundaries correspond to the transitional boundary between phases. Motivated by this geometry problem, we show that in a strictly convex domain, stable critical points of the original variational problem have a connected, thin transition layer separating the two phases. This relates to work in [GM] where special geometries such as cylindrical domains were treated, and is analogous to the results in [CHo] which show that in a convex domain, stable critical points of the corresponding unconstrained problem are constant. The proof of connectivity employs tools from geometric measure theory including the co‐area formula and the isoperimetric inequality on manifolds. The thinness of the transition layer follows from a separate calculation establishing spatial decay of solutions to the pure phases.