Connectivity of phase boundaries in strictly convex domains

We consider equilibria arising in a model for phase transitions which correspond to stable critical points of the constrained variational problem (ProQuest: Formulae and/or non-USASCII text omitted; see image) Here W is a double-well potential and (ProQuest: Formulae and/or non-USASCII text omitted; see image) is a strictly convex domain. For ε 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.[PUBLICATION ABSTRACT]