A continuum model of a two-phase crystal-crystal system is constructed in which the structure of the interface between the phases is determined by energy minimization, rather than by being specified a priori. The interfacial structure is parameterized by a variable Ĥ corresponding to the jump in the surface deformation gradient (or strain) at the interface, so that coherence is defined locally by the condition Ĥ = 0. The energy of the system is taken to be the sum of the bulk and interfacial energies, where the interfacial energy density fxs depends on Ĥ. In order to explore how the equilibrium interfacial structure depends on the function fxs(Ĥ), a model system consisting of an elastic film on a rigid substrate is studied, and the interfacial energy density is taken to be nonconvex with a sharp minimum associated with coherence. In this case, it can be shown that the energy of the system is driven to its infimum by separating the interface into coherent and incoherent regions, which may be viewed as a continuum analog to a partially coherent interface. Further, this solution only appears above a certain critical thickness of the film, in agreement with misfit dislocation models of partially coherent interfaces.