For a potential function equation image that attains its global minimum value at two disjoint compact connected submanifolds N± in equation image, we discuss the asymptotics, as ϵ [RIGHTWARDS ARROW] 0, of minimizers uϵ of the singular perturbed functional equation image under suitable Dirichlet boundary data equation image. In the expansion of Eϵ (uϵ) with respect to equation image, we identify the first-order term by the area of the sharp interface between the two phases, an area-minimizing hypersurface Γ, and the energy cmath image of minimal connecting orbits between N+ and N, and the zeroth-order term by the energy of minimizing harmonic maps into N± both under the Dirichlet boundary condition on ∂Ω and a very interesting partially constrained boundary condition on the sharp interface Γ. © 2012 Wiley Periodicals, Inc.