Phase transition for potentials of high-dimensional wells

For a potential function F : R{double-struck} ^k → R{double-struck} ₊ that attains its global minimum value at two disjoint compact connected submanifolds N ^± in R{double-struck} ^k, we discuss the asymptotics, as ε{lunate} → 0, of minimizers u _ε{lunate} of the singular perturbed functional E _ε(u)=∫ _ω (|∇u| ²+ 1/∈2 F(u))dx under suitable Dirichlet boundary data g _∈ : ∂Ω → R{double-struck} ^k. In the expansion of E _ε{lunate} (u _ε{lunate}) with respect to ${1 \over \varepsilon }$, 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 c0F 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.