Summary: For a potential function \(F: \mathbb{R}^k\to \mathbb{R}_+\) that attains its global minimum value at two disjoint compact connected submanifolds \(N^{\pm}\) in \(\mathbb{R}^k\), we discuss the asymptotics, as \(\varepsilon\to 0\), of minimizers \(u_\varepsilon\) of the singular perturbed functional \[ {\mathbf E}_{\varepsilon}(u)= \int_\Omega\Biggl(|\nabla u|^2+{1\over \varepsilon^2} F(u)\Biggr)\,dx \]
under suitable Dirichlet boundary data \(g_\varepsilon: \partial\Omega\to\mathbb{R}^k\). In the expansion of \({\mathbf E}_{\varepsilon} (u_{\varepsilon})\) 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 \(\Gamma\), and the energy \(c^F_0\) of minimal connecting orbits between \(N^+\) and \(N^-\), and the zeroth-order term by the energy of minimizing harmonic maps into \(N^{\pm}\) both under the Dirichlet boundary condition on \(\partial\Omega\) and a very interesting partially constrained boundary condition on the sharp interface \(\Gamma\).