Phase transition for potentials of high‐dimensional wells

Source abstract

Abstract For a potential function $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}F:\R^k \to\R _ +$ that attains its global minimum value at two disjoint compact connected submanifolds N ± in $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}\R^k$ , we discuss the asymptotics, as ϵ → 0, of minimizers u ϵ of the singular perturbed functional ${\bf E}_\varepsilon (u) = \int_\Omega {(|\nabla u|^2 + {1 \over {\varepsilon ^2 }}F(u))} dx$ under suitable Dirichlet boundary data $\font\open=msbm10 at 10pt\def\R{\hbox{\open R}}g_\varepsilon :\partial \Omega \to\R ^k$ . In the expansion of E ϵ ( u ϵ ) 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 c 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.