Phase transition for potentials of high‐dimensional wells

Abstract

AbstractFor 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 uϵ 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 Eϵ (uϵ) 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.