Phase transition for potentials of high-dimensional wells

Fanghua Lin, Xing Bin Pan, Changyou Wang

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)833-888
Number of pages56
JournalCommunications on Pure and Applied Mathematics
Issue number6
StatePublished - Jun 2012

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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