Abstract
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 language | English (US) |
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Pages (from-to) | 833-888 |
Number of pages | 56 |
Journal | Communications on Pure and Applied Mathematics |
Volume | 65 |
Issue number | 6 |
DOIs | |
State | Published - Jun 2012 |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics