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) |
|---|---|
| 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 |
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ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
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Phase transition for potentials of high-dimensional wells. / Lin, Fang-Hua; Pan, Xing Bin; Wang, Changyou.
In: Communications on Pure and Applied Mathematics, Vol. 65, No. 6, 06.2012, p. 833-888.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Phase transition for potentials of high-dimensional wells
AU - Lin, Fang-Hua
AU - Pan, Xing Bin
AU - Wang, Changyou
PY - 2012/6
Y1 - 2012/6
N2 - 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.
AB - 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=84858795932&partnerID=8YFLogxK
U2 - 10.1002/cpa.21386
DO - 10.1002/cpa.21386
M3 - Article
VL - 65
SP - 833
EP - 888
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
SN - 0010-3640
IS - 6
ER -