# Phase transition for potentials of high-dimensional wells

Fang-Hua Lin, Xing Bin Pan, Changyou Wang

Research output: Contribution to journalArticle

### 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) 833-888 56 Communications on Pure and Applied Mathematics 65 6 https://doi.org/10.1002/cpa.21386 Published - Jun 2012

### Fingerprint

High-dimensional
Phase Transition
Phase transitions
Boundary conditions
Connecting Orbits
Zeroth
Harmonic Maps
Global Minimum
Term
Potential Function
Energy
Minimizer
Submanifolds
Dirichlet Boundary Conditions
Dirichlet
Hypersurface
Disjoint
Orbits
First-order

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

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 journalArticle

Lin, Fang-Hua ; Pan, Xing Bin ; Wang, Changyou. / Phase transition for potentials of high-dimensional wells. In: Communications on Pure and Applied Mathematics. 2012 ; Vol. 65, No. 6. pp. 833-888.
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