Phase transition for potentials of high-dimensional wells
Article first published online: 6 JAN 2012
Copyright © 2012 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 65, Issue 6, pages 833–888, June 2012
How to Cite
Lin, F., Pan, X.-B. and Wang, C. (2012), Phase transition for potentials of high-dimensional wells. Comm. Pure Appl. Math., 65: 833–888. doi: 10.1002/cpa.21386
- Issue published online: 21 MAR 2012
- Article first published online: 6 JAN 2012
- Manuscript Received: APR 2011
For a potential function that attains its global minimum value at two disjoint compact connected submanifolds N± in , we discuss the asymptotics, as ϵ 0, of minimizers uϵ of the singular perturbed functional under suitable Dirichlet boundary data . In the expansion of Eϵ (uϵ) with respect to , 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.