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Transition layer for the heterogeneous Allen–Cahn equation
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Mahmoudi, Fethi, Malchiodi, A. and Wei, Juncheng (2008) Transition layer for the heterogeneous Allen–Cahn equation. Annales de l'Institut Henri Poincare (C) Non Linear Analysis , Vol.25 (No.3). pp. 609-631. doi:10.1016/j.anihpc.2007.03.008 ISSN 0294-1449.
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Official URL: http://dx.doi.org/10.1016/j.anihpc.2007.03.008
Abstract
We consider the equation
(1)
where Ω is a smooth and bounded domain in Rn, ν the outer unit normal to ∂Ω, and a a smooth function satisfying −1<a(x)<1 in . We set K, Ω+ and Ω− to be respectively the zero-level set of a, {a>0} and {a<0}. Assuming ∇a≠0 on K and a≠0 on ∂Ω, we show that there exists a sequence εj→0 such that Eq. (1) has a solution uεj which converges uniformly to ±1 on the compact sets of Ω± as j→+∞. This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for n=2
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Annales de l'Institut Henri Poincare (C) Non Linear Analysis | ||||
Publisher: | Elsevier Masson | ||||
ISSN: | 0294-1449 | ||||
Official Date: | 2008 | ||||
Dates: |
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Volume: | Vol.25 | ||||
Number: | No.3 | ||||
Page Range: | pp. 609-631 | ||||
DOI: | 10.1016/j.anihpc.2007.03.008 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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