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A three‐layered minimizer in R 2 for a variational problem with a symmetric three‐well potential
Author(s) -
Bronsard Lia,
Gui Changfeng,
Schatzman Michelle
Publication year - 1996
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199607)49:7<677::aid-cpa2>3.0.co;2-6
Subject(s) - mathematics , mathematical analysis , calculus (dental) , pure mathematics , orthodontics , medicine
Let W be a potential on R 2 which is equivariant by the symmetry group of the equilateral triangle and has three minima. We show that the elliptic system $$-\Delta U + D W (U)^T = 0,$$ possesses a nontrivial smooth solution U : R 2 → R 2 . Here DW (U) T is the transpose of the derivative DW ( U ). The natural energy of the problem is unbounded and compactness techniques cannot be applied. The proof depends on careful energy estimates and asymptotics for several one‐dimensional problems and for two‐dimensional problems on bounded domains. © 1996 John Wiley & Sons, Inc.