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- Title
LAYERED SOLUTIONS TO THE VECTOR ALLEN-CAHN EQUATION IN R². MINIMIZERS AND HETEROCLINIC CONNECTIONS.
- Authors
Fusco, Giorgio
- Abstract
Let W : Rm → R be a nonnegative potential with exactly two nondegenerate zeros a- ≠ = a+ ∈ Rm. We assume that there are N ≥ 1 distinct heteroclinic orbits connecting a- to a+ represented by maps u1,..., uN that minimize the one-dimensional energy ... . We first consider the problem of characterizing the minimizers u : Rn → Rm of the energy ... Under a nondegeneracy condition on uj, j = 1.,,,.N and in two space dimensions, we prove that, provided it remains away from a- and a+ in corresponding half spaces S- and S+, a bounded minimizer u : R² → Rm is necessarily an heteroclinic connection between suitable translates u-(•-η-) and u+(•-η+) of some ... . Then we focus on the existence problem and assuming N = 2 and denoting u-,u+ the representations of the two orbits connecting a- to a+ we give a new proof of the existence (first proved in [32]) of a solution u : R² → Rm of Δu= Wu(u); that connects certain translates of ... .
- Subjects
NON-degenerate perturbation theory; TIME-independent perturbation theory; QUANTUM perturbations; LINEAR Stark effect; ORBITS (Astronomy)
- Publication
Communications on Pure & Applied Analysis, 2017, Vol 16, Issue 5, p1807
- ISSN
1534-0392
- Publication type
Article
- DOI
10.3934/cpaa.2017088