Coincidence criteria and junction obstructions for two partially segregated elliptic systems

Abstract

We compare two singularly perturbed elliptic systems with the same partial-segregation constraint but different limiting mechanisms. System~A is determined by harmonic-difference identities and admits an explicit lower-envelope limit, whereas System~B converges, along subsequences, to a constrained Dirichlet-energy minimizer. We show that the System~A profile need not be stationary for the limiting variational problem. We establish conditional coincidence criteria, give explicit data for which the interfaces differ, and derive a planar triple-junction obstruction. In particular, coincidence with the standard variational cone requires the triangle formed by the harmonic gradients to be equilateral. This condition is necessary but not sufficient.

arXiv commentSubstantially revised: new title and abstract; the earlier general coincidence claim has been replaced by conditional coincidence criteria, noncoincidence examples, and a planar triple-junction obstruction