Generic configurations in 2D strongly competing systems
arXiv:2411.19867 · doi:10.2422/2036-2145.202412_011
The paper analyzes planar segregation models for many competing species and proves that, generically, the domain splits into regions meeting only at triple junctions, using harmonic map techniques and the Hopf differential.
Abstract
We study a problem modelling segregation of an arbitrary number of competing species in planar domains. The solutions give rise to a well known free boundary problem with the domain partitioning itself into subdomains occupied by different species. In principle, several of them can coexist in a neighborhood of any point. However, we show that {\it generically} the domain partitions into subdomains with only triple junctions, meaning that at most three populations meet at the free boundary. Our main tools are the use of the formalism of harmonic maps into singular spaces and the introduction of a complex structure via the Hopf differential.
to appear on Annali Scuola Normale Superiore di Pisa - Classe di Scienze