On some singularly perturbed elliptic systems modeling partial segregation: uniform H\"older estimates and basic properties of the limits

TL;DR

This paper establishes uniform H"older regularity for solutions of singularly perturbed elliptic systems with ternary interactions, revealing new geometric patterns and properties of the limiting partially segregated profiles.

Contribution

It proves uniform H"older estimates for a class of competition-diffusion systems with three-way interactions, advancing understanding of their regularity and limit behavior.

Findings

Uniform H"older estimates hold throughout the approximation process.

Limiting profiles are partially segregated, leading to new geometric phenomena.

The results apply to models of ultracold gases and multicomponent liquids.

Abstract

We prove uniform H\"older estimates in a class of singularly perturbed competition-diffusion elliptic systems, with the particular feature that the interactions between the components occur three by three (ternary interactions). These systems are associated to the minimization of Gross-Pitaevski energies modeling ternary mixture of ultracold gases and other multicomponent liquids and gases. We address the question whether this regularity holds uniformly throughout the approximation process up to the limiting profiles, answering positively. A very relevant feature of limiting profiles in this process is that they are only partially segregated, giving rise to new phenomena of geometric pattern formation and optimal regularity.