Identical Free Boundaries in two partially Segregated Systems

TL;DR

This paper demonstrates that two different singularly perturbed elliptic systems modeling phase segregation share identical free boundary geometries in their limits, highlighting the influence of structural properties over solution values.

Contribution

It proves that different models of phase segregation have the same free boundary configurations in the limit, emphasizing the role of structural properties.

Findings

Limiting free boundaries are identical despite different formulations.

Interface geometry depends on structural properties, harmonicity, and boundary data.

Numerical experiments support the theoretical results.

Abstract

We compare two singularly perturbed elliptic systems modeling partially phase segregation. Although the formulations are fundamentally different, we prove that their limiting configurations have identical free boundaries. The result shows that interface geometry depends only on basic structural properties of the limit segregation, harmonicity in positivity sets, and boundary data, while the limiting solution values may differ. Numerical experiments confirm the theoretical findings.