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Identical Free Boundaries in two partially Segregated Systems

Farid Bozorgnia

TL;DR

The paper addresses whether two distinct singularly perturbed partial segregation models with $m=3$ components yield the same limiting interface geometry. By leveraging Gamma-convergence, harmonic extensions $h_{ij}$, and explicit constructions for System A, alongside variational arguments and interior harmonicity for System B, the authors show that the free boundaries coincide and the domain partition is identical in the limit, though the limiting component values can differ. The key contribution is the demonstration that interface geometry is universal and determined solely by boundary data via the harmonic level sets $\{h_{ij}=0\}$, independent of the formulation. Numerical simulations in two dimensions corroborate the theoretical result, validating the robustness of interface localization across models and guiding practical computations. The findings illuminate the geometric robustness of segregation patterns and provide a framework for analyzing similar multi-component systems under different perturbation schemes.

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.

Identical Free Boundaries in two partially Segregated Systems

TL;DR

The paper addresses whether two distinct singularly perturbed partial segregation models with components yield the same limiting interface geometry. By leveraging Gamma-convergence, harmonic extensions , and explicit constructions for System A, alongside variational arguments and interior harmonicity for System B, the authors show that the free boundaries coincide and the domain partition is identical in the limit, though the limiting component values can differ. The key contribution is the demonstration that interface geometry is universal and determined solely by boundary data via the harmonic level sets , independent of the formulation. Numerical simulations in two dimensions corroborate the theoretical result, validating the robustness of interface localization across models and guiding practical computations. The findings illuminate the geometric robustness of segregation patterns and provide a framework for analyzing similar multi-component systems under different perturbation schemes.

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.
Paper Structure (7 sections, 7 theorems, 41 equations, 3 figures, 2 tables)

This paper contains 7 sections, 7 theorems, 41 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. Basic Facts and Background
  4. Main Results
  5. Interface Characterization
  6. Geometric Structure of the Partition
  7. Numerical Simulation

Key Result

Lemma 3.1

BA Let $u^{\varepsilon} = (u^{\varepsilon}_1, u^{\varepsilon}_2, u^{\varepsilon}_3)$ be a minimizer of $E_{\varepsilon}$. Then any limit point $u = (u_1, u_2, u_3)$ in $L^2(\Omega)^3$ as $\varepsilon \to 0$ is a minimizer of the constrained Dirichlet energy:

Figures (3)

  • Figure 1: Direct construction versus penalization method.
  • Figure 2: Direct construction versus penalization method.
  • Figure 3: Direct construction versus penalization method.

Theorems & Definitions (18)

  • Definition 2.2
  • Lemma 3.1
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • Lemma 4.5
  • ...and 8 more