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Gamma Convergence of Partially Segregated Elliptic Systems

Farid Bozorgnia, Avetik Arakelyan

TL;DR

The paper proves that penalized energies modeling partially segregated, three-component elliptic systems in planar domains Gamma-converge to a constrained Dirichlet energy as the penalty parameter vanishes. The authors establish a liminf inequality via compactness and lower semicontinuity, and construct recovery sequences by a detailed geometric decomposition into bulk, interface, and triple-junction regions, including boundary layers and curvature corrections. They treat Type I–III interfaces with TanH-type transition profiles and Type IV triple junctions with polar-coordinate sector constructions, showing interface energies scale as $O(\sqrt{\varepsilon})$ and junction energies as $O(\varepsilon^{1/4})$, respectively. A global partition of unity assembles the recovery sequences while preserving the constraint $u_1^\varepsilon u_2^\varepsilon u_3^\varepsilon=0$, thereby yielding the desired limsup inequality. Together, these results provide a rigorous variational bridge between penalized and constrained formulations for three-phase segregation and pave the way for numerical methods addressing partial segregation in multi-species or multi-phase systems.

Abstract

We study partially segregated elliptic systems through the use of penalized energy functionals. These systems arise from the minimization of Gross-Pitaevskii-type energies that capture the behavior of multi-component ultracold gas mixtures and other systems involving multiple interacting fluid or gas species. In the case when the domain is planar, i.e., in $\mathbb{R}^2$, our main result is the Gamma convergence of penalized energy to the constrained Dirichlet energy with strict segregation. The proof combines lower semicontinuity arguments with a recovery sequence construction based on geometric decompositions near interfaces and triple junctions. This establishes a rigorous variational link between the penalized and constrained formulations.

Gamma Convergence of Partially Segregated Elliptic Systems

TL;DR

The paper proves that penalized energies modeling partially segregated, three-component elliptic systems in planar domains Gamma-converge to a constrained Dirichlet energy as the penalty parameter vanishes. The authors establish a liminf inequality via compactness and lower semicontinuity, and construct recovery sequences by a detailed geometric decomposition into bulk, interface, and triple-junction regions, including boundary layers and curvature corrections. They treat Type I–III interfaces with TanH-type transition profiles and Type IV triple junctions with polar-coordinate sector constructions, showing interface energies scale as and junction energies as , respectively. A global partition of unity assembles the recovery sequences while preserving the constraint , thereby yielding the desired limsup inequality. Together, these results provide a rigorous variational bridge between penalized and constrained formulations for three-phase segregation and pave the way for numerical methods addressing partial segregation in multi-species or multi-phase systems.

Abstract

We study partially segregated elliptic systems through the use of penalized energy functionals. These systems arise from the minimization of Gross-Pitaevskii-type energies that capture the behavior of multi-component ultracold gas mixtures and other systems involving multiple interacting fluid or gas species. In the case when the domain is planar, i.e., in , our main result is the Gamma convergence of penalized energy to the constrained Dirichlet energy with strict segregation. The proof combines lower semicontinuity arguments with a recovery sequence construction based on geometric decompositions near interfaces and triple junctions. This establishes a rigorous variational link between the penalized and constrained formulations.

Paper Structure

This paper contains 10 sections, 8 theorems, 127 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. The Minimization Problem and Known Results
  4. $\Gamma$ - Convergence Analysis
  5. Lower Semicontinuity
  6. Construction of Recovery Sequence
  7. Analysis of Interfaces of Type I - III
  8. Analysis of Interfaces of Type IV: Junction Points
  9. Explicit Partition of Unity and Constraint Verification
  10. Conclusion

Key Result

Theorem 2.1

Let $u^{\varepsilon} = (u_{1}^{\varepsilon}, u_{2}^{\varepsilon}, u_{3}^{\varepsilon})$ be a solution of eq:penalized_system for fixed $\varepsilon > 0$. Suppose that assumptions (h1) and (h2) hold. Then for every compact set $K \subset\subset \Omega$ and for every $\alpha \in (0, \frac{3}{4})$, Moreover, as $\varepsilon \to 0$, we have (up to a subsequence):

Figures (1)

  • Figure 1: Schematic picture of the junction $p_j$ with sectors $S_1,S_2,S_3$, small transparent boundary sectors on top, and junction region ball $B_{\sqrt{\varepsilon}}(p_j)$. The transparent purple sectors are the transition regions. The bulk regions are defined as the remaining chunks of $S_i$ sectors, without transition regions.

Theorems & Definitions (17)

  • Theorem 2.1: Optimal uniform bounds in Hölder spaces soave2024partial1soave2024partial2
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6: Existence for Constrained Problem
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 7 more