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Regularity of two-phase free boundary minimizers in periodic media

Farhan Abedin, William M Feldman

TL;DR

This work develops a comprehensive regularity theory for two-phase free boundary minimizers in periodic media. By linking the original heterogeneous functional \\mathcal{J}(u,U) to its homogenized form \\mathcal{J}_0 and proving a one-step improvement of flatness for non-degenerate approximate minimizers, the authors derive a large-scale Lipschitz estimate and a Liouville property for entire minimizers. The analysis combines approximate viscosity solutions, harmonic replacement techniques, and a quantitative homogenization framework (upscaling and downscaling) to obtain sharp energy and flatness control across scales, culminating in large-scale Lipschitz regularity and plane-like classifications. The results extend classical Alt-Caffarelli-Friedman-type regularity to periodic media, with significant implications for free boundary problems in heterogeneous environments and potential applications in materials science and fluid interfaces.

Abstract

We study the regularity of minimizers of a two-phase energy functional in periodic media. Our main result is a large scale Lipschitz estimate. We also establish improvement-of-flatness for non-degenerate minimizers, which is a key ingredient in the proof of the Lipschitz estimate. As a consequence, we obtain a Liouville property for entire non-degenerate minimizers.

Regularity of two-phase free boundary minimizers in periodic media

TL;DR

This work develops a comprehensive regularity theory for two-phase free boundary minimizers in periodic media. By linking the original heterogeneous functional \\mathcal{J}(u,U) to its homogenized form \\mathcal{J}_0 and proving a one-step improvement of flatness for non-degenerate approximate minimizers, the authors derive a large-scale Lipschitz estimate and a Liouville property for entire minimizers. The analysis combines approximate viscosity solutions, harmonic replacement techniques, and a quantitative homogenization framework (upscaling and downscaling) to obtain sharp energy and flatness control across scales, culminating in large-scale Lipschitz regularity and plane-like classifications. The results extend classical Alt-Caffarelli-Friedman-type regularity to periodic media, with significant implications for free boundary problems in heterogeneous environments and potential applications in materials science and fluid interfaces.

Abstract

We study the regularity of minimizers of a two-phase energy functional in periodic media. Our main result is a large scale Lipschitz estimate. We also establish improvement-of-flatness for non-degenerate minimizers, which is a key ingredient in the proof of the Lipschitz estimate. As a consequence, we obtain a Liouville property for entire non-degenerate minimizers.

Paper Structure

This paper contains 40 sections, 52 theorems, 403 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Acknowledgments
  4. Improvement of flatness for $\mathcal{J}_0$ approximate minimizers.
  5. Approximate energy minimality and approximate viscosity solutions
  6. Energy difference formulae
  7. Viscosity solution property of minimizers
  8. Viscosity solution property of approximate minimizers
  9. Harmonic approximation and consequences
  10. Free boundary Harnack inequality and Hölder regularity of flat correction
  11. Free Boundary Harnack Property
  12. Proof of Theorem \ref{['t.holder-est-w']}
  13. Asymptotic expansion for flat approximate viscosity solutions
  14. Formal asymptotic expansion of flat solutions and the transmission problem
  15. Rigorous asymptotic expansion of flat solutions
  16. ...and 25 more sections

Key Result

Theorem 1.1

Suppose that $u$ minimizes $\mathcal{J}$ over $u + H^1_0(B_{R})$. Then The constant $C$ depends only on the universal parameters $\Lambda$ and $d$.

Figures (1)

  • Figure 1: Set-up to apply Lemma \ref{['l.intbyparts2']}. Left figure: $u$, $u_\mu$ and $\Gamma_\mu$ are displayed. Middle figure: set up to apply (\ref{['e.intbyparts2-supersoln']}), the dashed set is $\{u > (u_\mu)_+\}$ -- note that $\bar{u}_\mu(x)$ is equal to $\alpha_+(P(x) - \mu)$ and hence is strictly superharmonic in this region. Right figure: set up to apply (\ref{['e.intbyparts2-subsoln']}), the dashed set is $\{-u_\mu > (-u)_+\}$ -- note that $-u_\mu$ is equal to $-\alpha_-(P(x) - \mu)$ and hence is strictly subharmonic in this region.

Theorems & Definitions (101)

  • Theorem 1.1: Large scale Lipschitz estimate for $\mathcal{J}$ minimizers
  • Theorem 1.2: Improvement of flatness for non-degenerate $\mathcal{J}$ minimizers
  • Remark 1.3
  • Theorem 1.4: Liouville property of minimizers on $\mathbb{R}^d$
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 91 more