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Lipschitz regularity of homogenization with continuous coefficients: Dirichlet problem

Sungjin Lee

TL;DR

This work establishes uniform Lipschitz regularity for solutions to elliptic systems in divergence form with rapidly oscillating 1-periodic coefficients under minimal, integral-type regularity assumptions. By employing a compactness method augmented with Dirichlet correctors and Green's function techniques, the authors prove interior Lipschitz estimates when the coefficient matrix $\mathbf{A}$ has Dini mean oscillation and the data are in Dini-type classes, and boundary Lipschitz estimates in $C^{1,\text{Dini}}$ domains with corresponding data regularity. The results extend the classical Avellaneda–Lin framework by replacing pointwise Hölder/VMO conditions with integral conditions (DMO and Dini continuity), aligning with the latest general regularity theory for homogenization. The analysis combines a one-step improvement and iterative scheme for interior estimates, a two-step boundary compactness argument using Dirichlet correctors, and a decomposition argument for the boundary problem, yielding a comprehensive Lipschitz theory in the homogenization regime.

Abstract

We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda and Lin [Comm. Pure Appl. Math. 40 (1987), pp. 803-847] by minimizing all regularity conditions of the given data to integral conditions. We remark that the coefficients of an elliptic operator have Dini mean oscillation, which corresponds to the results of the latest general regularity theory.

Lipschitz regularity of homogenization with continuous coefficients: Dirichlet problem

TL;DR

This work establishes uniform Lipschitz regularity for solutions to elliptic systems in divergence form with rapidly oscillating 1-periodic coefficients under minimal, integral-type regularity assumptions. By employing a compactness method augmented with Dirichlet correctors and Green's function techniques, the authors prove interior Lipschitz estimates when the coefficient matrix has Dini mean oscillation and the data are in Dini-type classes, and boundary Lipschitz estimates in domains with corresponding data regularity. The results extend the classical Avellaneda–Lin framework by replacing pointwise Hölder/VMO conditions with integral conditions (DMO and Dini continuity), aligning with the latest general regularity theory for homogenization. The analysis combines a one-step improvement and iterative scheme for interior estimates, a two-step boundary compactness argument using Dirichlet correctors, and a decomposition argument for the boundary problem, yielding a comprehensive Lipschitz theory in the homogenization regime.

Abstract

We study uniform Lipschitz regularity estimates for elliptic systems in divergence form with continuous coefficients, based on rapidly oscillating periodic coefficients derived from homogenization theory. We extend a result by Avellaneda and Lin [Comm. Pure Appl. Math. 40 (1987), pp. 803-847] by minimizing all regularity conditions of the given data to integral conditions. We remark that the coefficients of an elliptic operator have Dini mean oscillation, which corresponds to the results of the latest general regularity theory.

Paper Structure

This paper contains 8 sections, 14 theorems, 156 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of Interior Lipschitz estimates
  4. Proof of boundary Lipschitz estimates
  5. Green's function and Dirichlet correctors
  6. Compactness method via Dirichlet correctors
  7. Uniform Lipschitz estimates
  8. Proof of Theorem \ref{['thm01']}

Key Result

Theorem 1.1

Assume the coefficients $\mathbf{A}=(a^{ij})$ of the operator $\mathcal{L}_\varepsilon$ satisfy the condition ellipticity, perodicity and are of Dini mean oscillation in $\mathbb{R}^n$. Let $u_\varepsilon\in W^{1,2}(B(x_0,2R);\mathbb{R}^m)$ be a weak solution of where $f:\Omega\rightarrow\mathbb{R}^{n\times m}$ are of Dini mean oscillation and $0<R\leq\frac{1}{2}R_0$. Then, we have where $C=C(n,

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1: One-step improvement
  • proof
  • Lemma 3.2: Iteration
  • proof
  • Lemma 3.3: Blow-up method
  • proof
  • ...and 13 more