Large-scale harmonic measures and nontangential maximal functions in periodic homogenization
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arXiv
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2026
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| _version_ | 1866912978683559936 |
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| author | Shen, Zhongwei Zhuge, Jinping |
| author_facet | Shen, Zhongwei Zhuge, Jinping |
| contents | In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $Ω$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(Ω)$ is a microscopic scale. Due to the irregularity of the coefficients at $\varepsilon$ scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an $\varepsilon$ distance away from the boundary. The $L^p$ estimates uniform in $\varepsilon$ are established for these nontangential maximal functions for the same and optimal ranges of $p$ as the Laplace operator in the Lipschitz or $C^1$ domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale $\mathcal{L}_\varepsilon$-harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21902 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Large-scale harmonic measures and nontangential maximal functions in periodic homogenization Shen, Zhongwei Zhuge, Jinping Analysis of PDEs 35B27, 35J25 In this paper, we consider the elliptic operators $\mathcal{L}_\varepsilon = -\nabla\cdot (A(X/\varepsilon) \nabla )$ with periodic coefficients in a bounded domain $Ω$ without any local smoothness assumption on $A = A(Y)$, where $\varepsilon \ll \text{diam}(Ω)$ is a microscopic scale. Due to the irregularity of the coefficients at $\varepsilon$ scale, we introduce the correct forms of the large-scale nontangential maximal functions for the Dirichlet, Neumann and regularity problems that measure the behaviors of solutions at an $\varepsilon$ distance away from the boundary. The $L^p$ estimates uniform in $\varepsilon$ are established for these nontangential maximal functions for the same and optimal ranges of $p$ as the Laplace operator in the Lipschitz or $C^1$ domains. With some additional regularity assumption on the coefficients, the large-scale estimates combined with the small-scale estimates recover the classical full-scale estimates of the nontangential maximal functions. Our proofs are based on the notion of large-scale $\mathcal{L}_\varepsilon$-harmonic measures, the periodic structure of operators in the transversal direction to the boundaries, and the homogenization tools, including convergence rates and large-scale regularity. |
| title | Large-scale harmonic measures and nontangential maximal functions in periodic homogenization |
| topic | Analysis of PDEs 35B27, 35J25 |
| url | https://arxiv.org/abs/2603.21902 |