Quantitative homogenization of elliptic equations with infinitely many scales
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910189142147072 |
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| author | Shen, Zhongwei Xu, Yao Zhuge, Jinping |
| author_facet | Shen, Zhongwei Xu, Yao Zhuge, Jinping |
| contents | In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with $\varepsilon_1>\varepsilon_2>\cdots$ and $\varepsilon_n \to 0$ as $n \to \infty$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\varepsilon$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02561 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantitative homogenization of elliptic equations with infinitely many scales Shen, Zhongwei Xu, Yao Zhuge, Jinping Analysis of PDEs 35B27 In this paper, we develop a general homogenization theory for elliptic equations with coefficients that oscillate periodically at infinitely many scales $\varepsilon = (\varepsilon_1, \varepsilon_2, \cdots) \in (0,1)^\infty$, with $\varepsilon_1>\varepsilon_2>\cdots$ and $\varepsilon_n \to 0$ as $n \to \infty$. Such problems arise naturally in the study of fractal materials and diffusion in fluids. Under suitable scale-separation assumptions, we prove a qualitative homogenization theorem and obtain optimal $L^2$ convergence rates. We also establish interior and boundary Lipschitz estimates that are uniform in $\varepsilon$. |
| title | Quantitative homogenization of elliptic equations with infinitely many scales |
| topic | Analysis of PDEs 35B27 |
| url | https://arxiv.org/abs/2605.02561 |