Quantitative homogenization of elliptic equations with infinitely many scales

Zhongwei Shen; Yao Xu; Jinping Zhuge

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arXiv · 2605.02561

Quantitative homogenization of elliptic equations with infinitely many scales

Zhongwei Shen·Yao Xu·Jinping Zhuge

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Abstract

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 $\varepsilon$.

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Zhongwei Shen, Yao Xu, Jinping Zhuge. 2026-05-04. Quantitative homogenization of elliptic equations with infinitely many scales. https://arxiv.org/abs/2605.02561

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