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- W4214679966 abstract "In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-text{div} big(A_varepsilon nabla u_{varepsilon} big) = 0$, where $A_varepsilon(x) = A(x, x/varepsilon_1,cdots, x/varepsilon_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(varepsilon_i)_{1le ile n}$ are scale parameters. We show that the $C^alpha$-H{o}lder continuity with any $alphain (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $(varepsilon_1, varepsilon_2, cdots, varepsilon_n) in (0,1]^n$ and particularly is independent of the ratios between $varepsilon_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $(varepsilon_i)_{1le ile n}$ still remains open. However, for special laminate structures, i.e., $A_varepsilon(x) = A(x,x_1/varepsilon_1, cdots, x_d/varepsilon_n)$, we show that the Lipschitz estimate is stable for arbitrary $(varepsilon_1, varepsilon_2, cdots, varepsilon_n) in (0,1]^n$. This is proved by a technique of reperiodization." @default.
- W4214679966 created "2022-03-02" @default.
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- W4214679966 date "2022-02-25" @default.
- W4214679966 modified "2023-10-17" @default.
- W4214679966 title "Compactness and stable regularity in multiscale homogenization" @default.
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- W4214679966 doi "https://doi.org/10.1007/s00208-022-02378-9" @default.
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