Compactness and stable regularity in multiscale homogenization

DOI10.1007/S00208-022-02378-9arXiv2112.02400MaRDI QIDQ6384818

Weisheng Niu, Jinping Zhuge

Publication date: 4 December 2021

Abstract: In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $$, where $A_{v}arepsilon(x)=A(x,x/varepsilo{n}_1,cdots,x/varepsilo{n}_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $(varepsilo{n}_i{)}_{1leilen}$ are scale parameters. We show that the $C^{a}lpha$-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 $(varepsilo{n}_1,varepsilo{n}_2,cdots,varepsilo{n}_n)in(0,1{]}^n$ and particularly is independent of the ratios between $varepsilo{n}_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $(varepsilo{n}_i{)}_{1leilen}$ still remains open. However, for special laminate structures, i.e., $A_{v}arepsilon(x)=A(x,x_1/varepsilo{n}_1,cdots,x_d/varepsilo{n}_n)$, we show that the Lipschitz estimate is stable for arbitrary $(varepsilo{n}_1,varepsilo{n}_2,cdots,varepsilo{n}_n)in(0,1{]}^n$. This is proved by a technique of reperiodization.

Mathematics Subject Classification ID

Smoothness and regularity of solutions to PDEs (35B65) Boundary value problems for second-order elliptic equations (35J25) A priori estimates in context of PDEs (35B45) Homogenization in context of PDEs; PDEs in media with periodic structure (35B27)

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