Compactness and stable regularity in multiscale homogenization

Abstract

In this paper we develop some new techniques to study the multiscale elliptic equations in the form of $-\text{div}(A_{\epsilon }\nabla u_{\epsilon })=0$, where $A_{\epsilon }(x)=A(x,x/{\epsilon }_1,\cdots ,x/{\epsilon }_n)$ is an $n$-scale oscillating periodic coefficient matrix, and $({\epsilon }_i{)}_{1\le i\le n}$ are scale parameters. We show that the $C^{\alpha }$-Hölder continuity with any $\alpha \in (0,1)$ for the weak solutions is stable, namely, the constant in the estimate is uniform for arbitrary $({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\in (0,1{]}^n$ and particularly is independent of the ratios between ${\epsilon }_i$'s. The proof uses an upgraded method of compactness, involving a scale-reduction theorem by $H$-convergence. The Lipschitz estimate for arbitrary $({\epsilon }_i{)}_{1\le i\le n}$ still remains open. However, for special laminate structures, i.e., $A_{\epsilon }(x)=A(x,x_1/{\epsilon }_1,\cdots ,x_d/{\epsilon }_n)$, we show that the Lipschitz estimate is stable for arbitrary $({\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n)\in (0,1{]}^n$. This is proved by a technique of reperiodization.

37 pages; comments are welcome