Compactness and stable regularity in multiscale homogenization

TL;DR

This paper introduces new techniques to analyze multiscale elliptic equations, demonstrating the stability of Hölder continuity estimates across multiple scales and establishing Lipschitz stability for certain laminate structures.

Contribution

The paper develops a novel compactness method and a scale-reduction theorem for multiscale homogenization, providing uniform regularity estimates independent of scale ratios.

Findings

Hölder continuity constants are uniform across all scale parameters.

Lipschitz estimates are stable for laminate structures across all scales.

New techniques involve scale-reduction and reperiodization methods.

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 $(\varepsilon_i)_{1\le i\le n}$ are scale parameters. We show that the $C^\alpha$-H\"{o}lder continuity with any $\alpha\in (0,1)$ 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 $(\varepsilon_i)_{1\le i\le n}$ 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.