Uniform Calderón-Zygmund estimates
in multiscale elliptic homogenization

In this paper, we study the uniform regularity for a family of linear elliptic equations or systems in divergence form in a bounded domain $\Omega\subset\mathbb{R}^{d}$,

where the coefficient matrix takes a form of $A_{\varepsilon}(x)=A(x/\varepsilon_{1},x/\varepsilon_{2},\cdots,x/\varepsilon_{n})$, and $(\varepsilon_{i})_{1\leq i\leq n}\in(0,1]^{n}$ are small parameters describing $n$ different oscillating scales of the coefficients. We assume that $A(y_{1},y_{2},\cdots,y_{n}):\mathbb{R}^{d\times n}\mapsto\mathbb{R}^{d\times d}$ is 1-periodic in each $y_{i}\in\mathbb{R}^{d}$ with $1\leq i\leq n$. Due to the symmetry of $y_{i}$’s, it is harmless to assume $\varepsilon_{1}\geq\varepsilon_{2}\geq\cdots\geq\varepsilon_{n}>0$. The full assumptions on $A$ will be given shortly.

The homogenization theorem (or quantitative convergence rates) and uniform regularity are the fundamental questions in homogenization theory. These two questions have been well answered for the equation (1.1) in the case of one oscillating scale (i.e., $n=1$); see recent monographs [She18, AK22] and references therein. For the case of multiple oscillating scales, we refer to our recent work [NZ23] for a brief survey. It is important to point out that before the work [NZ23], the homogenization theorem and uniform regularity were considered closely interconnected and always appear together, in the sense that the uniform regularity can be derived as a consequence of convergence rates either by a compactness method [AL87, KLS13] or a quantitative method (excess decay iteration [GNO20, AS16, She17] or a real-variable argument [CP98]). For the equation (1.1) with $n\geq 2$, we have the qualitative homogenization theorem or a quantitative convergence rate to a deterministic homogenized equation only if the scales $(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})$ are separated or well-separated [AB96], respectively. Recall that the scales $(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})\in(0,1]^{n}$ are well-separated if $\varepsilon_{i+1}\lesssim\varepsilon_{i}^{\alpha_{i}}$ for some $\alpha_{i}>1$. Consequently, under this scale-separation condition, the uniform regularity estimates for (1.1) were established in [NSX20] by a quantitative method. Yet, as pointed out by Avellaneda in [Ave96] (see also [GH03]) such an assumption is not adequate for treating the most general problems of transport and diffusion in self-similar (particularly random) media. The main contribution of [NZ23] is that, without any separation condition on $(\varepsilon_{i})_{1\leq i\leq n}$, we obtained the uniform $C^{\alpha}$ estimate for any $\alpha\in(0,1)$ by the compactness method combined with a novel scale-reduction process. This result was unexpected since it shows for the first time that the uniform regularity holds stably even without homogenization, and in some sense the averaging effect always takes places across all the mesoscopic scales.

This paper is a continuation of our previous work [NZ23]. It was conjectured in [NZ23] that with $f=0$ in (1.1) and without any scale-separation condition, the solution admits uniform Lipschitz estimate. In this paper, we partially solve the conjecture by proving a slightly weaker result, namely, the gradient $\nabla u_{\varepsilon}$ is uniformly bounded in $L^{p}$ spaces for any $p<\infty$. In particular, this recovers the uniform $C^{\alpha}$ estimate in [NZ23] in view of the Sobolev embedding theorem. Moreover, our proof is quantitative (in contrast to the compactness method in [NZ23]) in the sense that the constant can be computed explicitly.

Before stating the main result, we list the assumptions on the coefficient matrix $A$:

• •

  Strong ellipticity condition: there exists $\Lambda\in(0,1]$ so that

  $$\Lambda|\xi|^{2}\leq\xi\cdot A(y_{1},\cdots,y_{n})\xi\leq\Lambda^{-1}|\xi|^{2}$$

  (1.2)

  for every $\xi\in\mathbb{R}^{d}$ and for any $(y_{1},\cdots,y_{n})\in\mathbb{R}^{d\times n}.$

• •

  Periodicity: for any $(y_{1},\cdots,y_{n})\in\mathbb{R}^{d\times n}$ and $(z_{1},\cdots,z_{n})\in\mathbb{Z}^{d\times n}$,

  $$A(y_{1}+z_{1},\cdots,y_{n}+z_{n})=A(y_{1},\cdots,y_{n}).$$

  (1.3)

• •

  Smoothness: there exists $\tau\in(0,1]$ and $L>0$ such that

  $$|A(y_{1},\cdots,y_{n})-A(y_{1}^{\prime},\cdots,y_{n}^{\prime})|\leq L\big{\{}|y_{1}^{\prime}-y_{1}|+\cdots+|y_{n}^{\prime}-y_{n}|\big{\}}^{\tau}$$

  (1.4)

  for any $(y_{1},\cdots,y_{n}),(y_{1}^{\prime},\cdots,y_{n}^{\prime})\in\mathbb{R}^{d\times n}$.

We now state our main result.

The same uniform estimate holds if the Dirichlet boundary condition in (1.5) is replaced by the compatible Neumann boundary condition

where $\nu$ is the unit outer normal vector of $\partial\Omega$.

The new ingredient in the proof of Theorem 1.1 is the Dirichlet’s theorem (see Theorem 3.1) on simultaneous Diophantine approximation from number theory, which enters into a new technique of reperiodization and allow us to quantitatively separate one scale from the rest $n-1$ scales; see Section 3.1 for details. Precisely, for a given sequence $(\varepsilon_{1},\varepsilon_{2},\cdots,\varepsilon_{n})$ in nonincreasing order and any number $Q>1$, we can find a new 1-periodic matrix $A^{\sharp}$, depending on $Q$ and the ratios $\varepsilon_{n}/\varepsilon_{i}$, such that $A_{\varepsilon}(x)$ can be rewritten as

with some $(\varepsilon_{1}^{\prime},\varepsilon_{2}^{\prime},\cdots,\varepsilon_{n}^{\prime})\in(0,\infty)^{n}$ and the last scale $\varepsilon_{n}^{\prime}$ is $Q$-separated from the rest $\varepsilon_{i}^{\prime}$’s (i.e., $\varepsilon^{\prime}_{i}\geq Q\varepsilon_{n}^{\prime}$ for all $1\leq i\leq n-1$). Theoretically, $Q$ can be chosen as large as $(\varepsilon_{n}/r)^{-\sigma}$ with some $\sigma>0$, yielding a well-separation condition. However, as a payoff the regularity of $A^{\sharp}(y_{1},y_{2},\cdots,y_{n})$ in $y_{n}$ will become worse as $Q$ increases, which leads to a small-scale estimate depending on $Q$ in a blow up argument. Fortunately, a sufficiently large $Q$ independent of $(\varepsilon_{i})_{1\leq i\leq n}$ will be just enough for our proof. With the $Q$-separation condition, we can perform a quantitative argument of the reiterated homogenization and approximate the solution of the original equation with $n$ oscillating scales by a solution of a new equation with at most $n-1$ oscillating scales. Hence, an inductive argument on the number of scales combined with a careful real-variable argument involving a double-averaging estimate will complete the proof.

It is crucial to point out that the proofs for $C^{\alpha}$ estimate (a type of Schauder estimates) and $L^{p}$ gradient estimate (Calderón-Zygmund estimate related to singular integrals) are essentially different, for the latter is stronger and more informative. As in [NZ23], the $C^{\alpha}$ estimate can be derived with a qualitative $H$-convergence theorem and a compactness method (the correctors are not needed; also see [She15, Theorem 3.1] or [SZ18, Theorem 6.1] for similar situations). However, the estimate of gradient must involve a quantitative convergence rate or the fine properties of correctors. While our proof of Theorem 1.1 also uses a scale-reduction theorem and an inductive argument on the number scales, it is exactly the Dirichlet’s theorem that helps us quantify the error in the scale-reduction process and provides much stronger implications and more information about the behavior of the solutions. In particular, it possibly explains the reason behind the occurrence of the mesoscopic averaging effect in arbitrary multiscale media.

As a direct consequence, Theorem 1.1 implies the uniform Calderón-Zygmund estimate for the operator $-\text{\rm div}(A(x/\varepsilon)\nabla)$ with arbitrary quasiperiodic coefficient $A$. We recall that $A(y)$ is said to be quasiperiodic if $A(y)=B(My)$, where $B(w)$ is $1$-periodic in $w\in\mathbb{R}^{N}$ with $1\leq N\in\mathbb{N}$ and $M$ is an $N\times d$ constant real matrix. Note that we do not have any restriction on the entries of $M$ and therefore $A$ is allowed to be periodic with a very degenerate periodic cell.

In the setting of quasiperiodic homogenization, the convergence rate was first obtained by Kozlov in [Koz78] under a Diophantine condition (a quantitative ergodicity condition): for each row $M_{i}$ of $M$, $|M_{i}\cdot z|\geq c_{0}|z|^{-\beta}$ for all $z\in\mathbb{Z}^{d}\setminus\{0\}$ and for some $c_{0},\beta>0$. It was observed in [AL91] that the uniform regularity is valid under Kozlov’s Diophantine condition. This condition was also required in recent work [She15, AGK16, BG19] applied to the case of quasiperiodic coefficients. In our Corollary 1.2, we do not need the Diophantine condition and the constant in the estimate is independent of the entries of the matrix $M$ (except for the dimension of $M$). This includes some typical degenerate cases, such as periodic coefficients with a degenerate periodic cell (e.g., a thin rectangle), quasiperiodic coefficients oscillating at two almost resonant frequencies (e.g., the ratio of the frequencies is a Liouville number), multiscale quasiperiodic coefficients, etc.

The proof of Corollary 1.2 based on Theorem 1.1 is simple and we include it here. In fact, it suffices to write $A(x/\varepsilon)$ into a periodic coefficient matrix with multiple oscillating scales and apply Theorem 1.1. To this end, let $M=(M_{ij})_{1\leq i\leq N,1\leq j\leq d}$. Let $y_{ij}\in\mathbb{R}^{d}$ with $1\leq i\leq N$ and $1\leq j\leq d$ be $Nd$ independent variables. Define a matrix with variables $y_{ij}$

Then $A^{+}$ is 1-periodic in each $y_{ij}$ since $B(w)$ is 1-periodic. Moreover, it is straightforward to verify

This reduces the quasiperiodic coefficient matrix with one oscillating scale into a periodic coefficient matrix with $Nd$ oscillating scales. Clearly, in this case, all the scales are possibly not well-separated. Moreover, if $B$ is uniformly elliptic and Hölder continuous, so is $A^{+}$. As a result, Corollary 1.2 follows readily from Theorem 1.1.

Finally, using a similar idea of scale separation, we can show the uniform large-scale or mesoscopic-scale Lipschitz estimates. In particular, in the case of two scales, i.e., $n=2$, for any $\alpha\in(0,1)$, we have for all $\varepsilon_{2}^{1-\alpha}\leq r\leq 1$,

where $C_{\alpha}$ depends only on $d,\Lambda,p,\tau,L$ and $\alpha$; see Theorem 5.1. For general cases $n\geq 3$, we have some mesoscopic-scale Lipschitz estimates near any given scale; see Theorem 5.2.

Organization. The rest of the paper is organized as follows. In Section 2, we recall some knowledge and known results from locally periodic homogenization and reiterated homogenization. In Section 3, we use the Dirichlet’s theorem and reperiodization technique to separate scales and reduce one scale by a uniform approximation. The main theorem is proved in Section 4 by a real-variable argument. In Section 5, we exploit the idea of scale separation to obtain the uniform large-scale or mesoscopic-scale Lipschitz estimates.

Acknowledgements. W. Niu is supported by NNSF of China (12371106, 11971031). J. Zhuge is partially supported by grants for Excellent Youth from the NSFC.