11institutetext: Thuyen Dang 22institutetext: University of Houston, Houston,
TX 77204 USA, 22email<EMAIL_ADDRESS>33institutetext: Yuliya Gorb
44institutetext: National Science Foundation, Alexandria, VA 22314 USA,
44email<EMAIL_ADDRESS>55institutetext: Silvia Jiménez Bolaños
66institutetext: Colgate University, Hamilton, NY 13346 USA, 66email:
<EMAIL_ADDRESS>
# Global gradient estimate for a divergence problem and its application to the
homogenization of a magnetic suspension

Thuyen Dang  Yuliya Gorb  and Silvia Jiménez Bolaños

###### Abstract

This paper generalizes the results obtained by the authors in
dangHomogenizationNondiluteSuspension2021  concerning the homogenization of a
non-dilute suspension of magnetic particles in a viscous flow. More
specifically, in this paper, a restrictive assumption on the coefficients of
the coupled equation, made in dangHomogenizationNondiluteSuspension2021 , that
significantly narrowed the applicability of the homogenization results
obtained, is relaxed and a new regularity of the solution of the fine-scale
problem is proven. In particular, we obtain a global $L^{\infty}$-bound for
the gradient of the solution of the scalar equation
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(x/\varepsilon\right)\nabla\varphi^{\varepsilon}(x)\right]=f(x)$,
uniform with respect to microstructure scale parameter $\varepsilon\ll 1$ in a
small interval $(0,\varepsilon_{0})$, where the coefficient $\mathbf{a}$ is
only _piecewise_ Hölder continuous. Thenceforth, this regularity is used in
the derivation of the effective response of the given suspension discussed in
dangHomogenizationNondiluteSuspension2021 .

##  1 Introduction

The purpose of this paper is to generalize the results obtained by the authors
in dangHomogenizationNondiluteSuspension2021 , where the rigorous analysis of
the homogenization of a particulate flow consisting of a non-dilute suspension
of a viscous Newtonian fluid with magnetizable particles was developed. Here,
the fluid is assumed to be described by the Stokes flow and the particles are
either paramagnetic or diamagnetic. The coefficients of the corresponding
partial differential equations are locally periodic and a one-way coupling
between the fluid domain and the particles is also assumed. Such one-way
coupling has been observed in nature, see
(davidsonIntroductionMagnetohydrodynamics2001, , Chapter 1). For details and
information about the applications and literature on the magnetic suspension,
we turn to dangHomogenizationNondiluteSuspension2021  and references cited
therein; however, the mathematical formulation of the considered problem is
given in Section 2.2 below. References on the effective viscosity of a
suspension without the coupling with magnetic field include
niethammerLocalVersionEinstein2020 ; gorbHomogenizationRigidSuspensions2014 ;
hoferSedimentationInertialessParticles2018 ;
mecherbetSedimentationParticlesStokes2019 ; gerard-
varetAnalysisViscosityDilute2020 ; duerinckxCorrectorEquationsFluid2021 ;
hainesProofEinsteinEffective2012 ; duerinckxSedimentationRandomSuspensions2021
; duerinckxEinsteinEffectiveViscosity2020 ;
duerinckxQuantitativeHomogenizationTheory2021 ;
berlyandFictitiousFluidApproach2009a ;
berlyandHomogenizedNonNewtonianViscoelastic2004 ;
desvillettesMeanfieldLimitSolid2008 .

In dangHomogenizationNondiluteSuspension2021 , a restrictive assumption about
the magnetic permeability of the suspension, denoted by $\mathbf{a}$, was
made. Here, the function $\mathbf{a}(\cdot)$ is locally periodic and elliptic,
where the latter means that $\lambda{\mathbf{I}}\leq\mathbf{a}(x)$ and
$\left\lVert\mathbf{a}\right\rVert_{L^{\infty}}\leq\Lambda$, for all
$x\in\Omega$, with the suspension domain $\Omega\subset\mathbb{R}^{d}$,
$d=2,3$, including both the ambient fluid and the particles, and
$\lambda,\Lambda>0$ given in (A2)-(A3) below. The assumption on the function
$\mathbf{a}$ made in dangHomogenizationNondiluteSuspension2021  is as follows:
for a given $s\in(4,6]$, there exists a small number
$\delta=\delta(\Lambda,d,\Omega)>0$, for which the magnetic permeability
$\mathbf{a}$ satisfies the following condition:

$\mathrm{ess}\sup\mathbf{a}-\mathrm{ess}\inf\mathbf{a}\leq\delta.$ (1)

As a consequence of (1), in dangHomogenizationNondiluteSuspension2021  we
obtained the following gradient estimate for the magnetic potential
$\varphi^{\varepsilon}$:

$\int_{\Omega}\left\lvert\nabla\varphi^{\varepsilon}\right\rvert^{s}\operatorname{d\\!}x\leq
C\int_{\Omega}\left\lvert\mathbf{k}\right\rvert^{s}\operatorname{d\\!}x,$ (2)

where the constant $C>0$ is independent of $\varepsilon$,
$\varphi^{\varepsilon}$ and $\mathbf{k}$; with $0<\varepsilon\ll 1$ the scale
of the microstructure, $\mathbf{k}\in H^{1}({\Omega},{\mathbb{R}}^{d})$
divergence-free, satisfying the compatibility condition
$\displaystyle\int_{\partial\Omega}\mathbf{k}\cdot\mathbf{n}_{\partial\Omega}\operatorname{d\\!}\mathrm{s}=0$,
and appearing in the Neumann boundary condition on $\partial\Omega$, the
boundary of the domain $\Omega$, given by:

$\left(\mathbf{a}\nabla\varphi^{\varepsilon}\right)\cdot\mathbf{n}_{\partial\Omega}=\mathbf{k}\cdot\mathbf{n}_{\partial\Omega},$
(3)

where $\mathbf{n}_{\partial\Omega}$ is the outward-pointing unit normal vector
to $\partial\Omega$. The regularity result (2) was then used in the derivation
of the effective (or homogenized) response of the given suspension that was
rigorously justified in dangHomogenizationNondiluteSuspension2021 .

The main goal of this paper is to relax the assumption (1) on the magnetic
permeability $\mathbf{a}$. To achieve this, we consider the Dirichlet boundary
condition given in (4b) below, rather than one given in (3), and obtain the
Lipschitz estimate (5), instead of (2), for the gradient of the magnetic
potential $\varphi^{\varepsilon}$, see Theorem 3.1 below. In this paper, we
are able to remove the condition (1) and have $\mathbf{a}$ only required to be
_piecewise Hölder continuous_. Such relaxation will be based on the following
observations:

  * •

The De Giorgi-Nash-Moser estimate (gilbargEllipticPartialDifferential2001, ,
Theorem 8.24) states that the solutions of scalar equations are Hölder
continuous.

  * •

If $1>\varepsilon\geq\varepsilon_{0}$, for some $\varepsilon_{0}>0$, the
uniform gradient bound (5) can be obtained by the result of Li and Vogelius
liGradientEstimatesSolutions2000 . The case when
$\varepsilon_{0}>\varepsilon>0$ is resolved by the compactness method, which
is discussed below.

  * •

Provided $\mathbf{a}$ is also symmetric (this assumption is only necessary for
the corrector results in Theorem 6.1), the gradients of the solutions of the
cell problems are in $L^{\infty}(Y)$.

The main tools used in the proof of this theorem are (i) the regularity
results of Li and Vogelius liGradientEstimatesSolutions2000 , and (ii) the
celebrated _compactness method_ , which was first used in homogenization in
the seminal works of Avellaneda and Lin avellanedaCompactnessMethodsTheory1987
; avellanedaCompactnessMethodsTheory1989 . Its machinery and applications in
homogenization are carefully explained in
prangeUniformEstimatesHomogenization2014 . In the context of homogenization,
this method utilizes compactness in order to gain an improved regularity from
a limiting equation via a proof by contradiction. This improvement of
regularity is iterated and then used in a blow-up argument. Usually, the
implementation of this method follows three steps, coined by Avellaneda and
Lin avellanedaCompactnessMethodsTheory1987 ;
avellanedaCompactnessMethodsTheory1989  titled as (i) “improvement”, (ii)
“iteration”, and (iii) “blow-up”.

The main contribution of this improved regularity result is that it will allow
us to significantly widen the range of applicability of the results obtained
in dangHomogenizationNondiluteSuspension2021 .

The outline of the paper is as follows. In Section 2, the main notations are
introduced and the formulation of the fine-scale problem is discussed. Theorem
3.1, which provides an improved gradient estimate for the magnetic potential,
is stated and discussed in Section 3. In Section 4, we obtain the interior
Lipschitz and Hölder estimates, which provide the foundation for the boundary
and corrector estimates discussed in Section 5. With all the results at hand,
we then present the proof of our main theorem, also in Section 5. In Section
6, the homogenization results are obtained and summarized in Theorem 6.1. The
conclusions are given in Section 7. The classical Schauder estimate is
recalled in Section 8.

##  2 Formulation

###  2.1 Notation

For a measurable set $A$ and a measurable function $f\colon A\to{\mathbb{R}}$,
we define by $\left\lvert A\right\rvert$ the measure of $A$ and
$\displaystyle\intbar_{A}f(x)\operatorname{d\\!}x\coloneqq\frac{1}{|A|}\int_{A}f(x)\operatorname{d\\!}x$.

Throughout this paper, the scalar-valued functions, such as the pressure $p$,
are written in usual typefaces, while vector-valued or tensor-valued
functions, such as the velocity $\mathbf{u}$ and the Cauchy stress tensor
$\boldsymbol{\sigma}$, are written in bold. Sequences are indexed by numeric
superscripts ($\phi^{i}$), while elements of vectors or tensors are indexed by
numeric subscripts ($x_{i}$). Finally, the Einstein summation convention is
used whenever applicable.

###  2.2 Set up of the problem

Consider $\Omega\subset{\mathbb{R}}^{d}$, for $d\geq 2$, a simply connected
and bounded domain of class $C^{1,\alpha},~{}0<\alpha<1$, and let
$Y\coloneqq(0,1)^{d}$ be the unit cell in ${\mathbb{R}}^{d}$. The unit cell
$Y$ is decomposed into:

$Y=Y_{s}\cup Y_{f}\cup\Gamma,$

where $Y_{s}$, representing the magnetic inclusion, and $Y_{f}$, representing
the fluid domain, are open sets in $\mathbb{R}^{d}$, and $\Gamma$ is the
closed $C^{1,\alpha}$ interface that separates them. Let
$i=(i_{1},\ldots,i_{d})\in{\mathbb{Z}}^{d}$ be a vector of indices and
$\\{\mathrm{e}^{1},\ldots,\mathrm{e}^{d}\\}$ be the canonical basis of
${\mathbb{R}}^{d}$. For a fixed small $\varepsilon>0,$ we define the dilated
sets:

$\displaystyle
Y^{\varepsilon}_{i}\coloneqq\varepsilon(Y+i),~{}~{}Y^{\varepsilon}_{i,s}\coloneqq\varepsilon(Y_{s}+i),~{}~{}Y^{\varepsilon}_{i,f}\coloneqq\varepsilon(Y_{f}+i),~{}~{}\Gamma^{\varepsilon}_{i}\coloneqq\partial
Y^{\varepsilon}_{i,s}.$

Typically, in homogenization theory, the positive number $\varepsilon\ll 1$ is
referred to as the size of the microstructure. The effective (or homogenized)
response of the given suspension corresponds to the case $\varepsilon=0$.

We denote by $\mathbf{n}_{i},~{}\mathbf{n}_{\Gamma}$ and
$\mathbf{n}_{\partial\Omega}$ the unit normal vectors to
$\Gamma^{\varepsilon}_{i}$ pointing outward $Y^{\varepsilon}_{i,s}$, on
$\Gamma$ pointing outward $Y_{s}$ and on $\partial\Omega$ pointing outward,
respectively; and also, we denote by $\operatorname{d\\!}\mathcal{H}^{d-1}$
the $(d-1)$-dimensional Hausdorff measure. In addition, we define the sets:

$\displaystyle I^{\varepsilon}\coloneqq\\{i\in{\mathbb{Z}}^{d}\colon
Y^{\varepsilon}_{i}\subset\Omega\\},~{}~{}\Omega_{s}^{\varepsilon}\coloneqq\bigcup_{i\in
I^{\varepsilon}}Y_{i,s}^{\varepsilon},~{}~{}\Omega_{f}^{\varepsilon}\coloneqq\Omega\setminus\Omega_{s}^{\varepsilon},~{}~{}\Gamma^{\varepsilon}\coloneqq\bigcup_{i\in
I^{\varepsilon}}\Gamma^{\varepsilon}_{i},$

see Figure 1.

[scale=.25]fig.eps

Figure 1: Reference cell $Y$ and domain $\Omega$.

The magnetic permeability $\mathbf{a}$ is a $d\times d$ matrix satisfying the
following conditions:

  1. (A1)

$Y-$periodicity: for all $z\in{\mathbb{R}}^{d}$, for all $m\in{\mathbb{Z}}$,
and for all $k\in\\{1,\ldots,d\\}$ we have:

$\mathbf{a}(z+m\mathrm{e}^{k})=\mathbf{a}(z).$

  2. (A2)

Boundedness and measurability: there exists $\Lambda>0$ such that:

$\left\lVert\mathbf{a}\right\rVert_{L^{\infty}\left({\mathbb{R}}^{d}\right)}\leq\Lambda.$

  3. (A3)

Ellipticity: there exists $\lambda>0$ such that for all
$\xi\in{\mathbb{R}}^{d}$, for all $x\in{\mathbb{R}}^{d}$, we have:

$\mathbf{a}(x)\xi\cdot\xi\geq\lambda\left\lvert\xi\right\rvert^{2}.$

Denote by $\mathfrak{M}(\lambda,\Lambda)$ the set of matrices that satisfy
(A2)-(A3), and $\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ the subset of
matrices in $\mathfrak{M}(\lambda,\Lambda)$ that also satisfy (A1).

##  3 Statement and discussion of the main result

The main result of this paper is summarized in the following theorem.

######  Theorem 3.1 (Global Lipschitz Estimate)

Let $\Omega$ be a bounded $C^{1,\alpha}$ domain, $g\in
C^{1,\alpha^{\prime}}(\partial\Omega)$ and $f\in L^{\infty}({\Omega})$, where
$0<\alpha^{\prime}<\alpha<1$. Suppose that
$\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ is piecewise
$C^{\alpha}$-continuous. There exists
$C=C(\alpha,\alpha^{\prime},\lambda,\Lambda,d,\Omega)>0$ such that, for all
$\varepsilon>0$, the (unique) solution $\varphi^{\varepsilon}$ of:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]$
$\displaystyle=f,\quad$ $\displaystyle\text{ in }\Omega$ (4a)
$\displaystyle\varphi^{\varepsilon}$ $\displaystyle=g,\quad$
$\displaystyle\text{ on }\partial\Omega,$ (4b)

satisfies:

$\displaystyle\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(\Omega)}$
$\displaystyle\leq C\left(\left\lVert
g\right\rVert_{C^{1,\alpha^{\prime}}(\partial\Omega)}+\left\lVert
f\right\rVert_{L^{\infty}({\Omega})}\right).$ (5)

###### Remark 1

For each $\varepsilon>0$, let $N_{\varepsilon}$ be the number of subdomains
inside $\Omega$ such that in each of them the function $\mathbf{a}$ is
$C^{\alpha}$-continuous. Denote those subdomains by $D_{m},~{}1\leq m\leq
N_{\varepsilon}$. Then, for
$0<\alpha^{\prime}<\min\left\\{\alpha,\frac{\alpha}{(\alpha+1)d}\right\\}$, by
(liGradientEstimatesSolutions2000, , Corollary 1.3), one has:

$\displaystyle\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(\Omega)}\leq
C\left(\left\lVert
g\right\rVert_{C^{1,\alpha^{\prime}}(\partial\Omega)}+\left\lVert
f\right\rVert_{L^{\infty}(\Omega)}\right),$ (6)

where $C$ depends on
$\Omega,d,\alpha,\alpha^{\prime},\lambda,\Lambda,\left\lVert\mathbf{a}\right\rVert_{C^{\alpha^{\prime}}(\overline{D_{m}},{\mathbb{R}}^{d\times
d})}$ and the $C^{1,\alpha}$-modulus of $\cup_{m=1}^{N_{\varepsilon}}\partial
D_{m}$ (defined in page 92 liGradientEstimatesSolutions2000 ). As
$\varepsilon\to 0,$ the number $N_{\varepsilon}$ increases, while the sizes of
the subdomains decrease, which leads to the blow-up of the
$C^{1,\alpha}$-modulus. Therefore, estimate (6) is not uniform in
$\varepsilon$.

However, if $\varepsilon_{0}\leq\varepsilon\leq 1$ for some constant
$\varepsilon_{0}>0$, then one can control the number, size, distance and
$C^{1,\alpha}$-modulus of the subdomains in $\Omega$, uniformly with respect
to $\varepsilon$. Note that the upper and lower bounds of those quantities are
positive and independent of $\varepsilon$. Thus, by
(liGradientEstimatesSolutions2000, , Corollary 1.3), there exists $C$
independent of $\varepsilon$ such that (5) holds when
$\varepsilon_{0}\leq\varepsilon\leq 1$. Therefore, Theorem 3.1 will be proven,
if one can specify a constant $\varepsilon_{0}>0$ such that (5) holds for
$0<\varepsilon<\varepsilon_{0}.$

Our proof of Theorem 3.1 follows the classical steps in regularity theory: (i)
derive an interior Lipschitz estimate, (ii) derive a boundary Lipschitz
estimate, and finally (iii) combine the estimates in (i) and (ii) to obtain
the global Lipschitz estimate. Step (i) is obtained via the compactness
method. For step (ii), we additionally need to establish the following
preliminary results:

  * •

Interior and boundary Hölder estimates, see Propositions 1, 2 and 3.

  * •

Estimates for the Green’s function, which are obtained using the Hölder bounds
above, see Proposition 5. The existence of the Green’s function for scalar
uniformly elliptic equations is established in
littmanRegularPointsElliptic1963 ; gruterGreenFunctionUniformly1982 ;
kenigEllipticEquationLuk1985 .

  * •

Estimates for the Dirichlet boundary corrector, see Proposition 6.

If the coefficient $\mathbf{a}$ is not Hölder continuous, the classical
results in avellanedaCompactnessMethodsTheory1987 ;
prangeUniformEstimatesHomogenization2014 ;
shenPeriodicHomogenizationElliptic2018  can not be applied directly.
Nevertheless, some of their proofs can be adapted for the case at hand. In
those situations, we will explicitly point out what needs to be modified in
their proofs, in order to relax the continuity assumption on the coefficient
matrix $\mathbf{a}$.

##  4 Interior estimates

We start with an estimate for homogenized equations, i.e. the equations with
constant coefficients, which are limits of some fine-scale problems.

###### Lemma 1

Let $\lambda>0,$ $\Lambda>0$, $\gamma>0$ and $0<\mu<\frac{1}{2}$ be fixed. For
each constant matrix $\mathbf{b}\in\mathfrak{M}(\lambda,\Lambda)$, $h\in
L^{d+\gamma}({B}(x_{0},1))$, with $\left\lVert
h\right\rVert_{L^{d+\gamma}({B}(x_{0},1))}\leq 1$, there exists
$\theta=\theta(\gamma,\mu,\lambda,\Lambda,d)>0$ such that if $\phi\in
H^{1}(B(x_{0},1))$ satisfies:

$\displaystyle-\operatorname{\mathrm{div}}\left(\mathbf{b}\nabla\phi\right)=h\text{
in }B(x_{0},1),$

then the following estimate holds:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta}\left\lvert\phi(x)-\phi(x_{0})-(x-x_{0})\cdot\intbar_{B(x_{0},\theta)}\nabla\phi(z)\operatorname{d\\!}z\right\rvert<\theta^{1+3\mu/4}.$
(7)

###### Proof

By the classical Schauder estimate for the scalar equation with constant
coefficients (Theorem 8.1), we have $\phi\in
C^{1,\mu}\left(B\left(x_{0},1/4\right)\right)$ and:

$\displaystyle\begin{split}\left\lVert\phi\right\rVert_{C^{1,\mu}\left(B\left(x_{0},1/4\right)\right)}&\leq
C(\gamma,\mu,\lambda,\Lambda,d)\left(\left\lVert
h\right\rVert_{L^{d+\gamma}\left(B(x_{0},3/8)\right)}+\left\lVert\phi\right\rVert_{H^{1}(B(x_{0},3/8))}\right)\\\
&\leq C(\gamma,\mu,\lambda,\Lambda,d)\left\lVert
h\right\rVert_{L^{d+\gamma}({B}(x_{0},1))}\\\ &\leq
C(\gamma,\mu,\lambda,\Lambda,d).\end{split}$ (8)

For $0<\theta<\frac{1}{4}$ and $\left\lvert x-x_{0}\right\rvert<\theta$, there
exist $z_{x}$ such that:

$\displaystyle\left\lvert\phi(x)-\phi(x_{0})-(x-x_{0})\cdot\intbar_{B(x_{0},\theta)}\nabla\phi(z)\operatorname{d\\!}z\right\rvert$
$\displaystyle\quad=\left\lvert\frac{x-x_{0}}{\left\lvert
B(x_{0},\theta)\right\rvert}\cdot\int_{B(x_{0},\theta)}\left(\nabla\phi(z_{x})-\nabla\phi(y)\right)\operatorname{d\\!}y\right\rvert$
$\displaystyle\quad\leq C(\gamma,\mu,\lambda,\Lambda,d)\left\lvert
x-x_{0}\right\rvert^{1+\mu}$ $\displaystyle\quad\leq
C(\gamma,\mu,\lambda,\Lambda,d)\theta^{1+\mu}.$

Choosing $\theta$ small enough so that
$C(\gamma,\mu,\lambda,\Lambda,d)\theta^{1+\mu}<\theta^{1+3\mu/4}$, we obtain
(7).

The fact that $\theta$ doesn’t depend on the matrix $\mathbf{b}$ or the source
term $h$ is crucial for the contradiction argument in Proposition 1 below. We
now state the interior Lipschitz estimate. Note that, here, $\mathbf{a}$ is
not necessarily Hölder continuous in the domain $\Omega$.

######  Proposition 1 (Interior Lipschitz Estimate I)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ and
$f\in L^{\infty}(\Omega)$. Fix $x_{0}\in\Omega$ and $R>0$, so that
$B(x_{0},R)\subset\Omega$. There exist
$\varepsilon_{0}=\varepsilon_{0}(\lambda,\Lambda,R,d)>0$ and
$C=C(\lambda,\Lambda,R,d)>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$ and for every weak solution
$\varphi^{\varepsilon}\in H^{1}(B(x_{0},R))$ of the equation
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]=f$
in $B(x_{0},R)$, the following estimate holds:

$\displaystyle\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},R/2))}\leq
C\left(\left\lVert\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},R))}+\left\lVert
f\right\rVert_{L^{\infty}({B}(x_{0},R))}\right).$ (9)

###### Proof

By dilation, we may assume that $R=1$. Fix $0<\mu<\frac{1}{2}$. We prove, by
the compactness method, that there exists
$\varepsilon_{0}=\varepsilon_{0}(\lambda,\Lambda,d,\mu)$ so that (9) holds for
all $0<\varepsilon<\varepsilon_{0}$. To do this, we only need to show that
there exists $C>0$, independent of $\varepsilon$, such that:

$\displaystyle\max\left\\{\left\lVert\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},1))},\left\lVert
f\right\rVert_{L^{\infty}({B}(x_{0},1))}\right\\}\leq 1\text{ implies
}\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},1/2))}\leq
C.$ (10)

Let $\boldsymbol{\omega}\coloneqq(\omega^{1},\ldots,\omega^{d})$, where
$\omega^{i}\in H^{1}_{\mathrm{per}}(Y)/{\mathbb{R}},~{}1\leq i\leq d,$ is the
solution of the cell problem:

$-\operatorname{\mathrm{div}}_{y}\left[{\mathbf{a}(y)}\left(\mathrm{e}^{i}+\nabla_{y}\omega^{i}(y)\right)\right]=0\text{
in }Y.$ (11)

  1. 1.

_Improvement._ In this step, we prove by contradiction that:

For fixed $0<\mu<\frac{1}{2}$, there exist $\theta$ and $\varepsilon^{*}$,
with $0<\theta<\frac{1}{4},~{}0<\varepsilon^{*}<1$, depending on
$\lambda,\Lambda,d$ and $\mu$, such that if
$\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,L)$, $f\in
L^{\infty}({B}(x_{0},1))$, $\varphi^{\varepsilon}\in H^{1}(B(x_{0},1))$
satisfy:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]$
$\displaystyle=f\text{ in }B(x_{0},1),$ (12a)
$\displaystyle\max\left\\{\left\lVert\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},1))},\left\lVert
f\right\rVert_{L^{\infty}({B}(x_{0},1))}\right\\}$ $\displaystyle\leq 1,$
(12b)

then, for all $0<\varepsilon<\varepsilon^{*}$, we have:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta}\left\lvert\varphi^{\varepsilon}(x)-\varphi^{\varepsilon}(x_{0})-\left[x-x_{0}+\varepsilon\boldsymbol{\omega}\left(\frac{x}{\varepsilon}\right)\right]\cdot\intbar_{B(x_{0},\theta)}\nabla\varphi^{\varepsilon}(z)\operatorname{d\\!}z\right\rvert\leq\theta^{1+\mu/2},$
(13)

where $\boldsymbol{\omega}$ solves (11).

Take $\theta$ as in (7) of Lemma 1. By contradiction, suppose there exist
sequences:

$\displaystyle\varepsilon_{n}\to
0,\quad\mathbf{a}_{n}\in\mathfrak{M}_{\mathrm{per}}(\lambda,L),\quad f_{n}\in
L^{\infty}\left(B(x_{0},1)\right),\text{ and }\varphi_{n}\in
H^{1}(B(x_{0},1))$

satisfying:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}_{n}\left(\frac{x}{\varepsilon_{n}}\right)\nabla\varphi_{n}\right]$
$\displaystyle=f_{n}\text{ in }B(x_{0},1),$ (14)
$\displaystyle\max\left\\{\left\lVert\varphi_{n}\right\rVert_{L^{\infty}(B(x_{0},1))},\left\lVert
f_{n}\right\rVert_{L^{\infty}({B}(x_{0},1))}\right\\}$ $\displaystyle\leq 1,$
(15)

such that:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta}\left\lvert\varphi_{n}(x)-\varphi_{n}(x_{0})-\left[x-x_{0}+\varepsilon_{n}\boldsymbol{\omega}\left(\frac{x}{\varepsilon_{n}}\right)\right]\cdot\intbar_{B(x_{0},\theta)}\nabla\varphi_{n}(z)\operatorname{d\\!}z\right\rvert>\theta^{1+\mu/2}.$
(16)

Let $\mathcal{A}_{n}\in\mathfrak{M}(\lambda,\Lambda)$ denote the effective
matrix corresponding to $\mathbf{a}_{n}$. By the Banach-Alaoglu theorem, the
Caccioppoli inequality, the Rellich-Kondrachov theorem and the Schauder
theorem (see, e.g. (giaquintaIntroductionRegularityTheory2012, , Theorem 4.4)
and (brezisFunctionalAnalysisSobolev2011, , Theorem 3.16, 6.4 and 9.16)),
there exist functions $\varphi_{0}\in L^{2}(B(x_{0},1)),$ $f_{0}\in
L^{\infty}(B(x_{0},1))$ and a constant matrix
$\mathbf{a}_{0}\in\mathfrak{M}(\lambda,\Lambda)$ such that, up to
subsequences, we have:

$\displaystyle\varphi_{n}$ $\displaystyle\rightharpoonup\varphi_{0}\text{ in
}L^{2}(B(x_{0},1))$ $\displaystyle f_{n}$
$\displaystyle\overset{*}{\rightharpoonup}f_{0}\text{ in
}L^{\infty}(B(x_{0},1))$ $\displaystyle\varphi_{n}$
$\displaystyle\rightharpoonup\varphi_{0}\text{ in }H^{1}(B(x_{0},1/2))$
$\displaystyle f_{n}$ $\displaystyle\to f_{0}\text{ in }H^{-1}(B(x_{0},1))$
$\displaystyle\mathcal{A}_{n}$ $\displaystyle\to\mathbf{a}_{0}.$

By (cioranescuIntroductionHomogenization1999, , Theorem 13.4 (iii)) or
(shenPeriodicHomogenizationElliptic2018, , Theorem 2.3.2), we have
$\varphi_{0}$ is the solution of:

$\displaystyle-\operatorname{\mathrm{div}}\left(\mathbf{a}_{0}\nabla\varphi_{0}\right)=f_{0}\text{
in }B(x_{0},1/2).$ (17)

Fix $x\in B(x_{0},1)$ and let $U\subset B(x_{0},1)$ be an open neighborhood of
$x$. By the De Giorgi-Nash-Moser Theorem
(gilbargEllipticPartialDifferential2001, , Theorem 8.24), there exists
$0<\beta=\beta(d,\lambda/\Lambda)<1$ such that:

$\displaystyle\left\lVert\varphi_{n}\right\rVert_{C^{\beta}(\overline{U})}\leq
C\left(\left\lVert\varphi_{n}\right\rVert_{L^{\infty}(B(x_{0},1))}+\left\lVert
f_{n}\right\rVert_{L^{\infty}({B}(x_{0},1))}\right)\leq 2C.$

By the Arzela-Ascoli Theorem, up to a subsequence, $\varphi_{n}$ uniformly
converges to $\varphi^{*}$ in $C(U)$ for some $\varphi^{*}$. Since
$\varphi_{n}\rightharpoonup\varphi_{0}$ in $L^{2}(B(x_{0},1))$, we conclude
that $\varphi^{*}=\varphi_{0}$ a.e. in $U$. Therefore,
$\lim_{n\to\infty}\varphi_{n}(x)=\varphi_{0}(x)$ a.e. in $B(x_{0},1)$. Letting
$n\to\infty$ in (15), the argument above and
(brezisFunctionalAnalysisSobolev2011, , Theorem 3.13) yield:

$\displaystyle\max\left\\{\left\lVert\varphi_{0}\right\rVert_{L^{\infty}(B(x_{0},1))},\left\lVert
f_{0}\right\rVert_{L^{\infty}({B}(x_{0},1))}\right\\}$ $\displaystyle\leq 1,$

which, together with (17), implies that (7) still holds for $\phi=\varphi_{0}$
(observe that, from (8), shrinking the domain from $B(x_{0},1)$ to
$B(x_{0},1/2)$ does not affect estimates), that is:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta}\left\lvert\varphi_{0}(x)-\varphi_{0}(x_{0})-(x-x_{0})\cdot\intbar_{B(x_{0},\theta)}\nabla\varphi_{0}(z)\operatorname{d\\!}z\right\rvert<\theta^{1+3\mu/4}.$
(18)

On the other hand, letting $n\to\infty$ in (16) and since
$\left\lVert\boldsymbol{\omega}\right\rVert_{L^{\infty}(Y)}<\infty$, we
obtain:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta}\left\lvert\varphi_{0}(x)-\varphi_{0}(x_{0})-(x-x_{0})\cdot\intbar_{B(x_{0},\theta)}\nabla\varphi_{0}(z)\operatorname{d\\!}z\right\rvert\geq\theta^{1+\mu/2},$

which contradicts (18), since $0<\theta<\frac{1}{4}.$

  2. 2.

_Iteration_ Let $0<\varepsilon<\varepsilon^{*}$. Direct evaluation yields
that:

$\displaystyle P^{\varepsilon}(x)\coloneqq$
$\displaystyle\frac{1}{\theta^{1+\mu/2}}\left\\{\varphi^{\varepsilon}(\theta
x)-\varphi^{\varepsilon}(\theta
x_{0})\vphantom{\intbar_{B(x_{0},\theta)}}\right.$
$\displaystyle\qquad\left.-\left[\theta(x-x_{0})+\varepsilon\boldsymbol{\omega}\left(\frac{\theta
x}{\varepsilon}\right)\right]\cdot\intbar_{B(x_{0},\theta)}\nabla\varphi^{\varepsilon}(z)\operatorname{d\\!}z\right\\}$

solves the following equation:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{\theta
x}{\varepsilon}\right)\nabla P^{\varepsilon}(x)\right]=\tilde{f}\text{ in
}B(x_{0},1),$ (19)

where $\tilde{f}\coloneqq\theta^{1-\mu/2}f(\theta x)$. Moreover, by (13) and
(12b), we have:

$\left\lVert P^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},1))}\leq
1\,\text{
and}\,\left\lVert\tilde{f}\right\rVert_{L^{\infty}({B}(x_{0},1))}\leq 1,$

so by using (13) again, we obtain:

$\displaystyle\sup_{\left\lvert x-x_{0}\right\rvert<\theta}\left\lvert
P^{\varepsilon}(x)-P^{\varepsilon}(x_{0})-\left[x-x_{0}+\frac{\varepsilon}{\theta}\boldsymbol{\omega}\left(\frac{\theta
x}{\varepsilon}\right)\right]\cdot\intbar_{B(x_{0},\theta)}\nabla
P^{\varepsilon}\right\rvert(z)\operatorname{d\\!}z\leq\theta^{1+\mu/2}.$ (20)

From (20) and scaling down, we have:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta^{2}}\left\lvert\varphi^{\varepsilon}(x)-\varphi^{\varepsilon}(x_{0})-\left(x-x_{0}\right)\cdot
a_{2}^{\varepsilon}+\varepsilon
b_{2}^{\varepsilon}\right\rvert\leq\theta^{2(1+\mu/2)},$

where:

$\displaystyle\begin{split}a_{2}^{\varepsilon}&\coloneqq\intbar_{B(x_{0},\theta)}\nabla\varphi^{\varepsilon}(z)\operatorname{d\\!}z+\theta^{\mu/2}~{}\intbar_{B(x_{0},\theta)}\nabla
P^{\varepsilon}(z)\operatorname{d\\!}z,\\\
b_{2}^{\varepsilon}(y)&\coloneqq\boldsymbol{\omega}\left(y\right)\cdot\left(\intbar_{B(x_{0},\theta)}\nabla\varphi^{\varepsilon}(z)\operatorname{d\\!}z+\theta^{\mu/2}\intbar_{B(x_{0},\theta)}\nabla
P^{\varepsilon}(z)\operatorname{d\\!}z\right),\text{ for
}y\coloneqq\frac{x}{\varepsilon}\in Y.\end{split}$ (21)

By the De Giorgi-Nash-Moser estimate and Caccioppoli inequality
(armstrongQuantitativeStochasticHomogenization2019, , Lemma C.2), there exists
a constant $C$, depending only on $\lambda,\Lambda$ and $d$, such that:

$\displaystyle\left\lVert\boldsymbol{\omega}\right\rVert_{L^{\infty}(Y)}\leq
C,\qquad\left\lvert\intbar_{B(x_{0},\theta)}\nabla\varphi^{\varepsilon}(z)\operatorname{d\\!}z\right\rvert\leq
C/\theta,\qquad\left\lvert\intbar_{B(x_{0},\theta)}\nabla
P^{\varepsilon}(z)\operatorname{d\\!}z\right\rvert\leq C/\theta.$

Therefore, (21) implies that:

$\displaystyle\left\lvert a_{2}^{\varepsilon}\right\rvert$
$\displaystyle\leq(C/\theta)\left(1+\theta^{\mu/2}\right),$
$\displaystyle\left\lVert b_{2}^{\varepsilon}\right\rVert_{L^{\infty}(Y)}$
$\displaystyle\leq(C/\theta)\left(1+\theta^{\mu/2}\right).$

Reiterating this process, we obtain that there exists
$C=C(\gamma,\lambda,\Lambda,d,\mu)>0$ such that:

$\displaystyle\begin{split}\left\lvert
a_{k}^{\varepsilon}\right\rvert&\leq(C/\theta)\left(1+\theta^{\mu/2}+\cdots+\theta^{(k-1)\mu/2}\right),\\\
\left\lVert
b_{k}^{\varepsilon}\right\rVert_{L^{\infty}(Y)}&\leq(C/\theta)\left(1+\theta^{\mu/2}+\cdots+\theta^{(k-1)\mu/2}\right),\end{split}$
(22)

and:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<\theta^{k}}\left\lvert\varphi^{\varepsilon}(x)-\varphi^{\varepsilon}(x_{0})-\left(x-x_{0}\right)\cdot
a_{k}^{\varepsilon}+\varepsilon
b_{k}^{\varepsilon}\right\rvert\leq\theta^{k(1+\mu/2)}.$ (23)

  3. 3.

_Blow-up_ Let
$\varepsilon_{0}\coloneqq\min\left\\{\varepsilon^{*},\frac{1}{5\sqrt{d}}\right\\}$
and $0<\varepsilon<\varepsilon_{0}.$

Choose $k$ such that $\theta^{k+1}\leq 4\varepsilon\sqrt{d}<\theta^{k}$. Then,
from (23), there exists $C=C(\theta,d)>0$ so that:

$\displaystyle\sup_{\left\lvert
x-x_{0}\right\rvert<4\varepsilon\sqrt{d}}\left\lvert\varphi^{\varepsilon}(x)-\varphi^{\varepsilon}(x_{0})-\left(x-x_{0}\right)\cdot
a_{k}^{\varepsilon}+\varepsilon b_{k}^{\varepsilon}\right\rvert\leq
C\varepsilon^{1+\mu/2},$ (24)

which, together with (22), leads to:

$\displaystyle\left\lVert\varphi^{\varepsilon}-\varphi^{\varepsilon}(x_{0})\right\rVert_{L^{\infty}(B(x_{0},4\varepsilon\sqrt{d}))}\leq
C\varepsilon.$ (25)

Denote by $z_{0}^{\varepsilon}$ the center of the cell $Y^{\varepsilon}_{i}$
containing $x_{0}$, and define:

$\displaystyle
v^{\varepsilon}(x)\coloneqq\frac{1}{\varepsilon}\left[\varphi^{\varepsilon}(\varepsilon
x+z_{0}^{\varepsilon})-\varphi^{\varepsilon}(\varepsilon
x_{0}+z_{0}^{\varepsilon})\right],\quad x\in\Omega.$

Then, $\nabla v^{\varepsilon}(x)=\nabla\varphi^{\varepsilon}\left(\varepsilon
x+z_{0}^{\varepsilon}\right)$ and, moreover, $v^{\varepsilon}$ solves:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(x+\frac{z_{0}^{\varepsilon}}{\varepsilon}\right)\nabla
v^{\varepsilon}(x)\right]=\varepsilon f(\varepsilon
x+z_{0}^{\varepsilon})\quad\text{ in }B\left(0,3\sqrt{d}\right).$ (26)

Observe that:

$\displaystyle\begin{split}\frac{1}{\varepsilon}\left(B\left(x_{0},\varepsilon\sqrt{d}\right)-z_{0}^{\varepsilon}\right)&\subset
B\left(0,2\sqrt{d}\right)\\\ &\subset
B\left(0,3\sqrt{d}\right)\subset\frac{1}{\varepsilon}\left(B\left(x_{0},4\varepsilon\sqrt{d}\right)-z_{0}^{\varepsilon}\right).\end{split}$
(27)

Applying (liGradientEstimatesSolutions2000, , Theorem 1.1) to (26), we obtain
that there exists a constant $C>0$, independent of $\varepsilon$ and $x_{0}$,
such that:

$\displaystyle\begin{split}&\left\lVert\nabla
v^{\varepsilon}\right\rVert_{L^{\infty}\left(\frac{1}{\varepsilon}\left(B\left(x_{0},\varepsilon\sqrt{d}\right)-z_{0}^{\varepsilon}\right)\right)}\\\
&\quad\leq\left\lVert\nabla
v^{\varepsilon}\right\rVert_{L^{\infty}\left(B\left(0,2\sqrt{d}\right)\right)}\\\
&\quad\leq C\left[\left\lVert
v^{\varepsilon}\right\rVert_{L^{\infty}\left(B\left(0,3\sqrt{d}\right)\right)}+\sup_{x\in
B\left(0,3\sqrt{d}\right)}\left\lvert\varepsilon f(\varepsilon
x+z_{0}^{\varepsilon})\right\rvert\right]\\\ &\quad\leq C\left[\left\lVert
v^{\varepsilon}\right\rVert_{L^{\infty}\left(\frac{1}{\varepsilon}\left(B\left(x_{0},4\varepsilon\sqrt{d}\right)-z_{0}^{\varepsilon}\right)\right)}+\left\lVert
f\right\rVert_{L^{\infty}\left(\frac{1}{\varepsilon}\left(B\left(x_{0},4\varepsilon\sqrt{d}\right)-z_{0}^{\varepsilon}\right)\right)}\right]\end{split}$
(28)

Scaling down (28) and using (25), we obtain:

$\displaystyle\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}\left(B\left(x_{0},\varepsilon\sqrt{d}\right)\right)}$
$\displaystyle\leq
C\left[\frac{1}{\varepsilon}\left\lVert\varphi^{\varepsilon}-\varphi^{\varepsilon}(x_{0})\right\rVert_{L^{\infty}\left(\left(B\left(x_{0},4\varepsilon\sqrt{d}\right)\right)\right)}+\left\lVert
f\right\rVert_{L^{\infty}\left(B\left(x_{0},4\varepsilon\sqrt{d}\right)\right)}\right]\leq
C,$

where $C>0$ is independent of $x_{0}$ and $\varepsilon$.

###### Remark 2

Under stronger smoothness assumptions on the coefficient $\mathbf{a}$, similar
estimates to (6) are proved in the literature. In particular, if $\mathbf{a}$
is in VMO$({\mathbb{R}}^{d})$, the real-variable method of L. Caffarelli and
I. Peral caffarelliW1PestimatesElliptic1998  yields an uniform
$W^{1,p}-$estimate; on the other hand, if $\mathbf{a}$ is Hölder continuous,
then one has the uniform Lipschitz estimate. Those results hold also for
elliptic systems, and even for Neumann boundary condition. We refer the reader
to shenPeriodicHomogenizationElliptic2018 ; shenBoundaryEstimatesElliptic2017
; shenW1EstimatesElliptic2008 ; kenigHomogenizationEllipticSystems2013 ;
liEstimatesEllipticSystems2003 ; avellanedaCompactnessMethodsTheory1989 ;
avellanedaCompactnessMethodsTheory1987 ;
prangeUniformEstimatesHomogenization2014  and references cited therein.

However, in this paper, we focus on the case when $\mathbf{a}$ is only
piecewise Hölder continuous. A similar argument as in the papers cited above,
together with the regularity theorem of Li and Vogelius
(liGradientEstimatesSolutions2000, , Theorem 1.1) yield the interior Lipschitz
estimate as showed in Proposition 1. Moreover, some additional care is needed
to ensure that the constant $C$ in (28) is independent of _both_ $\varepsilon$
and $x_{0}$. In the Blow-up step of the proof above, one may try to let

$\displaystyle
s^{\varepsilon}(x)\coloneqq\frac{1}{\varepsilon}\varphi^{\varepsilon}(\varepsilon
x+x_{0})$ (29)

so that

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(x\right)\nabla
s^{\varepsilon}(x)\right]=\varepsilon f(\varepsilon x+x_{0}),$ (30)

then by applying (liGradientEstimatesSolutions2000, , Theorem 1.1), one
obtains

$\left\lVert\nabla
s^{\varepsilon}\right\rVert_{L^{\infty}\left(B\left(0,\frac{1}{2\varepsilon_{0}}\right)\right)}\leq
C^{\prime}\left[\left\lVert
s^{\varepsilon}\right\rVert_{L^{\infty}\left(B\left(0,\frac{1}{\varepsilon_{0}}\right)\right)}+\left\lVert
f\right\rVert_{L^{\infty}\left(B\left(0,\frac{1}{\varepsilon_{0}}\right)\right)}\right].$

However, _$C^{\prime}$ indeed depends on $x_{0}$_. The reason is that, when
one shifts $x_{0}$ in the scaling (29), one also changes the
$C^{1,\alpha}$-modulus of the subdomains, where the latter, in the context of
our problem, are generated by taking intersections of the ball centered at
$x_{0}$ and the heterogenous domain. In short, we do not have the uniform
control of the subdomains when using the scaling (29) for arbitrary $x_{0}$,
see Figure 2. In order to circumvent the dependence on $x_{0}$, we use a
different scaling and combine with a geometric argument, as demonstrated in
the proof of Proposition 1.

[scale=.450]Cablowup-mag.eps

Figure 2: As the center $x_{0}$ of the ball
$B\left(x_{0},\frac{1}{\varepsilon_{0}}\right)$ slides on the line $d$ to the
right, the subdomain $D_{I}$ shrinks to 0, which makes the $C^{1,\alpha}$
modulus to become unbounded (liGradientEstimatesSolutions2000, , page 93).
Moreover, in some cases, it is possible that a cusp also appears at some
points (point $A$ on the zoomed in figure above).

The following result follows from Proposition 1, the De Giorgi-Nash-Moser
estimate and a change of variable.

######  Proposition 2 (Interior Lipschitz Estimate II)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ and
$f\in L^{\infty}(\Omega)$. Fix $x_{0}\in\Omega$, $R>0$ so that
$B(x_{0},R)\subset\Omega$. There exist
$\varepsilon_{0}=\varepsilon_{0}(\lambda,\Lambda,d)>0$ and
$C=C(\lambda,\Lambda,d)>0$ such that, for all $0<\varepsilon<\varepsilon_{0}$,
the weak solution $\varphi^{\varepsilon}\in H^{1}(B(x_{0},R))$ of the equation
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]=f$
in $B(x_{0},R)$ satisfies:

$\displaystyle\left\lvert\nabla\varphi^{\varepsilon}(x_{0})\right\rvert\leq
C^{\prime}\left[\left(\intbar_{B(x_{0},R)}\left\lvert\nabla\varphi^{\varepsilon}(z)\right\rvert^{2}\operatorname{d\\!}z\right)^{\frac{1}{2}}+R\sup_{x\in
B(x_{0},R)}\left\lvert f(x)\right\rvert\right].$ (31)

###### Proof

Without loss of generality, we assume $x_{0}=0$. By Proposition 1, with $R=1$
and considering
$\varphi^{\varepsilon}-\intbar_{B(0,1)}\varphi^{\varepsilon}(z)\operatorname{d\\!}z$,
which solves
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]=f$
in $B(0,1)$, we have that there exist $\varepsilon_{0}>0$ and $C^{\prime}>0$,
depending only on $\lambda,\Lambda,d$, such that:

$\displaystyle\begin{split}\left\lvert\nabla\varphi^{\varepsilon}(0)\right\rvert&\leq\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(0,1/4))}\\\
&\leq
C^{\prime}\left(\left\lVert\varphi^{\varepsilon}-\intbar_{B(0,1)}\varphi^{\varepsilon}(z)\operatorname{d\\!}z\right\rVert_{L^{\infty}(B(0,1/2))}+\left\lVert
f\right\rVert_{L^{\infty}(B(0,1/2))}\right)\\\ &\leq
C^{\prime}\left(\left\lVert\varphi^{\varepsilon}-\intbar_{B(0,1)}\varphi^{\varepsilon}(z)\operatorname{d\\!}z\right\rVert_{L^{2}(B(0,1))}+\left\lVert
f\right\rVert_{L^{\infty}(B(0,1))}\right)\\\ &\leq
C^{\prime}\left(\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{2}(B(0,1))}+\left\lVert
f\right\rVert_{L^{\infty}(B(0,1))}\right)\\\ &\leq
C^{\prime}\left[\left(\intbar_{B(0,1)}\left\lvert\nabla\varphi^{\varepsilon}(z)\right\rvert^{2}\operatorname{d\\!}z\right)^{\frac{1}{2}}+\sup_{x\in
B(0,1)}\left\lvert f(x)\right\rvert\right],\end{split}$ (32)

where we have used the De Giorgi-Nash-Moser estimate and Poincaré’s
inequality.

For $R>0$ and $x\in B(0,1)$, let $v^{\varepsilon}(x)\coloneqq
R^{-1}\varphi^{\varepsilon}(Rx),$ then $\nabla
v^{\varepsilon}(x)=\nabla\varphi^{\varepsilon}(Rx)$ and:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{b}\left(\frac{x}{\varepsilon}\right)\nabla
v^{\varepsilon}(x)\right]=Rf(Rx),$

where $\mathbf{b}(z)\coloneqq\mathbf{a}(Rz).$ We have
$\mathbf{b}\in\mathfrak{M}(\lambda,\Lambda)$ is $R^{-1}Y-$periodic. Note that
the proof of Proposition 1 does not depend on the period, hence, (32) holds
for $v^{\varepsilon}$ in particular:

$\displaystyle\begin{split}\left\lvert\nabla\varphi^{\varepsilon}(0)\right\rvert&=\left\lvert\nabla
v^{\varepsilon}(0)\right\rvert\\\ &\leq
C^{\prime}\left[\left(\intbar_{B(0,1)}\left\lvert\nabla
v^{\varepsilon}(x)\right\rvert^{2}\operatorname{d\\!}x\right)^{\frac{1}{2}}+R\sup_{x\in
B(0,1)}\left\lvert f(Rx)\right\rvert\right]\\\
&=C^{\prime}\left[\left(\intbar_{B(0,1)}\left\lvert\nabla\varphi^{\varepsilon}(Rx)\right\rvert^{2}\operatorname{d\\!}x\right)^{\frac{1}{2}}+R\sup_{x\in
B(0,1)}\left\lvert f(Rx)\right\rvert\right].\end{split}$ (33)

By a change of variable in (33), we obtain (31).

We recall the interior Hölder estimate, adapted from
(prangeUniformEstimatesHomogenization2014, , Proposition 1) (or
(avellanedaCompactnessMethodsTheory1987, , Lemma 9)) that will be used to
obtain the boundary Hölder estimate in the next section.

######  Proposition 3 (Interior Hölder Estimate)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ and
$f\in L^{d+\gamma}(\Omega)$, for some $\gamma>0$. Fix $x_{0}\in\Omega$, $R>0$
such that $B(x_{0},R)\subset\Omega$. Let
$0<\mu\coloneqq\frac{\gamma}{d+\gamma}<1$. There exists
$C=C(\gamma,\lambda,\Lambda,p,R,d)>0$ such that, for all $\varepsilon>0$, the
weak solution $\varphi^{\varepsilon}\in H^{1}(B(x_{0},R))$ of the equation
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]=f$
in $B(x_{0},R)$ satisfies:

$\displaystyle\left[\varphi^{\varepsilon}\right]_{C^{0,\mu}(B(x_{0},R/2))}\leq
C\left(\left\lVert\varphi^{\varepsilon}\right\rVert_{L^{2}(B(x_{0},R))}+\left\lVert
f\right\rVert_{L^{d+\gamma}(B(x_{0},R))}\right),$ (34)

where $\left[h\right]_{C^{0,\mu}(A)}\coloneqq\sup_{x\neq y\in
A}\frac{\left\lvert h(x)-h(y)\right\rvert}{\left\lvert
x-y\right\rvert^{\mu}}$.

The proof of Proposition 3 is similar to
(prangeUniformEstimatesHomogenization2014, , Proposition 1). Indeed, a closer
look at the proof of (prangeUniformEstimatesHomogenization2014, , Proposition
1) reveals that the Hölder continuity assumption on $\mathbf{a}$ is needed
only for the classical Schauder estimate for elliptic systems to hold.
However, this paper is devoted to the scalar case, and we use the De Giorgi-
Nash-Moser Theorem (gilbargEllipticPartialDifferential2001, , Theorem 8.24),
for which the assumption $\mathbf{a}$ is bounded is sufficient.

###### Remark 3

For the case of elliptic systems, the De Giorgi-Nash-Moser Theorem does not
hold, see counterexamples by De Giorgi, Giusti and Miranda, and others, c.f.
(giaquintaIntroductionRegularityTheory2012, , Section 9.1), and references
cited therein. Because of that, this paper is concerned with the scalar case
only.

##  5 Boundary estimates, Green functions, Dirichlet correctors and proof of
main theorem

The following result is adapted from (avellanedaCompactnessMethodsTheory1987,
, Section 2.3) and (shenPeriodicHomogenizationElliptic2018, , Section 5.2).

######  Proposition 4 (Boundary Hölder Estimate)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$, and
$\Omega$ is a $C^{1}$-domain. Fix $x_{0}\in\partial\Omega$,
$0<r<\operatorname{diam}(\Omega)$ and $0<\mu<1$. Let $g\in
C^{0,1}\left(B(x_{0},r)\cap\partial\Omega\right).$ There exist
$\varepsilon_{0}=\varepsilon_{0}(\mu,\lambda,\Lambda,d)>0$ and
$C=c(\mu,\lambda,\Lambda,d)>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$, every weak solution $\varphi^{\varepsilon}\in
H^{1}(B(x_{0},r))$ of the equation:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]$
$\displaystyle=0\text{ in }B(x_{0},r)\cap\Omega,$
$\displaystyle\varphi^{\varepsilon}$ $\displaystyle=g\text{ on
}B(x_{0},r)\cap\partial\Omega,$

satisfies:

$\displaystyle\begin{split}&\left[\varphi^{\varepsilon}\right]_{C^{0,\mu}(B(x_{0},r/2)\cap\Omega)}\\\
&\quad\leq
Cr^{-\mu}\left[\left(\intbar_{B(x_{0},r)\cap\Omega}\left\lvert\varphi^{\varepsilon}(z)\right\rvert^{2}\operatorname{d\\!}z\right)^{\frac{1}{2}}+\left\lvert
g(x_{0})\right\rvert+r\left\lVert
g\right\rVert_{C^{0,1}\left(B(x_{0},r)\cap\partial\Omega\right)}\right].\end{split}$
(35)

The proof of Proposition 4 follows the proof of
(shenPeriodicHomogenizationElliptic2018, , Theorem 5.2.1) with minor
modifications. In (shenPeriodicHomogenizationElliptic2018, , Theorem 5.2.1),
the assumption that $\mathbf{a}\in\mathrm{VMO}({\mathbb{R}}^{d})$ is used only
in two places: 1) when $\varepsilon\geq\varepsilon_{0}$, which is beyond of
the scope of this particular theorem, and 2) to obtain the interior Hölder
estimate, which we already relaxed in Proposition 3.

Thanks to Proposition 3 and Proposition 4, we can drop the assumption that
$\mathbf{a}\in\mathrm{VMO}({\mathbb{R}}^{d})$ of Theorem 5.4.1-2 and Lemma
5.4.5 in shenPeriodicHomogenizationElliptic2018 . The results are summarized
in the following proposition.

######  Proposition 5 (Green’s functions)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,\Lambda)$ and
$\Omega$ is a $C^{1}$-domain. Fix $0<\mu,\sigma,\sigma_{1}<1$ and let
$\delta(x)\coloneqq\operatorname{\mathrm{dist}}(x,\partial\Omega).$ Then,
there exist $\varepsilon_{0}=\varepsilon_{0}(\mu,\lambda,\Lambda,d)>0$ and
$C=C(\lambda,\Lambda,\sigma,\sigma_{1},\Omega)>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$, the Green’s functions $G^{\varepsilon}(x,y)$
exist and satisfy the following:

$\displaystyle\left\lvert
G^{\varepsilon}(x,y)\right\rvert\leq\begin{cases}C\frac{1}{\left\lvert
x-y\right\rvert^{d-2}}&\text{ if }d\geq 3,\\\
C\left[1+\ln\left(\frac{r_{0}}{\left\lvert
x-y\right\rvert}\right)\right]&\text{ if }d=2.\end{cases}$ (36a)
$\displaystyle\left\lvert
G^{\varepsilon}(x,y)\right\rvert\leq\begin{cases}\frac{C\delta(x)^{\sigma}}{\left\lvert
x-y\right\rvert^{d-2+\sigma}}&\text{ if }\delta(x)<\frac{1}{2}\left\lvert
x-y\right\rvert,\\\ \frac{C\delta(y)^{\sigma_{1}}}{\left\lvert
x-y\right\rvert^{d-2+\sigma_{1}}}&\text{ if }\delta(y)<\frac{1}{2}\left\lvert
x-y\right\rvert,\\\
\frac{C\delta(x)^{\sigma}\delta(y)^{\sigma_{1}}}{\left\lvert
x-y\right\rvert^{d-2+\sigma+\sigma_{1}}}&\text{ if
}\delta(x)<\frac{1}{2}\left\lvert x-y\right\rvert\text{ or
}\delta(y)<\frac{1}{2}\left\lvert x-y\right\rvert,\end{cases}$ (36b)
$\displaystyle\int_{\Omega}\left\lvert\nabla_{y}G^{\varepsilon}(x,y)\right\rvert\delta(y)^{\sigma-1}\operatorname{d\\!}y\leq
C\delta(x)^{\sigma},$ (36c)

where $x,y\in\Omega,x\neq y$ and $r_{0}\coloneqq\operatorname{diam}(\Omega).$

As a consequence, for $0<c<1$ and $g\in C^{0,1}(\Omega)$, there exists
$C=C(\lambda,\Lambda,d)>0$ such that, for any $x_{0}\in\partial\Omega$, for
any $\varepsilon$ satisfying $c\varepsilon\leq\min\\{c\varepsilon_{0},r\\}\leq
r<r_{0}\coloneqq\operatorname{diam}(\Omega)$, and for any solution
$\varphi^{\varepsilon}$ of the Dirichlet problem
$-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]=0$
in $\Omega$, $\varphi^{\varepsilon}=g$ on $\partial\Omega$, the following
estimate holds:

$\displaystyle\left(\intbar_{B(x_{0},r)\cap\Omega}\left\lvert\nabla\varphi^{\varepsilon}\right\rvert^{2}\right)^{\frac{1}{2}}\leq
C\left[\left\lVert\nabla
g\right\rVert_{L^{\infty}(\Omega)}+\varepsilon^{-1}\left\lVert
g\right\rVert_{L^{\infty}(\Omega)}\right].$ (37)

We now define the boundary Dirichlet corrector: For $1\leq i\leq d,$ let
$\Phi^{i,\varepsilon}\in H^{1}(\Omega)$ be the solution of the problem:

$\displaystyle\begin{split}-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\Phi^{i,\varepsilon}(x)\right]&=0\text{
for }x\in\Omega,\\\ \Phi^{i,\varepsilon}(z)&=z_{i}\text{ for
}z\in\partial\Omega.\end{split}$ (38)

The following proposition provides a bound on the boundary Dirichlet
corrector.

###### Proposition 6

Let $\Omega$ be a bounded $C^{1,\alpha}$-domain. Suppose that
$\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,L)$ is piecewise
$C^{\alpha}$-continuous. Then, for all $\varepsilon>0$, the solution
$\Phi^{i,\varepsilon}$ of (38) satisfies:

$\displaystyle\left\lVert\nabla\Phi^{i,\varepsilon}\right\rVert_{L^{\infty}(\Omega)}\leq
C,$ (39)

where constant $C$ depends only on $\lambda,\Lambda$ and $\Omega$.

The proof is similar to (shenPeriodicHomogenizationElliptic2018, , Theorem
5.4.4). One only needs to use three observations:

  * •

The case $c\varepsilon\geq\min\\{c\varepsilon_{0},r\\}$ follows from
(liGradientEstimatesSolutions2000, , Theorem 1.2), by the same argument used
in Remark 1.

  * •

Let $\boldsymbol{\omega}=(\omega^{1},\omega^{2},\ldots,\omega^{d})$ be the
solutions of the cell problems (11). Given only that $\mathbf{a}$ is piecewise
$C^{\alpha}$-continuous, then $\nabla\boldsymbol{\omega}$ is bounded in
$L^{\infty}(Y,{\mathbb{R}}^{d\times d})$, see the first paragraph in the proof
of (francfortEnhancementElastodielectricsHomogenization2021, , Theorem 3.2) or
(zhangInteriorHolderGradient2013a, , Corollary 3.5).

  * •

The interior Lipschitz estimate in Proposition 2 only requires $\mathbf{a}$ is
piecewise Hölder continuous.

We now combine Proposition 6, Proposition 2 and
(liGradientEstimatesSolutions2000, , Theorem 1.2) to obtain a discontinuous
coefficient-version of (shenPeriodicHomogenizationElliptic2018, , Theorem
5.5.1).

######  Proposition 7 (Boundary Lipschitz Estimate)

Suppose that $\mathbf{a}\in\mathfrak{M}_{\mathrm{per}}(\lambda,L)$ is
piecewise $C^{\alpha}$-continuous, and $\Omega$ is a $C^{1,\alpha}$-domain.
Fix $x_{0}\in\partial\Omega$, $0<r<\operatorname{diam}(\Omega)$ and $0<\mu<1$.
Let $g\in C^{1,\alpha}\left(B(x_{0},r)\cap\partial\Omega\right).$ There exist
$\varepsilon_{0}=\varepsilon_{0}(\mu,\lambda,\Lambda,d,\Omega)>0$ and
$C=C(\mu,\lambda,\Lambda,d,\Omega)>0$ such that, for all
$0<\varepsilon<\varepsilon_{0}$, the weak solution $\varphi^{\varepsilon}\in
H^{1}(B(x_{0},r))$ of the equation:

$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]$
$\displaystyle=0\text{ in }B(x_{0},r)\cap\Omega,$
$\displaystyle\varphi^{\varepsilon}$ $\displaystyle=g\text{ on
}B(x_{0},r)\cap\partial\Omega,$

satisfies:

$\displaystyle\begin{split}&\left\lVert\nabla\varphi^{\varepsilon}\right\rVert_{L^{\infty}(B(x_{0},r/2)\cap\partial\Omega)}\\\
&\leq
C\left[r^{-1}\left(\intbar_{B(x_{0},r)\cap\Omega}\left\lvert\varphi^{\varepsilon}\right\rvert^{2}\right)^{\frac{1}{2}}+r^{\alpha}\left\lVert\nabla_{\tan}g\right\rVert_{C^{0,\alpha}(B(x_{0},r)\cap\partial\Omega)}\right.\\\
&\quad\qquad\left.{}{+}{}\left\lVert\nabla_{\tan}g\right\rVert_{L^{\infty}(B(x_{0},r)\cap\partial\Omega)}+r^{-1}\left\lVert
g\right\rVert_{L^{\infty}(B(x_{0},r)\cap\partial\Omega)}\vphantom{\left(\intbar_{B(x_{0},r)\cap\Omega}\left\lvert\varphi^{\varepsilon}\right\rvert^{2}\right)^{\frac{1}{2}}}\right].\end{split}$
(40)

The estimate (5) of Theorem 3.1 is a consequence of Proposition 2 and
Proposition 7, by an argument similar to
(shenPeriodicHomogenizationElliptic2018, , Theorem 5.6.2).

##  6 Application to magnetic suspensions

In this section, we apply the regularity results obtained above to the
rigorous homogenization procedure discussed in
dangHomogenizationNondiluteSuspension2021 . For that, we first recap the
formulation of the fine-scale problem and the homogenization result itself. We
begin by introducing the definition of two-scale convergence, which will be
used below.

###### Definition 1

A sequence $\\{v^{\varepsilon}\\}_{\varepsilon>0}$ in $L^{2}(\Omega)$ is said
to _two-scale converge_ to $v=v({x},{y})$, with $v\in L^{2}(\Omega\times Y)$,
if and only if:

$\displaystyle\lim_{\varepsilon\to
0}\int_{\Omega}v^{\varepsilon}(x)\psi\left(x,\frac{x}{\varepsilon}\right)\operatorname{d\\!}x=\frac{1}{\left\lvert
Y\right\rvert}\int_{\Omega}\int_{Y}v(x,y)\psi(x,y)\operatorname{d\\!}y\operatorname{d\\!}x,$

for any test function $\psi=\psi({x},{y})$, with
$\psi\in\mathcal{D}(\Omega,C_{\mathrm{per}}^{\infty}(Y))$, see
allaireHomogenizationTwoscaleConvergence1992 ;
cioranescuIntroductionHomogenization1999 ;
nguetsengGeneralConvergenceResult1989 . In this case, we write
$v^{\varepsilon}\xrightharpoonup{~{}2~{}}v$.

Let the kinematic viscosity be denoted by $\nu=\frac{\eta}{\rho_{f}}$, where
$\eta>0$ and $\rho_{f}>0$ are the fluid viscosity and the fluid density,
respectively. The dimensionless quantities that appear in this problem are the
(hydrodynamic) _Reynolds number_ ${R}_{{e}}=UL/\nu$ , the _Froude number_
${F}_{{r}}=U/\sqrt{FL}$ and the _coupling parameter_
$S=\frac{B^{2}}{\rho_{f}\Lambda U^{2}},$ where $L,U,B$ and $F$ are the
characteristic scales corresponding to length, fluid velocity, magnetic field
and body density force, respectively. Moreover, $\Lambda>0$ is defined in
(A2).

From now on, we suppose $\Omega$ is $C^{3,\alpha}$, which is needed for the
corrector result below. Suppose further that $\mathbf{g}\in
H^{1}({\Omega},{\mathbb{R}}^{d})$, $k\in C^{1,\alpha}(\partial{\Omega})$ and
$f\in L^{\infty}(\Omega)$. Let $\mathbf{u}^{\varepsilon}$ and
$p^{\varepsilon}$ be the fluid velocity and the fluid pressure, respectively.
Also, in a space free of current, the magnetic field strength is given by
$\mathbf{H}^{\varepsilon}=\nabla\varphi^{\varepsilon}$, for some magnetic
potential $\varphi^{\varepsilon}(x)$. Let $\mathbf{u}^{\varepsilon}\in
H_{0}^{1}(\Omega,{\mathbb{R}}^{d})$, $p^{\varepsilon}\in
L^{2}(\Omega)/{\mathbb{R}}$, and $\varphi^{\varepsilon}\in H^{1}(\Omega)$ be
the solution of the following boundary value problem:

$\displaystyle-\operatorname{\mathrm{div}}\left[\boldsymbol{\sigma}(\mathbf{u}^{\varepsilon},p^{\varepsilon})+\boldsymbol{\tau}(\varphi^{\varepsilon})\right]$
$\displaystyle=\frac{1}{{F}_{{r}}^{2}}\mathbf{g},$ $\displaystyle\text{ in
}\Omega_{f}^{\varepsilon}$ (41a)
$\displaystyle\operatorname{\mathrm{div}}\mathbf{u}^{\varepsilon}$
$\displaystyle=0,$ $\displaystyle\text{ in }\Omega_{f}^{\varepsilon}$ (41b)
$\displaystyle{\mathbb{D}}(\mathbf{u}^{\varepsilon})$ $\displaystyle=0,$
$\displaystyle\text{ in }\Omega_{s}^{\varepsilon}$ (41c)
$\displaystyle-\operatorname{\mathrm{div}}\left[\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\varphi^{\varepsilon}\right]$
$\displaystyle=f$ $\displaystyle\text{ in }\Omega,$ (41d)

together with the balance equations:

$\displaystyle\int_{\Gamma_{i}^{\varepsilon}}\left[\boldsymbol{\sigma}(\mathbf{u}^{\varepsilon},p^{\varepsilon})+\boldsymbol{\tau}(\varphi^{\varepsilon})\right]\mathbf{n}_{i}\operatorname{d\\!}\mathcal{H}^{d-1}$
$\displaystyle=0,$ (42a)
$\displaystyle\int_{\Gamma_{i}^{\varepsilon}}\left(\left[\boldsymbol{\sigma}(\mathbf{u}^{\varepsilon},p^{\varepsilon})+\boldsymbol{\tau}(\varphi^{\varepsilon})\right]\mathbf{n}_{i}\right)\times\mathbf{n}_{i}\operatorname{d\\!}\mathcal{H}^{d-1}$
$\displaystyle=0,$ (42b)

and boundary conditions:

$\displaystyle\mathbf{u}^{\varepsilon}$ $\displaystyle=0,\text{ on
}\partial\Omega,$ (43a) $\displaystyle\varphi^{\varepsilon}$
$\displaystyle=k,\text{ on }\partial\Omega,$ (43b)

where:

$\displaystyle\boldsymbol{\sigma}(\mathbf{u}^{\varepsilon},p^{\varepsilon})$
$\displaystyle\coloneqq\frac{2}{{R}_{{e}}}{\mathbb{D}}(\mathbf{u}^{\varepsilon})-p^{\varepsilon}{\mathbf{I}},$
(44a) $\displaystyle{\mathbb{D}}(\mathbf{u}^{\varepsilon})$
$\displaystyle\coloneqq\frac{\nabla\mathbf{u}^{\varepsilon}+\nabla^{\top}\mathbf{u}^{\varepsilon}}{2},$
(44b) $\displaystyle\boldsymbol{\tau}(\varphi^{\varepsilon})$
$\displaystyle\coloneqq
S\mathbf{a}\left(\frac{x}{\varepsilon}\right)\left(\nabla\varphi^{\varepsilon}\otimes\nabla\varphi^{\varepsilon}-\frac{1}{2}\left\lvert\nabla\varphi^{\varepsilon}\right\rvert^{2}{\mathbf{I}}\right).$
(44c)

are the rate of strain, the Cauchy stress and the Maxwell stress tensors,
respectively. For the detailed derivation and the physical meaning of the
equations above, we refer the readers to
dangHomogenizationNondiluteSuspension2021  and references therein. Observe
that, in the context of this paper, we consider the Dirichlet boundary
condition (43b), instead of a Neumann boundary condition (3) in
dangHomogenizationNondiluteSuspension2021 , to relax the regularity assumption
on the magnetic permeability needed in
dangHomogenizationNondiluteSuspension2021 . Then, the weak formulation for
(41d) and (43b) is given by:

$\displaystyle\begin{split}\int_{\Omega}&\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla\left(\varphi^{\varepsilon}-k\right)\cdot\nabla\xi\operatorname{d\\!}x\\\
&\quad=-\int_{\Omega}\mathbf{a}\left(\frac{x}{\varepsilon}\right)\nabla
k\cdot\nabla\xi\operatorname{d\\!}x+\int_{\Omega}f\xi\operatorname{d\\!}x,\quad\forall\xi\in
H_{0}^{1}(\Omega).\end{split}$ (45)

One immediately has that
$\left\lVert\varphi^{\varepsilon}\right\rVert_{H^{1}(\Omega)}\leq
C\left(\left\lVert k\right\rVert_{H^{1/2}(\partial\Omega)}+\left\lVert
f\right\rVert_{L^{q}(\Omega)}\right)$, which implies that
$\varphi^{\varepsilon}$ is two-scale convergent. Choosing a test function as
in (dangHomogenizationNondiluteSuspension2021, , Lemma 3.7), we obtain the
cell problem (46) and the first two effective equations defined in (50) below.

Moreover, Theorem 3.1 ensures that $\nabla\varphi^{\varepsilon}$ is uniformly
bounded in $L^{\infty}(\Omega,{\mathbb{R}}^{d})$, with respect to
$\varepsilon\in(0,\varepsilon_{0})$. Therefore, we obtain the existence,
uniqueness and a priori bounds for $\mathbf{u}^{\varepsilon}$ and
$p^{\varepsilon}$ as in (dangHomogenizationNondiluteSuspension2021, ,
Corollary 3.11). Here, we have relaxed the restrictive assumption (1) made in
dangHomogenizationNondiluteSuspension2021  and we can use our results in the
case when the constant magnetic permeability is anisotropic, namely, when
$\mathbf{a}$ is a matrix.

To carry on with the homogenization formulation, for $1\leq i,j\leq d,$ denote
by $\mathbf{U}^{ij}$ the vector defined by $\mathbf{U}^{ij}_{k}\coloneqq
y_{j}\delta_{ik}$. Consider $\omega^{i}\in
H^{1}_{\mathrm{per}}(Y)/{\mathbb{R}}$, the solution of:

$-\operatorname{\mathrm{div}}_{y}\left[{\mathbf{a}(y)}\left(\mathrm{e}^{i}+\nabla_{y}\omega^{i}(y)\right)\right]=0\text{
in }Y.$ (46)

Also, consider $\boldsymbol{\chi}^{ij}\in
H^{1}_{\mathrm{per}}(Y,{\mathbb{R}}^{d})/{\mathbb{R}}$ and $q^{ij}\in
L^{2}(Y)/{\mathbb{R}}$, solving:

$\displaystyle\begin{split}\operatorname{\mathrm{div}}_{y}\left[{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\chi}^{ij}\right)+q^{ij}{\mathbf{I}}\right]&=0\text{
in }Y_{f},\\\ \operatorname{\mathrm{div}}_{y}\boldsymbol{\chi}^{ij}&=0\text{
in }Y,\\\
{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\chi}^{ij}\right)&=0\text{
in }Y_{s},\\\
\int_{\Gamma}\left[{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\chi}^{ij}\right)-q^{ij}{\mathbf{I}}\right]\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\\\
\int_{\Gamma}\left[{\mathbb{D}}_{y}\left(\mathbf{U}^{ij}-\boldsymbol{\chi}^{ij}\right)-q^{ij}{\mathbf{I}}\right]\mathbf{n}_{\Gamma}\times\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\end{split}$
(47)

and consider $\boldsymbol{\xi}^{ij}\in
H^{1}_{\mathrm{per}}(Y,{\mathbb{R}}^{d})/{\mathbb{R}}$ and $r^{ij}\in
L^{2}(Y)/{\mathbb{R}}$, solving:

$\displaystyle\begin{split}\operatorname{\mathrm{div}}_{y}\left[{\mathbb{D}}_{y}\left(\boldsymbol{\xi}^{ij}\right)+r^{ij}{\mathbf{I}}+\boldsymbol{\tau}^{ij}\right]&=0\text{
in }Y_{f},\\\ \operatorname{\mathrm{div}}_{y}\boldsymbol{\xi}^{ij}&=0\text{ in
}Y,\\\ {\mathbb{D}}_{y}\left(\boldsymbol{\xi}^{ij}\right)&=0\text{ in
}Y_{s},\\\
\int_{\Gamma}\left[{\mathbb{D}}_{y}\left(\boldsymbol{\xi}^{ij}\right)+r^{ij}{\mathbf{I}}+\boldsymbol{\tau}^{ij}\right]\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0,\\\
\int_{\Gamma}\left[{\mathbb{D}}_{y}\left(\boldsymbol{\xi}^{ij}\right)+r^{ij}{\mathbf{I}}+\boldsymbol{\tau}^{ij}\right]\mathbf{n}_{\Gamma}\times\mathbf{n}_{\Gamma}\operatorname{d\\!}\mathcal{H}^{d-1}&=0.\end{split}$
(48)

We also define:

$\displaystyle\begin{split}\mathcal{A}_{jk}&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\mathbf{a}(y)(\mathrm{e}^{k}+\nabla\omega^{k}(y))\cdot(\mathrm{e}^{j}+\nabla\omega^{j}(y))\operatorname{d\\!}y,\\\
\mathcal{N}^{ij}_{mn}&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}{\mathbb{D}}_{y}(\mathbf{U}^{ij}-\boldsymbol{\chi}^{ij}):{\mathbb{D}}_{y}(\mathbf{U}^{mn}-\boldsymbol{\chi}^{mn})\operatorname{d\\!}y,\\\
\boldsymbol{\tau}_{\mathrm{ref}}^{ij}&\coloneqq{\mathbf{a}(y)}\left[(\mathrm{e}^{i}+\nabla_{y}\omega^{i})\otimes(\mathrm{e}^{j}+\nabla_{y}\omega^{j})-\frac{1}{2}(\mathrm{e}^{i}+\nabla_{y}\omega^{i})\cdot(\mathrm{e}^{j}+\nabla_{y}\omega^{j}){\mathbf{I}}\right],~{}y\in
Y,\\\ \mathcal{B}^{ij}&\coloneqq\frac{1}{\left\lvert
Y\right\rvert}\int_{Y}\left({\mathbb{D}}_{y}(\boldsymbol{\xi}^{ij})+\boldsymbol{\tau}^{ij}\right)\operatorname{d\\!}y,\end{split}$
(49)

where $\mathcal{A}$ is the _effective magnetic permeability_ , which is
symmetric and elliptic. The tensor
$\mathcal{N}\coloneqq\left\\{\mathcal{N}^{ij}_{mn}\right\\}_{1\leq i,j,m,n\leq
d}$ is the _effective viscosity_ , and it is a fourth rank tensor. Moreover,
$\mathcal{N}$ is symmetric, i.e.
$\mathcal{N}^{ij}_{mn}=\mathcal{N}^{mn}_{ij}=\mathcal{N}^{ji}_{mn}=\mathcal{N}^{ij}_{nm}$,
and it satisfies the Legendre-Hadamard condition (or strong ellipticity
condition), i.e. there exist $\beta>0$ such that, for all
${\zeta},\eta\in{\mathbb{R}}^{d}$, one has
$\mathcal{N}^{ij}_{mn}\zeta_{i}\zeta_{m}\eta_{j}\eta_{n}\geq\beta\left\lvert{\zeta}\right\rvert^{2}\left\lvert{\eta}\right\rvert^{2}$.
The matrix $\boldsymbol{\tau}_{\mathrm{ref}}$ is the _Maxwell stress tensor_
on $Y$, and $\mathcal{B}$ is the _effective coupling matrix_.

By the same argument as in Theorem 3.5, Lemma 3.9 and Lemma 3.14 of
dangHomogenizationNondiluteSuspension2021 , the following result holds:

###### Theorem 6.1

Let $(\varphi^{\varepsilon},\mathbf{u}^{\varepsilon},p^{\varepsilon})\in
H^{1}(\Omega)\times H_{0}^{1}(\Omega,{\mathbb{R}}^{d})\times
L_{0}^{2}(\Omega)$ be the solution of (41). Then

$\displaystyle\varphi^{\varepsilon}$
$\displaystyle\rightharpoonup\varphi^{0}\text{ in }H^{1}(\Omega),$
$\displaystyle\mathbf{u}^{\varepsilon}$
$\displaystyle\rightharpoonup\mathbf{u}^{0}\text{ in
}H_{0}^{1}(\Omega,{\mathbb{R}}^{d}),$ $\displaystyle p^{\varepsilon}$
$\displaystyle\rightharpoonup\pi^{0}\text{ in }L_{0}^{2}(\Omega),$

where $\varphi^{0},\mathbf{u}^{0}$ and $\pi^{0}$ are solutions of:

$\displaystyle\begin{aligned}
-\operatorname{\mathrm{div}}\left(\mathcal{A}\nabla\varphi^{0}\right)&=f&&\text{
in }\Omega,\\\ \varphi^{0}&=k&&\text{ on }\partial\Omega,\\\
\operatorname{\mathrm{div}}\left[\frac{2}{{R}_{{e}}}\mathcal{N}^{ij}{\mathbb{D}}\left(\mathbf{u}^{0}\right)_{ij}-{\pi^{0}}+S\mathcal{B}^{ij}\frac{\partial\varphi^{0}}{\partial
x_{i}}\frac{\partial\varphi^{0}}{\partial
x_{j}}\right]&=\frac{1}{{F}_{{r}}^{2}}\mathbf{g}&&\text{ in }\Omega,\\\
\operatorname{\mathrm{div}}\mathbf{u}^{0}&=0&&\text{ in }\Omega\\\
\mathbf{u}^{0}&=0&&\text{ on }\partial\Omega,\end{aligned}$ (50)

with $\mathcal{A},\mathcal{N}^{ij},$ $\mathcal{B}^{ij}$, $1\leq i,j\leq d$,
defined in (49). Moreover, the first-order correctors satisfy:

$\displaystyle\lim_{\varepsilon\to
0}\left\lVert\nabla\varphi^{\varepsilon}(\cdot)-\nabla\varphi^{0}(\cdot)-\nabla_{y}\varphi^{1}\left(\cdot,\frac{\cdot}{\varepsilon}\right)\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d})}$
$\displaystyle=0,$ $\displaystyle\lim_{\varepsilon\to
0}\left\lVert{\mathbb{D}}(\mathbf{u}^{\varepsilon})(\cdot)-{\mathbb{D}}(\mathbf{u}^{0})(\cdot)-{\mathbb{D}}_{y}(\mathbf{u}^{1})\left(\cdot,\frac{\cdot}{\varepsilon}\right)\right\rVert_{L^{2}(\Omega,{\mathbb{R}}^{d\times
d})}$ $\displaystyle=0,$

where:

$\displaystyle\varphi^{1}(x,y)$
$\displaystyle\coloneqq\omega^{i}(y)\frac{\partial\varphi^{0}}{\partial
x_{i}}(x),$ $\displaystyle\mathbf{u}^{1}(x,y)$
$\displaystyle\coloneqq-{\mathbb{D}}\left(\mathbf{u}^{0}(x)\right)_{ij}\boldsymbol{\chi}^{ij}(y)+S\frac{\partial\varphi^{0}}{\partial
x_{i}}(x)\frac{\partial\varphi^{0}}{\partial
x_{j}}(x)\boldsymbol{\xi}^{ij}(y).$

##  7 Conclusions

This paper concerns a homogenized description of a non-dilute suspension of
magnetic particles in a viscous flow. The results demonstrated in this paper
generalize the ones obtained by the authors in
dangHomogenizationNondiluteSuspension2021 , where a more restrictive
assumption on the magnetic permeability (1) was used and a Neumann boundary
condition (3) was imposed instead of the Dirichlet condition (4b). Theorem 6.1
above demonstrates the _effective response_ of a viscous fluid with a locally
periodic array of paramagnetic/diamagnetic particles suspended in it, given by
the system of equations (41). The effective equations are described by (50),
with the effective coefficients given in (49). These effective quantities
depend only on the instantaneous position of the particles, their geometry,
and the magnetic and flow properties of the original suspension described by
(41). Using the tools introduced in liGradientEstimatesSolutions2000  and the
compactness method, an improved regularity estimate for the gradient of the
magnetic potential of the original fine-scale problem (41) was obtained, see
Theorem 3.1. This theorem allows us to drop the restrictive assumption (1)
mentioned above. Comparing to the classical results on regularity of this
type, we do require the coefficient matrix belongs to a VMO-space, see e.g.
avellanedaCompactnessMethodsTheory1987 ;
prangeUniformEstimatesHomogenization2014 ;
shenPeriodicHomogenizationElliptic2018 . Recently, in
(francfortEnhancementElastodielectricsHomogenization2021, , Proposition 3.1),
the authors obtained an $L^{q}$-bound of the gradient of the solution of the
scalar divergence equation, uniform with respect to $\varepsilon$, for
$q<\infty$. Our result, in Theorem 3.1, shows that the gradient bound actually
holds for the case $q=\infty$.

## Acknowledgements

The authors would like to thank Antoine Gloria for bringing up the results in
francfortEnhancementElastodielectricsHomogenization2021  to their attention.
The first author would also like to thank Christophe Prange for allowing him
to audit his class, in which he first learned about the compactness method
used in this paper. The work of the first author was partially supported by
the NSF through grant DMS-1350248. The work of the third author was supported
by NSF grant DMS-2110036. This material is based upon work supported by and
while serving at the National Science Foundation for the second author Yuliya
Gorb. Any opinion, findings, and conclusions or recommendations expressed in
this material are those of the authors and do not necessarily reflect views of
the National Science Foundation.

##  8 Appendix

######  Theorem 8.1 (Interior Schauder Estimates
gilbargEllipticPartialDifferential2001 ; hanEllipticPartialDifferential2011 )

Let $\mathbf{b}\in\mathfrak{M}(\lambda,\Lambda)$ be a constant matrix and
$w\in H^{1}(\Omega)$ be a weak solution of:

$\displaystyle\mathbf{b}_{ij}D_{i}D_{j}w=f+\sum_{i=1}^{d}D_{i}f_{i}.$

For every $\alpha\in(0,1)$, there exists a uniform constant
$C=C(\alpha,d,\lambda,\Lambda)$ such that if
$\Omega^{\prime}\subset\subset\Omega$, with
$\delta=\operatorname{\mathrm{dist}}(\Omega^{\prime},\partial\Omega)$, then
the following estimates hold:

  * (i)

If $f\in L^{p}(\Omega)$, $f_{i}\in L^{q}(\Omega)$ and
$\alpha=1-\frac{d}{q}=2-\frac{d}{p}\in(0,1)$, then $w\in
C^{\alpha}(\Omega^{\prime})$ and:

$\displaystyle\left\lVert w\right\rVert_{C^{\alpha}(\Omega^{\prime})}\leq
C\delta^{-\frac{d}{2}+1-\alpha}\left(\left\lVert
f\right\rVert_{L^{p}(\Omega)}+\sum_{i=1}^{d}\left\lVert
f_{j}\right\rVert_{L^{q}(\Omega)}+\left\lVert
w\right\rVert_{H^{1}(\Omega)}\right).$

  * (ii)

If $f\in L^{p}(\Omega),$ $\alpha=1-\frac{d}{p}\in(0,1)$ and $f_{i}\in
C^{\alpha}(\Omega),$ then $\nabla w\in C^{\alpha}(\Omega^{\prime})$ and:

$\displaystyle\left\lVert\nabla
w\right\rVert_{C^{\alpha}(\Omega^{\prime})}\leq
C\delta^{-\frac{d}{2}-\alpha}\left(\left\lVert
f\right\rVert_{L^{p}(\Omega)}+\sum_{i=1}^{d}\left\lVert
f_{i}\right\rVert_{C^{\alpha}(\Omega)}+\left\lVert
w\right\rVert_{H^{1}(\Omega)}\right).$

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