A Common Approach to Singular Perturbation and Homogenization II: Semilinear
Elliptic Systems

Nikolay N. Nefedov (Moscow) and Lutz Recke (Berlin)

###### Abstract

We consider perodic homogenization of boundary value problems for second-order
semilinear elliptic systems in 2D of the type

$\mathop{\rm div}\nolimits\left(A(x/\varepsilon)\nabla
u(x)\right)=b(x,u(x))\mbox{ for }x\in\Omega.$

For small $\varepsilon>0$ we prove existence of weak solutions
$u=u_{\varepsilon}$ as well as their local uniqueness for
$\|u-u_{0}\|_{\infty}\approx 0$, where $u_{0}$ is a given non-degenerate weak
solution to the homogenized boundary value problem, and we estimate the rate
of convergence to zero of $\|u_{\varepsilon}-u_{0}\|_{\infty}$ for
$\varepsilon\to 0$.

Our assumptions are, roughly speaking, as follows: The map $y\mapsto A(y)$ is
bounded, measurable and $\mathbb{Z}^{2}$-periodic, the maps $b(\cdot,u)$ are
bounded and measurable, the maps $b(x,\cdot)$ are $C^{1}$-smooth, and $\Omega$
is a bounded Lipschitz domain in $\mathbb{R}^{2}$. Neither global solution
uniqueness is supposed nor growth restriction of $b(x,\cdot)$ nor
$W^{2,2}$-regularity of $u_{0}$, and cross-diffusion is allowed.

The main tool of the proofs is an abstract result of implicit function theorem
type which in the past has been applied to singularly perturbed nonlinear ODEs
and elliptic and parabolic PDEs and, hence, which permits a common approach to
existence, local uniqueness and error estimates for singularly perturbed
problems and and for homogenization problems.

##  1 Introduction

In this paper we present an abstract result of implicit function theorem type
(see Section 2), which in the past has been applied to singularly pertubed
nonlinear ODEs and PDEs in [5, 6, 7, 18, 20, 21, 22] and, in Part I [19], to
periodic homogenization of quasilinear ODE systems. In the present paper we
apply it to periodic homogenization of Dirichlet problems for 2D semilinear
elliptic PDE systems of the type

$\left.\begin{array}[]{l}\partial_{x_{i}}\Big{(}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\Big{)}=b^{\alpha}(x,u(x))\mbox{
for }x\in\Omega,\\\ u^{\alpha}(x)=0\mbox{ for
}x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n$ (1.1)

as well as of other boundary value problems for those systems (see Section 4).
Here and in the following repeated indices are to be summed over
$\alpha,\beta,\gamma,\ldots=1,\ldots,n$ and $i,j,k,\ldots=1,2$, and
$\varepsilon>0$ is the small homogenization parameter. We assume that

$\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^{2}$, (1.2)

and

$a_{ij}^{\alpha\beta}\in L^{\infty}(\mathbb{R}^{2})\mbox{ and
}a_{ij}^{\alpha\beta}(\cdot+z)=a_{ij}^{\alpha\beta}\mbox{ for all
}z\in\mathbb{Z}^{2},$ (1.3)

and

$u\in\mathbb{R}^{n}\mapsto b^{\alpha}(\cdot,u)\in L^{\infty}(\Omega)\mbox{ is
$C^{1}$-smooth,}$ (1.4)

and that there exists $a>0$ such that for all $\varphi\in
C^{\infty}(\mathbb{R}^{2};\mathbb{R}^{n})$ with compact support we have

$\int_{\mathbb{R}^{2}}a_{ij}^{\alpha\beta}(y)\partial_{y_{i}}\varphi^{\alpha}(y)\partial_{y_{j}}\varphi^{\beta}(y)dy\geq
a\int_{\mathbb{R}^{2}}\partial_{y_{i}}\varphi^{\alpha}(y)\partial_{y_{i}}\varphi^{\alpha}(y)dy.$
(1.5)

The components of the homogenized diffusion tensor are

$\hat{a}^{\alpha\beta}_{ij}:=\int_{[0,1]^{2}}\left(a^{\alpha\beta}_{ij}(y)+a^{\alpha\gamma}_{ik}(y)\partial_{y_{k}}v^{\gamma\beta}_{j}(y)\right)dy,$
(1.6)

where the correctors $v^{\alpha\beta}_{j}$ are defined by the cell problems

$\left.\begin{array}[]{l}\partial_{y_{i}}\left(a^{\alpha\beta}_{ij}(y)+a^{\alpha\gamma}_{ik}(y)\partial_{y_{k}}v^{\gamma\beta}_{j}(y)\right)=0\mbox{
for }y\in\mathbb{R}^{2},\\\
v^{\alpha\beta}_{j}(\cdot+z)=v^{\alpha\beta}_{j}\mbox{ for
}z\in\mathbb{Z}^{2},\;\displaystyle\int_{[0,1]^{2}}v^{\alpha\beta}_{j}(y))dy=0,\end{array}\right\\}\alpha,\beta=1,\ldots,n;\;j=1,2.$
(1.7)

It is well-known (as a consequence of assumption (1.5) and the Lax-Milgram
lemma, cf. [23, Section 2.2 and Lemma 2.2.4]) that the problem (1.7) is
uniquely weakly solvable and that the homogenized diffusion coefficients
$\hat{a}^{\alpha\beta}_{ij}$ satisfy the coercivity condition (1.5) as well,
i.e.

$\int_{\mathbb{R}^{2}}\hat{a}_{ij}^{\alpha\beta}\partial_{y_{i}}\varphi^{\alpha}(y)\partial_{y_{j}}\varphi^{\beta}(y)dy\geq
a\int_{\mathbb{R}^{2}}\partial_{y_{i}}\varphi^{\alpha}(y)\partial_{y_{i}}\varphi^{\alpha}(y)dy$
(1.8)

for all $\varphi\in C^{\infty}(\mathbb{R}^{2};\mathbb{R}^{n})$ with compact
support.

Let us formulate our main result. It concerns existence and local uniqueness
of weak solutions $u=u_{\varepsilon}$ to (1.1) with $\varepsilon\approx 0$,
which are close to a given non-degenerate weak solution $u=u_{0}$ to the
homogenized problem

$\left.\begin{array}[]{l}\hat{a}^{\alpha\beta}_{ij}\partial_{x_{i}}\partial_{x_{j}}u^{\beta}(x)=b^{\alpha}(x,u(x))\mbox{
for }x\in\Omega,\\\ u^{\alpha}(x)=0\mbox{ for
}x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n$ (1.9)

as well as the rate of convergence to zero for $\varepsilon\to 0$ of the
homogenization error $\|u_{\varepsilon}-u_{0}\|_{\infty}$. Here and in what
follows we denote

$\|u\|_{\infty}:=\max_{\alpha=1,\ldots,n}\mbox{ess
sup}\\{|u^{\alpha}(x)|:x\in\Omega\\}$ (1.10)

for $u\in L^{\infty}(\Omega;\mathbb{R}^{n})$. As usual, a vector function
$u\in W_{0}^{1,2}(\Omega;\mathbb{R}^{n})\cap
L^{\infty}(\Omega;\mathbb{R}^{n})$ is called weak solution to the boundary
value problem (1.1) if it satisfies the variational equation

$\int_{\Omega}\Big{(}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)+b^{\alpha}(x,u(x))\varphi^{\alpha}(x)\Big{)}dx=0\mbox{
for all }\varphi\in W_{0}^{1,2}(\Omega;\mathbb{R}^{n}),$

and similarly for (1.9) and for its linearization (1.11) and for the cell
problem (1.7).

###### Theorem 1.1

Suppose (1.2)-(1.5), and let $u=u_{0}$ be a weak solution to (1.9) such that
the linearized homogenized boundary value problem

$\left.\begin{array}[]{l}\hat{a}^{\alpha\beta}_{ij}\partial_{x_{i}}\partial_{x_{j}}u^{\beta}(x)=\partial_{u^{\gamma}}b^{\alpha}(x,u_{0}(x))u^{\gamma}(x)\mbox{
for }x\in\Omega,\\\ u^{\alpha}(x)=0\mbox{ for
}x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n$ (1.11)

does not have weak solutions $u\not=0$. Then the following is true:

(i) There exist $\varepsilon_{0}>0$ and $\delta>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0}]$ there exists exactly one weak solution
$u=u_{\varepsilon}$ to (1.1) with $\|u-u_{0}\|_{\infty}\leq\delta$. Moreover,

$\|u_{\varepsilon}-u_{0}\|_{\infty}\to 0\mbox{ for }\varepsilon\to 0.$

(ii) If $u_{0}\in W^{2,p_{0}}(\Omega;\mathbb{R}^{n})$ with certain $p_{0}>2$,
then for any $p>2$ we have

$\|u_{\varepsilon}-u_{0}\|_{\infty}=O(\varepsilon^{1/p})\mbox{ for
}\varepsilon\to 0.$ (1.12)

###### Remark 1.2

It is well-known that the assumptions (1.2)-(1.5) do not imply that $u_{0}\in
W^{2,p_{0}}(\Omega;\mathbb{R}^{n})$ with certain $p_{0}>2$, in general. But
several sufficient conditions for that are known, for example if the boundary
$\partial\Omega$ is $C^{1,1}$-smooth and if $\hat{a}^{\alpha\beta}_{ij}=0$ for
$\alpha>\beta$ (no cross-diffusion). We conjecture that (1.12) is not true
anymore, in general, if $u_{0}\notin W^{2,2}(\Omega;\mathbb{R}^{n})$.

###### Remark 1.3

We conjecture that Theorem 1.1 remains to be true for any space dimension if
the elliptic system is triangular, i.e. $\hat{a}^{\alpha\beta}_{ij}=0$ for
$\alpha>\beta$ (in particular, for scalar elliptic equations). The reason for
that is the following: In the present paper, which concerns space dimension
two, we use K. Grögers result [10] about maximal regularity of boundary value
problems for elliptic systems with non-smooth data in the pair of Sobolev
spaces $W_{0}^{1,p}(\Omega)$ and $W^{-1,p}(\Omega)$ with $p\approx 2$ as well
as the continuous embedding $W^{1,p}(\Omega)\hookrightarrow
L^{\infty}(\Omega)$ for $p>2$ in the case of space dimension two. But there
exists a replecement of these results for triangular systems with any space
dimension (cf. [9, 11]), where the Sobolev spaces are replaced by appropriate
Sobolev-Campanato spaces which for any space dimension are continuously
embedded into $L^{\infty}(\Omega)$. Remark that in [9, 10, 11] more general
types of boundary conditions are allowed, for example mixed Dirichlet-Robin
boundary conditions. Hence, we expect that Theorem 1.1 (as well as its version
with any space dimension if $\hat{a}^{\alpha\beta}_{ij}=0$ for $\alpha>\beta$)
is true also for those boundary conditions.

###### Remark 1.4

In many applications the reaction functions $b^{\alpha}$ are of the type

$b^{\alpha}(x,u)=\sum_{l=1}^{m}c_{l}^{\alpha}(x)d_{l}^{\alpha}(u)\mbox{ with
}c_{l}^{\alpha}\in L^{\infty}(\Omega),\;d_{l}^{\alpha}\in
C^{1}(\mathbb{R}^{n}),$

and those satisfy assumption (1.4).

###### Remark 1.5

The integral condition (1.5) is often referred as $V$-ellipticity or
$V$-coercivity, and it follows from the Legendre condition

$a_{ij}^{\alpha\beta}(y)\xi_{i}^{\alpha}\xi_{j}^{\beta}\geq\mbox{\rm
const}\;\xi_{i}^{\alpha}\xi_{i}^{\alpha}\mbox{ for almost all
}y\in\mathbb{R}^{2}\mbox{ and all }\xi\in\mathbb{R}^{2n},$

and it implies the Legendre-Hadamard condition

$a_{ij}^{\alpha\beta}(y)\xi_{i}\xi_{j}\eta^{\alpha}\eta^{\beta}\geq\mbox{\rm
const}\;\xi_{i}\xi_{i}\eta^{\alpha}\eta^{\alpha}\mbox{ for almost all
}y\in\mathbb{R}^{2}\mbox{ and all
}\xi\in\mathbb{R}^{2},\;\eta\in\mathbb{R}^{n}.$

If $m=1$ or if the coefficients $a_{ij}^{\alpha\beta}$ are constant (as in
(1.9)), then (1.5) is equivalent to the Legendre-Hadamard condition.

###### Remark 1.6

For $L^{\infty}$ estimates of $u_{\varepsilon}-u_{0}$ for scalar linear
elliptic PDEs see, e.g. [2, Chapter 2.4] and [17] and for linear elliptic
systems [13, Theorem 3.4], [23, Theorem 7.5.1], [25, Theorem 1.7] and [27,
Theorem 1.5]. For $L^{p}$ homogenization error estimates for scalar linear
elliptic PDEs see, e.g. [17] and [26, Theorem 1.1] and for linear elliptic
systems [23, Theorem 7.5.1].

What concerns existence and local uniqueness for nonlinear elliptic
homogenization problems (without assumption of global uniqueness) we know only
the result [4] for scalar semilinear elliptic PDEs of the type $\mbox{\rm
div}\,a(x/\varepsilon)\nabla u(x)=f(x)g(u(x))$, where the nonlinearity $g$ is
supposed to have a sufficiently small local Lipschitz constant (on an
appropriate bounded interval). Let us mention also [14, 15], where existence
and local uniqueness for a homogenization problem for the linear Poisson
equation with nonlinear Robin boundary conditions in a periodically perforated
domain is shown. There the specific structure of the problem (no highly
oscillating diffusion coefficients) allows to apply the classical implicit
function theorem.

For periodic homogenization of linear elliptic PDEs (with small homogenization
parameter $\varepsilon$), which are singularly perturbed (with small singular
perturbation parameter $\delta$) see [24].

Our paper is organized as follows: In Section 2 we consider abstract nonlinear
parameter depending equations of the type

$F_{\varepsilon}(u)=0.$ (1.13)

Here $\varepsilon>0$ is the parameter. We prove a result on existence and
local uniqueness of a family of solutions
$u=u_{\varepsilon}\approx\bar{u}_{\varepsilon}$ to (1.13) with
$\varepsilon\approx 0$, where $\bar{u}_{\varepsilon}$ is a family of
approximate solutions to (1.13), i.e. a family with
$F_{\varepsilon}(\bar{u}_{\varepsilon})\to 0$ for $\varepsilon\to 0$, and we
estimate the norm of the error $u_{\varepsilon}-\bar{u}_{\varepsilon}$ by the
norm of the discrepancy $F_{\varepsilon}(\bar{u}_{\varepsilon})$. This type of
generalized implicit function theorems has been successfully applied to
singularly perturbed nonlinear ODEs and PDEs and to homogenization of
nonlinear ODEs. Contrary to the classical implicit function theorem it is not
supposed that the linearized operators $F^{\prime}_{\varepsilon}(u)$ converge
for $\varepsilon\to 0$ in the uniform operator norm. And, indeed, in the
applications to singularly perturbed problems as well as to homogenization
problems they do not converge for $\varepsilon\to 0$ in the uniform operator
norm (cf. Remark 3.1 below). Hence, the present paper introduces an
application to semilinear elliptic PDE systems of a common approach to
existence, local uniqueness and error estimates for singularly perturbed
problems and for homogenization problems. Another application to periodic
homogenization of quasilinear ODE systems of the type

$a(x,x/\varepsilon,u(x),u^{\prime}(x))^{\prime}=f(x,x/\varepsilon,u(x))$

has been presented in Part I [19].

In Section 3 we prove Theorem 1.1 by means of the results of Section 2. Here
the main work is to construct appropriate families of approximate solutions to
(1.1) with $\varepsilon\approx 0$ with small discrepancies in appropriate
function space norms. In order to apply implicit function theorems mainly one
needs isomorphism properties of the linearized operators. In the setting of
Section 3 they follow from K. Grögers result [10] about maximal regularity of
boundary value problems for elliptic systems with non-smooth data in the pair
of Sobolev spaces $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ and
$W^{-1,p}(\Omega;\mathbb{R}^{n})$ with $p\approx 2$. In order to apply
implicit function theorems one needs also $C^{1}$-smoothness of the appearing
nonlinear superposition operators. In the setting of Section 3 these operators
have to be well-defined and $C^{1}$-smooth on the Sobolev spaces
$W^{1,p}(\Omega;\mathbb{R}^{n})$ with $p>2$, but $p\approx 2$, and therefore
we have to suppose that the space dimension is two.

##  2 An abstract result of implicit function theorem type

Let $U$ and $V$ be a Banach spaces with norms $\|\cdot\|_{U}$ and
$\|\cdot\|_{V}$, respectively. For $\varepsilon>0$ let be given

$\bar{u}_{\varepsilon}\in U\mbox{ and }F_{\varepsilon}\in C^{1}(U;V).$

We consider the abstract equation

$F_{\varepsilon}(u)=0.$ (2.1)

Roughly speaking, we will show the following: If the elements
$\bar{u}_{\varepsilon}$ satisfy (2.1) approximately for $\varepsilon\approx
0$, i.e. if $\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}\to 0$ for
$\varepsilon\to 0$, and if they are non-degenerate solutions (cf. assumption
(2.2) below), then for $\varepsilon\approx 0$ there exists exactly one
solution $u=u_{\varepsilon}$ to (2.1) with
$\|u-\bar{u}_{\varepsilon}\|_{U}\approx 0$, and
$\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{U}=O(\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V})$
for $\varepsilon\to 0$. For that we do not suppose any convergence of the
operators $F_{\varepsilon}$ or $F^{\prime}_{\varepsilon}(u)$ or of the
elements $\bar{u}_{\varepsilon}$ for $\varepsilon\to 0$. Remark that in the
classical implicit function theorem one cannot omit, in general, the
assumption, that $F_{\varepsilon}^{\prime}(u)$ converges for $\varepsilon\to
0$ with respect to the uniform operator norm (cf. [12, Section 3.6]).

###### Theorem 2.1

Suppose that

$\left.\begin{array}[]{l}\mbox{there exist $\varepsilon_{0}>0$ and $c>0$ such
that for all $\varepsilon\in(0,\varepsilon_{0}]$ the operators
$F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon})$ are}\\\ \mbox{Fredholm of
index zero from }$U$\mbox{ into }$V$,\mbox{and
}\|F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon}))u\|_{V}\geq
c\|u\|_{U}\mbox{ for all }u\in U\end{array}\right\\}$ (2.2)

and

$\sup_{\|v\|_{U}\leq
1}\|(F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon}+u)-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon}))v\|_{V}\to
0\mbox{ for }\varepsilon+\|u\|_{U}\to 0$ (2.3)

and

$\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}\to 0\mbox{ for }\varepsilon\to
0.$ (2.4)

Then there exist $\varepsilon_{1}\in(0,\varepsilon_{0}]$ and $\delta>0$ such
that for all $\varepsilon\in(0,\varepsilon_{1}]$ there exists exactly one
solution $u=u_{\varepsilon}$ to (2.1) with
$\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{U}\leq\delta$. Moreover, for all
$\varepsilon\in(0,\varepsilon_{1}]$ we have

$\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{U}\leq\frac{2}{c}\|F_{\varepsilon}(\bar{u}_{\varepsilon}))\|_{V}.$
(2.5)

Proof  Take $\varepsilon\in(0,\varepsilon_{0}]$. Because of assumption (2.2)
the linear operator $F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})$ is an
isomorphism from $U$ onto $V$, and, hence, equation (2.1) is equivalent to the
fixed point problem

$u=G_{\varepsilon}(u):=u-F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon}))^{-1}F_{\varepsilon}(u).$

Take $u_{1},u_{2}\in U$. Then

$\displaystyle\|G_{\varepsilon}(u_{1})-G_{\varepsilon}(u_{2})\|_{V}$
$\displaystyle=$
$\displaystyle\left\|F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon})^{-1}\int_{0}^{1}\left(F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(su_{1}+(1-s)u_{2})\right)ds(u_{1}-u_{2})\right\|_{V}$
(2.6) $\displaystyle\leq$ $\displaystyle\frac{1}{c}\,\max_{0\leq s\leq
1}\|\left(F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(su_{1}+(1-s)u_{2})\right)(u_{1}-u_{2})\|_{V}.$

Here we used that (2.2) yields that
$c\|F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon})^{-1}v\|_{U}\leq\|v\|_{V}$
for all $v\in V$.

Denote ${\cal B}^{r}_{\varepsilon}:=\\{u\in
U:\;\|u-\bar{u}_{\varepsilon}\|_{U}\leq r\\}$. If $u_{1},u_{2}\in{\cal
B}^{r}_{\varepsilon}$, then also $su_{1}+(1-s)u_{2}\in{\cal
B}^{r}_{\varepsilon}$ for all $s\in[0,1]$. Therefore it follows from (2.3) and
(2.6) that there exist $r_{0}>0$ and $\varepsilon_{1}\in(0,\varepsilon_{0}]$
such that for all $\varepsilon\in(0,\varepsilon_{1}]$ the maps
$G_{\varepsilon}$ are strictly contractive with contraction constant $1/2$ on
the closed balls ${\cal B}^{r_{0}}_{\varepsilon}$. Moreover, for all
$\varepsilon\in(0,\varepsilon_{1}]$ and $u\in{\cal B}^{r_{0}}_{\varepsilon}$
we have

$\displaystyle\left\|G_{\varepsilon}(u)-\bar{u}_{\varepsilon}\right\|_{U}$
$\displaystyle\leq$
$\displaystyle\left\|G_{\varepsilon}(u)-G_{\varepsilon}(\bar{u}_{\varepsilon})\right\|_{U}+\left\|G_{\varepsilon}(\bar{u}_{\varepsilon})-\bar{u}_{\varepsilon}\right\|_{U}$
$\displaystyle\leq$
$\displaystyle\frac{r_{0}}{2}+\left\|F_{\varepsilon}^{\prime}(\bar{u}_{\varepsilon}))^{-1}F_{\varepsilon}(\bar{u}_{\varepsilon})\right\|_{U}\leq\frac{r_{0}}{2}+\frac{1}{c}\left\|F_{\varepsilon}(\bar{u}_{\varepsilon})\right\|_{V},$

and (2.4) yields that $G_{\varepsilon}$ maps ${\cal B}^{r_{0}}_{\varepsilon}$
into itself if $\varepsilon_{1}$ is taken sufficiently small.

Now, Banach’s fixed point principle yields the existence and uniqueness
assertions of Theorem 2.1, and the estimate (2.5) follows as above:

$\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{U}\leq\|G_{\varepsilon}(u_{\varepsilon})-G_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}+\|G_{\varepsilon}(\bar{u}_{\varepsilon})-\bar{u}_{\varepsilon}\|_{U}\leq\frac{1}{2}\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{V}+\frac{1}{c}\left\|F_{\varepsilon}(\bar{u}_{\varepsilon})\right\|_{V}.$

###### Remark 2.2

In [5, 6, 7, 18, 20, 21, 22]) various versions of Theorem 2.1 are presented.
They differ slightly according to which problems they are applied (ODEs or
elliptic or parabolic PDEs, stationary or time-periodic solutions, semilinear
or quasilinear problems, smooth or nonsmooth data, one- or multi-dimensional
perturbation parameter $\varepsilon$).

For another result of the type of Theorem 2.1 and its applications to
semilinear elliptic PDE systems with numerically determined approximate
solutions see [3, Theorem 2.1].

If one applies Theorem 2.1, for example to boundary value problems for
elliptic PDEs, then different choices of function spaces $U$ and $V$ and of
their norms $\|\cdot\|_{U}$ and $\|\cdot\|_{V}$ and of the family
$\bar{u}_{\varepsilon}$ of approximate solutions are appropriate. Criteria for
these choices often are the following: The family $\bar{u}_{\varepsilon}$ of
should be ”simple” (for example, $\bar{u}_{\varepsilon}$ should be
$\varepsilon$-independent or given more less explicit in closed formulas, or
to determine $\bar{u}_{\varepsilon}$ numerically should be much cheeper than
to determine the exact solution $u_{\varepsilon}$ numerically), and the rate
of convergence to zero of $\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}$ for
$\varepsilon\to 0$ should be high. The norm $\|\cdot\|_{V}$ should be weak and
the norm $\|\cdot\|_{U}$ should be strong such that the error estimate (2.5)
is strong. But at the same time the norm $\|\cdot\|_{U}$ should be weak such
that the domain of local uniqueness, which contains all $u\in U$ with
$\|u-\bar{u}_{\varepsilon}\|_{U}\leq\delta$, is large. These criteria are
contradicting, of course. Hence, in any application of Theorem 2.1 the choices
of $U$, $V$, $\|\cdot\|_{U}$, $\|\cdot\|_{V}$ and $\bar{u}_{\varepsilon}$ are
compromises according to the requirements of the application.

One way to find such compromises is described in Corollary 2.3 below. It
delivers existence and local uniqueness of solutions $u=u_{\varepsilon}$ to
the equation $F_{\varepsilon}(u)=0$ with $\varepsilon\approx 0$ and
$\|u-u_{0}\|\approx 0$, where $\|F_{\varepsilon}(u_{0})\|_{V}$ does not
converge to zero for $\varepsilon\to 0$, in general, and where the space $U$
is not complete with respect to the norm $\|\cdot\|$, in general. The price
for that is that the estimate (2.11) of the error $u_{\varepsilon}-u_{0}$ is
with respect to the weaker norm $\|\cdot\|$, only.

###### Corollary 2.3

Suppose (2.4). Further, let be given $u_{0}\in U$ and a norm $\|\cdot\|$ in
$U$ such that

$\left.\begin{array}[]{l}\mbox{there exist $\varepsilon_{0}>0$ and $c>0$ such
that for all $\varepsilon\in(0,\varepsilon_{0}]$ the operators
$F_{\varepsilon}^{\prime}(u_{0})$ are}\\\ \mbox{Fredholm of index zero from
}$U$\mbox{ into }$V$,\mbox{and }\|F_{\varepsilon}^{\prime}(u_{0}))u\|_{V}\geq
c\|u\|_{U}\mbox{ for all }u\in U\end{array}\right\\}$ (2.7)

and

$\displaystyle d:=\sup\\{\|u\|:\;u\in U,\|u\|_{U}\leq 1\\}<\infty,$ (2.8)
$\displaystyle\|\bar{u}_{\varepsilon}-u_{0}\|\to 0\mbox{ for }\varepsilon\to
0,$ (2.9) $\displaystyle\sup_{\|v\|_{U}\leq
1}\|(F^{\prime}_{\varepsilon}(u_{0}+u)-F^{\prime}_{\varepsilon}(u_{0}))v\|_{V}\to
0\mbox{ for }\varepsilon+\|u\|\to 0.$ (2.10)

Then there exist $\varepsilon_{1}\in(0,\varepsilon_{0}]$ and $\delta>0$ such
that for all $\varepsilon\in(0,\varepsilon_{1}]$ there exists exactly one
solution $u=u_{\varepsilon}$ to (2.1) with $\|u-u_{0}\|\leq\delta$, and

$\|u_{\varepsilon}-u_{0}\|\leq\|\bar{u}_{\varepsilon}-u_{0}\|+\frac{4d}{c}\|F_{\varepsilon}(\bar{u}_{\varepsilon}))\|_{V}.$
(2.11)

Proof  Because of assumption (2.9) the condition (2.10) is equivalent to

$\sup_{\|v\|_{U}\leq
1}\|(F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon}+u)-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon}))v\|_{V}\to
0\mbox{ for }\varepsilon+\|u\|\to 0,$

and because of assumption (2.8) these two equivalent conditions are stronger
that condition (2.3). And similarly, because of assumptions (2.9) and (2.10)
the condition (2.7) is equivalent to condition (2.2) (with another
$\varepsilon_{0}$ in (2.2) than in (2.7) and with arbitrary smaller $c$ in
(2.2) than in (2.7), for example $c/2$). Hence, Theorem 2.1 yields the
existence assertion of Corollary 2.3 and the error estimate

$\|u_{\varepsilon}-u_{0}\|\leq\|\bar{u}_{\varepsilon}-u_{0}\|+d\|u_{\varepsilon}-\bar{u}_{\varepsilon}\|_{U}\leq\|\bar{u}_{\varepsilon}-u_{0}\|+\frac{4d}{c}\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}.$

Now let us prove the local uniqueness assertion of Corollary 2.3. Take
$\varepsilon\in(0,\varepsilon_{1}]$ and a solution $u\in U$ to (2.1). Then

$0=F_{\varepsilon}(u)=F_{\varepsilon}(\bar{u}_{\varepsilon})+F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})(u-\bar{u}_{\varepsilon})+\int_{0}^{1}\left(F^{\prime}_{\varepsilon}(su+(1-s)\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})\right)(u-\bar{u}_{\varepsilon})ds,$

i.e.

$u-\bar{u}_{\varepsilon}=-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})^{-1}\left(F_{\varepsilon}(\bar{u}_{\varepsilon})+\int_{0}^{1}\left(F^{\prime}_{\varepsilon}(su+(1-s)\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})\right)(u-\bar{u}_{\varepsilon})ds\right),$

i.e.

$\|u-\bar{u}_{\varepsilon}\|_{U}\leq\frac{1}{c}\left(\|F_{\varepsilon}(\bar{u}_{\varepsilon})\|_{V}+\max_{0\leq
s\leq
1}\|\left(F^{\prime}_{\varepsilon}(su+(1-s)\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})\right)(u-\bar{u}_{\varepsilon})\|_{V}\right).$
(2.12)

But (2.10) yields that

$\max_{0\leq s\leq
1}\|\left(F^{\prime}_{\varepsilon}(su+(1-s)\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})\right)(u-\bar{u}_{\varepsilon})\|_{V}=o(\|u-\bar{u}_{\varepsilon}\|_{U})\mbox{
for }\varepsilon+\|u-\bar{u}_{\varepsilon}\|\to 0.$

Therefore (2.9) implies that

$\max_{0\leq s\leq
1}\|\left(F^{\prime}_{\varepsilon}(su+(1-s)\bar{u}_{\varepsilon})-F^{\prime}_{\varepsilon}(\bar{u}_{\varepsilon})\right)(u-\bar{u}_{\varepsilon})\|_{V}=o(\|u-\bar{u}_{\varepsilon}\|_{U})\mbox{
for }\varepsilon+\|u-u_{0}\|\to 0.$

Hence, if $\varepsilon$ and $\|u-u_{0}\|$ are sufficiently small, then (2.12)
yields that $\|u-\bar{u}_{\varepsilon}\|_{U}\leq\delta$, and the local
uniqueness assertion of Theorem (2.1) implies that $u=u_{\varepsilon}$.

###### Remark 2.4

In most of the applications of Corollary 2.3 to PDEs the element $u_{0}$ and
the norm $\|\cdot\|$ are a priori given, and one has to choose Banach spaces
$U$ and $V$ such that the PDE problem is equivalent to an abstract equation
$F_{\varepsilon}(u)=0$ with Fredholm maps $F_{\varepsilon}\in C^{1}(U;V)$, and
one has to construct a family $\bar{u}_{\varepsilon}$ with the properties
(2.2), (2.4) and (2.8)-(2.10). But at the beginning one does not know if
existence and local uniqueness for $\varepsilon\approx 0$ and
$\|u-u_{0}\|\approx 0$ is true or not for the given PDE problem. If not, then
one is trying to choose and to construct something, which does not exist.

For example, in Theorem 1.1, which is the result of an application of
Corollary 2.3 to problem (1.1), the spaces $U$ and $V$ and the family
$\bar{u}_{\varepsilon}$ are hidden only, they do not appear in the formulation
of Theorem 1.1.

##  3 Proof of Theorem 1.1

In this section we will prove Theorem 1.1 by means of Corollary 2.3. For that
we use the objects of Theorem 1.1: The bounded Lipschitz domain
$\Omega\subset\mathbb{R}^{2}$, the diffusion coefficient functions
$a^{\alpha\beta}_{ij}\in L^{\infty}(\mathbb{R}^{2})$ with (1.3) and (1.5), the
reaction functions $b^{\alpha}:\Omega\times\mathbb{R}^{n}\to\mathbb{R}$ with
(1.4), the correctors $v^{\alpha\beta}_{i}\in W^{1,2}_{\rm
loc}(\mathbb{R}^{2};\mathbb{R}^{n})$, which are defined by the cell problems
(1.7), the homogenized diffusion coefficients
$\hat{a}^{\alpha\beta}_{ij}\in\mathbb{R}$, which are defined in (1.6) and
which satisfy (1.8), and the weak solution $u_{0}\in
W_{0}^{1,2}(\Omega;\mathbb{R}^{n})$ to the homogenized boundary value problem
(1.9).

As usual, for $p\geq 2$ we denote by $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ the
closure with respect to the norm

$\|u\|_{1,p}:=\left(\sum_{\alpha=1}^{n}\sum_{i=1}^{2}\int_{\Omega}|\partial_{x_{i}}u^{\alpha}(x)|^{p}dx\right)^{1/p}$

of the set of all $C^{\infty}$-maps $u:\Omega\to\mathbb{R}^{n}$ with compact
support. And
$W^{-1,p}(\Omega;\mathbb{R}^{n}):=W_{0}^{1,q}(\Omega;\mathbb{R}^{n})^{*}$ is
the dual space to $W_{0}^{1,q}(\Omega;\mathbb{R}^{n})$ with $1/p+1/q=1$ with
dual space norm

$\|f\|_{-1,p}:=\sup\\{\langle f,\varphi\rangle_{1,q}:\;\varphi\in
W_{0}^{1,q}(\Omega;\mathbb{R}^{n}),\|\varphi\|_{1,q}\leq 1\\},$

where $\langle\cdot,\cdot\rangle_{1,q}:W^{-1,p}(\Omega;\mathbb{R}^{n})\times
W_{0}^{1,q}(\Omega;\mathbb{R}^{n})\to\mathbb{R}$ is the dual pairing.

Further, we introduce linear bounded operators
$A_{0}:W_{0}^{1,2}(\Omega;\mathbb{R}^{n})\to W^{-1,2}(\Omega;\mathbb{R}^{n})$
and, for $\varepsilon>0$,
$A_{\varepsilon}:W_{0}^{1,2}(\Omega;\mathbb{R}^{n})\to
W^{-1,2}(\Omega;\mathbb{R}^{n})$ by

$\left.\begin{array}[]{l}\displaystyle\langle
A_{0}u,\varphi\rangle_{1,2}:=\int_{\Omega}\hat{a}_{ij}^{\alpha\beta}\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx,\\\
\displaystyle\langle
A_{\varepsilon}u,\varphi\rangle_{1,2}:=\int_{\Omega}a_{ij}^{\alpha\beta}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx,\end{array}\right\\}\mbox{
for all }u,\varphi\in W_{0}^{1,2}(\Omega;\mathbb{R}^{n}).$ (3.1)

Because of assumption (1.3) and the Hölder inequality we have the following:
For any $p\geq 2$ the restrictions of $A_{0}$ and $A_{\varepsilon}$ to
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ map $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$
into $W^{-1,p}(\Omega;\mathbb{R}^{n})$, and

$\|A_{0}u\|_{-1,p}+\|A_{\varepsilon}u\|_{-1,p}\leq\mbox{const}\;\|u\|_{1,p}\mbox{
for all }u\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n}),$

where the constant does not depend on $\varepsilon$, $p$ and $u$.

For $\varepsilon>0$ and $u\in W_{0}^{1,2}(\Omega;\mathbb{R}^{n})$ define
$\tilde{u}_{\varepsilon}(y):=u(\varepsilon y)$, if $\varepsilon y\in\Omega$,
and $\tilde{u}_{\varepsilon}(y):=0$, if $\varepsilon y\notin\Omega$. Then
(1.5) implies that

$\displaystyle\langle A_{\varepsilon}u,u\rangle_{2}$ $\displaystyle=$
$\displaystyle\int_{\Omega}a_{ij}^{\alpha\beta}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx=\frac{1}{\varepsilon}\int_{\mathbb{R}^{2}}a_{ij}^{\alpha\beta}(y)\partial_{y_{j}}\tilde{u}_{\varepsilon}^{\beta}(y)\partial_{y_{i}}\tilde{u}^{\alpha}(y)dy$
$\displaystyle\geq$
$\displaystyle\frac{a}{\varepsilon}\int_{\mathbb{R}^{2}}\partial_{y_{i}}\tilde{u}_{\varepsilon}^{\alpha}(y)\partial_{y_{i}}\tilde{u}^{\alpha}(y)dy=a\|u\|^{2}_{1,2}$

and similarly for $A_{0}$. Therefore K. Gröger’s results [10, Theorems 1 and
2] imply that there exists $p_{1}>2$ such that for all $\varepsilon>0$ and all
$p\in[2,p_{1}]$ the linear operators $A_{0}$ and $A_{\varepsilon}$ are
bijective from $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ onto
$W^{-1,p}(\Omega;\mathbb{R}^{n})$ and that

$\|A_{0}^{-1}f\|_{1,p}+\|A_{\varepsilon}^{-1}f\|_{1,p}\leq\mbox{const}\;\|f\|_{-1,p}\mbox{
for all }f\in W^{-1,p}(\Omega;\mathbb{R}^{n}),$ (3.2)

where the constant does not depend on $\varepsilon$, $p$ and $f$. In
particular, we have

$u_{0}\in W_{0}^{1,p_{1}}(\Omega;\mathbb{R}^{n})$ (3.3)

because $u_{0}$ is a weak solution to the homogenized boundary value problem
(1.9) with diffusion coefficients satisfying (1.8).

###### Remark 3.1

(i) Estimates of the type (3.2) often are called Meyers’ estimates (or
estimates of Gröger-Meyers type) because of the initiating paper [16] of N.G.
Meyers.

(ii) It is easy to verify that the linear operators $A_{\varepsilon}$ do not
converge for $\varepsilon\to 0$ in the uniform operator norm in ${\cal
L}(W_{0}^{1,p}(\Omega;\mathbb{R}^{n});W^{-1,p}(\Omega;\mathbb{R}^{n}))$ for
certain $p\geq 2$, in general (see [8, Remark 8.4] and Lemma 3.3 below).

Finally, we introduce a superposition operator
$B:L^{\infty}(\Omega;\mathbb{R}^{n})\to L^{\infty}(\Omega;\mathbb{R}^{n})$ by

$[B(u)](x):=\left(b^{1}(x,u(x)),\ldots,b^{n}(x,u(x))\right)\mbox{ for almost
all }x\in\Omega.$ (3.4)

Here and in what follows we consider the function space
$L^{\infty}(\Omega;\mathbb{R}^{n})$ with the norm $\|\cdot\|_{\infty}$
(defined in (1.10)), as usual. Remark that for any $p>2$ the operator $B$ can
be considered as a map from $W^{1,p}(\Omega;\mathbb{R}^{n})$ into
$W^{-1,p}(\Omega;\mathbb{R}^{n})$ because the space
$W^{1,p}(\Omega;\mathbb{R}^{n})$ is continuously embedded into the space
$L^{\infty}(\Omega;\mathbb{R}^{n})$ (because the dimension of $\Omega$ is
two), and the space $L^{\infty}(\Omega;\mathbb{R}^{n})$ is continuously
embedded into the space
$W^{-1,p}(\Omega;\mathbb{R}^{n})=W_{0}^{-1,q}(\Omega;\mathbb{R}^{n})^{*}$ via

$\langle u,\varphi\rangle_{1,q}:=\int_{\Omega}u(x)\varphi(x)dx\mbox{ for }u\in
L^{\infty}(\Omega;\mathbb{R}^{n})\mbox{ and }\varphi\in
W_{0}^{1,q}(\Omega;\mathbb{R}^{n}).$

Because of assumption (1.4) we have that the nonlinear operator $B$ is
$C^{1}$-smooth from $L^{\infty}(\Omega;\mathbb{R}^{n})$ into
$L^{\infty}(\Omega;\mathbb{R}^{n})$ and, hence, from
$W^{1,p}(\Omega;\mathbb{R}^{n})$ into $W^{-1,p}(\Omega;\mathbb{R}^{n})$, and

$[B^{\prime}(u)v](x):=\left(\partial_{u^{\gamma}}b^{1}(x,u(x)v^{\gamma}(x)),\ldots,\partial_{u^{\gamma}}b^{n}(x,u(x))v^{\gamma}(x)\right)\mbox{
for almost all }x\in\Omega,$

and for all $u\in L^{\infty}(\Omega;\mathbb{R}^{n})$ we have

$\left.\begin{array}[]{l}\displaystyle\lim_{\|w\|_{\infty}\to
0}\|B(v+w)-B(v)\|_{-1,p}=0\mbox{ uniformly with respect to
}\|v-u\|_{\infty}\leq 1,\\\ \displaystyle\lim_{\|v\|_{\infty}\to
0}\|(B^{\prime}(u+v)-B^{\prime}(u))w\|_{-1,p}=0\mbox{ uniformly with respect
to }\|w\|_{1,p}\leq 1.\end{array}\right\\}$ (3.5)

Moreover, using the notation (3.1) and (3.4) we get

$A_{0}u_{0}+B(u_{0})=0.$ (3.6)

Now we introduce the abstract setting of Corollary 2.3 for the boundary value
problem (1.1). We take the $p_{0}$ from assertion (ii) in Theorem 1.1 and the
$p_{1}$ from above and fix $p$ and $q$ as follows:

$2<p\leq\min\\{p_{0},p_{1}\\},\;q:=\frac{p}{p-1}.$ (3.7)

The Banach spaces $U$ and $V$ and their norms are defined by

$\displaystyle
U:=W_{0}^{1,p}(\Omega;\mathbb{R}^{n}),\;V:=W^{-1,p}(\Omega;\mathbb{R}^{n})=W_{0}^{1,q}(\Omega;\mathbb{R}^{n})^{*},$
$\displaystyle\|\cdot\|_{U}:=\|\cdot\|_{1,p},\;\|\cdot\|:=\|\cdot\|_{\infty},\;\|\cdot\|_{V}:=\|\cdot\|_{-1,p}.$

Because the space dimension is two, the assumption (2.8) of Corollary 2.3 is
satisfied in this setting.

Further, the $C^{1}$-smooth operators $F_{\varepsilon}:U\to V$ of Theorem 2.1
are defined by

$F_{\varepsilon}(u):=A_{\varepsilon}u+B(u).$

With these choices a vector function $u$ is a weak solution to the boundary
value problem (1.1) if and only if $u$ belongs to the function space $U$ and
satisfies the operator equation $F_{\varepsilon}(u)=0.$ Here we used [10]
again.

Finally, we define the exceptional approximate solution $u_{0}\in U$ of
Corollary 2.3 to be the solution $u_{0}$ of the homogenized boundary value
problem (1.9), which is given in Theorem 1.1.

In order to prove Theorem 1.1 we have choose the family
$\bar{u}_{\varepsilon}\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ such that the
assumption (2.2) is satisfied in the setting introduced above, i.e. that there
exist $\varepsilon_{0}>0$ and $c>0$ such that

$\left.\begin{array}[]{l}\mbox{for all $\varepsilon\in(0,\varepsilon_{0}]$ the
operators $A_{\varepsilon}+B^{\prime}(u_{0})$ are Fredholm}\\\ \mbox{of index
zero from }W_{0}^{1,p}(\Omega;\mathbb{R}^{n})\mbox{ into
}W^{-1,p}(\Omega;\mathbb{R}^{n}),\mbox{ and }\\\
\|(A_{\varepsilon}+B^{\prime}(u_{0}))u\|_{-1,p}\geq c\|u\|_{1,p}\mbox{ for all
}u\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n}),\end{array}\right\\}$ (3.8)

that the assumption and (2.9) is satisfied in the setting introduced above,
i.e.

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}\to 0\mbox{ for }\varepsilon\to 0,$
(3.9)

that the assumption and (2.10) is satisfied in the setting introduced above,
i.e.

$\lim_{\varepsilon+\|u\|_{\infty}\to
0}\|(B^{\prime}(u_{0}+u)-B^{\prime}(u_{0}))v\|_{-1,p}=0\mbox{ uniformly with
respect to }\|v\|_{1,p}\leq 1,$ (3.10)

and, for proving assertion (i) of Theorem 1.1, that

$\|A_{\varepsilon}\bar{u}_{\varepsilon}+B(\bar{u}_{\varepsilon})\|_{-1,p}=o(1)\mbox{
for }\varepsilon\to 0,$ (3.11)

and, for proving assertion (ii) of Theorem 1.1, that

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}+\|A_{\varepsilon}\bar{u}_{\varepsilon}+B(\bar{u}_{\varepsilon})\|_{-1,p}=O(\varepsilon^{1/p})\mbox{
for }\varepsilon\to 0.$ (3.12)

and

$\lim_{\|u\|_{\infty}\to
0}\|(B^{\prime}(u_{0}+u)-B^{\prime}(u_{0}))v\|_{-1,p}=0\mbox{ uniformly with
respect to }\|v\|_{1,p}\leq 1$ (3.13)

and

$\|A_{\varepsilon}\bar{u}_{\varepsilon}+B(u_{0})\|_{-1,p}=o(1)\mbox{ for
}\varepsilon\to 0,$ (3.14)

and

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}+\|A_{\varepsilon}\bar{u}_{\varepsilon}+B(u_{0})\|_{-1,p}=O(\varepsilon^{1/p})\mbox{
for }\varepsilon\to 0.$ (3.15)

Moreover, (3.13) is true because of (3.5), and because of (3.6) the conditions
(3.14) and (3.15) are equivalent to

$\|A_{\varepsilon}\bar{u}_{\varepsilon}-A_{0}u_{0}\|_{-1,p}=o(1)\mbox{ for
}\varepsilon\to 0,$ (3.16)

and

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}+\|A_{\varepsilon}\bar{u}_{\varepsilon}-A_{0}u_{0}\|_{-1,p}=O(\varepsilon^{1/p})\mbox{
for }\varepsilon\to 0.$ (3.17)

Hence, we have to verify (3.8), and for proving assertion (i) of Theorem 1.1
we have to construct a family $\bar{u}_{\varepsilon}\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ with (3.9) and (3.16), and for proving
assertion (ii) of Theorem 1.1 we have construct a family
$\bar{u}_{\varepsilon}\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ with (3.17).

###  3.1 Construction of approximate solutions with (3.9) and (3.16)

In this subsection we will construct a family $\bar{u}_{\varepsilon}\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ with (3.9) and (3.16). For that we will do
some calulations which are well-known in homogenization theory (cf., e.g. [23,
Chapter 3.2]), we present them for the convenience of the reader.

For $\varepsilon>0$ we set

$\Omega_{\varepsilon}:=\left\\{x\in\Omega:\;\inf_{y\in\partial\Omega}\left((x_{1}-y_{1})^{2}+(x_{2}-y_{2})^{2}\right)<\varepsilon^{2}\right\\}.$

It follows that

$\mbox{meas }\Omega_{\varepsilon}=O(\varepsilon)\mbox{ for }\varepsilon\to 0,$
(3.18)

where $\mbox{meas }\Omega_{\varepsilon}$ is the two-dimensional Lebesque
measure of $\Omega_{\varepsilon}$. Further, we take a family
$\eta_{\varepsilon}$ of cut-of functions of size $\varepsilon$, i.e. of
$C^{\infty}$-functions $\Omega\to\mathbb{R}$ such that

$\left.\begin{array}[]{l}0\leq\eta_{\varepsilon}(x)\leq 1\mbox{ for
}x\in\Omega,\\\ \eta_{\varepsilon}(x)=1\mbox{ for
}x\in\Omega\setminus\Omega_{2\varepsilon},\\\ \eta_{\varepsilon}(x)=0\mbox{
for }x\in\Omega_{\varepsilon},\\\
\sup\left\\{\varepsilon\,|\partial_{x_{i}}\eta_{\varepsilon}(x)|:\;\varepsilon>0,\;x\in\Omega,\;i=1,2\right\\}<\infty.\end{array}\right\\}$
(3.19)

Finally, we take a mollifier function, i.e. a $C^{\infty}$-function
$\rho:\mathbb{R}^{2}\to\mathbb{R}$ such that

$\rho(x)\geq 0\mbox{ and }\rho(x)=\rho(-x)\mbox{ for all
}x\in\mathbb{R}^{2},\;\rho(x)=0\mbox{ for }x_{1}^{2}+x_{2}^{2}\geq
1,\int_{\mathbb{R}^{2}}\rho(x)dx=1,$

and for $\delta>0$ we define linear smoothing operators
$S_{\delta}:L^{1}(\Omega)\to C^{\infty}(\mathbb{R}^{2})$ by

$[S_{\delta}u](x):=\int_{\Omega}\rho_{\delta}(x-y)u(y)dy\mbox{ with
}\rho_{\delta}(x):=\rho(x/\delta)/\delta^{2}.$

###### Lemma 3.2

(i) For all $r\geq 1$ and $u\in L^{r}(\Omega)$ we have

$\lim_{\delta\to 0}\int_{\Omega}|u(x)-[S_{\delta}u](x)|^{r}dx=0$ (3.20)

(ii) For all $r\geq 1$ there exists $c_{r}>0$ such that for all $\delta>0$,
$i=1,2$ and $u\in L^{r}(\Omega)$ we have

$\displaystyle\int_{\Omega}\left|[S_{\delta}u](x)\right|^{r}dx$
$\displaystyle\leq$ $\displaystyle\int_{\Omega}\left|u(x)\right|^{r}dx,$
(3.21)
$\displaystyle\int_{\Omega}\left|[\partial_{x_{i}}S_{\delta}u](x)\right|^{r}dx$
$\displaystyle\leq$
$\displaystyle\frac{c_{r}}{\delta^{r}}\int_{\Omega}\left|u(x)\right|^{r}dx,$
(3.22) $\displaystyle\sup_{x\in\Omega}\left|[S_{\delta}u](x)\right|^{r}$
$\displaystyle\leq$
$\displaystyle\frac{c_{r}}{\delta^{2}}\int_{\Omega}\left|u(x)\right|^{r}dx.$
(3.23)

Proof  Assertion (i) is proved e.g. in [1, Lemma 1.1.1]. In order to prove
assertion (ii) take $\delta>0$, $r,s>1$ with $1/r+1/s=1$, and take $u\in
L^{r}(\Omega)$. Then the Hölder inequality implies that for all $x\in\Omega$
we have

$|[S_{\delta}u](x)|=\left|\int_{\Omega}u(y)\rho_{\delta}(x-y)^{1/r}\rho_{\delta}(x-y)^{1/s}dy\right|\leq\left(\int_{\Omega}|u(y)|^{r}\rho_{\delta}(x-y)dy\right)^{1/r}.$

Here we used that
$\int_{\mathbb{R}^{2}}\rho_{\delta}(x-y)dy=\int_{\mathbb{R}^{2}}\rho(z)dz=1.$
It follows that

$|[S_{\delta}u](x)|^{r}\leq\frac{1}{\delta^{2}}\int_{\Omega}|u(y)|^{r}\rho((x-y)/\delta)dy\leq\mbox{const
}\frac{1}{\delta^{2}}\int_{\Omega}|u(y)|^{r}dy,$

i.e. (3.23) is proved. Further, we have

$\int_{\Omega}|[S_{\delta}u](x)|^{r}dx\leq\int_{\Omega}\int_{\Omega}|u(y)|^{r}\rho_{\delta}(x-y))dydx=\int_{\Omega}\int_{\Omega}\rho_{\delta}(x-y))dx|u(y)|^{r}dy\leq\int_{\Omega}|u(y)|^{r}dy,$

i.e. (3.21) is proved. Finally, because of
$\int_{\mathbb{R}^{2}}|\partial_{x_{i}}\rho_{\delta}(x-y)|dx=\frac{1}{\delta^{3}}\int_{\mathbb{R}^{2}}|\partial_{x_{i}}\rho((x-y)/\delta)|dy=\frac{1}{\delta}\int_{\mathbb{R}^{2}}|\partial_{x_{i}}\rho(z)|dz$
one gets similarly

$\displaystyle|[\partial_{x_{i}}S_{\delta}u](x)|$ $\displaystyle\leq$
$\displaystyle\left(\int_{\Omega}|u(y)|^{r}|\partial_{x_{i}}\rho_{\delta}(x-y)|dy\right)^{1/r}\left(\int_{\Omega}|\partial_{x_{i}}\rho_{\delta}(x-y)|dy\right)^{1/s}$
$\displaystyle\leq$ $\displaystyle\mbox{ const
}\frac{1}{\delta^{1/s}}\left(\int_{\Omega}|u(y)|^{r}|\partial_{x_{i}}\rho_{\delta}(x-y)|dy\right)^{1/r}$

and, hence,

$\displaystyle\int_{\Omega}\left|[\partial_{x_{i}}S_{\delta}u](x)\right|^{r}dx$
$\displaystyle\leq$ $\displaystyle\mbox{ const
}\frac{1}{\delta^{r/s}}\int_{\Omega}|u(y)|^{r}\int_{\Omega}|\partial_{x_{i}}\rho_{\delta}(x-y)|dx\;dy$
$\displaystyle\leq$ $\displaystyle\mbox{ const
}\frac{1}{\delta^{r}}\int_{\Omega}|u(y)|^{r}dy,$

i.e. (3.22) is proved.

It is well-known (cf., e.g. [23, Chapter 2.2]) that, if the exponent $p_{1}>2$
is taken sufficiently small, we have

$v^{\alpha\beta}_{i}\in W_{\rm loc}^{1,p_{1}}(\mathbb{R}^{2}).$ (3.24)

Using this and (3.3) and (3.7), we define, for $\varepsilon>0$, linear
operators $K_{\varepsilon}:W^{1,p}(\Omega;\mathbb{R}^{n})\to
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ by

$[K_{\varepsilon}u]^{\alpha}(x):=\varepsilon\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}](x)v_{k}^{\alpha\gamma}(x/\varepsilon)\mbox{
with }\delta_{\varepsilon}:=\varepsilon^{1/4}.$ (3.25)

###### Lemma 3.3

For all $u\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ we have

$\lim_{\varepsilon\to 0}\langle
A_{\varepsilon}(u+K_{\varepsilon}u)-A_{0}u,\varphi\rangle_{1,q}=0\mbox{
uniformly with respect to }\|\varphi\|_{1,q}\leq 1,$ (3.26)

and for all $\varphi\in W_{0}^{1,q}(\Omega;\mathbb{R}^{n})$ we have

$\lim_{\varepsilon\to 0}\langle
A_{\varepsilon}(u+K_{\varepsilon}u)-A_{0}u,\varphi\rangle_{1,q}=0\mbox{
uniformly with respect to }\|u\|_{1,p}\leq 1.$ (3.27)

Proof  For $\alpha,\beta=1,\ldots,n$ and $i,j,k=1,2$ we define
$\mathbb{Z}^{2}$-periodic functions $b_{ij}^{\alpha\beta}\in L_{\rm
loc}^{p}(\mathbb{R}^{2})$ and $c_{ij}^{\alpha\beta}\in W_{\rm
loc}^{2,p}(\mathbb{R}^{2})$ and $\phi_{ijk}^{\alpha\beta}\in W_{\rm
loc}^{1,p}(\mathbb{R}^{2})$ (the functions $\phi_{ijk}^{\alpha\beta}$ are
sometimes are called dual or flux correctors) by

$b_{ij}^{\alpha\beta}(y):=a_{ij}^{\alpha\beta}(y)+a^{\alpha\gamma}_{ik}(y)\partial_{y_{k}}v_{j}^{\gamma\beta}(y)-\hat{a}^{\alpha\beta}_{ij}$
(3.28)

and

$\Delta
c_{ij}^{\alpha\beta}(y)=b_{ij}^{\alpha\beta}(y),\;\int_{[0,1]^{2}}c_{ij}^{\alpha\beta}(y)dy=0$
(3.29)

and

$\phi_{ijk}^{\alpha\beta}(y):=\partial_{y_{i}}c_{jk}^{\alpha\beta}(y)-\partial_{y_{j}}c_{ik}^{\alpha\beta}(y).$
(3.30)

From (1.6) and (3.28) follows that
$\int_{[0,1]^{2}}b^{\alpha\beta}_{ij}(y)dy=0$, therefore problem (3.29) is
uniquely strongly solvable with respect to $c^{\alpha\beta}_{ij}$. Further,
from (1.7) follows that $\partial_{y_{i}}b_{ij}^{\alpha\beta}=0$. Hence,
(3.29) implies that $\partial_{y_{i}}c_{ij}^{\alpha\beta}=0$. Therefore (3.29)
and (3.30) yield that

$\partial_{y_{i}}\phi_{ijk}^{\alpha\beta}=b_{jk}^{\alpha\beta}\mbox{ and
}\phi_{ijk}^{\alpha\beta}=-\phi_{kji}^{\alpha\beta}.$ (3.31)

Take a test function $\varphi\in C^{\infty}(\Omega;\mathbb{R}^{n})$. Using
(3.31) we get

$\varepsilon\partial_{x_{k}}\left(\phi_{ijk}^{\alpha\beta}(x/\varepsilon)\partial_{x_{i}}\varphi^{\beta}(x)\right)=b_{ij}^{\alpha\beta}(x/\varepsilon)\partial_{x_{i}}\varphi^{\beta}(x)$
(3.32)

(this is [23, formula (3.1.5)]).

Now, we take $u\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$, insert (3.25) into
$\langle A_{\varepsilon}(u+K_{\varepsilon}u)-A_{0}u,\varphi\rangle_{1,q}$ and
calculate as follows:

$\displaystyle\langle
A_{\varepsilon}(u+K_{\varepsilon}u)-A_{0}u,\varphi\rangle_{1,q}$
$\displaystyle=\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}\left(u^{\beta}+\varepsilon\eta_{\varepsilon}[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}]v_{k}^{\beta\gamma}(x/\varepsilon)\right)-\hat{a}^{\alpha\beta}_{ij}\partial_{x_{j}}u^{\beta}\right)\partial_{x_{i}}\varphi^{\alpha}dx$
$\displaystyle=\int_{\Omega}\left(\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\partial_{x_{j}}u^{\beta}+a^{\alpha\beta}_{ij}(x/\varepsilon)\eta_{\varepsilon}[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}]\partial_{y_{j}}v_{k}^{\beta\gamma}(x/\varepsilon)\right)\partial_{x_{i}}\varphi^{\alpha}dx$
$\displaystyle\;\;\;\;\;\;+\varepsilon\int_{\Omega}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}(\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}](x))v_{k}^{\beta\gamma}(x/\varepsilon)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle=\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}+a^{\alpha\gamma}_{ik}(x/\varepsilon)\partial_{y_{k}}v_{j}^{\gamma\beta}(x/\varepsilon)\right)\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u^{\beta}(x)-\eta_{\varepsilon}(x))[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\varepsilon\int_{\Omega}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}(\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}](x))v_{k}^{\beta\gamma}(x/\varepsilon)\partial_{x_{i}}\varphi^{\alpha}(x)dx.$
(3.33)

We insert (3.28) and (3.32) into (3.33), integrate by parts and use that
$\phi_{kij}^{\alpha\beta}(x/\varepsilon)\partial_{x_{k}}\partial_{x_{i}}\varphi^{\alpha}(x)=0$
(cf. (3.31)) and get

$\displaystyle\langle
A_{\varepsilon}(u+K_{\varepsilon}u)-A_{0}u,\varphi\rangle_{1,q}$
$\displaystyle=\int_{\Omega}b_{ij}^{\alpha\beta}(x/\varepsilon)\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u^{\beta}(x)-\eta_{\varepsilon}(x))[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\varepsilon\int_{\Omega}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}(\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{k}}u^{\gamma}](x))v_{k}^{\beta\gamma}(x/\varepsilon)\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle=\varepsilon\int_{\Omega}\left(-\phi_{kij}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\partial_{x_{k}}(\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x))\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u^{\beta}(x)-\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx.$
(3.34)

Remark that no boundary integrals appeared after the integration by parts
because of the cut-off functions $\eta_{\varepsilon}$ (no matter if the test
function $\varphi$ vanishes on $\partial\Omega$ or not).

Let us estimate the the right-hand side of (3.34). Because of (3.3) and (3.24)
and the Hölder inequality we have

$\displaystyle\left|\varepsilon\int_{\Omega}\left(-\phi_{kij}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\partial_{x_{k}}\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const
}\varepsilon\left(\sum_{i=1}^{2}\int_{\Omega_{\varepsilon}}|\partial_{x_{i}}\eta_{\varepsilon}(x)|^{p}dx\right)^{1/p}\sum_{\beta=1}^{n}\sum_{j=1}^{2}\|[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}]\|_{\infty}\|\varphi\|_{1,q}$
$\displaystyle\leq\mbox{const
}\frac{\varepsilon^{1/p}}{\delta_{\varepsilon}^{2/p}}\|u\|_{1,p}\|\varphi\|_{1,q}=\mbox{const
}\varepsilon^{1/2p}\|u\|_{1,p}\|\varphi\|_{1,q}$ (3.35)

(here we used (3.18), (3.19) and (3.23)) and

$\displaystyle\left|\varepsilon\int_{\Omega}\left(-\phi_{kij}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\eta_{\varepsilon}(x)[\partial_{x_{k}}S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const
}\frac{\varepsilon}{\delta_{\varepsilon}}\|u\|_{1,p}\|\varphi\|_{1,q}=\mbox{const
}\varepsilon^{3/4}\|u\|_{1,p}\|\varphi\|_{1,q}$ (3.36)

(here we used (3.22)) and

$\displaystyle\left|\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(1-\eta_{\varepsilon}(x)\right)[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const}\left(\int_{\Omega_{\varepsilon}}|1-\eta_{\varepsilon}(x)|^{p}dx\right)^{1/p}\sum_{\beta=1}^{n}\sum_{j=1}^{2}\|[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}]\|_{\infty}\|\varphi\|_{W^{1,q}}$
$\displaystyle\leq\mbox{const
}\frac{\varepsilon^{1/p}}{\delta_{\varepsilon}^{2/p}}\|u\|_{1,p}\|\varphi\|_{1,q}=\mbox{const
}\varepsilon^{1/2p}\|u\|_{1,p}\|\varphi\|_{1,q}$ (3.37)

(here we used (3.18) and (3.23)), where the constants do not depend on
$\varepsilon$, $u$ and $\varphi$. Further, we have

$\displaystyle\left|\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u^{\beta}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const}\left(\sum_{\beta=1}^{n}\sum_{j=1}^{2}\int_{\Omega}\left|\partial_{x_{j}}u^{\beta}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right|^{p}dx\right)^{1/p}\|\varphi\|_{1,q},$
(3.38)

where the constant does not depend on $\varepsilon$, $u$ and $\varphi$ again.
But the right-hand side of (3.1) is $o(1)$ for $\varepsilon\to 0$ uniformly
with respect to $\|\varphi\|_{1,q}\leq 1$ (cf. (3.20)). Hence, (3.26) is
proved.

In order to prove (3.27) we change the estimate (3.1) as follows:

$\displaystyle\left|\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u^{\beta}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const}\int_{\Omega}\left|\left(\partial_{x_{j}}u^{\beta}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{j}}u^{\beta}](x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)\right|dx$
$\displaystyle=\mbox{const}\int_{\Omega}\left|\partial_{x_{j}}u^{\beta}(x)\left(\partial_{x_{i}}\varphi^{\alpha}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{i}}\varphi^{\alpha}](x)\right)\right|dx$
$\displaystyle\leq\mbox{const}\left(\sum_{\alpha=1}^{n}\sum_{i=1}^{2}\int_{\Omega}\left|\partial_{x_{i}}\varphi^{\alpha}(x)-[S_{\delta_{\varepsilon}}\partial_{x_{i}}\varphi^{\alpha}](x)\right|^{q}dx\right)^{1/q}\|u\|_{1,p},$
(3.39)

where the constants do not depend on $\varepsilon$, $u$ and $\varphi$ again.
This time the right-hand side of (3.1) is $o(1)$ for $\varepsilon\to 0$
uniformly with respect to $\|u\|_{1,p}\leq 1$. Hence, (3.27) is proved.

Now we are prepared to define the needed family $\bar{u}_{\varepsilon}\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$, which satisfies (3.9) and (3.16), as
follows:

$\bar{u}_{\varepsilon}^{\alpha}(x):=u^{\alpha}_{0}(x)+[K_{\varepsilon}u_{0}]^{\alpha}(x)=u^{\alpha}_{0}(x)+\varepsilon\eta_{\varepsilon}(x)[S_{\delta_{\varepsilon}}\partial_{x_{k}}u_{0}^{\gamma}](x)v_{k}^{\alpha\gamma}(x/\varepsilon)\mbox{
with }\delta_{\varepsilon}:=\varepsilon^{1/4}.$ (3.40)

Because of (3.3), (3.24), (3.7) and (3.23) we have that

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}\leq\mbox{const
}\frac{\varepsilon}{\delta_{\varepsilon}^{2/p}}\|u_{0}\|_{1,p}\leq\mbox{const
}\varepsilon^{(2p-1)/2p},$

where the constants do not depend on $\varepsilon$. Hence, (3.9) is satisfied.

Further, condition (3.16) follows from (3.26).

###  3.2 Construction of approximate solutions with (3.17)

In this subsection we will construct a family $\bar{u}_{\varepsilon}\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ with (3.17) under the assumption that

$u_{0}\in W^{2,p_{0}}(\Omega;\mathbb{R}^{n}).$ (3.41)

Because of (3.41) in the definition of the family $\bar{u}_{\varepsilon}$ we
do not need the smoothing operators $S_{\delta}$, i.e. this time we define

$\bar{u}_{\varepsilon}^{\alpha}(x):=u^{\alpha}_{0}(x)+\varepsilon\eta_{\varepsilon}(x)\partial_{x_{k}}u_{0}^{\gamma}(x)v_{k}^{\alpha\gamma}(x/\varepsilon),$
(3.42)

and because of (3.24) and (3.41) we have that

$\|\bar{u}_{\varepsilon}-u_{0}\|_{\infty}=O(\varepsilon)\mbox{ for
}\varepsilon\to 0.$

In order to verify (3.17) we proceed as in (3.34):

$\displaystyle\int_{\Omega}\Big{(}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}\bar{u}_{\varepsilon}^{\beta}(x)-\hat{a}^{\alpha\beta}_{ij}\partial_{x_{j}}u^{\beta}_{0}(x)\Big{)}\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle=\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}\left(u^{\beta}_{0}+\varepsilon\eta_{\varepsilon}\partial_{x_{k}}u_{0}^{\gamma}v_{k}^{\beta\gamma}(x/\varepsilon)\right)-\hat{a}^{\alpha\beta}_{ij}\partial_{x_{j}}u^{\beta}_{0}\right)\partial_{x_{i}}\varphi^{\alpha}dx$
$\displaystyle=\varepsilon\int_{\Omega}\left(-\phi_{ijk}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\partial_{x_{k}}(\eta_{\varepsilon}(x)\partial_{x_{j}}u_{0}^{\beta}(x))\partial_{x_{i}}\varphi^{\alpha}(x)dx$
$\displaystyle\;\;\;\;\;\;+\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(\partial_{x_{j}}u_{0}^{\beta}(x)-\eta_{\varepsilon}(x)\partial_{x_{j}}u_{0}^{\beta}(x)\right)\partial_{x_{i}}\varphi^{\alpha}(x)dx.$

And as in (3.35)-(3.37) one estimates as follows:

$\displaystyle\left|\varepsilon\int_{\Omega}\left(-\phi_{ijk}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\partial_{x_{k}}\eta_{\varepsilon}(x)\partial_{x_{j}}u_{0}^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const
}\varepsilon\left(\sum_{i=1}^{2}\int_{\Omega_{\varepsilon}}|\partial_{x_{i}}\eta_{\varepsilon}(x)|^{p}dx\right)^{1/p}\sum_{\beta=1}^{n}\sum_{j=1}^{2}\|\partial_{x_{j}}u^{\beta}_{0}\|_{\infty}\|\varphi\|_{1,q}\leq\mbox{const
}\varepsilon^{1/p}\|\varphi\|_{1,q}$

and

$\left|\varepsilon\int_{\Omega}\left(-\phi_{ijk}^{\alpha\beta}(x/\varepsilon)+a^{\alpha\gamma}_{ik}(x/\varepsilon)v_{j}^{\gamma\beta}(x/\varepsilon)\right)\eta_{\varepsilon}(x)\partial_{x_{k}}\partial_{x_{j}}u_{0}^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|\leq\mbox{const
}\varepsilon\|\varphi\|_{1,q}$

and

$\displaystyle\left|\int_{\Omega}\left(a^{\alpha\beta}_{ij}(x/\varepsilon)-\hat{a}^{\alpha\beta}_{ij}\right)\left(1-\eta_{\varepsilon}(x)\right)\partial_{x_{j}}u_{0}^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)dx\right|$
$\displaystyle\leq\mbox{const}\left(\int_{\Omega_{\varepsilon}}|1-\eta_{\varepsilon}(x)|^{p}dx\right)^{1/p}\|\varphi\|_{1,q}\leq\mbox{const
}\varepsilon^{1/p}\|\varphi\|_{1,q},$

where the constants do not depend on $\varepsilon$ and $\varphi$. Hence,
(3.17) is proved.

###  3.3 Verification of (3.8)

The linear operators $A_{\varepsilon}$ are isomorphisms from
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ onto $W^{-1,p}(\Omega;\mathbb{R}^{n})$
(cf. (3.2)), and the linear operators $B^{\prime}(u_{0})$ are bounded from
$L^{\infty}(\Omega;\mathbb{R}^{n})$ into $L^{\infty}(\Omega;\mathbb{R}^{n})$
and, hence, compact from $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ into
$W^{-1,p}(\Omega;\mathbb{R}^{n})$. Hence, condition (3.8) is satisfied if
there exists $\varepsilon_{0}>0$ such that

$\inf\left\\{\|(A_{\varepsilon}+B^{\prime}(u_{0}))u\|_{-1,p}:\;\varepsilon\in(0,\varepsilon_{0}],u\in
U,\|u\|_{1,p}=1\right\\}>0.$

Suppose that this is not true. Then there exist sequences
$\varepsilon_{1},\varepsilon_{2},\ldots>0$ and $u_{1},u_{2},\ldots\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ such that $\varepsilon_{l}\to 0$ for
$l\to\infty$ and

$\lim_{l\to\infty}\|(A_{\varepsilon_{l}}+B^{\prime}(u_{0}))u_{l}\|_{-1,p}=0,$
(3.43)

but

$\|u_{l}\|_{1,p}=1\mbox{ for all }l.$ (3.44)

Because $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ is reflexive and because it is
compactly embedded into $L^{\infty}(\Omega;\mathbb{R}^{n})$, without loss of
generality we may assume that there exists $u_{*}\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ such that the sequence
$u_{1},u_{2},\ldots$ converges to $u_{*}$ weakly in
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ and

$\lim_{l\to\infty}\|u_{l}-u_{*}\|_{\infty}=0.$ (3.45)

From (3.45) follows that $\|B^{\prime}(u_{0})(u_{l}-u_{*})\|_{\infty}\to 0$
and, hence, that $\|B^{\prime}(u_{0})(u_{l}-u_{*})\|_{-1,p}\to 0$ for
$l\to\infty$. Therefore (3.43) implies that

$\lim_{l\to\infty}\|A_{\varepsilon_{l}}u_{l}+B^{\prime}(u_{0})u_{*}\|_{-1,p}=0.$
(3.46)

###### Lemma 3.4

There exists $l_{0}\in\mathbb{N}$ such that for any $l\in\mathbb{N}$ with
$l\geq l_{0}$ there exits $w_{l}\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ such
that

$u_{l}=w_{l}+K_{\varepsilon_{l}}w_{l}$ (3.47)

and

$\lim_{l\to\infty}\langle f,w_{l}-u_{*}\rangle_{1,p}=0\mbox{ for all }f\in
W^{-1,q}(\Omega;\mathbb{R}^{n}).$ (3.48)

Proof  The operators $K_{\varepsilon_{l}}$ are bounded from
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ into $W^{2,p}(\Omega;\mathbb{R}^{n})\cap
W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ (cf. (3.21), (3.22) and (3.25)) and,
hence, compact from $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ into
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$. Therefore the operators
$I+K_{\varepsilon_{l}}$ are Fredholm of index zero from
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ into
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$.

Let us show that for large $l$ the operators $I+K_{\varepsilon_{l}}$ are
injective (and, hence, bijective from $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$
onto $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$). Suppose the contrary. Then without
loss of generality we may assume that there exists a sequence
$\bar{w}_{1},\bar{w}_{2},\ldots\in W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ such
that

$\bar{w}_{l}+K_{\varepsilon_{l}}\bar{w}_{l}=0\mbox{ and
}\|\bar{w}_{l}\|_{\infty}=1\mbox{ for all }l.$ (3.49)

It follows that
$A_{\varepsilon_{l}}(\bar{w}_{l}+K_{\varepsilon_{l}}\bar{w}_{l})=0$ for all
$l$, and because of (3.27) we get that for any $\varphi\in
W_{0}^{1,q}(\Omega;\mathbb{R}^{n})$ we have that

$0=\lim_{l\to\infty}\langle
A_{\varepsilon_{l}}(\bar{w}_{l}+K_{\varepsilon_{l}}\bar{w}_{l})-A_{0}\bar{w}_{l},\varphi\rangle_{1,q}=-\lim_{l\to\infty}\langle
A_{0}\bar{w}_{l},\varphi\rangle_{1,q},$

i.e. the sequence $A_{0}\bar{w}_{1},A_{0}\bar{w}_{2},\ldots$ converges to zero
weakly in $W^{-1,p}(\Omega;\mathbb{R}^{n})$. But (1.8) yields that $A_{0}$ is
an isomorphism from $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ onto
$W^{-1,p}(\Omega;\mathbb{R}^{n})$, hence the sequence
$\bar{w}_{1},\bar{w}_{2},\ldots$ converges to zero weakly in
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$ and, therefore, strongly in
$L^{\infty}(\Omega;\mathbb{R}^{n})$. But this contradicts to (3.49).

Finally, let us prove (3.48). Take a test function $\varphi\in
W_{0}^{1,q}(\Omega;\mathbb{R}^{n})$. Because of (3.27), (3.46) and (3.47) we
have

$-\langle B^{\prime}(u_{0})u_{*},\varphi\rangle_{1,q}=\lim_{l\to\infty}\langle
A_{\varepsilon_{l}}u_{l},\varphi\rangle_{1,q}=\lim_{l\to\infty}\langle
A_{\varepsilon_{l}}(w_{l}+K_{\varepsilon_{l}}w_{l}),\varphi\rangle_{1,q}=\lim_{l\to\infty}\langle
A_{0}w_{l},\varphi\rangle_{1,q}.$ (3.50)

Hence, the sequence $A_{0}w_{1},A_{0}w_{2},\ldots$ converges weakly in
$W^{-1,p}(\Omega;\mathbb{R}^{n})$, therefore the sequence $w_{1},w_{2},\ldots$
converges weakly in $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$. In particular, the
sequence $w_{1},w_{2},\ldots$ is bounded in
$W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$. Hence, (3.23), (3.25) and (3.45) yield
that

$\displaystyle\|w_{l}-u_{*}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\|u_{l}-u_{*}\|_{\infty}+\|K_{\varepsilon_{l}}w_{l}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\mbox{const
}\left(\|u_{l}-u_{*}\|_{\infty}+\varepsilon_{l}\sum_{\gamma=1}^{n}\sum_{i=1}^{2}\|S_{\delta_{l}}\partial_{x_{i}}w^{\gamma}_{l}\|_{\infty}\right)$
$\displaystyle\leq$ $\displaystyle\mbox{const
}\left(\|u_{l}-u_{*}\|_{\infty}+\varepsilon_{l}^{(2p-1)/2p}\|w_{l}\|_{1,p}\right)\to
0\mbox{ for }l\to\infty.$

Therefore the weak $W_{0}^{1,p}(\Omega;\mathbb{R}^{n})$-limit of the sequence
$w_{1},w_{2},\ldots$ equals to its strong
$L^{\infty}(\Omega;\mathbb{R}^{n})$-limit, that is $u_{*}$.

Now, because of (3.48) and (3.50) we have that
$\langle(A_{0}+B^{\prime}(u_{0}))u_{*},\varphi\rangle_{1,q}=0$ for all
$\varphi\in W_{0}^{1,q}(\Omega;\mathbb{R}^{n})$, i.e. that $u_{*}$ is a weak
solution to the linearized boundary value problem (1.11). Hence, by assumption
of Theorem 1.1, we get that $u_{*}=0$. Therefore (3.46) implies that
$\|A_{\varepsilon_{l}}u_{l}\|_{W^{-1,p}}\to 0$ for $l\to\infty$. But this
contradicts to (3.2) and (3.44).

##  4 Nonlinear natural boundary conditions

In this section we show that similar to Theorem 1.1 results are true also for
nonlinear natural boundary conditions, i.e. we consider the boundary vale
problem

$\left.\begin{array}[]{l}\partial_{x_{i}}\Big{(}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\Big{)}=b^{\alpha}(x,u(x))\mbox{
for }x\in\Omega,\\\
a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\nu_{i}(x)=b_{0}^{\alpha}(x,u(x))\mbox{
for }x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n,$ (4.1)

where $\nu=(\nu_{1},\nu_{2}):\partial\Omega\to\mathbb{R}^{2}$ is the outer
unit normal vector field on the boundary $\partial\Omega$, and
$u\in\mathbb{R}^{n}\mapsto b_{0}(\cdot,u)\in
L^{\infty}(\partial\Omega;\mathbb{R}^{n})$ is $C^{1}$-smooth. The reason, why
similar to Theorem 1.1 results are true also for the boundary value problem
(4.1), is easy to explain: In (3.34) we did not needed that the test functions
$\varphi$ satisfy zero boundary conditions.

Our assumptions on the domain $\Omega$, the diffusion coefficients
$a^{\alpha\beta}_{ij}$ and the reaction terms $b^{\alpha}$ are as in Section
1. Also the homogenized diffusion coefficients are as in Section 1, i.e.
defined in (1.6) via the cell problems (1.7). Hence, the homogenized boundary
value problem, corresponding to (4.1), is

$\left.\begin{array}[]{l}\hat{a}^{\alpha\beta}_{ij}\partial_{x_{i}}\partial_{x_{j}}u^{\beta}(x)=b^{\alpha}(x,u(x))\mbox{
for }x\in\Omega,\\\
\hat{a}^{\alpha\beta}_{ij}\partial_{x_{j}}u^{\beta}(x)\nu_{i}(x)=b^{\alpha}_{0}(x,u(x))\mbox{
for }x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n.$ (4.2)

A vector function $u\in C^{1}(\overline{\Omega};\mathbb{R}^{n})$ called weak
solution to the boundary value problem (4.1) if it satisfies the variational
equation

$\displaystyle\int_{\Omega}\Big{(}a^{\alpha\beta}_{ij}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi^{\alpha}(x)+b^{\alpha}(x,u(x))\varphi^{\alpha}(x)\Big{)}dx$
$\displaystyle=\int_{\partial\Omega}b^{\alpha}(x,u(x))\varphi^{\alpha}(x)d\Gamma(x)\mbox{
for all }\varphi\in W^{1,2}(\Omega;\mathbb{R}^{n}),$

where $\Gamma$ is the Lebesgue measure on $\partial\Omega$, and similarly for
(4.2) and for its linearization

$\left.\begin{array}[]{l}\hat{a}^{\alpha\beta}_{ij}\partial_{x_{i}}\partial_{x_{j}}u^{\beta}(x)=\partial_{u^{\gamma}}b^{\alpha}(x,u_{0}(x))u^{\gamma}(x)\mbox{
for }x\in\Omega,\\\
\hat{a}^{\alpha\beta}_{ij}\partial_{x_{j}}u^{\beta}(x)\nu_{i}(x)=\partial_{u^{\gamma}}b_{0}^{\alpha}(x,u_{0}(x))u^{\gamma}(x)\mbox{
for }x\in\partial\Omega,\end{array}\right\\}\alpha=1,\ldots,n.$ (4.3)

For these problems we get, similar to Theorem 1.1, the following

###### Theorem 4.1

Suppose (1.2)-(1.5), and let $u=u_{0}$ be a weak solution to (4.2) such that
(4.3) does not have weak solutions $u\not=0$. Then the following is true:

(i) There exist $\varepsilon_{0}>0$ and $\delta>0$ such that for all
$\varepsilon\in(0,\varepsilon_{0}]$ there exists exactly one weak solution
$u=u_{\varepsilon}$ to (4.1) with $\|u-u_{0}\|_{\infty}\leq\delta$. Moreover,

$\|u_{\varepsilon}-u_{0}\|_{\infty}\to 0\mbox{ for }\varepsilon\to 0.$ (4.4)

(ii) If $u_{0}\in W^{2,p_{0}}(\Omega;\mathbb{R}^{n})$ with certain $p_{0}>2$,
then for all $p>2$ we have

$\|u_{\varepsilon}-u_{0}\|_{\infty}=O(\varepsilon^{1/p})\mbox{ for all
}\varepsilon\to 0.$ (4.5)

The proof of Theorem 4.1 is similar to that of Theorem 1.1. We indicate only
the few small differences. One has to apply Corollary 2.3 again, but now in
the following setting:

$U:=W^{1,p}(\Omega;\mathbb{R}^{n}),\;V:=W^{1,q}(\Omega;\mathbb{R}^{n})^{*}\mbox{
with }1/p+1/q=1,$

where

$\displaystyle\|u\|_{U}:=\|u\|_{1,p}:=\left(\sum_{\alpha=1}^{n}\int_{\Omega}\left(|u^{\alpha}(x)|^{p}+\sum_{i=1}^{2}|\partial_{x_{i}}u^{\alpha}(x)|^{p}\right)dx\right)^{1/p},\;\|\cdot\|:=\|\cdot\|_{\infty},$
$\displaystyle\|f\|_{V}:=\sup\\{\langle f,\varphi\rangle_{1,q}:\;\varphi\in
W^{1,q}(\Omega;\mathbb{R}^{n}),\;\|\varphi\|_{1,q}\leq 1\\},$

and $\langle\cdot,\cdot\rangle_{1,q}:W^{1,q}(\Omega;\mathbb{R}^{n})^{*}\times
W^{1,q}(\Omega;\mathbb{R}^{n})\to\mathbb{R}$ is the dual pairing, again. The
$C^{1}$-smooth operators $F_{\varepsilon}:U\to V$ of Theorem 2.1 are defined
by

$F_{\varepsilon}(u):=A_{\varepsilon}u+B(u),$

again, where the linear operators
$A_{\varepsilon}:W^{1,p}(\Omega;\mathbb{R}^{n})\to
W^{1,q}(\Omega;\mathbb{R}^{n})^{*}$ and the nonlinearity
$B:C(\overline{\Omega};\mathbb{R}^{n})\to W^{1,q}(\Omega;\mathbb{R}^{n})^{*}$
are defined by

$\displaystyle\langle A_{\varepsilon}u,\varphi\rangle_{1,q}$ $\displaystyle:=$
$\displaystyle\int_{\Omega}\left(a_{ij}^{\alpha\beta}(x/\varepsilon)\partial_{x_{j}}u^{\beta}(x)\partial_{x_{i}}\varphi(x)+u^{\alpha}(x)\varphi^{\alpha}(x)\right)dx,$
$\displaystyle\langle B(u),\varphi\rangle_{1,q}$ $\displaystyle:=$
$\displaystyle\int_{\Omega}\left(b^{\alpha}(x,u(x))-u^{\alpha}(x)\right)\varphi^{\alpha}(x)dx-\int_{\partial\Omega}b_{0}^{\alpha}(x,u(x))\varphi^{\alpha}(x)d\Gamma(x)$

for all $\varphi\in W^{1,q}(\Omega;\mathbb{R}^{n})$, and similarly
$A_{0}:W^{1,p}(\Omega;\mathbb{R}^{n})\to W^{1,q}(\Omega;\mathbb{R}^{n})^{*}$.

For proving the error estimates (4.4) and (4.5) we use again the families of
approximate solutions (3.40) and (3.42), respectively.

###### Remark 4.2

Let us draw attention to the following technical detail:

In Section 3, i.e. in the case of Dirichlet boundary conditions, the reason
for using the cut-off functions $\eta_{\varepsilon}$ in (3.40) and (3.42) was
that the approximate solutions $\bar{u}_{\varepsilon}$ should satisfy the
Dirichlet boundary conditions. There the cut-off functions
$\eta_{\varepsilon}$ are not needed to avoid boundary integrals in (3.34)
after partial integration, because there the test function $\varphi$ could
have compact support.

But now, in Section 4, i.e. in the case of Robin boundary conditions, the
reason for using the cut-off functions $\eta_{\varepsilon}$ in (3.40) and
(3.42) is to avoid boundary integrals in (3.34) after partial integration,
because now the test function $\varphi$ is not allowed to have compact
support. Here the cut-off functions $\eta_{\varepsilon}$ are not needed for
certain boundary condition to be satisfied (because the approximate solutions
$\bar{u}_{\varepsilon}$ are not obliged to satisfy any boundary conditions).

This technical detail is important because of the following: If it would be
possible to improve the choices (3.40) and (3.42) of the approximate solutions
appropriately (for example, to avoid the use of cut-off functions), then,
perhaps, it would be possible to prove better than (4.4) and (4.5) error
estimates.

## References

  * [1] V. Barbu, Partial Differential Equations and Boundary Value Problems. Mathematics and its Applications vol. 441, Kluwer Academic Publishers, 1998.
  * [2] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications vol. 3, North-Holland, 1978.
  * [3] M. Breden, R. Castelli, Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof. J. Differ. Equations 264 (2018), 6418–6458.
  * [4] R. Bunoiu, R. Precup, Localization and multiplicity in the homogenization of nonlinear problems. Adv. Nonlinear Anal. 9 (2020), 292–304.
  * [5] V.F. Butuzov, N.N. Nefedov, O.E. Omel’chenko, L. Recke, Time-periodic boundary layer solutions to singularly perturbed parabolic problems. J. Differ. Equations 262 (2017), 4823–4862.
  * [6] V.F. Butuzov, N.N. Nefedov, O.E. Omel’chenko, L. Recke, Boundary layer solutions to singularly perturbed quasilinear systems. Discrete Cont. Dyn. Syst., Series B 27 (2022), 4255–4283.
  * [7] V.F. Butuzov, N.N. Nefedov, O.E. Omel’chenko, L. Recke, K.R. Schneider, An implicit function theorem and applications to nonsmooth boundary layers. In: Patterns of Dynamics, ed. by P. Gurevich, J. Hell, B. Sandstede, A. Scheel, Springer Proc. in Mathematics & Statistics vol. 205, Springer, 2017, 111–127.
  * [8] D. Cioranescu, P. Donato, An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications vol. 17, Oxford University Press, 1999.
  * [9] J. A. Griepentrog, L. Recke, Linear elliptic boundary value problems with non-smooth data: Normal solvability in Sobolev-Campanato spaces. Math. Nachr. 225 (2001), 39–74.
  * [10] K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second-order elliptic differential equations. Math. Ann. 283 (1989), 679–687.
  * [11] K. Gröger, L. Recke, Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data. NoDEA, Nonlinear Differ. Equ. Appl. 13 (2006), 263–285.
  * [12] S.G. Katz, H.R. Parks, The Implicit Function Theorem. History, Theory, and Applications. Birkhäuser, 2002.
  * [13] C. Kenig, F. Lin, Z. Shen, Periodic homogenization of Green and Neumann functions. Commun. Pure Appl. Math. 67 (2014), 1219–1262.
  * [14] M. Lanza de Cristoforis, P. Musolino, Two-parameter homogenization for a nonlinear periodic Robin problem for a Poisson equation: a functional analytic approach. Rev. Mat. Complut. 31 (2018), 63–110.
  * [15] M. Lanza de Cristoforis, P. Musolino, Asymptotic behaviour of the energy integral of a two-parameter homogenization problem with nonlinear periodic Robin boundary conditions. Proc. Edinb. Math. Soc. II. Ser. 62 (2019), 985–1016.
  * [16] N.G. Meyers, An $L^{p}$-estimate for the gradient of solutions of second-order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963), 189–206.
  * [17] Wen Ming He, Jun Zhi Cui, Error estimate of the homogenization solution for elliptic problems with small periodic coefficients on $L^{\infty}(\Omega)$. Science China Mathematics 53 (2010), 1231–1252.
  * [18] N.N. Nefedov, A.O. Orlov, L. Recke, K.R. Schneider, Nonsmooth regular perturbations of singularly perturbed problems. J. Differ. Equations 375 (2023), 206–236.
  * [19] N.N. Nefedov, L. Recke, A common approach to singular perturbation and homogenization I: Quasilinear ODE systems. arXiv:2309.15611.
  * [20] O.E. Omel’chenko, L. Recke, Existence, local uniqueness and asymptotic approximation of spike solutions to singularly perturbed elliptic problems. Hiroshima Math. J. 45 (2015), 35–89.
  * [21] L. Recke, Use of very weak approximate boundary layer solutions to spatially nonsmooth singularly perturbed problems. J. Math. Anal. Appl. 506 (2022), Article ID 125552.
  * [22] L. Recke, O.E. Omel’chenko, Boundary layer solutions to problems with infinite dimensional singular and regular perturbations. J. Differ. Equations 245 (2008), 3806–3822.
  * [23] Zongwei Shen, Periodic Homogenization of Elliptic Systems. Operator Theory: Advances and Applications vol. 269, Birkhäuser, 2018.
  * [24] V. Tewary, Combined effects of homogenization and singular perturbation: A Bloch wave approach. Netw. Heterog. Media 16 (2021), 427-458.
  * [25] Wei Wang, Ting Zhang, Homogenization theory of elliptic systems with lower order terms for dimension two. Commun. Pure Applied Analysis 22 (2023), 787–824.
  * [26] Qiang Xu, Convergence rates for general elliptic homogenization problems in Lipschitz domains. SIAM J. Math. Anal. 48 (2016), 3742–3788.
  * [27] Qiang Xu, Uniform regularity estimates in homogenization theory of elliptic systems with lower order terms. J. Math. Anal. Appl. 438 (2016), 1066–1107.