\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 290, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/290\hfil 
 Asymptotic behavior of the longitudinal permeability]
{Asymptotic behavior of the longitudinal permeability of a periodic
array of \\ thin cylinders}

\author[P. Musolino, V. Mityushev \hfil EJDE-2015/290\hfilneg]
{Paolo Musolino, Vladimir Mityushev}

\address{Paolo Musolino \newline
Department of Mathematics, University of Padova, Italy}
\email{musolinopaolo@gmail.com}

\address{Vladimir Mityushev \newline
Department of Computer Science and Computational Methods,
Pedagogical University of Cracow, Poland}
\email{mityu@up.krakow.pl}

\thanks{Submitted June 17, 2015. Published November 20, 2015.}
\subjclass[2010]{76D30, 76D05, 35J05, 35J25, 31B10, 45A05}
\keywords{Longitudinal permeability; asymptotic expansion; rectangular array;
\hfill\break\indent  singularly perturbed domain; integral equations; logarithmic capacity}

\begin{abstract}
 We consider a Newtonian fluid flowing at low Reynolds numbers
 along a spatially periodic array of cylinders of diameter proportional
 to a small nonzero parameter $\epsilon$. Then for $\epsilon \neq 0$ and
 close to $0$ we denote by  $K_{II}[\epsilon]$ the longitudinal permeability.
 We are interested in studying the asymptotic behavior of $K_{II}[\epsilon]$
 as $\epsilon$ tends to $0$. We analyze $K_{II}[\epsilon]$ for $\epsilon$
 close to $0$ by an approach based on functional analysis and potential theory,
 which is alternative to that of asymptotic analysis. We prove that
 $K_{II}[\epsilon]$ can be written as the sum of a logarithmic term and a
 power series in $\epsilon^2$. Then, for small $\epsilon$, we provide an
 asymptotic expansion of the longitudinal  permeability in terms of the sum
 of a logarithmic function of the square of the capacity of the cross section
 of the cylinders and a term which does not depend of the shape of the unit
 inclusion (plus a small remainder).
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}\label{introd}

In this article, we study the asymptotic behavior of the longitudinal 
permeability of a periodic array of cylinders, when a Newtonian fluid is 
flowing at low Reynolds numbers around the cylinders. We assume that the 
diameter of the cylinders is proportional to a small nonzero parameter 
$\epsilon$ and that the driving pressure gradient is parallel to the cylinders. 
Then the velocity field has only one non-zero component which, 
by the Stokes equations, satisfies a Poisson equation (cf. problem \eqref{bvpe}). 
By means of the longitudinal component of the velocity field, one can define  
for each $\epsilon$ close to 0 the longitudinal permeability $K_{II}[\epsilon]$. 
Here, we are interested in studying the behavior of $K_{II}[\epsilon]$ as 
$\epsilon$ approaches the degenerate value $\epsilon=0$, in correspondence 
of which the diameter of the cylinders collapses.

Several authors have studied the longitudinal permeability of arrays of 
cylinders by exploiting different techniques.  Emersleben \cite{Em25} 
considered a square array of contours of a constant value of a special 
Epstein zeta function.  Happel \cite{Ha59} provided an analysis for the dilute 
case on the basis that two concentric cylinders can serve as the model 
for fluid moving through an assemblage of cylinders.
In the seminal paper \cite{Has59}, Hasimoto investigated the viscous flow 
past a cubic array of spheres, in particular for the case when the radius 
of the spheres is small compared to the mutual distance. He also applied his 
results to the two-dimensional flow past a square array of circular cylinders. 
His techniques are based on the construction of a spatially periodic fundamental 
solution for the Stokes' system and apply to specific shapes 
(circular/spherical obstacles and square/cubic arrays). 
Schmid \cite{Sc66} investigated the longitudinal laminar flow in an infinite 
square array of circular cylinders.

Sangani and Yao \cite {SaYa88b} extended a method described in \cite{SaYa88a} 
in order treat the problem of determining the permeability of random arrays 
of infinitely long cylinders. In \cite{SaYa88b} the transverse and 
longitudinal permeabilities averaged over several configurations of random 
arrays of circular cylinders are expressed as a function of the area fraction 
of the cylinders.   Adler and the second author in \cite{MiAd02a, MiAd02b}  
considered the longitudinal permeability of periodic rectangular arrays of 
circular cylinders. By methods of complex variable, they transformed the 
boundary value problem defining the permeability into a functional equation. 
Then they expressed the solution to such a functional equation in terms of 
a series of the radius of the cylinders, which can then be exploited to 
derive a formula for the longitudinal permeability as a logarithmic term 
and a power series in the radius of the cylinder. More precisely, first 
in \cite{MiAd02a} they considered the case of a single circular cylinder in 
the unit cell and then in \cite{MiAd02b} they turned to study the case of an 
arbitrary (finite) distribution of circular cylinders inside the periodicity cell.


In this article instead we investigate the asymptotic behavior of the longitudinal 
permeability of a rectangular array of thin cylinders,  corresponding to 
the so called dilute case. Our method allows to treat general shapes of
 the cross section of the cylinders without restrictions to circular cylinders. 
Our aim is twofold. From one side, we extend the results of \cite{MiAd02a, MiAd02b} 
to very general shapes and we prove rigorously that  the longitudinal 
permeability can be represented as the sum of a logarithmic term and a 
power series of the square of the `size parameter' of the cylinders. 
This is deduced by an analyticity result for a singularly perturbed 
boundary value problem. Such a result is based on an operator theoretical 
reformulation of the problem and on the Implicit Function Theorem in Banach spaces. 
 From the other side,  we provide and justify an asymptotic expansion of the 
longitudinal  permeability in terms of the square of the logarithmic capacity 
of the cross section of the cylinders. Once more, since there is no particular 
restriction on the shape of the cylinders, this expression allows comparisons 
between different geometric assumptions.

In contrast, for example, with the functional equation method of 
\cite{MiAd02a, MiAd02b}, here we exploit a potential theoretical method 
to investigate to behavior of the longitudinal permeability in the dilute case. 
Integral equation techniques have indeed revealed to be a powerful tool 
to analyze the asymptotic behavior of a wide class of quantities of physical 
interest. For example, in Vogelius and Volkov \cite{VoVo00}  the time-harmonic 
Maxwell equation is considered and the leading order boundary perturbations 
are derived. Ammari, Kang and Touibi \cite{AmKaTo05}  derived asymptotic 
expansions of the effective electrical conductivity of periodic dilute 
composites in terms of the volume fraction occupied by the inclusions. 
In Ammari, Kang, and Lee \cite{AmKaLi06} asymptotic formulas for the effective 
parameters of a two-phase elastic medium consisting of materials of different 
elastic properties are shown. Analogous results for the effective viscosity 
properties of dilute suspensions of arbitrarily shaped particles are obtained 
in Ammari, Garapon, Kang, and Lee \cite{AmGaKaLe12}. For an extensive list 
of applications of this method, we refer, e.g., to the monograph by
 Ammari and Kang \cite{AmKa07}.

Furthermore, the potential theoretical method proposed by Lanza de Cristoforis 
for real analytic continuation properties of  solutions of singularly perturbed 
linear or nonlinear problems has been used to analyzed different problems of 
concrete nature. As an example, in \cite{DaMu13} the effective conductivity 
of periodic two-phase dilute composites with nonideal contact condition was 
investigated, whereas in \cite{LaMu14} a quasi-linear heat transmission 
problem in a dilute composite was studied.

To introduce the mathematical problem, we fix 
\[
l \in ]0,+\infty[\, , \quad Q\equiv]0,l[\times ]0,1/l[\,,\quad 
q\equiv \begin{pmatrix}
l &   0
\\
0          & 1/l
\end{pmatrix}\, .
\]
Hence, the area $|Q|_2$ of the cell $Q$ holds unity.  
We denote by $q^{-1}$ the inverse matrix of $q$. Clearly,
$
q {\mathbb{Z}}^{2}\equiv  \{qz:z\in{\mathbb{Z}}^{2}\}
$
is the set of vertices of a periodic subdivision of ${\mathbb{R}}^{2}$ 
corresponding to the fundamental periodicity cell $Q$. We also take
\begin{equation}
\parbox{10cm}
{$\alpha \in ]0,1[$ and a bounded open connected subset $\Omega$ of 
$\mathbb{R}^{2}$ of  class  $C^{1,\alpha}$
such that $\mathbb{R}^{2}\setminus{\operatorname{cl}}\Omega$ is  connected   
and  that  $0\in \Omega$.}
\label{dom}
\end{equation}
The symbol $\operatorname{cl}$ denotes the closure. Moreover, we fix
\[% \label{e0}
\text{$p \in Q$ and $\epsilon_{0}\in]0,+\infty[$ such that
 $p+\epsilon {\operatorname{cl}}\Omega\subseteq Q
$ for all $\epsilon\in]-\epsilon_{0},\epsilon_{0}[$}\,.
\]
To shorten our notation, we set
\[
\Omega_{p,\epsilon}\equiv p+\epsilon\Omega\quad\forall\epsilon\in{\mathbb{R}}\,.
\]
Then we introduce the periodic domains
\[
{\mathbb{S}}[\Omega_{p,\epsilon}]\equiv \cup_{z\in {\mathbb{Z}}^{2}}\left(
qz+\Omega_{p,\epsilon} \right)\,, \quad 
{\mathbb{S}}[\Omega_{p,\epsilon}]^{-}\equiv
{\mathbb{R}}^{2}\setminus \cup_{z\in {\mathbb{Z}}^{2}}
\left(qz+\operatorname{cl}\Omega_{p,\epsilon} \right) \quad \forall \epsilon\in ]
-\epsilon_{0},\epsilon_{0}[ \,.
\]
If $\epsilon \in ]-\epsilon_0,\epsilon_0[$, the set 
$\operatorname{cl}{\mathbb{S}}[\Omega_{p,\epsilon}] \times \mathbb{R}$ 
represents an infinite array of parallel cylinders. 
Instead, the set  ${\mathbb{S}}[\Omega_{p,\epsilon}]^{-} \times \mathbb{R}$ 
is the region where a Newtonian fluid of viscosity $\mu$ is flowing at 
low Reynolds number. Then we assume that the driving pressure gradient 
is constant and parallel to the cylinders. As a consequence, by a standard 
argument based on the particular geometry of the problem 
(cf., e.g., Adler \cite[Ch.~4]{Ad92}, Sangani and Yao \cite{SaYa88b}, 
and \cite{MiAd02a, MiAd02b}), one reduces the Stokes system to a Poisson 
equation for the non-zero component of the velocity field. Since we are 
working with dimensionless quantities, we may assume that the viscosity of 
the fluid and the pressure gradient are both set equal to $1$. 
For a more complete discussion on spatially periodic structures, 
we refer to Adler \cite[Ch.~4]{Ad92}. Accordingly, if 
$\epsilon \in ]-\epsilon_0,\epsilon_0[ \setminus \{0\}$, we consider the 
following Dirichlet problem for the Poisson equation:
\begin{equation} \label{bvpe}
\begin{gathered}
\Delta u=1  \quad \text{in } {\mathbb{S}}[\Omega_{p,\epsilon}]^{-}\,,
\\
u(x+qz)=u(x)\quad  \forall x \in \operatorname{cl}\mathbb{S}
 [\Omega_{p,\epsilon}]^{-}\,, \forall z \in\mathbb{Z}^2\,,\\
u(x)=0\quad \forall x\in  \partial\Omega_{p,\epsilon}\,.
\end{gathered}
\end{equation}
If $\epsilon \in ]-\epsilon_0,\epsilon_0[ \setminus \{0\}$, then the solution 
of problem \eqref{bvpe} in the space 
$C^{1,\alpha}_{q}(\operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^{-})$ of 
$q$-periodic functions in $\operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^-$ 
of class $C^{1,\alpha}$ (cf.~Subsection \ref{per}) is unique and we denote it 
by $u[\epsilon]$.  From the physical point of view, the function $u[\epsilon]$ 
represents the non-zero component of the velocity field (cf. \cite[\S 2]{MiAd02a}).
 By means of the function $u[\epsilon]$, we can introduce the effective permeability 
$K_{II}[\epsilon]$ which we define as the integral of the opposite of the 
flow velocity over the unit cell 
(cf.~Adler \cite{Ad92}, \cite[\S 3]{MiAd02a}), i.e.,
\[
K_{II}[\epsilon]\equiv - \int_{Q\setminus \Omega_{p,\epsilon}}u[\epsilon](x)\, dx 
\quad \forall \epsilon \in ]-\epsilon_0,\epsilon_0[\setminus\{0\}\, ,
\]
and we pose the following question:
\begin{equation}\label{question}
\text{What  can be said on the map
$\epsilon\mapsto K_{II}[\epsilon]$ when $\epsilon$ is close to $0$?}
\end{equation}

Questions of this type have long been investigated with the methods of 
asymptotic analysis, with the aim of providing  asymptotic expansions 
for the quantities of interest. Those techniques have been exploited to 
analyze a large class of singular perturbation problems by several authors. 
For example, in Maz'ya, Nazarov, and Plamenevskij \cite{MaNaPl00} one can 
find the method of compound asymptotic expansions which allows the analysis 
of general Douglis-Nirenberg elliptic boundary value problems in domains 
with holes and corners. Maz'ya, Movchan, and Nieves \cite{MaMoNi13} studied 
the asymptotic treatment of the Green's kernel in domains with small holes. 
Bonnaillie-No\"el, Dambrine, Tordeux, and Vial  \cite{BoDaToVi09} exploited 
the method of multiscale asymptotic expansions to study the behavior of the 
solution of the Poisson equation in a domain with moderately close holes. 
Bonnaillie-No\"el, Lacave, and Masmoudi \cite{BoLaMa} investigated  the effect
 of small inclusions of size $\epsilon$ on the behavior of an ideal fluid 
governed by the 2D Euler system. Problems in perforated domains have been 
considered also in the frame of shape optimization. For example, in Novotny 
and Soko\l owsky \cite{NoSo13}, the authors apply the topological derivative
 (i.e., the first term of the asymptotic
expansion of a given shape functional with respect to a small parameter that 
measures the size of singular domain perturbations) to problems in elasticity 
and heat diffusion. Furthermore,  boundary value problems in periodic domains 
have been  analyzed with the method of functional equations (cf., \emph{e.g.},
Castro, Kapanadze, and Pesetskaya \cite{CaKaPe14}, Castro, Pesetskaya, and 
Rogosin \cite{CaPeRo09},  Kapanadze, Mishuris, and Pesetskaya \cite{KaMiPe14}, 
and the works of the second-named author and his collaborators Adler, Dryga{\'s}, 
and Rylko  \cite{DrMi09, Mi01,MiAd02a, MiAd02b, MiRy13}). Such a method is based 
on complex variable techniques and has revealed to be a useful tool for specific 
shapes of the inclusions/obstacles.

Here, instead,  we answer the question in \eqref{question} by showing that
\begin{equation}\label{eq:Krep}
K_{II} [\epsilon]=-\frac{\log |\epsilon|}{2 \pi}+a_0+a_1 \epsilon^2+\epsilon^4 
J[\epsilon]
\end{equation}
for $\epsilon\neq 0$ close to 0, where $a_0$, $a_1$ are suitable real numbers 
(that we compute in terms of the solution of explicit boundary value problems 
and integral equations) and $J$ is a real analytic function defined in a
neighborhood of $0$.
We observe that our approach does have its advantages.  
Indeed, equality \eqref{eq:Krep} and the parity of $J$ 
(cf. Proposition \ref{prop:J}) imply that
\begin{equation}
\label{eq:Kser}
K_{II} [\epsilon]=-\frac{\log |\epsilon|}{2 \pi}+a_0+a_1 \epsilon^2
+\epsilon^4\sum_{j=0}^{+\infty}b_j \epsilon^{2j},
\end{equation}
for a suitable sequence of coefficients $\{b_j\}_{j=0}^{+\infty}$, 
where the series converges absolutely on a whole neighborhood of the 
degenerate value $\epsilon=0$. Such an approach has been previously 
exploited by Lanza de Cristoforis, Dalla Riva and the first-named author 
to analyze several singularly perturbed boundary value problems in periodically 
perforated domains. This method has shown to be a flexible tool to analyze 
problems of different nature. Indeed, linear and nonlinear boundary value
 problems for the Laplace equation have been studied for example 
in \cite{LaMu13, Mu12}. In \cite{DaMu14},  a nonlinear traction problem in
linearized elastostatics is studied. In \cite{DaMu13} the functional analytic 
approach is exploited to investigate the asymptotic behavior of the effective 
conductivity of a dilute composite in presence of a thermal resistance at the 
interface. This approach has also allowed to study quasi-linear transmission 
problems. Indeed, by this method, the asymptotic behavior of the temperature 
distribution of a two-phase composite has been analyzed under the assumption 
that the thermal conductivities of the materials depend nonlinearly upon the 
temperature (see \cite{LaMu14}).

Then we show that by equation \eqref{eq:Kser} we can deduce that the right-hand 
side of \eqref{eq:Kser} is equal to
\[
-\frac{\log \rho}{4 \pi}+c+O(\rho)\quad \rho\equiv  
(\operatorname{cap}(\epsilon\Omega))^2\, .
\]
The quantity $\rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$ is 
the square of the logarithmic capacity of the inclusion 
(cf., e.g., Goluzin \cite[Ch.~VII,\S 3]{Go69} and Kirsch \cite[\S 2]{Ki05}), 
i.e., of the cross section of the cylinder. Since the area of the cell is
 normalized to unity,  $\rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$ 
is a measure of the concentration of inclusions in the host medium. 
Moreover, if we set $\rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$ 
then the function $K_{II}[\epsilon]$ equals the sum of $-\log \rho/(4 \pi)$ 
and of a real analytic function of $\rho$ at zero. In the present paper,
 it is established that  the coefficient $c$ does not depend on the shape 
of $\Omega$. In particular,
\[
c=-R_{q,2}(0)+\int_{Q}S_{q,2}(x)\, dx \, ,
\]
where $S_{q,2}$ and $R_{q,2}$ are as in \eqref{eq:Sq2} and \eqref{eq:Rq2}, 
respectively. The dimension of permeability is equal to the surface, i.e., 
length$^2$ (cf.~Adler \cite{Ad92}).

This article is organized as follows. 
Section \ref{prel} is a section of preliminaries and notation. 
In Section \ref{asy} we prove our main result on the asymptotic behavior 
of $K_{II}[\epsilon]$.
In Section \ref{sec:low} we exploit conformal maps to further investigate 
$K_{II}[\epsilon]$.  Finally, in Section \ref{conc} we present some remarks 
and conclusions.

\section{Preliminaries}\label{prel}

\subsection*{Notation}

Let  $\mathcal{O}\subseteq {\mathbb {R}}^{2}$. 
Then $\operatorname{cl}\mathcal{O}$ denotes the
closure of $\mathcal{O}$ and $\partial\mathcal{O}$ denotes the boundary 
of $\mathcal{O}$.  For all $R>0$, $ x\in{\mathbb{R}}^{2}$,
$x_{j}$ denotes the $j$-th coordinate of $x$, $x^T$ the transpose vector,
$| x|$ denotes the Euclidean modulus of $ x$ in
${\mathbb{R}}^{2}$, and 
${\mathbb{B}}_{2}( x,R) \equiv \{y\in{\mathbb{R}}^{2}: | x- y|<R\}$. 
Let $\Omega$ be an open subset of ${\mathbb{R}}^{2}$.
If $f\in C^{m}(\Omega)$, then $Df$ denotes the gradient
$\big(\frac{\partial f} {\partial x_1}, \frac{\partial f} {\partial
x_2}\big)$  and $D^2f$ denotes the Hessian matrix.
For a multi-index  $\eta\equiv
(\eta_{1},\eta_{2})\in{\mathbb{N}}^{2}$ we set $|\eta |\equiv
\eta_{1}+\eta_{2}  $. Then $D^{\eta} f$ denotes
$\frac{\partial^{|\eta|}f}{\partial
x_{1}^{\eta_{1}}\partial x_{2}^{\eta_{2}}}$.    The
subspace of $C^{m}(\Omega )$ of those functions $f$ whose derivatives 
$D^{\eta }f$ of order $|\eta |\leq m$ can be extended with continuity to
$\operatorname{cl}\Omega$  is  denoted $C^{m}(\operatorname{cl}\Omega )$.
The subspace of $C^{m}(\operatorname{cl}\Omega ) $  whose
functions have $m$-th order derivatives which are
uniformly H\"{o}lder continuous  with exponent  $\alpha\in
]0,1[$ is denoted $C^{m,\alpha} (\operatorname{cl}\Omega )$. 
The subspace of $C^{m}(\operatorname{cl}\Omega ) $ of those functions $f$ 
such that $f_{|{\operatorname{cl}}(\Omega\cap{\mathbb{B}}_{2}(0,R))}\in
C^{m,\alpha}({\operatorname{cl}}(\Omega\cap{\mathbb{B}}_{2}(0,R)))$ 
for all $R\in]0,+\infty[$ is denoted 
$C^{m,\alpha}_{{\mathrm{loc}}}(\operatorname{cl}\Omega ) $.  
Now let $\Omega $ be a bounded
open subset of  ${\mathbb{R}}^{2}$. Then $C^{m}(\operatorname{cl}\Omega)$
and $C^{m,\alpha }({\operatorname{cl}}\Omega)$ are endowed with their 
usual norm and are well known to be
Banach spaces. Similarly, we define the space  
$C^{m}(\operatorname{cl}\Omega, \mathbb{R}^2)$
and $C^{m,\alpha }({\operatorname{cl}}\Omega, \mathbb{R}^2)$ of functions 
from $\operatorname{cl}\Omega$ to $\mathbb{R}^2$ of class $C^m$ and $C^{m,\alpha}$, 
respectively. Then we denote by $C^{m,\alpha }({\operatorname{cl}}
\Omega,\mathcal{O})$ the subset of those functions 
$ f \in C^{m,\alpha }({\operatorname{cl}}
\Omega, \mathbb{R}^2)$ such $f(\operatorname{cl}\Omega)\subseteq \mathcal{O}$.
We say that a bounded open subset $\Omega$ of ${\mathbb{R}}^{2}$ is of class
$C^{m}$ or of class $C^{m,\alpha}$, if ${\operatorname{cl}} \Omega$ is a
manifold with boundary imbedded in
${\mathbb{R}}^{2}$ of class $C^{m}$ or $C^{m,\alpha}$, respectively. 
We define the spaces $C^{k,\alpha}(\partial\Omega)$ for 
$k\in\{0,\dots,m\}$ by exploiting the local parametrizations  
(cf., e.g., Gilbarg and Trudinger \cite[\S 6.2]{GiTr83}).  
For standard properties of functions in Schauder spaces, we refer the 
reader to Gilbarg and
Trudinger~\cite{GiTr83}.
We denote by $\nu_{\Omega}$
the outward unit normal to $\partial\Omega$, by $d\sigma$ the arc length element 
on $\partial\Omega$, and by $|\Omega|_2$ the two-dimensional measure of $\Omega$. 
Finally, for the definition and properties of real analytic operators, 
we refer,  e.g., to Deimling \cite[p.~150]{De85}.

\subsection*{Spaces of bounded and periodic functions}\label{per}

 If $\Omega$ is an arbitrary open subset of ${\mathbb{R}}^{2}$, 
$k\in {\mathbb{N}}$, $\beta\in]0,1]$, we set
\[
C^{k}_{b}({\operatorname{cl}}\Omega)\equiv
\{
u\in C^{k}({\operatorname{cl}}\Omega):
D^{\gamma}u\ {\mathrm{is\ bounded}}\ \forall\gamma\in {\mathbb{N}}^{2}\
{\mathrm{such\ that}}\ |\gamma|\leq k
\}\,,
\]
and we endow $C^{k}_{b}({\operatorname{cl}}\Omega)$ with its usual  norm 
$\|u\|_{ C^{k}_{b}({\operatorname{cl}}\Omega) }\equiv
\sum_{|\gamma|\leq k}\sup_{x\in {\operatorname{cl}}\Omega }|D^{\gamma}u(x)|$  
for all $u\in C^{k}_{b}({\operatorname{cl}}\Omega)$.
Then we set
\[
C^{k,\beta}_{b}({\operatorname{cl}}\Omega)\equiv
\{u\in C^{k,\beta}({\operatorname{cl}}\Omega):
D^{\gamma}u\ {\mathrm{is\ bounded}}\ \forall\gamma\in {\mathbb{N}}^{2}\
{\mathrm{such\ that}}\ |\gamma|\leq k \}\,.
\]
We endow $C^{k,\beta}_{b}({\operatorname{cl}}\Omega)$ with its usual  norm 
$\|u\|_{ C^{k,\beta}_{b}({\operatorname{cl}}\Omega) }\equiv
\sum_{|\gamma|\leq k}\sup_{x\in {\operatorname{cl}}\Omega }|D^{\gamma}u(x)|
+\sum_{|\gamma| = k}|D^{\gamma}u: {\operatorname{cl}}\Omega |_{\beta}
$  for all $u\in C^{k,\beta}_{b}({\operatorname{cl}}\Omega)$, where 
$|D^{\gamma}u: {\operatorname{cl}}\Omega |_{\beta}$ denotes the 
$\beta$-H\"{o}lder constant of $D^{\gamma}u$. Next we turn to periodic domains. 
If $\Omega_Q$ is an arbitrary subset of ${\mathbb{R}}^{2}$  such that
 ${\operatorname{cl}}\Omega_Q\subseteq Q$, we set
\[
{\mathbb{S}} [\Omega_Q]\equiv
\cup_{z\in{\mathbb{Z}}^{2} }(qz+\Omega_Q)=q{\mathbb{Z}}^{2}+\Omega_Q\,,
\quad
{\mathbb{S}} [\Omega_Q]^{-}\equiv {\mathbb{R}}^{2}
\setminus{\operatorname{cl}}{\mathbb{S}} [\Omega_Q]\,.
\]
 Then a function $u$  from ${\operatorname{cl}}{\mathbb{S}}[\Omega_Q]$ 
or from ${\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-}$ to ${\mathbb{R}}$
is $q$-periodic if $u(x+q z)=u(x)$ for all $x$ in the domain of definition 
of $u$ and for all $z \in \mathbb{Z}^2$.
If $\Omega_Q$ is an open subset of ${\mathbb{R}}^{2}$  such that 
${\operatorname{cl}}\Omega_Q\subseteq Q$ and if
$k\in {\mathbb{N}}$ and $\beta\in]0,1[$, then we denote by 
$C^{k}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q] )$, 
$C^{k,\beta}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q] )$, 
$C^{k}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-})$, and 
$C^{k,\beta}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-} )$ 
the subsets of the $q$-periodic functions belonging to 
$C^{k}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, to 
$C^{k,\beta}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, to 
$C^{k}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^- )$, and to
 $C^{k,\beta}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^-)$, 
respectively. We regard the sets 
$C^{k}_{q}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, 
$C^{k,\beta}_{q}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, 
$C^{k}_{q}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^-)$, and 
$C^{k,\beta}_{q}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^-)$ 
as Banach subspaces of the space
$C^{k}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, 
 $C^{k,\beta}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}])$, 
$C^{k}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^-)$,  and 
$C^{k,\beta}_{b}({\operatorname{cl}}{\mathbb{S}}[{\Omega_Q}]^-)$, respectively.

\subsection*{Preliminaries of potential theory}

To investigate problem \eqref{bvpe} by a potential theoretical approach, 
we need to introduce a periodic analogue of the fundamental solution of 
the Laplace operator. As is well known there exists a $q$-periodic tempered 
distribution $S_{q,2}$ such that
\[
\Delta S_{q,2}=\sum_{z\in {\mathbb{Z}}^{2}}\delta_{qz}-1\,,
\]
where $\delta_{qz}$ denotes the Dirac distribution with mass in $qz$.
The distribution $S_{q,2}$ is determined  up to an additive constant, 
and we can take
\begin{equation}\label{eq:Sq2}
S_{q,2}(x)\equiv-\sum_{ z\in {\mathbb{Z}}^{2}\setminus\{0\} }
\frac{1}{     4\pi^{2}|q^{-1}z|^{2}   }e^{2\pi i (q^{-1}z)\cdot x}
\,,
\end{equation}
where the series converges in the sense of distributions on $\mathbb{R}^{2}$ 
(cf., e.g., Hasimoto \cite{Has59}, \cite[Thm.~2.1]{Mu12}).
Then, $S_{q,2}$ is real analytic in 
${\mathbb{R}}^{2}\setminus q{\mathbb{Z}}^{2}$
 and is locally integrable in ${\mathbb{R}}^{2}$. Now, let $S_{2}$ be the function 
from ${\mathbb{R}}^{2}\setminus\{0\}$ to ${\mathbb{R}}$ defined by
\[
S_{2}(x)\equiv  \frac{1}{2 \pi}\log |x| \quad   \forall x\in
{\mathbb{R}}^{2}\setminus\{0\}\, .
\]
$S_{2}$ is well known to be the
fundamental solution of the Laplace operator on the plane. 
Then  $S_{q,2}-S_{2}$ is analytic in 
$({\mathbb{R}}^{2}\setminus q{\mathbb{Z}}^{2})\cup\{0\}$ and we find convenient 
to set
\begin{equation}\label{eq:Rq2}
R_{q,2}\equiv S_{q,2}-S_{2}\quad\text{in }
({\mathbb{R}}^{2}\setminus q{\mathbb{Z}}^{2})\cup\{0\}\,
\end{equation}
(see also \cite{BeMi01}). Moreover, one has
\begin{equation}\label{eq:Rq2par}
R_{q,2}(x)=R_{q,2}(-x) \quad \forall x \in 
({\mathbb{R}}^{2}\setminus q{\mathbb{Z}}^{2})\cup\{0\}\, .
\end{equation}
As a consequence,
\[
D^\beta R_{q,2}(0)=0 \quad \text{for all $\beta \in \mathbb{N}^2$ such 
that $|\beta|$ is odd}\, .
\]
As observed in \cite[p.~336]{MiAd02a}, we note that in the case $Q\equiv ]0,1[^2$ 
(or equivalently $l\equiv1$), we have
\[
S_{q,2}(x_1,x_2)=\frac{1}{2 \pi}\log |\sigma (x_1+i x_2)|
-\frac{1}{4\pi}\left(cx_1^2+(2\pi-c)x_2^2\right) 
\]
for all $x\equiv(x_1,x_2) \in \mathbb{R}^2 \setminus \mathbb{Z}^2$,
where $i$ is the imaginary unit, and the constant $c$ and the Weierstrass 
function $\sigma(\cdot)$ are derived in \cite[Appendix A]{MiAd02a}.

Let $\alpha \in ]0,1[$ and let $\tilde{\Omega}$ be a bounded open subset 
of $\mathbb{R}^2$ of class $C^{1,\alpha}$. 
If $\mu\in C^{1,\alpha}(\partial \tilde{\Omega})$, we  define the classical 
double layer potential by setting
\[
w[\partial\tilde{\Omega},\mu](x)\equiv
-\int_{\partial\tilde{\Omega}}DS_{2}(x-y)\cdot
 \nu_{\tilde{\Omega}}(y)\mu(y)\,d\sigma_{y}
\quad\forall x\in {\mathbb{R}}^{2}\,.
\]
As is well known,
$w[\partial\tilde{\Omega},\mu]_{| \tilde{\Omega}}$ 
admits a continuous extension to ${\operatorname{cl}}\tilde{\Omega}$, 
which we denote by $w^{+}[\partial\tilde{\Omega},\mu]$ and 
$w[\partial\tilde{\Omega},\mu]_{|{\mathbb{R}^2 \setminus 
\operatorname{cl}\tilde{\Omega}}}$ admits a continuous extension to 
$\mathbb{R}^2 \setminus \tilde{\Omega}$, which we denote by 
$w^{-}[\partial\tilde{\Omega},\mu]$. Moreover,  
$w^{+}[\partial\tilde{\Omega},\mu]\in C^{1,\alpha}({\operatorname{cl}}\tilde{\Omega})$
and $w^{-}[\partial\tilde{\Omega},\mu] 
\in C^{1,\alpha}_{\rm loc}(\mathbb{R}^2 \setminus \tilde{\Omega})$
(cf., e.g., Miranda~\cite{Mi65},   Lanza de Cristoforis and Rossi 
\cite[Thm.~3.1]{LaRo04}).

Analogously, we introduce the periodic simple layer potential. 
Let  $\alpha\in]0,1[$ and let $\Omega_Q$ be a bounded open subset of 
${\mathbb{R}}^{2}$ of class $C^{1,\alpha}$ such that 
${\operatorname{cl}}\Omega_Q\subseteq Q$. 
If $\mu\in C^{0,\alpha}(\partial\Omega_Q)$, we set
\[
v_{q}[\partial\Omega_Q,\mu](x)\equiv
\int_{\partial\Omega_Q}S_{q,2}(x-y)\mu(y)\,d\sigma_{y}
\quad\forall x\in {\mathbb{R}}^{2}\,.
\]
The function
$v^{+}_{q}[\partial\Omega_Q,\mu]\equiv v_{q}[\partial\Omega_Q,
\mu]_{|{\operatorname{cl}}{\mathbb{S}}[\Omega_Q]}$ belongs to 
$C^{1,\alpha}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q])$, 
and the function
$v^{-}_{q}[\partial\Omega_Q,\mu]\equiv v_{q}[\partial\Omega_Q,
\mu]_{|{\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-}}$ 
belongs to $C^{1,\alpha}_{q}
({\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-})$. 
Similarly, we introduce the periodic double layer potential. 
If $\mu\in C^{1,\alpha}(\partial\Omega_Q)$, we set
\[
w_{q}[\partial\Omega_Q,\mu](x)\equiv
-\int_{\partial\Omega_Q}DS_{q,2}(x-y)\cdot \nu_{\Omega_Q}(y)\mu(y)\,d\sigma_{y}
\quad\forall x\in {\mathbb{R}}^{2}\,.
\]
The restriction $w_{q}[\partial\Omega_Q,\mu]_{|{\mathbb{S}}[\Omega_Q]}$ 
can be extended uniquely to an element $w^{+}_{ q }[\partial\Omega_Q,\mu]$ 
of $C_{q}^{1,\alpha}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q])$, and the  
restriction $w_{q}[\partial\Omega_Q,\mu]_{|{\mathbb{S}}[\Omega_Q]^{-}}$ 
can be extended uniquely to an element $w_{ q }^{-}[\partial\Omega_Q,\mu]$ 
of $C^{1,\alpha}_{q}({\operatorname{cl}}{\mathbb{S}}[\Omega_Q]^{-})$, and we have
$w^{\pm }_{q}[\partial\Omega_Q,\mu]=\pm\frac{1}{2}\mu+
w_{q}[\partial\Omega_Q,\mu]$ on $\partial \Omega_Q$. Moreover,
\begin{equation}\label{perpot}
w_{q}[\partial\Omega_Q,\mu](x)=-
\sum_{j=1}^{2}\frac{\partial}{\partial x_{j}}
\int_{\partial\Omega_Q}
S_{q,2}(x-y)(\nu_{\Omega_Q}(y))_{j}\mu(y)\,d\sigma_{y}\, ,
\end{equation}
for all  $x\in{\mathbb{R}}^{2}\setminus\partial{\mathbb{S}}[\Omega_Q]$.
 Here $(\nu_{\Omega_Q}(\cdot))_{j}$ denotes the $j$-th component of
 $\nu_{\Omega_Q}(\cdot)$.


\section{Asymptotic behavior of $K_{II}[\epsilon]$}\label{asy}

As a first thing, we face the issue of transforming the boundary 
value problem \eqref{bvpe} for the Poisson equation into a problem 
for the Laplace equation. To do so, we need to introduce a periodic 
function $B_\epsilon$ such that $\Delta B_\epsilon=1$. We introduce such 
a function in the following lemma, whose validity follows immediately 
by \cite[Thm.~2.1]{Mu12}.

\begin{lemma} \label{lem:newt}
Let $\epsilon\in]-\epsilon_0,\epsilon_{0}[ \setminus \{0\}$.
 Let $B_\epsilon$ be the function from $\mathbb{R}^2\setminus (p+q\mathbb{Z}^2)$ 
to ${\mathbb{R}}$ defined by
\[
B_\epsilon(x)\equiv-S_{q,2}(x-p)+\frac{\log |\epsilon|}{2 \pi}
\quad\forall x\in \mathbb{R}^2\setminus (p+q\mathbb{Z}^2)\,.
\]
 Then
\begin{gather*}
{B_{\epsilon}}_{|
{\operatorname{cl}}{\mathbb{S}}[\Omega_{p,\epsilon}]^{-}
}\in  C^{1,\alpha}_{q}(
{\operatorname{cl}}{\mathbb{S}}[\Omega_{p,\epsilon}]^{-})
\,, \quad \Delta B_{\epsilon}= 1 \quad 
\text{in  ${\mathbb{S}}[\Omega_{p,\epsilon}]^{-}$}\,,\\
 B_{\epsilon}(p+\epsilon t)= -S_2(t)- R_{q,2}(\epsilon t)  \quad 
\forall t \in \partial \Omega\, .
\end{gather*}
\end{lemma}

By means of the function $B_\epsilon$ introduced in Lemma \ref{lem:newt}, 
we are in the position to convert the homogeneous Dirichlet problem 
\eqref{bvpe} for the Poisson equation into a non-homogeneous Dirichlet 
problem for the Laplace equation. 
If $\epsilon \in ]-\epsilon_0,\epsilon_0[\setminus \{0\}$, 
we denote by $u_\#[\epsilon]$ the unique solution in 
$C^{1,\alpha}_q(\operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^-)$ 
of the  auxiliary boundary value problem
\begin{equation} \label{auxbvpe}
\begin{gathered}
\Delta u=0 \quad \text{in } {\mathbb{S}}[\Omega_{p,\epsilon}]^{-}\,,
\\
u(x+qz)=u(x)\quad  \forall x \in 
\operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^{-}\, ,
 \forall z \in\mathbb{Z}^2\,,\\
u(x)=-B_\epsilon (x) \quad  \forall x\in  \partial\Omega_{p,\epsilon}\,.
\end{gathered}
\end{equation}
 Clearly,
\[
u[\epsilon]= u_\#[\epsilon]  +B_\epsilon \quad 
\text{in $\operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^-$}\,,
\]
and accordingly
\begin{equation}\label{eq:sumint}
K_{II}[\epsilon]= - \int_{Q \setminus \Omega_{p,\epsilon}}u_\#[\epsilon]\, dx  
-\int_{Q \setminus \Omega_{p,\epsilon}} B_\epsilon\, dx\, .
\end{equation}
Now  for each $\epsilon \in ]-\epsilon_0,\epsilon_0[$, we define the function 
$\Gamma[\epsilon] \in C^{1,\alpha}(\partial \Omega)$, by setting
\begin{equation}\label{eq:Gamma}
\Gamma[\epsilon](t)\equiv  S_2(t)+R_{q,2}(\epsilon t) \quad 
\forall t \in \partial \Omega \, .
\end{equation}
As a consequence, if $\epsilon \in ]-\epsilon_0,\epsilon_0[ \setminus \{0\}$, 
the Dirichlet condition in problem \eqref{auxbvpe} can be rewritten as
\[
u(x)=\Gamma[\epsilon]\bigl((x-p)/\epsilon\bigr)  \quad 
\forall x \in \partial \Omega_{p,\epsilon}\,.
\]
Now we would like to exploit the results of \cite{Mu13} to study the behavior 
of  the integral of $u_\#[\epsilon]$ in the perforated cell 
$Q\setminus \Omega_{p,\epsilon}$. Indeed, in \cite{Mu13} real analytic 
continuation properties for the solution of a Dirichlet problem in 
$\mathbb{S}[\Omega_{p,\epsilon}]^-$ upon $\epsilon$ and the Dirichlet 
datum have been shown. Hence, in order to exploit those results, 
in the following lemma we  verify the analytic dependence of $\Gamma[\epsilon]$ 
upon $\epsilon$.

\begin{lemma}\label{lem:Gamma}
The map $\Gamma$ from $]-\epsilon_0,\epsilon_0[$ to 
$C^{1,\alpha}(\partial \Omega)$, which takes $\epsilon$ to the function of 
the variable $t \in \partial \Omega$ defined by \eqref{eq:Gamma}, 
is real analytic.
\end{lemma}

\begin{proof}
We first note that if $\epsilon \in ]-\epsilon_0,\epsilon_0$[, then 
$\epsilon t \in (\mathbb{R}^2 \setminus q \mathbb{Z}) \cup \{0\}$ for all 
$t \in \operatorname{cl}\Omega$. Clearly, the map from $]-\epsilon_0,\epsilon_0[$ 
to $C^{1,\alpha}(\operatorname{cl}\Omega,(\mathbb{R}^2 \setminus q \mathbb{Z}) 
\cup \{0\} )$ which takes $\epsilon$ to the function $\epsilon t$ of the variable 
$t \in \operatorname{cl}\Omega$ is real analytic. Then by the analyticity 
of $R_{q,2}$ in $(\mathbb{R}^2 \setminus q\mathbb{Z}^2) \cup \{0\}$ 
and by analyticity results for the composition operator (cf., e.g., 
Valent~\cite[Thm.~5.2, p.~44]{Va88}), we deduce that the map from 
$]-\epsilon_0,\epsilon_0[$ to $C^{1,\alpha}(\operatorname{cl}\Omega)$ 
which takes $\epsilon$ to the function $R_{q,2}(\epsilon t)$ of the variable 
$t \in \operatorname{cl}\Omega$ is real analytic. Finally, by the continuity 
of the trace operator from $C^{1,\alpha}(\operatorname{cl}\Omega)$ to 
$C^{1,\alpha}(\partial \Omega)$ we immediately deduce the validity of the lemma. 
\end{proof}


In the following proposition, we provide an integral formulation of the 
auxiliary problem \eqref{auxbvpe}. In order to do so, we need to 
introduce the space 
$C^{1,\alpha}(\partial \Omega)_0\equiv \{\theta \in C^{1,\alpha}
(\partial \Omega) \colon \int_{\partial \Omega}\theta \, d\sigma=0\}$.

\begin{proposition}\label{prop:Lmbd}
Let $\Lambda$ be the map from 
$]-\epsilon_0,\epsilon_0[\times C^{1,\alpha}(\partial \Omega)_0 \times \mathbb{R}$ 
to $C^{1,\alpha}(\partial \Omega)$ defined by
\begin{align*}
\Lambda[\epsilon,\theta,\xi]&\equiv-\frac{1}{2}\theta(t)
 -\int_{\partial\Omega}DS_{2}(t-s) \cdot \nu_{\Omega}(s)\theta(s)\,d\sigma_{s}\\
&\quad -\epsilon \int_{\partial\Omega}DR_{q,2}(\epsilon(t-s)) \cdot 
 \nu_{\Omega}(s)\theta(s)\,d\sigma_{s}+\xi-\Gamma[\epsilon](t) \quad 
\forall t \in \partial \Omega\, ,
\end{align*}
for all $(\epsilon, \theta,\xi) \in ]-\epsilon_0,
\epsilon_0[\times C^{1,\alpha}(\partial \Omega)_0\times \mathbb{R}$. 
Then the following three statements hold:

(i) If $\epsilon \in ]-\epsilon_0,\epsilon_0[$, then there exists a unique 
pair $(\theta,\xi)$ in $C^{1,\alpha}(\partial \Omega)_0 \times \mathbb{R}$ 
such that $\Lambda[\epsilon,\theta,\xi]=0$, and we denote such a pair by 
$(\theta_\epsilon,\xi_\epsilon)$. In particular,
$\theta_0$ is the unique function in $C^{1,\alpha}(\partial \Omega)_0$ such that
\begin{gather}\label{eq:Lmbd1a}
-\frac{1}{2}\theta_0(t)-\int_{\partial\Omega}DS_{2}(t-s) \cdot 
\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s}=S_2(t)-\lim_{s\to \infty}H_0(s) \quad 
\forall t \in \partial \Omega\, ,\\
\label{eq:Lmbd1}
\xi_0 = \lim_{t\to \infty}H_0(t)+R_{q,2}(0)\,,
\end{gather}
where $H_0$  is the unique function in 
$C^{1,\alpha}_{\rm loc}(\mathbb{R}^2\setminus\Omega)$ such that
\begin{gather*}
\Delta H_0(t)=0\quad \forall t \in \mathbb{R}^2\setminus\operatorname{cl}\Omega\,,\\
H_{0}(t)=S_2(t)\quad \forall t\in\partial\Omega\,,\\
\sup_{t\in\mathbb{R}^2\setminus\Omega}|H_0(t)|<+\infty\,.
\end{gather*}
Moreover,
\begin{equation}\label{eq:Lmbd1b}
w^-[\partial \Omega,\theta_0]=H_0-\lim_{t\to \infty}H_0(t) \quad 
\text{in $\mathbb{R}^2 \setminus \Omega$}\, .
\end{equation}

(ii) There exist $\epsilon_1 \in ]0,\epsilon_0[$ and a real analytic 
map $(\Theta,\Xi)$ from $]-\epsilon_1,\epsilon_1[$ to 
$C^{1,\alpha}(\partial \Omega)_0 \times \mathbb{R}$  such that 
$(\Theta[\epsilon],\Xi[\epsilon])=(\theta_\epsilon,\xi_\epsilon)$ for all 
$\epsilon \in ]-\epsilon_1,\epsilon_1[$. Moreover,
\begin{equation}\label{eq:Lmbd1c}
(\Theta[\epsilon],\Xi[\epsilon])=(\Theta[-\epsilon],\Xi[-\epsilon]) \quad 
\forall \epsilon \in ]-\epsilon_1,\epsilon_1[\, .
\end{equation}

(iii) If $\epsilon \in ]-\epsilon_1,\epsilon_1[\setminus \{0\}$, then
\[
u_\#[\epsilon](x)=w_q^-[\partial \Omega_{p,\epsilon},
\Theta[\epsilon]((\cdot-p)/\epsilon)](x)+\Xi[\epsilon] \quad 
\forall x \in \operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^-\, .
\]
As a consequence,
\[
u_\#[\epsilon](x)=-\epsilon \int_{\partial \Omega}DS_{q,2}(x-p-\epsilon s) 
\cdot \nu_{\Omega}(s)\Theta[\epsilon](s)\, d\sigma_s+\Xi[\epsilon] \quad 
\forall x \in \mathbb{S}[\Omega_\epsilon]^{-}\, .
\]
\end{proposition}

\begin{proof}
We first consider statement (i). If $\epsilon \in ]-\epsilon_0,\epsilon_0[$, 
the existence and uniqueness of a pair $(\theta,\xi)$ in 
$C^{1,\alpha}(\partial \Omega)_0 \times \mathbb{R}$ such that 
$\Lambda[\epsilon,\theta,\xi]=0$ follows by \cite[Lem.~3.4, Prop.~A.5]{Mu12}. 
By \cite[Lem.~3.2]{Mu13}, we deduce that
\[
\xi_0=\int_{\partial \Omega}\Gamma[0]\tau_0\, d\sigma\, ,
\]
where $\tau_0$ is the unique solution in $C^{0,\alpha}(\partial \Omega)$ 
of the  problem
\begin{equation}\label{tau0}
 \begin{gathered}
-\frac{1}{2}\tau(t)+\int_{\partial\Omega}DS_{2}(t-s)
\cdot \nu_{\Omega}(t)\tau(s)\,d\sigma_{s}=0 \quad \forall t \in \partial \Omega\, ,\\
\int_{\partial \Omega}\tau \, d\sigma=1\, . 
 \end{gathered}
\end{equation}
Next we note that
\[
\int_{\partial \Omega}\Gamma[0] \tau_0 \, d\sigma
= \int_{\partial \Omega}S_2(t) \tau_0(t) \, d\sigma_t+ R_{q,2}(0) \, .
\]
Then by classical potential theory (cf. \cite[\S 7]{DaMuRo}), one shows that
\[
\int_{\partial \Omega}S_2\tau_0 \, d\sigma = \lim_{t\to \infty}H_0(t)\, ,
\]
and, as a consequence, the validity of \eqref{eq:Lmbd1} follows. 
Then by equality \eqref{eq:Lmbd1} one verifies that $\theta_0$ is the 
unique solution of \eqref{eq:Lmbd1a}. By classical potential theory, 
we deduce that $w^-[\partial \Omega,\theta_0]$ and $H_0-\lim_{t \to \infty}H_0(t)$ 
are harmonic in $\mathbb{R}^2 \setminus \operatorname{cl}\Omega$, 
coincide on $\partial \Omega$, and vanish at infinity. Accordingly, the 
validity of \eqref{eq:Lmbd1b} follows. 

Statement (ii) follows by \cite[Prop.~3.14]{Mu12} and Lemma \ref{lem:Gamma}. 
In particular, equality \eqref{eq:Lmbd1c} follows by \eqref{eq:Rq2par} and 
\eqref{eq:Gamma}. Finally, statement (iii) is a consequence of the theorem of 
change of variables in integrals and of standard results for the periodic 
double layer potential (cf.~\cite[Thm.~2.3, Lem.~3.4, and Rem.~3.15]{Mu12}).
\end{proof}


By Proposition \ref{prop:Lmbd}, we can provide an asymptotic expansion 
of $\Xi[\epsilon]$ for $\epsilon$ close to $0$.

\begin{lemma}\label{lem:Xi}
There exist $\epsilon_2 \in ]0,\epsilon_1[$ and a real analytic function 
$\tilde{\Xi}$ from $]-\epsilon_2,\epsilon_2[$ to $\mathbb{R}$ such that
\[
\begin{split}
\Xi[\epsilon]&=  \lim_{t\to \infty}H_0(t)+R_{q,2}(0) 
+ \epsilon^2\Big(\int_{\partial \Omega}\int_{\partial\Omega} 
 (t-s)^{T}D^2 R_{q,2}(0)\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} 
\tau_0(t)\, d\sigma_t\\&+\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
\frac{D^\beta R_{q,2}(0)}{\beta !} \int_{\partial \Omega}t^\beta\tau_0(t) \, 
d\sigma_t \Big)+ \epsilon^4 \tilde{\Xi}[\epsilon] \quad 
\forall \epsilon \in ]-\epsilon_2,\epsilon_2[ \, ,
\end{split}
\]
where $\tau_0$ is the unique solution in $C^{0,\alpha}(\partial \Omega)$ 
of \eqref{tau0}. Here, if $\beta=(\beta_1,\beta_2) \in \mathbb{N}^2$ and 
$t=(t_1,t_2) \in \mathbb{R}^2$, then $t^\beta \equiv t_1^{\beta_1}t_2^{\beta_2}$.
\end{lemma}

\begin{proof}
By Proposition \ref{prop:Lmbd}, we deduce the existence of 
$\epsilon_2 \in ]0,\epsilon_1[$, $\xi_\ast \in \mathbb{R}$, and of a real 
analytic function $\tilde{\Xi}$ from $]-\epsilon_2,\epsilon_2[$ to 
$\mathbb{R}$ such that 
$\Xi[\epsilon]=\xi_0+\epsilon^2 \xi_\ast +\epsilon^4\tilde{\Xi}[\epsilon]$ 
for all $\epsilon \in ]-\epsilon_2,\epsilon_2[$. As a consequence,
\begin{align*}
&-\frac{1}{2}\Theta[\epsilon](t)-\int_{\partial\Omega}DS_{2}(t-s) 
 \cdot \nu_{\Omega}(s)\Theta[\epsilon](s)\,d\sigma_{s}\\
&-\epsilon \int_{\partial\Omega}DR_{q,2}(\epsilon(t-s)) 
 \cdot \nu_{\Omega}(s)\Theta[\epsilon](s)\,d\sigma_{s}
 + \lim_{t\to \infty}H_0(t)+R_{q,2}(0) +\epsilon^2 \xi_\ast
 + \epsilon^4 \tilde{\Xi}[\epsilon]\\
&=S_2(t)+R_{q,2}(\epsilon t) \quad \forall t \in \partial \Omega\, , 
\forall \epsilon \in ]-\epsilon_2,\epsilon_2[\, .
\end{align*}
By the analyticity of $R_{q,2}$ in 
$(\mathbb{R}^2 \setminus q\mathbb{Z}^2) \cup \{0\}$ and by equality 
\eqref{eq:Rq2par}, we deduce the existence of $\tilde{\epsilon} \in ]0,\epsilon_2[$ 
and of a bounded function $\tilde{R}_1$ from
 $]-\tilde{\epsilon},\tilde{\epsilon}[\times \partial \Omega$ to $\mathbb{R}$ 
such that
\[
R_{q,2}(\epsilon t)=R_{q,2}(0)
+ \epsilon^2 \sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
\frac{D^\beta R_{q,2}(0)}{\beta !}t^\beta+\epsilon^4\tilde{R}_1(\epsilon,t) 
\quad \forall t \in \partial \Omega\, ,\forall \epsilon \in ]
-\tilde{\epsilon},\tilde{\epsilon}[\, .
\]
Similarly, by possibly taking a smaller $\tilde{\epsilon}$, we can assume 
that there exists a bounded function $\tilde{R}_2$ from 
$]-\tilde{\epsilon},\tilde{\epsilon}[\times \partial \Omega \times \partial \Omega$ 
to $\mathbb{R}$ such that
\[
DR_{q,2}(\epsilon (t-s))\cdot \nu_{\Omega}(s)= \epsilon (t-s)^{T}D^2 R_{q,2}
(\epsilon (t-s))\nu_{\Omega}(s)+\epsilon^2\tilde{R}_2(\epsilon,t,s)
\]
for all $t,s \in \partial \Omega$ and all 
$\epsilon \in ]-\tilde{\epsilon},\tilde{\epsilon}[$.
As a consequence
\begin{equation}\label{eq:Xi1}
\begin{split}
&-\frac{1}{2}\Theta[\epsilon](t)-\int_{\partial\Omega}DS_{2}(t-s) 
 \cdot \nu_{\Omega}(s)\Theta[\epsilon](s)\,d\sigma_{s}\\
&-\epsilon^2 \int_{\partial\Omega}  (t-s)^{T}D^2 R_{q,2}(\epsilon (t-s))
 \nu_{\Omega}(s)\Theta[\epsilon](s)\,d\sigma_{s}\\
&-\epsilon^3 \int_{\partial\Omega}  \tilde{R}_{2}(\epsilon,t,s)
 \Theta[\epsilon](s)\,d\sigma_{s}
 + \lim_{t\to \infty}H_0(t)+R_{q,2}(0) +\epsilon^2 \xi_\ast
 +\epsilon^4\tilde{\Xi}[\epsilon]\\
&=S_2(t)+R_{q,2}(0)+\epsilon^2 \sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
\frac{D^\beta R_{q,2}(0)}{\beta !}t^\beta+\epsilon^4\tilde{R}_1(\epsilon,t) 
\end{split}
\end{equation}
for all $t \in \partial \Omega$ and all 
$\epsilon \in ]-\tilde{\epsilon},\tilde{\epsilon}[$.
Now let $\tau_0$ be as in the proof of Proposition \ref{prop:Lmbd}. 
By the classical Fredholm Theory, we have
\[
\int_{\partial \Omega}\Big(-\frac{1}{2}\Theta[\epsilon](t)-
\int_{\partial\Omega}DS_{2}(t-s) \cdot \nu_{\Omega}(s)
\Theta[\epsilon](s)\,d\sigma_{s}\Big) \tau_0(t)\, d\sigma_t=0\, ,
\]
for all $\epsilon \in ]-\tilde{\epsilon},\tilde{\epsilon}[$. Moreover,
\[
\lim_{t\to \infty}H_0(t)+R_{q,2}(0)
=\int_{\partial \Omega}S_2(t)\tau_0(t)\, d\sigma_t+R_{q,2}(0)\, .
\]


Hence, by multiplying \eqref{eq:Xi1} by $\epsilon^{-2} \tau_0(t)$ 
and integrating on $\partial \Omega$ with respect to the variable $t$, we obtain
\begin{align*}
&- \int_{\partial \Omega}\int_{\partial\Omega}  (t-s)^{T}D^2 R_{q,2}
 (\epsilon (t-s))\nu_{\Omega}(s)\Theta[\epsilon](s)\,d\sigma_{s} 
 \tau_0(t)\, d\sigma_t\\ 
&- \epsilon\int_{\partial \Omega}\int_{\partial\Omega}   
 \tilde{R}_{2}(\epsilon,t,s)\Theta[\epsilon](s)\,d\sigma_{s}\tau_0(t)\, d\sigma_t
 +  \xi_\ast+\epsilon^2 \tilde{\Xi}[\epsilon]\\
&=  \sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2} 
 \frac{D^\beta R_{q,2}(0)}{\beta !}\int_{\partial \Omega}  
  t^\beta \tau_0(t)\,  d\sigma_t\\ 
&\quad + \epsilon^2 \int_{\partial \Omega}\tilde{R}_1(\epsilon,t)\tau_0(t)
\, d\sigma_t \quad \forall t \in \partial \Omega\, , \forall \epsilon \in 
]-\tilde{\epsilon},\tilde{\epsilon}[\, .
\end{align*}
Thus, by letting $\epsilon$ tend to $0$, we deduce
\begin{align*}
\xi_\ast 
&=\int_{\partial \Omega}\int_{\partial\Omega}  (t-s)^{T}D^2 R_{q,2}(0)
\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} \tau_0(t)\, d\sigma_t\\
&\quad +\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
 \frac{D^\beta R_{q,2}(0)}{\beta !}\int_{\partial \Omega}  
t^\beta \tau_0(t)\,  d\sigma_t\, .
\end{align*}
Hence, the validity of the statement follows.
\end{proof}


\begin{remark} \label{rmk3.5} \rm
Let  $\tau_0$ be the unique solution in $C^{0,\alpha}(\partial \Omega)$ 
of \eqref{tau0}. By classical potential theory (cf. \cite[\S 7]{DaMuRo}),
 one verifies that
\[
\int_{\partial \Omega}t^\beta\tau_0(t) \, d\sigma_t 
= \lim_{t\to \infty}H_{2,\beta}(t) \quad \forall \beta \in \mathbb{N}^2\, ,
 |\beta|=2 \, ,
\]
where if $\beta \in \mathbb{N}^2$, $|\beta|=2$ the function  $H_{2,\beta}$  
is the unique element of $C^{1,\alpha}_{\rm loc}(\mathbb{R}^2\setminus\Omega)$ 
such that
\begin{gather*}
\Delta H_{2,\beta}(t)=0\quad \forall t \in 
 \mathbb{R}^2\setminus\operatorname{cl}\Omega\,,\\
H_{2,\beta}(t)=t^\beta \quad \forall t\in\partial\Omega\,,\\
\sup_{t\in\mathbb{R}^2\setminus\Omega}|H_{2,\beta}(t)|<+\infty\,.
\end{gather*}
\end{remark}

In the following proposition, we study the behavior of
 $\int_{Q \setminus \Omega_{p,\epsilon}}u_\#[\epsilon]\, dx$.

\begin{proposition}\label{prop:a-intaux}
There exist $\epsilon_3 \in ]0,\epsilon_2[$ and a real analytic function 
$J_{\#,1}$ from $]-\epsilon_3,\epsilon_3[$ to $\mathbb{R}$, such that
\begin{equation}\label{eq:a-int1aux}
\begin{split}
&\int_{Q \setminus \Omega_{p,\epsilon}}u_\#[\epsilon](x)\, dx\\
&=  \lim_{t\to \infty}H_0(t)+R_{q,2}(0) 
 + \epsilon^2 \Big(- |\Omega|_2 \big( \lim_{t\to \infty}H_0(t)+R_{q,2}(0)\big)\\
&\quad +\int_{\partial \Omega}\int_{\partial\Omega}  (t-s)^{T}D^2 R_{q,2}(0)
 \nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} \tau_0(t)\, d\sigma_t\\
&\quad +\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}\frac{D^\beta R_{q,2}(0)}{\beta !}
\int_{\partial \Omega}t^\beta\tau_0(t) \, d\sigma_t\\
&\quad+\sum_{j=1}^2 \int_{\partial \Omega}
\Bigl(\int_{\partial \Omega}S_2(t-s)\theta_0(s)(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)
(\nu_{\Omega}(t))_j\,d\sigma_t\Big)
 + \epsilon^4 J_{\#,1}[\epsilon] 
\end{split}
\end{equation}
for all $\epsilon \in ]-\epsilon_3,\epsilon_3[\setminus \{0\}$,
where $\tau_0$ is the unique solution in $C^{0,\alpha}(\partial \Omega)$ 
of \eqref{tau0}. Here $(\nu_{\Omega}(\cdot))_{j}$ denotes the $j$-th 
component of $\nu_{\Omega}(\cdot)$.
\end{proposition}

\begin{proof}
We proceed as in the proof of \cite[Thm.~4.4]{Mu13}. 
Let $\epsilon \in ]-\epsilon_2,\epsilon_2[\setminus \{0\}$. Clearly,
\begin{align*}
&\int_{Q \setminus \operatorname{cl}\Omega_{p,\epsilon}}u_\#[\epsilon](x)\,dx\\
&=\int_{Q \setminus \operatorname{cl}\Omega_{p,\epsilon}}w^-_q
 \bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]
 ((\cdot-p)/\epsilon)\bigr](x)\,dx\\
&+\big(\lim_{t\to \infty}H_0(t)+R_{q,2}(0)\big)
  \bigl(1-\epsilon^2|\Omega|_2\bigr)\\
& + \epsilon^2 \Big(\int_{\partial \Omega}\int_{\partial\Omega} 
  (t-s)^{T}D^2 R_{q,2}(0)\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} 
 \tau_0(t)\, d\sigma_t\\
&+\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}\frac{D^\beta R_{q,2}(0)}{\beta !} 
 \int_{\partial \Omega}  t^\beta \tau_0(t)\,  d\sigma_t \Big) 
 \bigl(1-\epsilon^2|\Omega|_2\bigr)
 + \epsilon^4 \tilde{\Xi}[\epsilon] \bigl(1-\epsilon^2|\Omega|_2\bigr) \, .
\end{align*}
By  \eqref{perpot}, we have
\begin{align*}
&w^-_q\bigl[\partial \Omega_{p,\epsilon},
 \Theta[\epsilon]((\cdot-p)/\epsilon)\bigr](x)\\
&=-	\sum_{j=1}^2 \frac{\partial}{\partial x_j}v^-_q
\bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]((\cdot-p)/\epsilon)
(\nu_{\Omega_{p,\epsilon}}(\cdot))_j\bigr](x)\quad \forall x 
\in \operatorname{cl} Q \setminus \operatorname{cl}\Omega_{p,\epsilon}\,.
\end{align*}
Now let $j \in \{1,2\}$. By the Divergence Theorem and the periodicity of 
the periodic simple layer potential, we have
\begin{align*}
&\int_{Q \setminus \operatorname{cl}\Omega_{p,\epsilon}}
 \frac{\partial}{\partial x_j}v^-_q
 \bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]
 ((\cdot-p)/\epsilon)(\nu_{\Omega_{p,\epsilon}}(\cdot))_j\bigr](x)\,dx\\
&=\int_{\partial Q}v^-_q\bigl[\partial \Omega_{p,\epsilon},
 \Theta[\epsilon]((\cdot-p)/\epsilon)(\nu_{\Omega_{p,\epsilon}}(\cdot))_j
 \bigr](x)(\nu_{Q}(x))_j \,d\sigma_x\\
&\quad -\int_{\partial \Omega_{p,\epsilon}}v^-_q
 \bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]((\cdot-p)/\epsilon)
 (\nu_{\Omega_{p,\epsilon}}(\cdot))_j\bigr](x)(\nu_{\Omega_{p,\epsilon}}(x))_j 
 \,d\sigma_x\\
&=-\epsilon \int_{\partial \Omega}v^-_q\bigl[\partial \Omega_{p,\epsilon},
 \Theta[\epsilon]((\cdot-p)/\epsilon)(\nu_{\Omega_{p,\epsilon}}(\cdot))_j\bigr]
(p+\epsilon t)(\nu_{\Omega}(t))_j \,d\sigma_t\,.
\end{align*}
Then we note that
\begin{align*}
&v^-_q\bigl[\partial \Omega_{p,\epsilon},
 \Theta[\epsilon]((\cdot-p)/\epsilon)(\nu_{\Omega_{p,\epsilon}}(\cdot))_j
 \bigr](p+\epsilon t)\\
&=\epsilon \int_{\partial \Omega}S_2(\epsilon(t-s))
 \Theta[\epsilon](s)(\nu_{\Omega}(s))_j\,d\sigma_s\\
&\quad +\epsilon \int_{\partial \Omega}R_{q,2}(\epsilon(t-s))
 \Theta[\epsilon](s)(\nu_{\Omega}(s))_j\,d\sigma_s \quad 
\forall t \in \partial \Omega\,.
\end{align*}
We now observe that if $\epsilon \neq 0$ and 
$x \in \mathbb{R}^2 \setminus \{0\}$ then we have
\begin{equation}\label{n:Sneps}
S_{2}(\epsilon x)= S_{2}(x)+\frac{1}{2\pi}\log|\epsilon|\,.
\end{equation}
Moreover, by the Divergence Theorem, we have
\begin{equation}\label{eq:int3aux}
\begin{split}
&\int_{\partial \Omega}\Bigl(\int_{\partial \Omega} 
 \Theta[\epsilon](s)(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)
 (\nu_{\Omega}(t))_j\,d\sigma_t \\
&=\Bigl(\int_{\partial \Omega}\Theta[\epsilon](s)(\nu_{\Omega}(s))_j 
 \,d\sigma_s\Bigr)\Bigl(\int_{\partial \Omega}(\nu_{\Omega}(t))_j\,d
 \sigma_t\Bigr)=0\,.
\end{split}
\end{equation}
Hence, by equalities \eqref{n:Sneps} and \eqref{eq:int3aux}, 
if $\epsilon \in ]-\epsilon_2,\epsilon_2[\setminus \{0\}$, we have
\begin{align*}
&\int_{Q \setminus \operatorname{cl}\Omega_{p,\epsilon}}w^-_q
 \bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]((\cdot-p)/\epsilon)\bigr](x)\,dx\\
&=\sum_{j=1}^2 \epsilon^2\Bigl[\int_{\partial \Omega}
 \Bigl(\int_{\partial \Omega}S_2(t-s)\Theta[\epsilon](s)(\nu_{\Omega}(s))_j 
 \,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t\\
&\quad +\int_{\partial \Omega}\Bigl(\int_{\partial \Omega}R_{q,2}
 (\epsilon(t-s))\Theta[\epsilon](s)(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)
 (\nu_{\Omega}(t))_j\,d\sigma_t\Bigr]\,.
\end{align*}
Thus we set
\begin{align*}
\tilde{J}_{\#,1}[\epsilon]
&\equiv\sum_{j=1}^2 \Bigl[\int_{\partial \Omega}
 \Bigl(\int_{\partial \Omega}S_2(t-s)\Theta[\epsilon](s)(\nu_{\Omega}(s))_j 
 \,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t\\
&\quad +\int_{\partial \Omega}\Bigl(\int_{\partial \Omega}
 R_{q,2}(\epsilon(t-s))\Theta[\epsilon](s)(\nu_{\Omega}(s))_j 
 \,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t\Bigr]\,,
\end{align*}
for all $\epsilon \in ]-\epsilon_2,\epsilon_2[$. 
Then the analyticity of $\Theta$, the continuity of the linear map 
from $C^{0,\alpha}(\partial \Omega)$ to $C^{1,\alpha}(\partial \Omega)$ 
which takes $f$ to the function $\int_{\partial \Omega}S_2(t-s)f(s)\,d\sigma_s$
 of the variable $t \in \partial \Omega$ (cf., e.g., Miranda~\cite{Mi65}, 
Lanza de Cristoforis and Rossi \cite[Thm.~3.1]{LaRo04}), the continuity 
of the pointwise product in Schauder spaces, standard properties of integral  
operators with real analytic kernels and with no singularity 
(cf., e.g., \cite[\S 4]{LaMu13a}), and standard calculus in Banach spaces 
imply that   $\tilde{J}_{\#,1}$ is a real analytic function from 
$]-\epsilon_2,\epsilon_2[$ to $\mathbb{R}$. Moreover, since 
$\int_{\partial \Omega}(\nu_{\Omega}(t))_j\, d\sigma_t=0$, we have
\[
\tilde{J}_{\#,1}[0]=\sum_{j=1}^2 \int_{\partial \Omega}
\Bigl(\int_{\partial \Omega}S_2(t-s)\theta_0(s)(\nu_{\Omega}(s))_j 
\,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t
\]
Furthermore, by \eqref{eq:Rq2par} and by \eqref{eq:Lmbd1c},
 $\tilde{J}_{\#,1}[\epsilon]=\tilde{J}_{\#,1}[-\epsilon]$ for all 
$\epsilon \in ]-\epsilon_2,\epsilon_2[$. As a consequence, there exist 
$\epsilon_3 \in ]0,\epsilon_2[$ and a real analytic function $\tilde{J}_\#$ 
from $]-\epsilon_3,\epsilon_3[$ to $\mathbb{R}$ such that
\begin{align*}
&\int_{Q \setminus \operatorname{cl}\Omega_{p,\epsilon}}w^-_q
\bigl[\partial \Omega_{p,\epsilon},\Theta[\epsilon]((\cdot-p)
 /\epsilon)\bigr](x)\,dx\\&=\epsilon^2\sum_{j=1}^2 
 \int_{\partial \Omega}\Bigl(\int_{\partial \Omega}S_2(t-s)
 \theta_0(s)(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t
 +\epsilon^4 \tilde{J}_\#[\epsilon]
\end{align*}
for all $\epsilon \in ]-\epsilon_3,\epsilon_3[\setminus \{0\}$.
Hence, if we set
\begin{align*}
J_{\#,1}[\epsilon]
&=\tilde{J}_\#[\epsilon] +  \tilde{\Xi}[\epsilon] 
\bigl(1-\epsilon^2|\Omega|_2\bigr) \\
&\quad  - |\Omega|_2 \Big(\int_{\partial \Omega}\int_{\partial\Omega} 
 (t-s)^{T}D^2 R_{q,2}(0)\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} 
\tau_0(t)\, d\sigma_t\\
&\quad +\sum_{\beta \in \mathbb{N}^2\, ,\, |\beta|=2}
 \frac{D^\beta R_{q,2}(0)}{\beta !} \int_{\partial \Omega}  
t^\beta \tau_0(t)\,  d\sigma_t \Big) \, ,
\end{align*}
for all $\epsilon \in ]-\epsilon_3,\epsilon_3[$, we immediately 
deduce that $J_{\#,1}$ is a real analytic function from 
$]-\epsilon_3,\epsilon_3[$ to $\mathbb{R}$ such that equality 
\eqref{eq:a-int1aux} holds, and thus the proof is complete.
\end{proof}

Then we turn to analyze the behavior of 
$\int_{Q \setminus \Omega_{p,\epsilon}} B_\epsilon(x)\, dx$ and we prove 
the following.


\begin{proposition}\label{prop:b-intaux}
There exist $\epsilon_4 \in ]0,\epsilon_0[$ and a real analytic function 
$J_{\#,2}$ from $]-\epsilon_4,\epsilon_4[$ to $\mathbb{R}$, such that
\begin{equation}\label{eq:b-int1aux}
\begin{aligned}
\int_{Q \setminus \Omega_{p,\epsilon}} B_\epsilon(x)\, dx
&=-\int_{Q}S_{q,2}(x) \, dx+\frac{\log |\epsilon|}{2 \pi}\\
&\quad + \epsilon^2 \Big(R_{q,2}(0)|\Omega|_2+  \int_{\Omega}S_{2}(t) \, dt
\Big)+ \epsilon^4 J_{\#,2}[\epsilon]\, ,
\end{aligned}
\end{equation}
for all $\epsilon \in ]-\epsilon_4,\epsilon_4[\setminus \{0\}$. 
\end{proposition}

\begin{proof}
We first note that
\begin{align*}
\int_{Q \setminus \Omega_{p,\epsilon}} B_\epsilon(x)\, dx
&= \int_{Q} B_\epsilon(x)\, dx - \int_{\Omega_{p,\epsilon}} B_\epsilon(x)\, dx\\
&= (1-\epsilon^2|\Omega|_2)\frac{\log |\epsilon|}{2 \pi}
 -\int_{Q}S_{q,2}(x-p) \, dx+\epsilon^2 \int_{\Omega}S_{q,2}(\epsilon t) \, dt\, ,
\end{align*}
for all $\epsilon \in ]-\epsilon_0,\epsilon_0[\setminus \{0\}$. 
Then we note that by the periodicity of $S_{q,2}$ we have 
$\int_{Q}S_{q,2}(x-p) \, dx=\int_{Q}S_{q,2}(x) \, dx$ 
(cf., e.g., Cioranescu and Donato \cite[Lem.~2.3, p.~27]{CiDo99}). Moreover,
\[
\int_{\Omega}S_{q,2}(\epsilon t)\, dt=\int_{\Omega}S_{2}(t) \, dt
 +\frac{|\Omega|_2\log |\epsilon|}{2 \pi}+ \int_{\Omega}R_{q,2}(\epsilon t) 
 \, dt\,  ,
\]
for all $\epsilon \in ]-\epsilon_0,\epsilon_0[\setminus \{0\}$.
 By the analyticity of $R_{q,2}$ in 
$(\mathbb{R}^2 \setminus q\mathbb{Z}^2) \cup \{0\}$ and
 by analyticity results for the composition operator 
(cf., e.g., Valent~\cite[Thm.~5.2, p.~44]{Va88}), the map from 
$]-\epsilon_0,\epsilon_0[$ to $C^0(\operatorname{cl}\Omega)$ 
which takes $\epsilon$ to the function $R_{q,2}(\epsilon t)$ 
of the variable $t\in \operatorname{cl}\Omega$ is real analytic. 
Hence, if we denote by $\tilde{J}$ the function from 
$]-\epsilon_0,\epsilon_0[$ to $\mathbb{R}$ which takes $\epsilon$ to
\[
\tilde{J}[\epsilon]\equiv \int_{\Omega}R_{q,2}(\epsilon  t) \, dt\, ,
\]
by standard calculus in Banach spaces, we deduce that $\tilde{J}$ is 
real analytic. Moreover, by \eqref{eq:Rq2par}, we have 
$\tilde{J}[\epsilon]=\tilde{J}[-\epsilon]$ for all 
$\epsilon \in ]-\epsilon_0,\epsilon_0[$ and
\[
\tilde{J}[0]= R_{q,2}(0)|\Omega|_2\, .
\]
As a consequence, there exist $\epsilon_4 \in ]0,\epsilon_0[$ and a real 
analytic function $J_{\#,2}$ from $]-\epsilon_4,\epsilon_4[$ to $\mathbb{R}$ 
such that
\[
\tilde{J}[\epsilon]= R_{q,2}(0)|\Omega|_2+\epsilon^2 J_{\#,2}[\epsilon] 
\quad \forall \epsilon \in ]-\epsilon_4,\epsilon_4[\, .
\]
 Hence, equality \eqref{eq:b-int1aux} holds and the proof is complete.
\end{proof}

By equality \eqref{eq:sumint} and Propositions \ref{prop:a-intaux} and 
\ref{prop:b-intaux}, we can immediately deduce the validity of our 
main result which concerns the behavior of $K_{II}[\epsilon]$.

\begin{theorem}\label{thm:K}
Let $\epsilon_3$, $H_0$ be as in Proposition \ref{prop:a-intaux}. 
Let $\epsilon_4$ be as in Theorem \ref{prop:b-intaux}. 
Let $\epsilon_\#\equiv \min \{\epsilon_3,\epsilon_4\}$. Let  $\tau_0$ be 
the unique solution in $C^{0,\alpha}(\partial \Omega)$ of \eqref{tau0}. 
Then there exists a real analytic function $J$ from 
$]-\epsilon_\#,\epsilon_\#[$ to $\mathbb{R}$ such that
\begin{equation}\label{eq:K}
\begin{split}
K_{II}[\epsilon]
&=-\frac{\log |\epsilon|}{2 \pi}-\lim_{t\to \infty}H_0(t)-R_{q,2}(0)
 +\int_{Q}S_{q,2}(x)\, dx\\&+ \epsilon^2 \Big( |\Omega|_2  
 \lim_{t\to \infty}H_0(t)- \int_{\Omega}S_{2}(t) \, dt\\
&\quad - \int_{\partial \Omega}\int_{\partial\Omega}  
 (t-s)^{T}D^2 R_{q,2}(0)\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} 
 \tau_0(t)\, d\sigma_t\\&-\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
 \frac{D^\beta R_{q,2}(0)}{\beta !}\int_{\partial \Omega} 
  t^\beta \tau_0(t)\,  d\sigma_t\\&-\sum_{j=1}^2 
 \int_{\partial \Omega}\Bigl(\int_{\partial \Omega}S_2(t-s)\theta_0(s)
(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)(\nu_{\Omega}(t))_j\,d\sigma_t\Big)
 +\epsilon^4 J[\epsilon]
\end{split}
\end{equation}
for all $\epsilon \in ]-\epsilon_\#,\epsilon_\#[\setminus \{0\}$.
\end{theorem}

Now we want to investigate parity properties of the function $J$
 of Theorem \ref{thm:K}. In order to do so, we need the following 
lemma on $K_{II}$.

\begin{lemma}\label{lem:parK}
We have
\[
K_{II}[\epsilon]=K_{II}[-\epsilon] \quad \forall 
\epsilon \in ]-\epsilon_0,\epsilon_0[\setminus \{0\}\, .
\]
\end{lemma}

\begin{proof}
If $\epsilon \in ]-\epsilon_0,\epsilon_0[\setminus \{0\}$ a simple 
computation shows that $u[\epsilon](x)=u[-\epsilon](-x+2p)$ for all 
$x \in \operatorname{cl}\mathbb{S}[\Omega_{p,\epsilon}]^-$. 
Then, by the Theorem of change of variable in integrals, we have
\begin{align*}
-K_{II}[\epsilon]=\int_{Q\setminus \Omega_{p,\epsilon}}u[\epsilon](x)\, dx
&=\int_{Q\setminus \Omega_{p,\epsilon}}u[-\epsilon](-x+2p)\, dx\\
&=\int_{-(Q\setminus \Omega_{p,\epsilon})+2p}u[-\epsilon](x)\, dx\\
&=\int_{(-Q+2p)\setminus \Omega_{p,-\epsilon}}u[-\epsilon](x)\, dx\, .
\end{align*}
Moreover, by the periodicity of $u[-\epsilon]$ we have
\[
\int_{(-Q+2p)\setminus \Omega_{p,-\epsilon}}u[-\epsilon](x)\, dx
=\int_{Q\setminus \Omega_{p,-\epsilon}}u[-\epsilon](x)\, dx=-K_{II}[-\epsilon]
\]
(cf., e.g., Cioranescu and Donato \cite[Lem.~2.3, p.~27]{CiDo99}). 
As a consequence, $K_{II}[\epsilon]=K_{II}[-\epsilon]$.
\end{proof}


By Theorem \ref{thm:K} and Lemma \ref{lem:parK}, we deduce the validity 
of the following statement.

\begin{proposition}\label{prop:J}
Let $\epsilon_5$, $J$ be as in Theorem \ref{thm:K}. Then
\[
J[\epsilon]=J[-\epsilon] \quad \forall \epsilon \in ]-\epsilon_5,\epsilon_5[\, .
\]
\end{proposition}


\section{Lower order coefficients in $K_{II}[\epsilon]$}\label{sec:low}

In this section, we compute via conformal mappings the quantity 
$\lim_{t\to  \infty}H_0(t)$, which is the only $\Omega$-dependent term 
in the zero order expansion of the quantity 
$K_{II}[\epsilon]+ (\log |\epsilon|)/(2\pi)$ 
for $\epsilon$ close to $0$. To do so, we identify $\mathbb{R}^2$ 
with the complex plane $\mathbb{C}$.
Therefore, we suppose that 
$\Omega^-\equiv \mathbb{C} \setminus \operatorname{cl}\Omega$ 
is the exterior domain which is outside the Jordan curve 
$\Gamma\equiv\partial \Omega$. Then, there exists a unique conformal map 
$f_{\Omega^-}$ of $\Omega^-$ onto the set 
$\mathbb{D}^-\equiv \{w \in \mathbb{C}\colon |w|>1\}$ normalized by the conditions
\[
f_{\Omega^-}(\infty)=\infty \, , \quad 
\lim_{z\to \infty} \frac{f_{\Omega^-}(z)}{z}\in ]0,+\infty[\, .
\]
The positive real number
\[
\operatorname{cap}(\Omega)\equiv\Big(\lim_{z\to \infty} 
\frac{f_{\Omega^-}(z)}{z}\Big)^{-1}
\]
is called the \textit{logarithmic capacity} of $\Omega$ and is equal
 to the \textit{transfinite diameter} of $\Omega$ 
(cf., e.g., Goluzin \cite[Ch.~VII,\S 3]{Go69} and Kirsch \cite[\S 2]{Ki05}). 
Then one verifies that the function which takes a point 
$z \in \operatorname{cl}\Omega^-$ to
\[
\frac{1}{2 \pi}\log \frac{|z|}{|f_{\Omega^-}(z)|}
\]
is harmonic in $\Omega^-$, bounded at infinity, and that
\[
\frac{1}{2 \pi}\log \frac{|z|}{|f_{\Omega^-}(z)|}
=\frac{1}{2 \pi}\log |z| \quad \forall z \in \partial \Omega\, .
\]
Thus
\[
H_0(z)=\frac{1}{2 \pi}\log \frac{|z|}{|f_{\Omega^-}(z)|} \quad
 \forall z \in \operatorname{cl}\Omega^-\, .
\]
In particular,
\begin{equation}\label{eq:limH}
\lim_{z \to \infty}H_0(z)=\frac{\log  \operatorname{cap}(\Omega)}{2\pi}\, .
\end{equation}
Application of \eqref{eq:limH} to formula \eqref{eq:K} allows us to deduce that
\[
K_{II}[\epsilon]=-\frac{\log |\epsilon|}{2 \pi}
-\frac{\log  \operatorname{cap}(\Omega)}{2\pi}-R_{q,2}(0)
+\int_{Q}S_{q,2}(x)\, dx+O(\epsilon^2)\, .
\]
which implies
\begin{equation}\label{eq:KIIa}
K_{II}[\epsilon]=-\frac{\log |\epsilon| \operatorname{cap}
(\Omega)}{2 \pi}-R_{q,2}(0)+\int_{Q}S_{q,2}(x)\, dx+O(\epsilon^2)\, .
\end{equation}
Thus, if we set $\rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$, 
then equation \eqref{eq:KIIa} can be rewritten as
\begin{equation}\label{eq:KIIb}
K_{II}[\epsilon]=-\frac{\log \rho}{4 \pi}-R_{q,2}(0)+\int_{Q}S_{q,2}(x)\, dx
+O(\rho)\quad \rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2\, .
\end{equation}


\section{Conclusions}\label{conc}

In the present paper, we prove that the longitudinal permeability of 
regular arrays of cylinders can be expanded as in \eqref{eq:Kser}.
 Moreover, it can be rewritten as a power series of the square of 
the logarithmic capacity of the inclusion 
$ \rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$ 
 plus a logarithmic term. Such quantity is a measure of the size of 
the section of the cylinder inside the unit cell. It is justified 
that the zero-th order term does not depend on the shape of cylinders.

The results can be extended to the case when the domain $\Omega$ is not 
connected. Arbitrary sets of $n$ disks were considered in \cite{MiAd02b}. 
In this case, the number $n$ enters in all the terms of the permeability 
including the logarithm. The dependence of the permeability on locations 
of circular holes can be analyzed following \cite{MiAd02b, MiRy13}.

We also note that by  Theorem \ref{thm:K} and Proposition \ref{prop:J},  
we can deduce the existence of $\epsilon'_\# \in ]0,\epsilon_\#[$ 
and of a sequence $\{b_j\}_{j=0}^{+\infty}$ of real numbers, such that
\begin{equation}\label{eq:conc}
\begin{split}
K_{II}[\epsilon]
&=-\frac{\log |\epsilon|}{2 \pi}-\lim_{t\to \infty}H_0(t)-R_{q,2}(0)
 +\int_{Q}S_{q,2}(x)\, dx \\
&\quad + \epsilon^2 
 \Big( |\Omega|_2  \lim_{t\to \infty}H_0(t)- \int_{\Omega}S_{2}(t) \, dt\\
&\quad  - \int_{\partial \Omega}\int_{\partial\Omega}  
(t-s)^{T}D^2 R_{q,2}(0)\nu_{\Omega}(s)\theta_0(s)\,d\sigma_{s} 
\tau_0(t)\, d\sigma_t\\
&\quad -\sum_{\beta \in \mathbb{N}^2\, ,|\beta|=2}
 \frac{D^\beta R_{q,2}(0)}{\beta !}\int_{\partial \Omega}  t^\beta \tau_0(t)\, 
 d\sigma_t\\
&\quad -\sum_{j=1}^2 \int_{\partial \Omega}\Bigl(\int_{\partial \Omega}
 S_2(t-s)\theta_0(s)(\nu_{\Omega}(s))_j \,d\sigma_s\Bigr)
 (\nu_{\Omega}(t))_j\,d\sigma_t\Big) \\
&\quad +\epsilon^4 \sum_{j=0}^{\infty}b_j \epsilon^{2j}
\quad \forall \epsilon \in ]-\epsilon'_\#,\epsilon'_\#[\setminus \{0\}\,,
\end{split}
\end{equation}
 where the series in the right hand side converges absolutely on 
$]-\epsilon'_\#,\epsilon'_\#[$. Formula \eqref{eq:conc} provides a constructive 
asymptotic expansion valid for  $\Omega$ as in \eqref{dom}. 
To prove rigorously a result of as in \eqref{eq:conc}, it is necessary 
to obtain an analyticity result for $K_{II}[\epsilon]+\frac{\log |\epsilon|}{2 \pi}$, 
which, to the best of our knowledge, is new. We emphasize that such 
an analysis holds for any shape of the cylinders with the only assumption \eqref{dom}.
Therefore we are not confined to specific shapes as, e.g.,  circles or ellipses.

Once an analyticity result of this type is shown, it is of interest to 
compute the coefficients $\{b_j\}_{j=0}^{+\infty}$. In \cite{DaMuRo},
 a completely constructive method has been shown to compute the coefficients 
for the solution of a Dirichlet problem for the Laplace equation in a planar 
domain with a small hole. The computation is based on the solutions of 
systems of integral equations. An approach of this type can be exploited also 
in the case of the present paper and can allow to provide explicit expression 
for \textit{all} the coefficients $b_j$. Once again, this would hold for all 
the domains which satisfy assumption \eqref{dom}. As in \cite{DaMuRo}, 
in the case of circular cylinders, one expects to have simplified expressions 
that recover certainly known formulas. Thus the present paper provides 
the theoretical background for this aim. However, for the sake of brevity,
 we decided not to perform this analysis here and this may be the object of 
future investigations by the authors. Moreover, we plan to investigate the 
dependence of the longitudinal permeability upon perturbations of the shape 
of the cross-section of the cylinders. The integral approach of this paper 
may be used to derive the shape differential and the topological derivative 
of the longitudinal permeability. These quantities can then be used to find 
optimal shapes.

Furthermore, one can see that the constant term in expansion \eqref{eq:KIIb} 
does not depend on the shape of $\Omega$.  Therefore, if $Q\equiv ]0,1[^2$, 
we can take the coefficient for the circular inclusions calculated by formula 
\cite[(41)]{Mi01} and \eqref{eq:KIIb} becomes
\begin{equation}\label{eq:KIIc}
K_{II}[\epsilon]= -\frac{\log \rho}{4 \pi}-\frac{1}{4\pi} 
( \log \pi+1.47644)+O(\rho),
\end{equation}
where $\rho\equiv  (\operatorname{cap}(\epsilon\Omega))^2$ is a measure 
of the concentration of the inclusions.

Formula \eqref{eq:KIIc} presents the asymptotic dependence of the macroscopic 
permeability $K_{II}$ on the shape of $\Omega_{p,\epsilon}\equiv p+\epsilon \Omega$ 
via the square of the logarithmic capacity, i.e., 
$\rho\equiv(\operatorname{cap}(\epsilon\Omega))^2=\epsilon^2 (\operatorname{cap}
(\Omega))^2$.  Previous constructive expansions for $K_{II}$ and for general 
macroscopic constants of porous media and composites (conductivity and elastic 
constants) were made by concentration or by a contrast parameter 
(cf.~Milton \cite{Mi02}, and  \cite{MiAd02a, MiAd02b, MiRy13}). 
Shapes parameters were introduced (see, for instance Landau, Lifshitz, 
and Pitaevskii \cite{LaLiPi82}) to describe the dependence of the macroscopic 
constants on the inclusion. Here we propose to use the logarithmic capacity 
of the cross section of the cylinder to describe the effective permeability. 
In this way, one obtains an approximate expression for the longitudinal 
permeability as a sum of a logarithmic function of the capacity and a term 
which does not depend on the shape of the unit inclusion (plus a small remainder).

\subsection*{Acknowledgements} 
 P.~Musolino was supported by ``Progetto di Ateneo: Singular perturbation 
problems for differential operators -- CPDA120171/12" - 
University of Padova. 
P.~Musolino is member of the Gruppo Nazionale per l'Analisi Matematica, 
la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di 
Alta Matematica (INdAM) and acknowledges the support of  
``INdAM GNAMPA Project 2015 - Un approccio funzionale analitico 
per problemi di perturbazione singolare e di omogeneizzazione". 
The authors wish to thank Prof.~P.M.~Adler for several useful discussions.


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