Small perturbations in the type of boundary conditions for an elliptic operator

E. Bonnetier1, C. Dapogny 2 and M.S. Vogelius 3

1 Institut Fourier, Université Grenoble-Alpes, BP 74, 38402 Saint-Martin-d'Hères Cedex, France.

2 Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.

3 Rutgers University,  Department of Mathematics, New Brunswick, NJ, USA.

In this article, we study the impact of a change in the type of boundary conditions of an elliptic boundary value problem.
In the context of the conductivity equation we consider a reference problem with mixed homogeneous Dirichlet and Neumann boundary conditions.
Two different perturbed versions of this “background” situation are investigated, when
(i) The homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition
on a “small” subset $\omega_\e$ of the Neumann boundary; and when
(ii) The homogeneous Dirichlet boundary condition is replaced by a homogeneous Neumann boundary condition on
a “small” subset $\omega_\e$ of the Dirichlet boundary.
The relevant quantity that measures the “smallness” of the subset $\omega_\e$ differs in the two cases: while it is
the harmonic capacity of $\omega_\e$ in the former case, we introduce a notion of “Neumann capacity” to handle
the latter.
In the first part of this work we derive representation formulas that catch the structure of the first non trivial term
in the asymptotic expansion of the voltage potential, for a general $\omega_\e$, under the sole assumption that it
is “small”  in the appropriate sense.
In the second part, we explicitly calculate the first non trivial term in the asymptotic expansion of the voltage potential, in the
particular geometric situation where the subset $\omega_\e$ is a vanishing surfacic ball.

§ GENERAL SETTING OF THE PROBLEM

Understanding the perturbations in physical fields caused by the presence of small inhomogeneities in a known ambient medium is  crucial for a variety of purposes.
For instance, it allows one to appraise the robustness of the behavior of a body with respect to alterations of its constituent material,
or to reconstruct “small” inclusions with unknown locations, shapes and properties
inside this body; see <cit.> for an overview of such applications.
From the mathematical point of view, this task translates into the asymptotic analysis of the solution $u_\e$ to a “physical” partial differential equation, whose defining domain or material coefficients are perturbed at a small scale, parametrized by the vanishing parameter $\e$.
Many instances of this general question have been investigated: beyond the model setting of the conductivity equation, addressed for instance in <cit.>,
let us mention the studies <cit.> in the context of the linearized elasticity system, or the works <cit.> devoted to the Maxwell equations.

Here we investigate, in the physical context of the conductivity equation, an interesting variant of the aforementioned  problems, namely the variant when the type of the boundary condition is changed on small sets.

Throughout this article, $\Omega \subset \R^d$ is a smooth, bounded domain ($d=2$ or $3$), whose boundary is decomposed as follows

\begin{equation}\label{eq.decpOm}
\partial \Omega = \overline{\Gamma_D} \cup \overline{\Gamma_N}, \text{ where } \Gamma_D, \: \Gamma_N \text{ are disjoint, non empty, open Lipschitz subsets of } \partial \Omega~,
\end{equation}

where we refer to <ref> below for the definition of an open Lipschitz subset of $\partial \Omega$.
The regions $\Gamma_D$ and $\Gamma_N$ correspond to homogeneous Dirichlet, and homogeneous Neumann conditions for the voltage potential, respectively; see <ref> for an illustration of this setting (in the case $d=3$).
The domain $\Omega$ is occupied by a medium with smooth isotropic conductivity $\gamma \in {\mathcal C}^\infty(\overline\Omega)$, satisfying the bounds

\begin{equation}\label{eq.ellconduc}
\forall x \in \Omega, \:\: \alpha \leq \gamma(x) \leq \beta,
\end{equation}

for some fixed constants $0 < \alpha \leq \beta$.
The “background” voltage potential $u_0$, in response to a smooth external source $f \in {\mathcal C}^\infty(\overline\Omega)$, is the unique
$H^1(\Omega)$ solution  to the mixed boundary value problem

\begin{equation}\label{eq.bg}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla u_0) = f & \text{in } \Omega, \\
u_0 = 0 & \text{on } \Gamma_D, \\
\gamma\frac{\partial u_0}{\partial n} = 0 & \text{on } \Gamma_N.
\end{array}
\right.
\end{equation}

We notice that, as a consequence of the classical regularity theory for elliptic partial differential equations,
$u_0$ is smooth except at the interface, $\Sigma$, between $\Gamma_D$ and $\Gamma_N$, where the boundary condition changes type;
see e.g. <cit.>.

In this paper we analyze perturbed versions of <ref>, where the boundary conditions are modified on a “small”, open Lipschitz subset  $\omega_\e$ of the boundary $\partial \Omega$.
More precisely, we are interested in two different situations:

* The case where the homogeneous Neumann boundary condition is replaced by a homogeneous Dirichlet boundary condition
on a “small” open Lipschitz subset $\omega_\e$, lying strictly inside the region $\Gamma_N$. In this situation, the voltage potential $u_\e$ is the unique $H^1(\Omega)$ solution to the boundary value problem

\begin{equation}\label{eq.uepsdir}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla u_\e) = f & \text{in } \Omega, \\
u_\e = 0 & \text{on } \Gamma_D \cup \omega_\e, \\
\gamma\frac{\partial u_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}.
\end{array}
\right.
\end{equation}

* The case where the homogeneous Dirichlet boundary condition on $\Gamma_D$ is replaced by a homogeneous Neumann boundary condition on a “small” open Lipschitz
subset $\omega_\e$, located strictly inside $\Gamma_D$. The voltage potential $u_\e$ is then the unique $H^1(\Omega)$  solution to the boundary value problem

\begin{equation}\label{eq.uepsneu}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla u_\e) = f & \text{in } \Omega, \\
u_\e = 0 & \text{on } \Gamma_D \setminus \overline{\omega_\e}, \\
\gamma\frac{\partial u_\e}{\partial n} = 0 & \text{on } \Gamma_N \cup \omega_\e.
\end{array}
\right.
\end{equation}

In either case, we assume that $\omega_\e$ lies “far” from the transition region $\Sigma$, in the sense that

\begin{equation}\label{assum.far}
\text{There exists a constant } \dmin >0 \text{ such that, for all } \e >0, \:\: \dist(\omega_\e, \Sigma) \geq \dmin.
\end{equation}

The considered setting when the Neumann region $\Gamma_N$ is perturbed by a “small” subset $\omega_\e$ bearing homogeneous Dirichlet boundary conditions.

Such problems show up in multiple physical applications.
The former situation, where homogeneous Neumann boundary conditions are replaced by Dirichlet boundary conditions, is sometimes referred to as the “narrow escape problem” in the literature. Originating from acoustics, it has recently attracted much attention due to its significance in the field of biology.
In this setting indeed, $\Omega$ represents a cavity whose boundary is reflecting except on the small absorbing window $\omega_\e \subset \partial \Omega$.
The particles inside $\Omega$ are guided by a Brownian motion; they may only leave through the region $\omega_\e$ and
the solution $u_\e$ to <ref> then represents their mean exit time.
We refer to <cit.> and the references therein for an overview of the physical relevance of this problem and for an account of recent developments.
In this context, the asymptotic behavior of $u_\e$ in the limit where $\e$ vanishes has been analyzed in <cit.>
by means of formal matched asymptotic expansions; the rigorous proofs of these results were later provided in <cit.> for “simple" sets $\omega_\e$.
Let us also mention the interesting variant of this “narrow escape problem”, tackled in <cit.>,
where the vanishing exit region $\omega_\e$ is connected to a thin, elongated channel, whose presence is modeled through the replacement of homogeneous Neumann conditions by Robin (and not Dirichlet) boundary conditions on $\omega_\e$.

The case <ref> where homogeneous Dirichlet boundary conditions are replaced by homogeneous Neumann boundary conditions
on the vanishing region $\omega_\e \subset \partial \Omega$ seems to have been more rarely considered.
Let us however mention the early investigations conducted in <cit.>,
where the asymptotics of the eigen elements of the Laplace operator are examined as the boundary condition passes from Dirichlet to Neumann type
on a small surfacic ball $\omega_\e \subset \partial \Omega$.
An analogous study is found in <cit.>, where, in three space dimensions,
the small subset $\omega_\e$ is shaped as a thin neighborhood of a curve on $\partial \Omega$.
An interesting physical motivation for this problem
was recently provided in the work <cit.>, devoted to the construction of metasurfaces capable of affecting such changes in boundary conditions. This mechanism was analyzed from the mathematical point of view, and the corresponding asymptotic behavior of $u_\e$ was derived in 2d in <cit.>, under the technical assumption that the boundary of $\Omega$ is completely flat in a neighborhood of the $\omega_\e$;
these results were then used in <cit.> so as to determine the optimal placement of such metasurfaces.

The present paper addresses both situations <ref>: our goal is to understand the asymptotic behavior of $u_\e$ as $\e \to 0$, a limiting regime in which
the small inclusions $\omega_\e$, where boundary conditions are changed, “vanish” in an appropriate sense. As we shall see, the relevant measure of “smallness” for the set $\omega_\e$ depends on which one  of the above situations we are in.
Our investigations go in two complementary directions. In the first part of this paper, we work from a quite abstract point of view,
making minimal assumptions about the inclusion set $\omega_\e$, apart from “smallness”.
We derive the general structure of the lowest order terms in the asymptotic expansion of the perturbation $u_\e-u_0$. In the second part of this paper, we consider a more specific situation
as far as the geometry of the inclusion set $\omega_\e$ is concerned: we assume that $\omega_\e$ is a surfacic ball of radius $\e$ on $\partial \Omega$.
In the two- and three-dimensional instances of <ref>,
we precisely calculate the lowest order terms in the asymptotic expansion of the perturbation  $u_\e-u_0$, thus offering four non trivial examples of our more abstract formulas.
As we shall see, our mathematical treatment of these four cases, based on an integral equation method,
displays some similarities but also important differences. To emphasize both aspects, we shall use the same notation for corresponding quantities, as far as possible.

This paper is organized as follows.
In <ref>, we recall some background material from functional analysis and potential theory,
which is essential for the rest of our investigation.
In <ref> we analyze, from an abstract point of view, the general structure of the (lowest order terms of the) perturbation $u_\e-u_0$, when the homogeneous Neumann boundary condition is replaced with a homogeneous Dirichlet boundary condition on a small subset $\omega_\e \Subset \Gamma_N$.
In <ref> we investigate the case  when the homogeneous Dirichlet boundary condition is replaced with a homogeneous Neumann boundary condition on a small subset $\omega_\e \Subset \Gamma_D$.
<ref> are then devoted to the explicit asymptotic expansion of $u_\e-u_0$ for  both scenarios in the particular case where $\omega_\e$ is a small surfacic ball,
lying in $\Gamma_N$ or $\Gamma_D$, respectively.
In <ref> we outline a few natural ideas for future work, suggested by the present study.
At the end this article are four appendices, collecting several useful results from the litterature, as well as some technical calculations in close connection with the topics discussed in the main parts of the text.

§ PRELIMINARY MATERIAL

We initiate our study by collecting some essential background material.
In <ref>, we outline classical results about fractional Sobolev spaces defined on the boundary of a smooth domain $\Omega$, or on a relatively open Lipschitz subset $\Gamma \subset \partial \Omega$;
in the latter case we emphasize the difference between the spaces $H^s(\Gamma)$ and $\widetilde H^s(\Gamma)$.
In <ref> we summarize the main properties of layer potential operators, and in <ref> we make a few remarks about the construction of fundamental solutions to boundary value problems with variable coefficients.
Finally, in <ref> we introduce and discuss the notion of $H^1$ capacity,
which turns out to be the relevant measure of smallness for sets supporting Dirichlet boundary conditions.

§.§ The Sobolev spaces $H^s(\partial \Omega)$, $H^s(\Gamma)$ and $\widetilde{H}^{s}(\Gamma)$

As is customary in the literature, for an arbitrary integer $n \geq 0$,
$H^n(\partial\Omega)$ stands for the Sobolev space of functions $u  \in L^2(\partial \Omega)$ defined on the boundary of $\Omega$,
whose tangential derivatives up to order $n$ also belong to $L^2(\partial \Omega)$, and the space $H^{-n}(\partial \Omega)$ is the topological dual of $H^n(\partial \Omega)$.

The definition of Sobolev spaces with fractional exponents on the closed hypersurface $\partial \Omega$,
or on an open Lipschitz subset $\Gamma \subset \partial \Omega$ gives rise to some subtleties,
which we briefly describe in this section, referring to <cit.> and <cit.> for more details.

Let us first consider Sobolev spaces of functions attached to the whole boundary $\partial \Omega$. Given a real number $0<s<1$, there are several equivalent ways of defining a norm on the fractional Sobolev Space $H^s(\partial\Omega)$; we use the following definition

\Vert v \Vert^2_{H^s(\partial\Omega)} = \Vert v \Vert_{L^2(\partial\Omega)}^2 + \int_{\partial\Omega}{\int_{\partial\Omega}{\frac{|v(x)-v(y)|^2}{|x-y|^{d-1+2s}}\:\d s(x)}\:\d s(y)}~.

Note that, in the literature the geodesic distance $d^{\partial \Omega}(x,y)$ between two points $x,y \in \partial \Omega$ is often used in place of the Euclidean one $|x-y|$ in the above formula.
However, since $\Omega$ is smooth and compact, the resulting norms are equivalent (with a constant depending on $\Omega$); see <ref>.

When $-1< s \leq 0$, $H^s(\partial \Omega)$ is the topological dual of $H^{-s}(\partial \Omega)$.

We next turn to Sobolev spaces $H^s(\Gamma)$, defined on a proper region $\Gamma \subset \partial \Omega$,
and to this end, we introduce a definition.

An open, connected subset $\Gamma \subset \partial \Omega$ is called a Lipschitz subdomain if locally at its boundary, $\Gamma$ consists of all points located on one side of the graph of a Lipschitz function. A Lipschitz subset $\Gamma \subset \partial \Omega$ is then defined to be the reunion of a finite number of Lipschitz subdomains, the closures of which do not intersect.

Let then $\Gamma \subset \partial \Omega$ be a Lipschitz subset of $\partial \Omega$.
For any real number $0<s<1$ we introduce the following two classes of Sobolev spaces on $\Gamma$:

* $\widetilde{H}^s(\Gamma)$ denotes the space of (restrictions to $\Gamma$ of) functions in $H^s(\partial \Omega)$ with compact support inside $\overline{\Gamma}$. This space is equipped with the norm $|| \cdot ||_{H^{s}(\partial \Omega)}$; it is the closure in $H^s(\partial \Omega)$ of the set of ${\mathcal C}^\infty$ functions on $\partial \Omega$ with compact support inside $\Gamma$. Equivalently, $u$ belongs to $\widetilde H^s(\Gamma)$ if and only if its extension by $0$ to all of $\partial \Omega$,
which we throughout the following still denote by $u$, belongs to $H^s(\partial \Omega)$.

* $H^s(\Gamma)$ is the space of the restrictions to $\Gamma$ of functions in $ H^s(\partial \Omega)$. This space is equipped with the norm:

\begin{equation}\label{eq.normHs}
\Vert v \Vert^2_{H^s(\Gamma)} = \Vert v \Vert_{L^2(\Gamma)}^2 + | v |^2_{H^s(\Gamma)}, \text{ where } | v |^2_{H^s(\Gamma)}:=\int_{\Gamma}{\int_{\Gamma}{\frac{|v(x)-v(y)|^2}{|x-y|^{d-1+2s}}\:\d s(x)}\:\d s(y)}~,
\end{equation}

which is equivalent to the quotient norm induced by that of $H^s(\partial \Omega)$, up to constants that may depend on $\Gamma$.

Let us point out a few facts about the relation between both types of spaces:

* When $0<s<1/2$, the spaces  $\widetilde{H}^s(\Gamma)$ and  $H^s(\Gamma)$ are identical, with equivalent norms.
On the other hand, when $\frac{1}{2} \leq s<1$,  $\widetilde{H}^s(\Gamma)$ is a proper subspace of  $H^s(\Gamma)$.

* When $\frac{1}{2}<s<1$, the space $\widetilde H^s(\Gamma)$ coincides with $H^s_0(\Gamma)$, the closure in $H^s(\Gamma)$ (for the natural norm <ref>) of the set of ${\mathcal C}^\infty$ functions with compact support $K \Subset \Gamma$.

For any real number $-1<s<0$, $H^s(\Gamma)$ is still defined as the space of restrictions to $\Gamma$ of distributions in $H^s(\partial \Omega)$ (equipped with the quotient norm). This space can be identified with the topological
dual of  $\widetilde{H}^{-s}(\Gamma)$, using as a pairing the natural extension of the $L^2(\Gamma)$ inner product, that we denote by:

$$ \langle u ,v \rangle, \:\: u \in H^s(\Gamma), \:\: v \in \widetilde H^{-s}(\Gamma).$$

Similarly, $\widetilde{H}^{s}(\Gamma)$  is the space of distributions in $H^s(\partial \Omega)$
with compact support inside $\overline\Gamma$. It is identified with
the dual space of  $H^{-s}(\Gamma)$, using the same pairing (with the same notation).

The case when $s= 1/2$ is particular: $\widetilde H^{1/2}(\Gamma)$ is a proper subspace of $H^{1/2}(\Gamma)$, with a strictly stronger norm, while the latter space, incidentally, coincides with $H^{1/2}_0(\Gamma)$. To better appraise this distinction between $\widetilde{H}^{1/2}(\Gamma)$ and $H^{1/2}(\Gamma)$, we calculate the norm $\Vert u \Vert_{\widetilde H^{1/2}(\Gamma)}= \Vert u \Vert_{H^{1/2}(\partial \Omega)}$ of an arbitrary function $u \in \widetilde{H}^{1/2}(\Gamma)$:

\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\Vert u \Vert ^2_{\widetilde H^{1/2}(\Gamma)}  &=&  || u||^2_{L^2(\Gamma)} + \int_{\Gamma}{\int_{\Gamma}{\frac{|u(x)-u(y)|^2}{|x-y|^{d}} \:\d s(x)}\:\d s(y)} + 2 \int_{\Gamma}{\rho_{\Gamma}(x) | u(x)|^2 \:\d s(x)} \\
&= &  \Vert u \Vert^2_{H^{1/2}(\Gamma)} + 2 \int_{\Gamma}{\rho_{\Gamma}(x) | u(x)|^2 \:\d s(x)}~.
\end{array}

The weight $\rho_{\Gamma}$ is here defined by

$$\forall x \in \Gamma,\quad \rho_{\Gamma}(x) = \int_{\partial \Omega \setminus \Gamma}{\frac{1}{|x-y|^d}\:\d s(y)}~.$$

The above norm on the space $\widetilde{H}^{1/2}(\Gamma)$ is stronger than that on $H^{1/2}(\Gamma)$, and in particular

\begin{equation}
\label{normcalc}
\left( \int_{\Gamma}{\rho_{\Gamma}(x) | u(x)|^2 \:\d s(x)}\right)^\frac12 \le \frac{1}{\sqrt 2} \Vert u \Vert _{\widetilde H^{1/2}(\Gamma)}~.
\end{equation}

The spaces with exponents $\pm \frac12$ are particularly relevant in the context of variational solutions to  boundary value problems like <ref>.
By a variational  solution to  <ref> we understand a function $u_0$ in the functional space

$$ H^1_{\Gamma_D}(\Omega) := \left\{ u \in H^1(\Omega), \:\: u = 0\text{ on } \Gamma_D \right\}$$

of $H^1(\Omega)$ functions with vanishing trace on $\Gamma_D$ (in other words $u_0|_{\Gamma_N} \in \widetilde H^{1/2}(\Gamma_N)$), and for which

\int_{\Omega}\gamma \nabla u_0 \cdot \nabla v~\d x = \int_\Omega f v~\d x~,

for all $v\in H^1_{\Gamma_D}(\Omega)$ (i.e., $v|_{\Gamma_N} \in \widetilde H^{1/2}(\Gamma_N)$). Using integration by parts, this identity asserts that:

\gamma \frac{\partial u_0}{\partial n} =0 \text{ as an element in } H^{-1/2}(\Gamma_N), \text{ and so } \gamma \frac{\partial u_0}{\partial n} \in \widetilde H^{-1/2}(\Gamma_D)~.

§.§ A short review of layer potentials

In the present section, we denote by $D \subset \R^d$ a smooth bounded domain, and
we briefly recall some background material about layer potential operators associated with $\partial D$;
we refer to <cit.> for more details about such operators.

Let $G(x,y)$ be the fundamental solution of the operator $-\Delta$ in the free space $\R^d$

\begin{equation}\label{eq.GreenLap}
G(x,y) = \left\{
\begin{array}{cl}
-\frac{1}{2\pi} \log|x -y | & \text{if } d= 2~, \\
\frac{1}{4\pi |x-y |} & \text{if } d= 3~.
\end{array}
\right.
\end{equation}

For $x \in \R^d$, the function $y \mapsto G(x,y)$ satisfies

$$ -\Delta_y G(x,y) = \delta_{y=x}~,$$

in the sense of distributions in $\R^d$, where $\delta_{y=x}$ is the Dirac distribution at $x$.

For a smooth density function $\phi \in {\mathcal C}^\infty(\partial D)$, the single layer potential associated with $\phi$ is defined by

\begin{equation}\label{eq.SD}
{\mathcal S}_D\phi(x) = \int_{\partial D}{G(x,y) \phi(y) \: \d s(y)}, \quad x \in \R^d \setminus \partial D,
\end{equation}

and the corresponding double layer potential is defined by

$$ {\mathcal D}_D\phi(x) = \int_{\partial D}{\frac{\partial G}{\partial n_y}(x,y) \phi(y) \: \d s(y)}, \quad x \in \R^d \setminus \partial D.$$

The operators ${\mathcal S}_D$ and ${\mathcal D}_D$ extend to bounded operators
from $H^{-1/2}(\partial D)$ into $H^1_{\text{loc}}(\R^d)$, and from $H^{1/2}(\partial D)$ into $H^1(D) \cup H^1_{\text{loc}}(\R^d \setminus \overline D)$, respectively. In addition, the functions ${\mathcal S}_D \phi$ and ${\mathcal D}_D \phi$ are both harmonic on $D$ and $\R^d \setminus \overline{D}$.
Of particular interest are their behavior at the interface $\partial D$.
Let us denote by

\begin{equation}\label{eq.oslimits}
g^\pm(x) := \lim\limits_{t \downarrow 0} g(x \pm t n(x)), \quad x \in \partial D
\end{equation}

the one-sided limits of a function $g$ which is smooth enough from either side of $\partial D$ and by $[g](x) := g^+(x) - g^-(x)$ the corresponding jump across $\partial D$.
The functions ${\mathcal S}_D \phi$ and ${\mathcal D}_D \phi$ satisfy the well-known jump relations:

\begin{equation}\label{eq.jumpSD}
[{\mathcal S}_D \phi] = 0 \text{ and } \left[ \frac{\partial }{\partial n}({\mathcal S}_D \phi)\right] = -\phi,
\end{equation}

\begin{equation}\label{eq.jumpDD}
[{\mathcal D}_D \phi] = \phi \text{ and } \left[ \frac{\partial }{\partial n}({\mathcal D}_D \phi)\right] = 0.
\end{equation}

The first and the last of these four jump relations allow to introduce the integral operators $S_D$ and $R_D$, defined for a smooth density function $\phi \in {\mathcal C}^\infty(\partial D)$ by:

$$ S_D \phi = ({\mathcal S}_D \phi) \lvert_{\partial D}, \quad S_D \phi(x)  = \int_{\partial D}{G(x,y) \phi(y) \: \d s(y)}~,~~~ x \in \partial D, $$

$$ R_D \phi =\frac{\partial }{\partial n} \left( {\mathcal D}_D \phi  \right), \quad R_D \phi(x) =  \underset{\eta \downarrow 0}{\text{\rm f.p.} }\int_{\partial D \setminus B_\eta(x) }{\frac{\partial ^2 G}{\partial n_x \partial n_y}(x,y) \phi(y) \: \d s(y)}~, ~~~ x\in \partial D, $$

where f.p. refers to a finite part integral in the sense of Hadamard.
These operators extend as bounded mappings $S_D : H^{-1/2}(\partial D) \to H^{1/2}(\partial D)$ and $R_D : H^{1/2}(\partial D) \to H^{-1/2}(\partial D)$.

Lastly, we recall the decay properties of the single and double layer potentials at infinity.
For a given density $\phi \in H^{-1/2}(\partial D)$, it follows from the explicit expression <ref> of the fundamental solution $G(x,y)$ that, for $d= 3$

\begin{equation}\label{eq.decayS3d}
{\mathcal S}_D \phi(x) = {\mathcal O}\left( \frac{1}{|x|}\right), \text{ and } \lvert \nabla {\mathcal S}_D \phi(x)\lvert = {\mathcal O}\left( \frac{1}{|x|^{2}}\right),
\end{equation}

where we have used the convenient notation ${\mathcal O}\left( \frac{1}{|x|^{2}}\right)$ to represent a function whose modulus is bounded by $\frac{C}{|x|}$ when $|x|$ is large enough,
for some constant $C>0$.

The case $d= 2$ is a little more subtle, and in general one only has

$$ {\mathcal S}_D \phi(x) = {\mathcal O}\left(|\log|x| |\right), \text{ and } \lvert \nabla {\mathcal S}_D \phi(x)\lvert = {\mathcal O}\left( \frac{1}{|x|}\right)~,$$

however, if $\int_{\partial D}{\phi \: \d s} = 0$, then it holds additionally

\begin{equation} \label{eq.decaySDmean0}
{\mathcal S}_D \phi(x) = {\mathcal O}\left( \frac{1}{|x|} \right), \text{ and } \lvert \nabla {\mathcal S}_D \phi(x)\lvert = {\mathcal O}\left( \frac{1}{|x|^2}\right)~.
\end{equation}

As far as the double layer potential is concerned, one has for $\phi \in H^{1/2}(\partial D)$ and $d= 2, 3$

\begin{equation}\label{eq.decayDD}
{\mathcal D}_D \phi(x) = {\mathcal O}\left( \frac{1}{|x|^{d-1}}\right), \text{ and } \lvert \nabla {\mathcal D}_D \phi(x)\lvert = {\mathcal O}\left( \frac{1}{|x|^{d}}\right)~.
\end{equation}

§.§ The fundamental solution $N(x,y)$ to the background equation <ref>

We turn our attention to the case when the reference problem under consideration is not the free-space Laplace equation, but rather the background boundary value problem <ref>.
The fundamental solution $N(x,y)$ to the latter is constructed from that $G(x,y)$ associated to the operator $-\Delta$ in the free space, given by <ref>, in a way which we now briefly describe. We refer, e.g., to <cit.> or <cit.> for similar results.

For any point $x \in \Omega$, the function $y \mapsto N(x,y)$ satisfies the following equation

\begin{equation}\label{fundsol}
\left\{
\begin{array}{cl}
-\dv_y(\gamma(y) \nabla_y N(x,y)) = \delta_{y=x} &\text{in } \Omega~,\\
N(x,y) = 0 & \text{for } y \in \Gamma_D~, \\
\gamma(y) \frac{\partial N}{\partial n_y}(x,y) = 0 & \text{for } y \in \Gamma_N~.
\end{array}
\right.
\end{equation}

This means that, for any function $\varphi \in {\mathcal C}^1(\overline\Omega)$ such that $\varphi = 0 $ on $\Gamma_D$, one has

$$ \varphi(x) = \int_\Omega{\gamma(y) \nabla_y N(x,y) \cdot \nabla \varphi (y) \: \d y }, \quad x \in \Omega.$$

By an easy adaptation of the proof of Lemma 2.36 in <cit.>, one sees that the function $N(x,y)$ is symmetric in its arguments. Furthermore, it is related to the fundamental solution $G(x,y)$ to the Laplace equation in free space via the relation

$$ N(x,y) = \frac{1}{\gamma(x)}G(x,y) + R(x,y)~,$$

where for given $x \in \Omega$, $y \mapsto R(x,y)$ is the solution to the equation

\left\{
\begin{array}{cl}
-\dv_y(\gamma(y) \nabla_y R(x,y)) =  \frac{1}{\gamma(x)}  \nabla \gamma(y) \cdot \nabla_y G(x,y)&\text{in } \Omega~,\\
R(x,y) = - \frac{1}{\gamma(x)} G(x,y) & \text{for } y \in \Gamma_D~, \\
\gamma(y) \frac{\partial R}{\partial n_y}(x,y) = -\frac{\gamma(y)}{\gamma(x)} \frac{\partial G}{\partial n_y}(x,y) & \text{for } y \in \Gamma_N~.
\end{array}
\right.

The precise functional characterization of $R(x,y)$ follows from standard elliptic regularity theory,
depending on the singularity of $G(x,y)$, see <cit.>.
Without entering into technicalities, let us just mention that, for fixed $x \in \Omega$,
the function $y \mapsto R(x,y)$ belongs (at least) to $H^1(\Omega)$.
Moreover, for every open subset $U \Subset \R^d \setminus (\Sigma \cup \{x\})$, it is of class ${\mathcal C}^\infty$ on $\overline \Omega \cap U$.

§.§ The capacity of a subset in $\R^d$

In one of the two scenarios studied in this article, namely when $\omega_\e$ accounts for Dirichlet boundary conditions being imposed inside the Neumann region $\Gamma_N$ (cf.  <ref>), the key quantity to measure the “smallness” of the set $\omega_\e$ will be the $H^1(\R^d)$ capacity.
For the convenience of the reader, we briefly recall the definition
and two simple results related to this notion, referring to <cit.> for further details.

The capacity $\capa(E)$ of an arbitrary subset $E \subset \R^d$ is defined by:

\begin{equation}\label{eq.cap}
\capa (E ) = \inf \left\{ || v ||^2_{H^1(\R^d)}, \:\: v(x) \geq 1 \text{ a.e. on an open neighborhood of } E  \right\}.
\end{equation}

A slightly different  formula for $\capa(E)$ is that of the following lemma.

For an arbitrary subset $E \subset \R^d$, it holds

\begin{equation}\label{eq.cap1}
\capa (E ) = \inf \left\{ || v ||^2_{H^1(\R^d)}, \:\: v(x) = 1 \text{ a.e. on an open neighborhood of } E  \right\}.
\end{equation}

On the one hand, it follows immediately from the definition <ref> that

$$\capa (E ) \leq \inf \left\{ || v ||^2_{H^1(\R^d)}, \:\: v(x) = 1 \text{ a.e. on an open neighborhood of } E  \right\}~.$$

Conversely, if $v_n \in H^1(\R^d)$ is a sequence of functions such that

$$ v_n \geq 1 \text{ a.e. on an open neighborhood of } E, \:\: || v_n ||^2_{H^1(\R^d)} \xrightarrow{n \to \infty} \capa(E)~,$$

then $w_n = \min(v_n,1)$ defines a sequence of functions in $H^1(\R^d)$ which satisfies

$$ || w_n ||_{H^1(\R^d) }^2 \leq  || v_n ||_{H^1(\R^d) }^2~;$$

see for instance <cit.>, Proposition 3.1.11. As a result,

$$ \inf \left\{ || v ||^2_{H^1(\R^d)}, \:\: v(x) = 1 \text{ a.e. on an open neighborhood of } E  \right\} \leq || w_n ||^2_{H^1(\R^d)} \leq || v_n ||^2_{H^1(\R^d)} \xrightarrow{n \to \infty} \capa(E)~,$$

which proves <ref>.

We now provide a useful lemma, whereby
the capacity of a subset $\omega$ of the boundary $\partial \Omega$ of a smooth domain $\Omega \subset \R^d$ can be estimated in terms of the energy norm of a function whose trace equals $1$ on $\omega$.

Let $\Omega$ be a smooth bounded domain in $\R^d$, $\omega$ be a Lipschitz open subset of $\partial\Omega$, and let $u$ be a function in $H^1(\R^d)$.
If $u=1$ on $\omega$ in the sense of traces in $H^{1/2}(\omega)$, then

$$ \capa(\omega) \leq || u ||^2_{H^1(\R^d)}.$$

Let $\calU$ be an open neighborhood of $\omega$ in $\R^d$, and let $\phi$ be a $C^{\infty}_c(\R^d)$ function which equals $1$ identically on $\calU$.
We decompose the function $u$ as:

u= \phi + u-\phi ~.

Since the set of functions $v \in H^1(\R^d)$ with vanishing trace on $\omega$ is exactly the closure of $C^{\infty}_c(\R^d\setminus \overline \omega)$ in $H^1(\R^d)$ (see <ref>), and since the trace of $u-\phi$ equals $0$ on $\omega$, there exists a sequence $v_n \in C^{\infty}_c(\R^d\setminus \overline \omega)$ such that

\Vert u -\phi -v_n  \Vert_{H^1(\R^d)} \le \frac{1}{n}~.

We now estimate

\begin{eqnarray*}
\Vert u \Vert_{H^1(\R^d)}&=& \Vert \phi + v_n +  u - \phi - v_n  \Vert_{H^1(\R^d)} \\
&\ge& \Vert  \phi +v_n \Vert_{H^1(\R^d)}- \Vert u - \phi -v_n \Vert_{H^1(\R^d)} \\
&\ge&\Vert  \phi +v_n \Vert_{H^1(\R^d)} -\frac{1}{n}~.
\end{eqnarray*}

The function $ \phi +v_n$ lies in $H^1(\R^d)$ (actually it lies in $C_c^\infty(\R^d)$) and it equals $1$ on an open neighborhood of $\omega$ in $\R^d$. From the definition of $ \capa(\omega)$ it follows that

\Vert  \phi +v_n \Vert_{H^1(\R^d)} \ge\left( \capa(\omega)\right)^{1/2} ~,

and so by combination with the previous estimate we get:

\Vert u \Vert_{H^1(\R^d)}\ge \left(\capa(\omega)\right)^{1/2} -\frac{1}{n}~.

By passing to the limit as $n \rightarrow \infty$ we arrive at the desired conclusion.

From the physical point of view, the capacity of a compact subset $E \subset \R^d$
is the total energy of the electric field in the whole ambient space $\R^d$, in the equilibrium regime where the potential is constant on $E$ (equal to $1$).
Different notions of capacity are found in the literature, depending on the kernel relating the charge distribution (i.e., the source term) to the induced potential.
A very natural notion of capacity is attached to the fundamental solution to the Laplace operator with homogeneous Dirichlet boundary conditions on a “ground surface” $A$ (where the potential is set to $0$); this concept is often associated with the name condenser capacity.
The version proposed in <ref> is convenient for our purpose, since it is somehow “universal” (it does not depend on the choice of a fixed grounding subset $A$ for the potential), and it  involves the Bessel kernel.
It is equivalent to the notion of condenser capacity, up to constants depending on the subset $A$; see for instance  <ref> for a result in this direction.
We refer to <cit.> for an extensive discussion of the concept of capacity; see also <cit.>.

In <ref> we shall be particularly interested in subsets of $\R^d$ of the form

\begin{equation}\label{eq.defDecapa}
\D_\e := \left\{ x = (x_1,\ldots,x_{d-1},0 ) \in \R^d, \:\: | x | < \e \right\}~.
\end{equation}

$\D_\e$ is a line segment of length $2\e$, when $d=2$, and a planar disk with radius $\e$, when $d=3$.
The following estimates of the capacity of $\D_\e$ will come in handy

\begin{equation}\label{eq.capaDe}
\capa(\D_\e) \leq \frac{C_2}{|\log \e|} \text{ when } d= 2, \quad \capa(\D_\e) \leq C_3\e \text{ when } d= 3,
\end{equation}

where $C_2$ and $C_3$ are universal constants; see for instance Chap. 2 in <cit.>.

§ REPLACING NEUMANN CONDITIONS BY DIRICHLET CONDITIONS ON A “SMALL SET"

In this section, we consider an arbitrary sequence $\omega_\e$ of open, Lipschitz subsets of the Neumann region $\Gamma_N$,
which are all “well-separated” from the Dirichlet region $\Gamma_D$ in the sense that the assumption <ref> holds.
The homogeneous Neumann boundary condition
satisfied on $\Gamma_N$ by the background potential $u_0$ (see <ref>)
is dropped on $\omega_\e$,
where it is replaced by a homogeneous Dirichlet condition. The perturbed potential $u_\e$ in this situation is the solution to the equation <ref>.

As we shall see, the potential $u_\e$ converges to $u_0$ as $\e \to 0$, when the set $\omega_\e$ vanishes in an appropriate sense.
In this general setting, where no additional hypothesis is made about $\omega_\e$,
our aim is to establish an abstract representation formula
for the first non-trivial term in the limiting asymptotics of $u_\e-u_0$.

§.§ Some preliminary estimates

We start with some a priori estimates related to modified versions of the perturbed boundary value problem <ref>.
The first of these results
is concerned with the unique solution $\chi_\e \in H^1(\Omega)$ to the problem

\begin{equation}\label{eq.chieps}
\left\{
\begin{array}{cl}
-\Delta \chi_\e = 0 & \text{in } \Omega~, \\
\chi_\e = 1 & \text{on } \omega_\e~, \\
\chi_\e = 0 & \text{on } \Gamma_D~, \\
\frac{\partial \chi_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}~.
\end{array}
\right.
\end{equation}

Let us recall that, by a solution to <ref> we understand a function $\chi_\e \in H^1_{\Gamma_D}(\Omega)$ such that $\chi_\e = 1$ on $\omega_\e$ and

\begin{equation}\label{eq.varfchiepsdir}
\forall v \in H^1_{\Gamma_D}(\Omega) \text{ with } v = 0 \text{ on } \omega_\e, \quad \int_\Omega{\nabla \chi_\e\cdot \nabla v \: \d x} = 0~.
\end{equation}

Let $\omega_\e$ be an open Lipschitz subset of the region $\Gamma_N \subset \partial \Omega$,
which lies “far” from $\Gamma_D$ in the sense that <ref> holds,
and let $\chi_\e$ be the solution to <ref>.
There exist two constants $ 0 < m \leq M$ which depend only on $\Omega$, $\Gamma_D$ and the lower bound $\dmin$ on the distance from $\omega_\e$ to $\Gamma_D$, but are otherwise independent of $\omega_\e$,  such that

\begin{equation}\label{eq.capas}
m \: \capa(\omega_\e) \leq || \chi_\e ||_{H^1(\Omega)}^2 \leq M \capa(\omega_\e)~.
\end{equation}

We start with the proof of the inequality $ m \:  \capa(\omega_\e) \leq || \chi_\e ||_{H^1(\Omega)}^2$.
Because of the smoothness of  $\Omega$, there exists an extension $\widetilde{\chi_\e} \in H^1(\R^d)$ to the whole space $\R^d$
such that

\begin{equation}\label{eq.poincarechiA}
|| \widetilde{\chi_\e} ||_{H^1(\R^d)} \leq C || \chi_\e||_{H^1(\Omega)}~,
\end{equation}

where the constant $C$ depends only on $\Omega$ (see e.g. Appendix A in <cit.>).
Using <ref> we may estimate the capacity of the subset $\omega_\e \subset \partial \Omega$, where the trace of $\widetilde{\chi_\e}$ equals $1$, by

$$ \capa(\omega_\e)  \leq  ||\widetilde{\chi_\e}||_{H^1(\R^d)}^2~.$$

This inequality, combined with <ref>, yields the desired result.

We now prove that $  || \chi_\e ||_{H^1(\Omega)}^2 \leq M \capa(\omega_\e)$. Let us first observe that,
due to a classical variation of the Poincaré inequality, there exists a constant $C >0$ which depends only on $\Omega$ and $\Gamma_D$, such that

\begin{equation}\label{eq.Poincarecapa}
|| \chi_\e ||_{H^1(\Omega)}^2 \leq C || \nabla \chi_\e ||^2_{L^2(\Omega)^d}~.
\end{equation}

Because of the separation assumption <ref>,
there exists a function $h \in C^1(\overline \Omega)$ such that, for any $\e >0$ small enough, one has

$$h(x)=1\text{ for } x \in \omega_\e, \:\: h(x)=0 \text{ for } x \in  \Gamma_D, \text{ and }\Vert h \Vert_{C^1(\overline \Omega)} \le C~,$$

where $C$ depends on $\dmin$, but is otherwise independent of $\omega_\e$.
For any function $\chi \in H^1(\R^d)$ such that $\chi =1$ on an open neighborhood of $\omega_\e$, we now have

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| \nabla \chi_\e ||^2_{L^2(\Omega)^d}  &=& \int_{\Omega}{\nabla \chi_\e \cdot \nabla \chi_\e \: \d x} \\[2ex]
%& =&  \int_{\partial \Omega}{\gamma \frac{\partial \chi_\e}{\partial n} \chi_\e \: \d s},  \\[2ex]
%&=& C \int_{\partial \Omega}{\gamma \frac{\partial \chi_\e}{\partial n} \chi h \: \d s},\\[2ex]
& =&  \int_{\Omega}{ \nabla \chi_\e \cdot \nabla (\chi h)\: \d x}\\[2ex]
&\leq& C || \nabla \chi_\e||_{L^2(\Omega)^d }|| \chi||_{H^1(\R^d) }~.
\end{array}

Here we have used the fact that $\chi_\e -\chi h$ vanishes on $\Gamma_D \cup \omega_\e$, together with the variational formulation <ref>,
to pass from the first line to the second. We immediately conclude that

|| \nabla \chi_\e ||^2_{L^2(\Omega)^d}  \leq C || \chi||^2_{H^1(\R^d) }~.

Since this holds for any function $\chi \in H^1(\R^d)$ which equals identically $1$ on an open neighborhood of $\omega_\e$,
the desired upper bound for $|| \chi_\e ||_{H^1(\Omega)}^2$  follows by taking the infimum over all such functions $\chi \in H^1(\R^d)$
and using the formula <ref> for the capacity, as well as the Poincaré inequality <ref>.

The second result in this section is concerned with solutions to a slight generalization of <ref>, where the prescribed Dirichlet data on $\omega_\e$ is given by a function $g$ (and not constantly equal to $1$) and the conductivity $1$ is replaced by $\gamma$. More precisely, we now consider the unique solution $v_\e \in H^1(\Omega)$ to the boundary value problem:

\begin{equation}\label{eq.veps}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla v_\e) = 0 & \text{in } \Omega, \\
v_\e = g & \text{on } \omega_\e, \\
v_\e = 0 & \text{on } \Gamma_D, \\
\gamma\frac{\partial v_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}~,
\end{array}
\right.
\end{equation}

where $g$ is a given function in $C^1(\overline\Omega)$. By a solution to <ref> we understand a function $v_\e \in H^1_{\Gamma_D}(\Omega)$
such that $v_\e = g$ on $\omega_\e$ and

\begin{equation}\label{eq.varfvedirth}
\forall v \in H^1_{\Gamma_D}(\Omega) \text{ with } v = 0 \text{ on } \omega_\e, \quad  \int_\Omega \gamma \nabla v_\e \cdot \nabla v \: \d x=0~.
\end{equation}

Suppose $d=2$ or $d=3$. Let $\omega_\e$ be an open Lipschitz subset of the region $\Gamma_N \subset \partial \Omega$,
satisfying <ref>, and let $v_\e \in H^1(\Omega)$ be the solution to <ref>.
There exists a constant $M$ which depends only on $\alpha$, $\beta$, the ellipticity constants of $\gamma$, $\Omega$, $\Gamma_D$ and $\dmin$,
but is otherwise independent of $\omega_\e$, such that

\begin{equation}\label{eq.H1estve}
|| v_\e ||_{H^1(\Omega)} \leq M || g ||_{{\mathcal C}^1(\overline \Omega)}  \capa(\omega_\e)^{\frac{1}{2}}~.
\end{equation}

In addition, $v_\e$ satisfies the following improved $L^2$ estimate

\begin{equation}\label{eq.L2estve}
|| v_\e ||_{L^2(\Omega)}  \leq M ||g ||_{{\mathcal C}^1(\overline \Omega)} \capa(\omega_\e)^{\frac{3}{4}}~.
\end{equation}

We first prove <ref>. Notice that, due to (a modified version of) the Poincaré inequality, it suffices to show that the term $|| \nabla v_\e ||_{L^2(\Omega)^d}^2$ satisfies the desired upper bound. To this end we introduce the solution $\chi_\e \in H^1(\Omega)$ to <ref>.
Since $v_\e - g \chi_\e = 0$ on $\omega_\e\cup \Gamma_D$, the variational formulation <ref> yields

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| \nabla v_\e ||_{L^2(\Omega)^d}^2 &\leq& C \int_{\Omega}{\gamma \nabla v_\e \cdot \nabla v_\e \: \d x} \\[2ex]
%&=& C \int_{\partial \Omega}{\gamma \frac{\partial v _\e}{\partial n} v_\e \: \d s} \\[2ex]
%&=&  C \int_{\partial \Omega}{\gamma \frac{\partial v _\e}{\partial n} g \chi_\e  \: \d s} \\[2ex]
&=&C \int_{\Omega}{\gamma \nabla v_\e \cdot \nabla (g \chi_\e )\: \d x} \\[2ex]
&\leq&  C \Vert g \chi_\e  \Vert_{H^{1}(\Omega)} \Vert  v _\e \Vert_{H^{1}( \Omega)} \\[2ex]
&\leq&C \Vert g \Vert_{C^1(\overline \Omega)} \Vert \chi_\e \Vert_{H^1(\Omega)}  \Vert  v_\e \Vert_{H^1(\Omega)}~.
\end{array}

Using the upper bound for $\Vert \chi_\e \Vert_{H^1(\Omega)}$ supplied by <ref>, and the Poincaré inequality for $v_\e$ we conclude that

|| \nabla v_\e ||_{L^2(\Omega)^d} \leq M \Vert g \Vert_{C^1(\overline \Omega)} \capa(\omega_\e)^{\frac{1}{2}} ~,

which is the desired estimate <ref>.

We proceed to prove  <ref>. To this end we rely on a variant of the “classical” Aubin-Nitsche trick <cit.>.
Let $w_\e$ denote the unique solution in  $H^1_{\Gamma_D}(\Omega)$  to the boundary value problem

\left\{\begin{array}{ccl}
-\dv(\gamma\nabla w_\e) = v_\e & \text{in } \Omega~, \\
w_\e = 0 & \text{on } \Gamma_D~, \\
\gamma\frac{\partial w_\e}{\partial n} = 0 & \text{on } \Gamma_N~,
\end{array}
\right.

or rather, in its variational form:

$$w_\e \in H^1_{\Gamma_D}(\Omega), \text{ and } \int_{\Omega}{\gamma \nabla w_\e \cdot \nabla v\: \d x} = \int_\Omega{v_\e v \: \d x}~~ \hbox{ for all } v \in H^1_{\Gamma_D}(\Omega)~.$$

Since <ref> holds, there exists a cut-off function $\eta \in {\mathcal C}^\infty_c(\R^d)$ with the property

$$\eta = 1 \text{ on a neighborhood of all the } \omega_\e  \text{ and } \eta = 0 \text{ on an open set } \calU \text{ of } \R^d \text{ with } \Gamma_D \Subset \calU~.$$

The key ingredient of the following derivation is that $w_\e$ shows improved regularity with respect to $v_\e$ (away from the interface between $\Gamma_D$ and $\Gamma_N$).
In particular, standard interior elliptic regularity results, discussed e.g. in <cit.>, give

$$ || \eta w_\e ||_{H^3(\Omega)} \leq C || v_\e ||_{H^1(\Omega)}~.$$

In addition, since $d=2$ or $3$, the classical Sobolev Embedding Theorem ensures that

H^{3}(\Omega) \subset C^{1}(\overline \Omega) ~~\text{ and for all } v \in H^3(\Omega), \:\: \Vert v \Vert_{C^{1}(\overline \Omega)} \le C \Vert v \Vert_{H^3(\Omega)}~,

see e.g. <cit.>.
It follows immediately from this and the previous regularity estimate that

\begin{equation}
\label{ellregwe}
\Vert  \eta w_\e  \Vert_{C^{1}(\overline \Omega)} \le C  || \eta w_\e ||_{H^3(\Omega)} \leq C || v_\e ||_{H^1(\Omega)}~.
\end{equation}

We now calculate

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\int_\Omega{v_\e ^2 \: \d x} &=&  \int_\Omega{\gamma \nabla w_\e \cdot \nabla v_\e \:\d x} \\[0.75em]
%&=&  \int_{\partial\Omega}{\gamma \frac{\partial v_\e}{\partial n} w_\e \:\d s}, \\
%&=&  \int_{\partial\Omega}{\gamma \frac{\partial v_\e}{\partial n} \chi_\e \eta w_\e \:\d s},
&=& \int_\Omega{\gamma \nabla (\chi_\e \eta w_\e) \cdot \nabla v_\e \:\d x},
\end{array}$$

where we have introduced the solution $\chi_\e  \in H^1(\Omega)$ to <ref>, as well as the fixed cut-off function $\eta$ from above. We have also used that $w_\e- \chi_\e \eta w_\e=0$ on $\omega_\e \cup \Gamma_D$ and the variational formulation <ref>.
It now follows that

\begin{equation}\label{eq.estvelem2prel}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| v_\e ||_{L^2(\Omega)}^2 &\leq& C ||  v_\e ||_{H^1(\Omega)}  || \chi_\e ||_{H^1(\Omega)} || \eta w_\e||_{C^{1}(\overline \Omega)} \\
&\leq & C   ||  v_\e ||_{H^1(\Omega)}^2 || \chi_\e ||_{H^1(\Omega)}~,
\end{array}
\end{equation}

where we have employed <ref> for the last inequality.
Finally, using the estimate

|| \chi_\e ||_{H^1(\Omega)} \leq M \capa(\omega_\e)^{\frac12}

from <ref>, together with the already established $H^1$ estimate <ref> for $v_\e$, it follows from <ref> that

$$|| v_\e ||_{L^2(\Omega)}^2  \leq C ||g ||_{{\mathcal C}^1(\overline \Omega)}^2 \capa(\omega_\e)^{\frac{3}{2}}~, $$

as desired.

We observe that the conclusions of <ref>, and their proofs, extend verbatim
to the case where the scalar conductivity $\gamma$ is replaced by a smooth conductivity matrix $A \in {\mathcal C}^\infty(\overline \Omega)^{d\times d}$ satisfying the bounds

\begin{equation}\label{eq.anisomat}
\forall \xi \in \R^d, \quad \alpha |\xi |^2  \leq A(x) \xi\cdot \xi \leq \beta |\xi|^2, \quad x \in \overline \Omega.
\end{equation}

More precisely, the $H^1$ and $L^2$ estimates <ref> still hold true when $v_\e$ is the solution to the following anisotropic counterpart of <ref>

\begin{array}{cl}
-\dv(A \nabla v_\e) = 0 & \text{in } \Omega, \\
v_\e = g & \text{on } \omega_\e, \\
v_\e = 0 & \text{on } \Gamma_D, \\
(A\nabla v_\e)\cdot n= 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}~.
\end{array}
\right.

§.§ The representation formula

The deviation $r_\e := u_\e - u_0$ between the perturbed potential and the background potential is the unique $H^1(\Omega)$ solution

\begin{equation}\label{eq.reps}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla r_\e) = 0 & \text{in } \Omega~, \\
r_\e = -u_0 & \text{on } \omega_\e~, \\
r_\e = 0 & \text{on } \Gamma_D~, \\
\gamma\frac{\partial r_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}~.
\end{array}
\right.
\end{equation}

Because of our separation assumption <ref>, there exists a smooth compact subset $K \Subset \Gamma_N$ such that
$\omega_\e \subset K$ for all $\e$.
Owing to local elliptic regularity estimates for the background problem <ref>,  we have

\Vert u_0\Vert_{C^1(K)} \le C \Vert f \Vert_{H^m(\Omega)}~,

for a sufficiently large integer $m$ (again, see e.g. <cit.>).
Hence, we may construct a $C^1$ function $g_0$ on all of  $\overline \Omega$ with the properties that

g_0 = -u_0 \text{ on } K~~, \text{ and } \Vert g_0 \Vert_{C^1(\overline \Omega)} \le C \Vert u_0\Vert_{C^1(K)} \le C \Vert f \Vert_{H^m(\Omega)}~.

With this notation, $r_\e $ is the unique $H^1(\Omega)$ solution to

\left\{
\begin{array}{cl}
-\dv(\gamma \nabla r_\e) = 0 & \text{in } \Omega~, \\
r_\e = g_0 & \text{on } \omega_\e~, \\
r_\e = 0 & \text{on } \Gamma_D~, \\
\gamma\frac{\partial r_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e}~.
\end{array}
\right.

As a straightforward consequence of <ref>, it follows that

\begin{equation}
\label{reh1est}
|| r_\e ||_{H^1(\Omega)} \leq C || f ||_{H^m(\Omega)}  \capa(\omega_\e)^{1/2}~,
\end{equation}

and we now search for the next term in the asymptotic expansion of $u_\e$. Our main result is the following.

Suppose $d=2$ or $d=3$, and suppose  $\omega_\e$ is a sequence of non-empty, open Lipschitz subsets of $\partial \Omega$, which are all contained in $\Gamma_N$ and well-separated from $\Gamma_D$ in the sense that <ref> holds. Let $u_\e$ denote the solution to <ref>.
Assume that  the capacity $\capa(\omega_\e)$ of $\omega_\e$ tends to $0$ as $\e \rightarrow 0$.
Then there exists a subsequence, still labeled by $\e$, and a non-trivial distribution $\mu$ in the dual space of ${\mathcal C}^1(\partial \Omega)$, such that
for any fixed point $x \in \Omega$, and any $\eta \in C^\infty(\partial \Omega)$ with $\eta =1$ on $\{ y \in \partial \Omega,~\dist(y,\Gamma_D)>\dmin/2\}$ and
$\eta =0$ on $\{ y \in \partial \Omega,~\dist(y,\Gamma_D)<\dmin/3\}$, it holds

\begin{equation}\label{eq.1storderdir}
u_\e(x) = u_0(x) - \capa(\omega_\e) \mu_y\left[\eta(y) \gamma(y) u_0(y) N(x,y)\right] + \o(\capa(\omega_\e))~, ~~\hbox{ as } \e \to 0~.
\end{equation}

The term $\o(\capa(\omega_\e))$ goes to zero faster than $\capa(\omega_\e)$ uniformly for $x \in K$, where $K$ is any compact subset of $\Omega$. The distribution $\mu$ depends only on the subsequence $\omega_\e$, $\Omega$, and $\Gamma_N$.

Introducing the fundamental solution $N(x,y)$ of the background operator defined in <ref>, we obtain for any $x\in \Omega$

$$ \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
r_\e(x) &=& \int_{\Omega}{r_\e(y) (-\dv_y(\gamma(y) \nabla_y N(x,y))) \: \d y} \\
&=& \int_{\Omega}{\gamma(y) \nabla r_\e(y) \cdot \nabla_y N(x,y) \: \d y} - \int_{\partial \Omega}{\gamma(y) \frac{\partial N}{\partial n_y}(x,y) r_\e(y) \: \d s(y)} \\
&=& \int_{\partial\Omega}{\gamma(y) \frac{\partial r_\e}{\partial n}(y) N(x,y) \: \d s(y)} - \int_{\partial \Omega}{\gamma(y) \frac{\partial N}{\partial n_y}(x,y) r_\e(y) \: \d s(y)}~.
\end{array}$$

Since $y \mapsto \gamma(y) \frac{\partial N}{\partial n_y}(x,y)$ vanishes on $\Gamma_N$ (i.e. as an element in $H^{-1/2}(\Gamma_N)$) and $r_\e$ vanishes on $\Gamma_D$ (i.e., $r_\e \in \widetilde{H}^{1/2}(\Gamma_N)$), the second integral in the above right-hand side equals $0$, and so

\begin{equation}\label{eq.rerepN}
r_\e(x) =  \int_{\partial \Omega}{\gamma(y) \frac{\partial r_\e}{\partial n} (y)N(x,y) \: \d s(y)}~.
\end{equation}

To proceed, we now use the same “compensated compactness”, or “clever integration by parts” technique as in <cit.>, see also <cit.>.
Let $\phi \in {\mathcal C}^1(\partial \Omega)$ be an arbitrary function, which vanishes on the set $\{ y \in \partial \Omega~:~\dist(y,\Gamma_D)<\dmin/3\}$. Since $\Omega$ is smooth, it is easy to  construct a function $\psi \in {\mathcal C}^1(\overline{\Omega})$ such that

\begin{equation}\label{eq.extphidir}
\psi = \phi \text{ on } \partial \Omega, \text{ and } || \psi ||_{{\mathcal C}^1(\overline{\Omega})} \leq C || \phi ||_{{\mathcal C}^1(\partial \Omega)}~,
\end{equation}

where the constant $C$ depends only on $\Omega$.
As before, let $\chi_\e$ denote the solution to <ref>. Since the function $(\phi- \chi_\e \psi)$ belongs to $H^{1/2}(\partial \Omega)$, and vanishes on $\Gamma_D \cup \omega_\e$, we have

$$  \int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n} \phi \: \d s}  =  \int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n}  \chi_\e \psi \: \d s}~.$$

An integration by parts now yields

\begin{equation}\label{eq.repdir1erest}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n} \phi \: \d s} & =&  \int_\Omega{\gamma \nabla r_\e \cdot \nabla (\chi_\e \psi) \: \d y}\\
&=&  \int_\Omega{\psi \gamma \nabla r_\e \cdot \nabla \chi_\e \: \d y} +  \int_\Omega{\chi_\e \gamma \nabla r_\e \cdot \nabla \psi  \: \d y}~.
\end{array}
\end{equation}

Using <ref> and the estimate <ref> applied to $\chi_\e$, we may control the second term in the above right-hand as follows

\begin{equation}\label{eq.ctr2termrepdir}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\left\lvert  \int_\Omega{\chi_\e \gamma \nabla r_\e \cdot \nabla \psi  \: \d y} \right\lvert &\leq& C || \chi_\e ||_{L^2(\Omega)} || r_\e||_{H^1(\Omega)}  || \psi ||_{{\mathcal C}^1(\overline{\Omega})} \\
& \leq &  C\capa(\omega_\e )^{\frac54} \Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~.
\end{array}
\end{equation}

A similar argument makes it possible to rewrite the first term in the right-hand side of <ref> as

\begin{eqnarray*}
\int_\Omega{\psi \gamma \nabla r_\e \cdot \nabla \chi_\e \: \d y} &=& \int_\Omega{ \nabla (\gamma \psi  r_\e) \cdot \nabla \chi_\e \: \d y} - \int_\Omega{  r_\e \nabla (\gamma \psi ) \cdot \nabla \chi_\e \: \d y} \\
&=&\int_\Omega{\nabla (\gamma \psi  r_\e) \cdot \nabla \chi_\e \: \d y} +\O(\capa(\omega_\e )^{\frac54}) \Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~.
\end{eqnarray*}

Inserting these two facts into <ref> we get

\int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n} \phi \: \d s} =  \int_\Omega{ \nabla (\gamma \psi r_\e) \cdot \nabla \chi_\e \: \d y}  \: + \:  \O(\capa(\omega_\e )^{\frac54}) \Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~,

and so, after another integration by parts

\int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n} \phi \: \d s} =  \int_{\partial \Omega}{ \frac{\partial \chi_\e}{\partial n} \gamma \phi r_\e \: \d s} \: + \: \O(\capa(\omega_\e )^{\frac54}) \Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~.

Since $\frac{\partial \chi_\e}{\partial n} = 0$ on $\Gamma_N \setminus \overline \omega_\e$,
and since $r_\e = -u_0 \chi_\e$ on $\omega_\e \cup \Gamma_D$, we may replace $r_\e$ with $-u_0 \chi_\e$ in the integral of the above right-hand side, thus obtaining

\begin{equation}
\label{eq.reapresipp}
\int_{\partial\Omega}{\gamma \frac{\partial r_\e}{\partial n} \phi \: \d s} =  -\int_{\partial \Omega}{ \frac{\partial \chi_\e}{\partial n} \chi_\e u_0 \gamma \phi \: \d s} \: + \: \O(\capa(\omega_\e )^{\frac54}) \Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~.
\end{equation}

Now let $\eta$ be a function as introduced in the statement of this theorem:

$$\eta \in C^\infty(\partial \Omega), \:\:  \eta =1 \text{ on } \{ y \in \partial \Omega,~\dist(y,\Gamma_D)>\dmin/2\} \text{ and }
\eta =0 \text{ on } \{ y \in \partial \Omega,~\dist(y,\Gamma_D)<\dmin/3\}.$$

Then, $N(x,\cdot) \eta(\cdot)$ is a $C^\infty$ function on $\partial \Omega$ which vanishes on the set $\{ y \in \partial \Omega~:~\dist(y,\Gamma_D)<\dmin/3\}$ and coincides with  $N(x,\cdot)$ on $\omega_\e \cup \Gamma_D$.
By a combination of <ref> and  <ref>, with $\phi(\cdot) =N(x,\cdot) \eta(\cdot)$, it follows that

\begin{eqnarray}
\label{finaleq}
r_\e(x) &=&\int_{\partial \Omega}{\gamma(y) \frac{\partial r_\e}{\partial n} (y)N(x,y) \: \d s(y)} \nonumber \\
&=& \int_{\partial \Omega}{\gamma(y) \frac{\partial r_\e}{\partial n} (y)N(x,y)\eta(y) \: \d s(y)} \\
&=&  -\int_{\partial \Omega} { \frac{\partial \chi_\e}{\partial n}(y) \chi_\e(y) u_0(y)\gamma(y)  N(x,y) \eta(y) \: \d s(y)}+  {\mathcal O}(\capa(\omega_\e )^{\frac54}) \Vert f\Vert_{H^m(\Omega)} \Vert N(x,\cdot)\eta(\cdot) \Vert_{C^1(\partial \Omega)}~ . \nonumber
\end{eqnarray}

Finally, the upper bound in <ref> reveals that $\capa(\omega_\e)>0$ (since the $\omega_\e$ are non-empty), and that  for any function $\phi \in {\mathcal C}^1(\partial \Omega)$

$$ \left\lvert \frac{1}{\capa(\omega_\e)}\int_{\partial\Omega}{\frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s}  \right\lvert = \left\lvert \frac{1}{\capa(\omega_\e)} \int_{\Omega}{\nabla \chi_\e \cdot \nabla ( \chi_\e \phi ) \: \d y}  \right\lvert \leq  C || \phi ||_{{\mathcal C}^1(\partial\Omega)}~.$$

It follows from the Banach-Alaoglu theorem that, up to extraction of a subsequence, which we still label by $\e$,
there exists a bounded linear functional $\mu$ on ${\mathcal C}^1(\partial \Omega)$ such that, for any $\phi \in {\mathcal C}^1(\partial\Omega)$:

$$ \frac{1}{\capa(\omega_\e)} \int_{\partial\Omega}{\frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s} \xrightarrow{\e \to 0} \mu(\phi)~.$$

The lower bound in <ref>, in combination with Poincaré's inequality, reveals that $\mu(1)>0$, in other words that $\mu$ is non-trivial.
A combination of <ref> and the above convergence result (with $\phi(\cdot) = u_0(\cdot) \gamma(\cdot) N(x,\cdot) \eta(\cdot )$) yields the desired representation formula

$$r_\e(x) = - \capa(\omega_\e) \mu_y\left[\eta(y) u_0(y)\gamma(y) N(x,y)\right] + \o(\capa(\omega_\e))~.$$

The uniformity of the convergence of the remainder term, when the point $x$ is confined to a fixed compact subset $K \Subset\Omega$, follows from the fact that the set of functions $\{ u_0(\cdot) \gamma(\cdot) N(x,\cdot) \eta(\cdot )\}_{x \in K} \subset {\mathcal C}^1(\partial \Omega)$ is compact in the ${\mathcal C}^1$ topology.

§.§ Properties of the limiting distribution $\mu$

The limiting distribution $\mu$ introduced in  <ref>
is a priori a distribution of order one on $\partial \Omega$, and as such it may depend on first-order derivatives of the argument function $\phi$.
We now show that this is not the case, and that $\mu$ is actually a non negative Radon measure on $\partial\Omega$.

The distribution $\mu$ in <ref> is a non-trivial, non negative Radon measure on $\partial \Omega$.
Moreover, the support of $\mu$ is contained in any compact subset $K \subset \partial \Omega$
such that $\omega_\e \subset K$ for $\e >0$ small enough.

We recall from the proof of <ref> that the distribution $\mu$ is defined by:

$$ \forall \phi \in {\mathcal C}^1(\partial \Omega), \:\: \mu(\phi) = \lim\limits_{\e \to 0}{\frac{1}{\capa(\omega_\e)}} \int_{\partial \Omega}{ \frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s}~,$$

where the limit is taken along a subsequence, and $\chi_\e \in H^1(\Omega)$ is the solution to the equation <ref>. Let $\phi$ be an arbitrary function $\phi \in {\mathcal C}^1(\partial \Omega)$. Since $\partial \Omega$ is smooth, it is easy to construct a function $\widetilde \psi \in {\mathcal C}^1(\overline \Omega)$
such that

\begin{equation}
\label{psidef}
\widetilde \psi = \phi \text{ on } \partial \Omega, \text{ and } || \widetilde \psi||_{C^0(\overline \Omega)} = || \phi ||_{C^0(\partial \Omega)}~.
\end{equation}

Green's formula then yields

\int_{\partial\Omega}{\frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s} = \int_\Omega{( \nabla \chi_\e \cdot \nabla \chi_\e )\widetilde \psi \: \d x} + \int_\Omega{(\nabla \chi_\e \cdot \nabla \widetilde \psi) \chi_\e \:\d x}~.

As in the proof of <ref> (see <ref>), the estimates of <ref> show that

\lim\limits_{\e \to 0}  \frac{1}{\capa(\omega_\e)} \int_\Omega{(\nabla \chi_\e \cdot \nabla \widetilde \psi) \chi_\e \:\d x} =0~,

and as a consequence

\begin{equation}
\label{mupart}
\mu(\phi) =\lim\limits_{\e \to 0} \frac{1}{\capa(\omega_\e)} \int_\Omega{(\nabla \chi_\e \cdot \nabla \chi_\e) \widetilde \psi \:\d x}~,
\end{equation}

for any function $\phi \in C^1(\partial \Omega)$, where $\widetilde \psi \in C^1(\overline \Omega)$ is related to $\phi$ by <ref>. On the other hand, using <ref>, there exists a constant $C >0$ such that

\begin{equation}\label{eq.boundedBAdir}
\forall \psi \in {\mathcal C}^0(\overline \Omega), \:\: \frac{1}{\capa(\omega_\e)} \left\lvert \int_\Omega ( \nabla \chi_\e \cdot \nabla \chi_\e) \psi \: \d x \right\lvert \leq C || \psi ||_{{\mathcal C}^0(\overline\Omega)}.
\end{equation}

Hence, using again the Banach-Alaoglu theorem, there exists a subsequence of the $\e$'s and a non negative Radon measure $\nu$ on $\overline\Omega$ such that

$$ \forall \psi \in {\mathcal C^0}(\overline\Omega), \:\: \frac{1}{\capa(\omega_\e)} \int_\Omega{( \nabla \chi_\e \cdot \nabla \chi_\e) \psi \: \d x} \rightarrow \int_\Omega{\psi \:\d \nu}~.$$

Combining this with <ref> we conclude that:

$$\mu(\phi) = \int_\Omega{\widetilde \psi \:\d \nu}$$

for any $\phi \in C^1(\partial \Omega)$, where $\widetilde \psi \in C^1(\overline\Omega)$ is related to $\phi$ by <ref>. Moreover,

\left| \int_\Omega{\widetilde \psi \:\d \nu} \right| \le C \Vert \widetilde \psi \Vert_{C^0(\overline \Omega)} = C || \phi ||_{C^0(\partial \Omega)},

and we have thus proved that, for any $\phi \in {\mathcal C}^1(\partial \Omega)$

|\mu(\phi)| \le C || \phi ||_{C^0(\partial \Omega)}~.

This shows that $\mu$ is a Radon measure on $\partial \Omega$, the non negativity of which follows from that of $\nu$.
Moreover, the proof of <ref> has already revealed that $\mu$ is non trivial since $\mu(1) > 0$.

Finally, let $K \Subset \partial \Omega$ be a compact subset of $\partial \Omega$ such that $\omega_\e \subset K$ for $\e >0$ small enough. Let
$\phi\in C^1(\partial \Omega)$ be an arbitrary function with support in the relatively open subset $U := \partial \Omega \setminus K$.
Then, $\chi_\e \phi$ belongs to $H^{1/2}(\partial \Omega)$ and vanishes on $\omega_\e \cup \Gamma_D$, so that

\int_{\partial\Omega}{\frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s}=0~.

It follows that

\mu(\phi) = \lim\limits_{\e \to 0}  \frac{1}{\capa(\omega_\e)} \int_{\partial\Omega}{\frac{\partial \chi_\e}{\partial n} \chi_\e \phi \: \d s} =0~.

Since this holds true for any $\phi\in C^1(\partial \Omega)$ with support in $U$, the desired result about the support of $\mu$ follows.

<ref> immediately leads to the following Corollary to <ref>.

Suppose $d=2$ or $d=3$. Let $\omega_\e$ be a sequence of non-empty, open Lipschitz subsets of $\partial \Omega$, which are all contained in $\Gamma_N$ and are well-separated from $\Gamma_D$ in the sense that <ref> holds. Let $u_\e$ denote the solution to <ref>.
Assume furthermore that  the capacity $\capa(\omega_\e)$ of $\omega_\e$ goes to $0$ as $\e \rightarrow 0$.
Then there exists a subsequence, still denoted by $\e$, and a non-trivial, non negative Radon measure $\mu$ on $\partial \Omega$, such that for any fixed point $x \in \Omega$

u_\e(x) = u_0(x) -\capa(\omega_\e) \int_{\partial \Omega}{u_0(y) \gamma(y) N(x,y) \:\d \mu(y)}  + \o(\capa(\omega_\e)).

The measure $\mu$ depends only on the subsequence $\omega_\e$, $\Omega$, and $\Gamma_N$.
The support of $\mu$ lies inside any compact subset $K \subset \partial \Omega$ containing the $\omega_\e$ for $\e >0$ small enough, and the term $\o(\capa(\omega_\e))$ goes to zero faster than $\capa(\omega_\e)$ uniformly (in $x$) on compact subsets of $\Omega$.

Let us comment about the physical meaning of the representation formula of <ref>.

* The first order term in this expansion arises as the superposition of the potentials $u_0(y) \gamma(y) N(x,y)$ created at $x$ by point sources (monopoles)
which are distributed on the “limiting location” of the vanishing subsets $\omega_\e$. The negative sign in front of this term indicates that these point sources have been replaced by a “ground” (homogeneous Dirichlet boundary condition) when passing from the background physical situation to the perturbed one.

* Assuming for simplicity that $f$ has compact support inside $\Omega$, the fact that the term $\o(\capa(\omega_\e))$ (in <ref>)  is uniformly small on compact subsets of $\Omega$ leads to the following asymptotic expansion for the compliance (or power consumption) of $\Omega$:

\int_{\Omega}f \, u_\e \: \d x = \int_{\Omega} f \, u_0 \: \d x -\capa(\omega_\e) \int_{\Omega} f(x) \int_{\partial \Omega}{\gamma(y) u_0(y) N(x,y) \:\d \mu(y)}\: \d x  + \o(\capa(\omega_\e))~.

Due to the symmetry of the fundamental solution (see <ref>), we have

$$ u_0(y) = \int_\Omega{N(x,y) f(x) \: \d x}~, $$

and this now implies

$$ \int_{\Omega}f \, u_\e \: \d x = \int_{\Omega} f \, u_0 \: \d x -\capa(\omega_\e) \int_{\partial\Omega} { \gamma(x) u_0^2(x)\: \d \mu(x)}  + \o(\capa(\omega_\e))~.

In particular, the emergence of a small Dirichlet region within the homogeneous Neumann zone $\Gamma_N$ always decreases the value of the compliance,
which is consistent with physical intuition,
since it amounts to enlarging the region of the boundary $\partial \Omega$ where the voltage potential is grounded.

§ REPLACING DIRICHLET CONDITIONS BY NEUMANN CONDITIONS ON A “SMALL SET"

We presently turn to the opposite situation of that considered in <ref>.
The considered sequence $\omega_\e$ of “small”, open Lipschitz subsets of $\partial \Omega$ is now included in $\Gamma_D$,
and it is well-separated from $\Gamma_N$ in the sense that <ref> holds.
The homogeneous Dirichlet boundary condition satisfied by the “background” voltage potential $u_0$ on $\Gamma_D$ (see <ref>)
is dropped on $\omega_\e$, where it is  replaced by a homogeneous Neumann boundary condition: the perturbed voltage potential $u_\e$
is then the solution to the equation <ref>.
Like in <ref>, without any further assumption on $\omega_\e$, we aim to derive a representation formula for $u_\e -u_0$ as $\e\to 0$.

Let us start by defining the quantity $e(\omega_\e)$ which will measure the “smallness” of a set $\omega_\e$ in the present setting.
When $\omega \subset \mathbb{R}^d$ is an arbitrary finite collection of disjoint Lipschitz hypersurfaces, we introduce:

\begin{equation} \label{eq.capaneu}
e(\omega) = \max\limits_{\kappa \in {\mathcal C}^\infty_c(\R^d), \atop \kappa(x) = \pm 1 \text{ for }  x\in\overline{\omega}} \Bigg\{  \int_{\R^d \setminus \overline\omega}{(z^2 + |\nabla z |^2 ) \: \d x}, \:\: z \in H^1(\R^d \setminus \overline \omega ) \text{ s.t. } \left\{
\begin{array}{cl}
- \Delta z + z = 0 & \text{in } \R^d \setminus \overline{\omega} , \\
\frac{\partial z}{\partial n} = \kappa & \text{on }\omega
\end{array}
\right. \Bigg\}~.
\end{equation}

In the above formulation, $n$ stands for any smooth unit normal vector field on (each connected component of) $\omega$, and the value of $e(\omega)$ does not depend on the choice of the particular direction(s) of $n$, due to the presence of the maximum.
More precisely, when $\omega$ has only one connected component, $e(\omega)$ is the energy of the unique $H^1(\R^d \setminus \overline \omega)$ solution $z$ to the equation

\begin{equation}\label{eq.funccapneu}
\left\{
\begin{array}{cl}
- \Delta z + z = 0 & \text{in } \R^d \setminus \overline{\omega}~ , \\
\frac{\partial z}{\partial n} = 1 & \text{on }\omega~,
\end{array}
\right.
\end{equation}

and the choice of an orientation for the normal vector $n$ to $\omega$ only affects the sign of $z$ and not the value of the energy $e(\omega)$.
When $\omega$ has several connected components, a direction for $n$ can be set independently on each connected component of $\omega$;
the possible choices for $\kappa$ in <ref> correspond to all possible configurations of the field $n$, and the quantity $e(\omega)$ captures the configuration with maximum energy.

In view of the discussion in <ref> (see notably <ref>), it is very tempting to interpret $e(\omega)$ as a sort of “capacity” of the set $\omega$,
which, in a Neumann context, measures the energy of the potential in an “equilibrium” situation where the current passing through $\omega$ is constant, with amplitude equal to $1$.

In spite of its intuitive physical interpretation, the quantity $e(\omega)$ is not very explicit,
since it involves the solution of a boundary value problem posed on the whole ambient space $\R^d$.
For this reason, we derive in <ref> several interesting surrogate quantities, depending only on the geometry of $\omega$,
which in some particular cases are equivalent to $e(\omega)$.

In <ref>, we shall conduct explicit calculations of the solution $u_\e$ to <ref>, in the particular case where the inclusion set $\omega_\e$ is a “surfacic ball” on $\partial \Omega$.
The following estimates for the “smallness” of the planar disk $\D_\e$ defined in <ref>, which follow straighforwardly from <ref> (see in particular <ref>), will be used repeatedly:

\begin{equation}\label{eq.capaneuDe}
e(\D_\e) \leq C_2 \e^2 \text{ if } d =2, \text{ and } e(\D_\e) \leq C_3 \e^3 \text{ if } d =3,
\end{equation}

for some universal constants $C_2$, $C_3$.

§.§ Preliminary estimates

We start with a preliminary result, which is analogous to <ref>,
and is essential for the derivation of our asymptotic representation formula. Let $\zeta_\e$ be the unique $H^1(\Omega)$ solution to

\begin{equation}\label{eq.zepsneum}
\left\{
\begin{array}{cl}
-\Delta \zeta_\e = 0 & \text{in } \Omega~, \\
\zeta_\e = 0 & \text{on } \Gamma_D \setminus \overline{\omega_\e}~, \\
\frac{\partial \zeta_\e}{\partial n} = 1 & \text{on } \omega_\e~,\\
\frac{\partial \zeta_\e}{\partial n} = 0 & \text{on } \Gamma_N~.
\end{array}
\right.
\end{equation}

The following lemma relates the energy of $\zeta_\e$ with the quantity $e(\omega_\e)$ defined in <ref>.

Let $\omega_\e$ be an open Lipschitz subset of the region $\Gamma_D \subset \partial \Omega$,
which lies “far” from $\Gamma_N$ in the sense that <ref> holds,
and let $\zeta_\e$ be the solution to <ref>.
There exist two constants $ 0 < m \leq M$, which depend only on $\Omega$, $\Gamma_D$ and the lower bound $\dmin$ on the distance from $\omega_\e$ to $\Gamma_N$, but are otherwise independent of $\omega_\e$,  such that

$$ m\: e(\omega_\e) \leq || \zeta_\e ||_{H^1(\Omega)}^2 \leq M\:  e(\omega_\e).$$

We start by looking at the right-hand inequality.
The latter is actually quite natural, since $\zeta_\e$ can be seen as arising from the solution $z$ to (an equation like) <ref>, for a suitable function $\kappa$, by “adding Dirichlet boundary conditions”. An adapted version of the Poincaré inequality for functions with vanishing trace on the set

$$ \left\{ x \in \Gamma_D, \:\: \dist(x,\Gamma_N) < \dmin /3 \right\}$$

reveals that there exists a constant $C >0$ which only depends on $\Omega$, $\Gamma_N$ and $\dmin$ such that

\begin{equation}\label{eq.poincareze}
|| \zeta_\e ||_{H^1(\Omega)} \leq  C || \nabla \zeta_\e ||_{L^2(\Omega)^d}~.
\end{equation}

Let $z_\e$ be the solution to <ref>,
where $n$ is the unit normal vector to $\partial \Omega$ pointing outward from $\Omega$ (in particular, it is normal to $\omega_\e$)
and $\kappa$ constantly equals $1$ on $\omega_\e$.
An integration by parts, using the boundary conditions satisfied by $\zeta_\e$ and $z_\e$, yields

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| \nabla \zeta_\e ||_{L^2(\Omega)^d}^2 &=&  \int_\Omega{\nabla \zeta_\e \cdot \nabla \zeta_\e \: \d x} \\[0.75em]
&=& \int_{\omega_\e}{\frac{\partial \zeta_\e}{\partial n} \zeta_\e \: \d s} \\[0.75em]
&=& \int_{\omega_\e}{\frac{\partial z_\e}{\partial n} \zeta_\e \: \d s} \\[0.75em]
&=& \int_{\partial \Omega}{\frac{\partial z_\e}{\partial n} (\eta \zeta_\e) \: \d s}~, \\
\end{array}

where $\eta$ is a smooth function such that

$$\eta \equiv 1 \text{ on } \left\{ x \in \Gamma_D, \:\: \dist(x,\Gamma_N) > \dmin /2 \right\} \text{ and }
\eta \equiv 0 \text{ on } \left\{ x \in \partial \Omega, \:\: \dist(x,\Gamma_N) < \dmin /3 \right\}~.$$

It follows that

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| \nabla \zeta_\e ||_{L^2(\Omega)^d}^2 &=&\int_\Omega{\Big( \nabla z_\e \cdot \nabla (\eta \zeta_\e) + z_\e \eta \zeta_\e \Big)\: \d x} \\
&\leq& C ||z_\e ||_{H^1(\Omega)}   ||\zeta_\e ||_{H^1(\Omega)}\\
&\leq& C ||z_\e ||_{H^1(\R^d \setminus \overline \omega_\e)}   ||\nabla \zeta_\e ||_{L^2(\Omega)^d}~,
\end{array}

where we have used the Poincaré inequality <ref>.
The desired inequality now follows from the definition <ref> of $e(\omega_\e)$ and repeated use of  the Poincaré inequality <ref>.
Let us now turn to the left-hand inequality.
To this end, let $z_\e$ be the $H^1(\R^d \setminus \overline \omega_\e)$ solution to (an equation like) <ref>,
where $\kappa$ is any ${\mathcal C}^\infty_c(\R^d)$ function taking values $1$ or $-1$ on $\omega_\e$, and $n$ again is chosen to be the unit normal to $\partial\Omega$, pointing outward $\Omega$.
The variational formulation associated to (an equation like) <ref> and an integration by parts immediately imply that

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| z_\e ||_{H^1(\R^d\setminus \overline{\omega_\e})}^2 &=&  \int_{\R^d \setminus \overline \omega_\e}{\Big( z_\e ^2 + \nabla z_\e \cdot \nabla z_\e \Big) \: \d x} \\
&=& -\int_{\omega_\e}{\kappa (z_\e^+ - z_\e^-) \: \d s}~. \\
\end{array}

Here we have denoted by $z_\e^+$ and $z_\e^-$ the one-sided traces of $z_\e$ on $\omega_\e$ from the exterior and  the interior of $\Omega$, respectively (see <ref>).
We obtain

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| z_\e ||_{H^1(\R^d\setminus \overline{\omega_\e})}^2 &\leq & \int_{\omega_\e}{\frac{\partial \zeta_\e}{\partial n} |z_\e^+ - z_\e^-| \: \d s}\\
&=& \int_{\partial \Omega}{\frac{\partial \zeta_\e}{\partial n}| z_\e^+ - z_\e^-| \: \d s}~, \\
\end{array}

where we have used the fact that $z_\e$ is continuous across $\partial \Omega$ (in the sense of traces) except on $\omega_\e$. Since $\Omega$ is smooth, there exists a bounded linear extension operator $E : H^{1/2}(\partial \Omega) \to H^1(\Omega)$ such that

$$\forall u \in H^{1/2}(\partial \Omega), \:\: || E u ||_{H^1(\Omega)} \leq C|| u ||_{H^{1/2}(\partial \Omega)}  \text{ and } E u = u \text{ on } \partial \Omega~,$$

for a constant $C$ which depends only on $\Omega$.
Based on the previous estimate we calculate

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| z_\e ||_{H^1(\R^d\setminus \overline{\omega_\e})}^2 &\leq& \int_{\partial \Omega}{\frac{\partial \zeta_\e}{\partial n}| E z_\e^+ - z_\e^-| \: \d s}\\
&=& \int_{\Omega}{\nabla \zeta_\e \cdot \nabla | E z_\e^+ - z_\e| \: \d x}\\
&\leq& C || \nabla \zeta_\e ||_{L^2(\Omega)^d} || Ez_\e^+ -z_\e ||_{H^1(\Omega)} \\
&\leq& C || \nabla \zeta_\e ||_{L^2(\Omega)^d} \left(|| z_\e^+ ||_{H^{1/2}(\partial \Omega)}+ ||z_\e ||_{H^1(\Omega)}\right)\\
&\leq& C || \nabla \zeta_\e ||_{L^2(\Omega)^d} || z_\e ||_{H^1(\R^d \setminus \overline \omega_\e)}~,
\end{array}

which finally results in the desired inequality

$$  || z_\e ||_{H^1(\R^d\setminus \overline{\omega_\e})} \leq C || \nabla \zeta_\e ||_{L^2(\Omega)^d}~.$$

Since this holds for any choice of the function $\kappa \in {\mathcal C}^\infty_c(\R^d)$ having values $1$ or $-1$ on $\omega_\e$,
the desired inequality follows by taking the maximum with respect to any such choice.

We now consider the $H^1(\Omega)$ solution $v_\e$ to the boundary value problem

\begin{equation}\label{eq.vepsneu}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla v_\e) = 0 & \text{in } \Omega~, \\
v_\e = 0 & \text{on } \Gamma_D \setminus \overline{\omega_\e}~, \\
\gamma\frac{\partial v_\e}{\partial n} = g & \text{on } \omega_\e~,\\
\gamma\frac{\partial v_\e}{\partial n} = 0 & \text{on } \Gamma_N~,
\end{array}
\right.
\end{equation}

where $g$ is a given ${\mathcal C}^0(\overline \Omega)$ function.
Our next result provides norm bounds for $v_\e$ in terms of the expression $e(\omega_\e)$.

Suppose $d=2$ or $d=3$. Let $\omega_\e$ be an open Lipschitz subset of the region $ \Gamma_D \subset \partial \Omega$, which lies “far" from $\Gamma_N$ in the sense that <ref> holds. There exists a constant $M$, which depends only on $\alpha$, $\beta$, the coercivity constants of $\gamma$, $\Omega$, $\Gamma_N$ and the lower bound $\dmin$ on the distance from $\omega_\e$ to $\Gamma_N$, but is otherwise independent of $\omega_\e$,  such that
the function $ v_\e$ in <ref> satisfies the following $H^1$ estimate

\begin{equation}\label{eq.H1estneu}
|| v_\e ||_{H^1(\Omega)} \leq M e(\omega_\e)^{\frac{1}{2}} \: || g ||_{{\mathcal C}^0(\overline \Omega)}~.
\end{equation}

In addition, the following “improved” $L^2$ estimate holds

\begin{equation}\label{eq.L2estneu}
|| v_\e ||_{L^2(\Omega)} \leq M e(\omega_\e)^{\frac{3}{4}} \: || g ||_{{\mathcal C}^0(\overline \Omega)}~.
\end{equation}

The quantity $e(\omega_\e)$ is that defined in <ref>.

We start by proving <ref>. Since $\omega_\e$ lies inside $\Gamma_D$ with $\dist(\omega_\e,\Gamma_N)>\dmin>0$,
a variant of the  Poincaré's inequality for functions whose trace vanishes on the fixed region $\{ x \in \Gamma_D, \:\: \dist(x,\Gamma_N)<\dmin \}$ yields the existence of a constant $C >0$,
depending only on $\partial\Omega$, $\Gamma_D$ and $\dmin$, such that

\begin{equation}\label{eq.poincarerepneu}
|| v_\e ||_{H^1(\Omega)} \leq C || \nabla v_\e ||_{L^2(\Omega)^d}~.
\end{equation}

We then calculate

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
|| \nabla v_\e ||_{L^2(\Omega)}^2 &\leq& C \int_\Omega{\gamma \nabla v_\e \cdot \nabla v_\e \: \d x}  \\[0.75em]
&=& C \int_{\partial \Omega}{\gamma \frac{\partial v_\e}{\partial n} v_\e \: \d s} \\[0.75em]
&=& C \int_{\omega_\e}{ g v_\e \: \d s}~.
\end{array}

An application of <ref> and introduction of the function $\zeta_\e$ – defined in <ref> and estimated in <ref> – now yields

\begin{eqnarray*}
|| v_\e ||_{H^1(\Omega)}^2  &\leq& C || g||_{{\mathcal C^0}(\overline \Omega)}  \int_{\omega_\e} | v_\e | \: \d s \\
&=& C || g||_{{\mathcal C^0}(\overline \Omega)}  \int_{\partial \Omega} | v_\e |  \frac{\partial \zeta_\e}{\partial n}\: \d s\\
&=& C || g||_{{\mathcal C^0}(\overline \Omega)} \int_{\Omega}  \nabla | v_\e | \cdot \nabla \zeta_\e \:\d x \\
&\leq& C || g||_{{\mathcal C^0}(\overline \Omega)} \Vert v_\e\Vert_{H^1(\Omega)} e(\omega_\e)^{\frac12}~,
\end{eqnarray*}

and the desired estimate <ref> follows.

Let us now consider the improved $L^2$ estimate <ref>.
To establish this, we proceed along the lines of the proof of <ref>. As in that proof, let $w_\e$ denote the unique $H^1(\Omega)$ solution to the boundary value problem

\left\{\begin{array}{ccl}
-\dv(\gamma\nabla w_\e) = v_\e & \text{in } \Omega~, \\
w_\e = 0 & \text{on } \Gamma_D~, \\
\gamma\frac{\partial w_\e}{\partial n} = 0 & \text{on } \Gamma_N~.
\end{array}
\right.

Taking advantage of the separation assumption <ref>, we may introduce a cut-off function $\eta \in {\mathcal C}^\infty_c(\R^d)$ with the property

$$\eta = 1 \text{ on a fixed neighborhood of all the } \omega_\e \text{ and } \eta = 0 \text{ on an open set } \calU \text{ in } \R^d \text{ with } \Gamma_N \Subset \calU~.$$

The function $w_\e$ shows improved regularity with respect to $v_\e$, away from the interface $\Sigma$ between the Dirichlet and Neumann regions $\Gamma_D$ and $\Gamma_N$.
More precisely, arguing as in the proof of <ref> (see in particular <ref>), one obtains that

\begin{equation}\label{ellregweneu}
\Vert  \eta w_\e  \Vert_{C^{1}(\overline \Omega)} \le C  || \eta w_\e ||_{H^3(\Omega)} \leq C || v_\e ||_{H^1(\Omega)}~.
\end{equation}

We now calculate

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\int_\Omega{v_\e ^2 \: \d x} &=&  -\int_{\Omega}\dv(\gamma\nabla w_\e)\, v_\e \: \d x\\
&=&\int_\Omega{\gamma \nabla w_\e \cdot \nabla v_\e \:\d x} -\int_{\omega_\e} \gamma \frac{\partial w_\e}{\partial n} v_\e \:\d s \\
&=& - \int_{\omega_\e}{\gamma \frac{\partial}{\partial n} (\eta w_\e) v_\e \:\d s}~. \\
\end{array}$$

Using the regularity estimate <ref> for $\eta w_\e$ and introducing the function $\zeta_\e$,
– defined in <ref>, and estimated in <ref> – we are now led to

\begin{eqnarray*}
\int_\Omega{v_\e ^2 \: \d x} &\leq& C \left\lvert\left\lvert  \frac{\partial (\eta w_\e)}{\partial n} \right\lvert\right\lvert_{C^0(\partial \Omega)}   \int_{\omega_\e}| v_\e|\: \d s  \\
&=& C \left\lvert\left\lvert  \frac{\partial (\eta w_\e)}{\partial n} \right\lvert\right\lvert_{C^0(\partial \Omega)}\int_{\partial \Omega} | v_\e |  \frac{\partial \zeta_\e}{\partial n}\: \d s\\
&=&C \left\lvert\left\lvert  \frac{\partial (\eta w_\e)}{\partial n} \right\lvert\right\lvert_{C^0(\partial \Omega)}\int_{\Omega} \nabla | v_\e | \cdot \nabla  \zeta_\e\: \d x\\
&\leq& C  || v_\e ||_{H^1(\Omega)}^2 e(\omega_\e)^{\frac12}~.
\end{eqnarray*}

In combination with the already established estimate <ref>, this yields

\int_\Omega{v_\e ^2 \: \d x} \leq\, C\, e(\omega_\e)^\frac32 || g ||^2_{{\mathcal C}^0(\overline \Omega)}~,

exactly as asserted in <ref>.

As in <ref> (see <ref>) close
inspection of the above proof reveals that both estimates <ref> still hold true when the function $v_\e$ from <ref> is replaced by the solution to the following anisotropic boundary value problem

\begin{array}{cl}
-\dv(A \nabla v_\e) = 0 & \text{in } \Omega, \\
v_\e = 0 & \text{on } \Gamma_D  \setminus \overline{\omega_\e}, \\
(A\nabla v_\e)\cdot n= g & \text{on } \omega_\e,\\
(A\nabla v_\e)\cdot n= 0 & \text{on } \Gamma_N~,\\
\end{array}
\right.

where $A \in {\mathcal C}(\overline\Omega)^{d\times d}$ is a smooth conductivity matrix satisfying the bounds <ref>.

§.§ The representation formula

One of our main results in this section is the following representation theorem.

Suppose that $d=2$ or $d=3$ and that $\omega_\e$ is a sequence of non-empty, open Lipschitz subsets of $\partial \Omega$, which are all contained in $\Gamma_D$ and well-separated from $\Gamma_N$ in the sense that <ref> holds. Let $u_\e$ denote the solution to <ref>. Assume that the quantity $e(\omega_\e)$, given by <ref>, goes to $0$ as $\e \rightarrow 0$. Then there exists a subsequence, still labeled by $\e$, and a non-trivial distribution $\mu$ in the dual space of ${\mathcal C}^1(\partial \Omega)$ such that
for any fixed point $x \in \Omega$, and any $\eta \in C^\infty(\partial \Omega)$ with  $\eta =1$ on $\{ y \in \partial \Omega,~\dist(y,\Gamma_N)>\dmin/2\}$ and
$\eta =0$ on $\{ y \in \partial \Omega,~\dist(y,\Gamma_N)<\dmin/3\}$

\begin{equation}\label{eq.repforneu}
u_\e (x) = u_0(x) + e(\omega_\e) \: \mu_y\left(\eta(y) \frac{\partial u_0}{\partial n}(y) \gamma(y) \frac{\partial N}{\partial n_y} (x,y) \right) + \o (e(\omega_\e))~.
\end{equation}

The term $\o(e(\omega_\e))$ goes to zero faster than $e(\omega_\e)$, uniformly for $x$ in any fixed compact subset $K$ of $\Omega$. The distribution $\mu$ depends only on the subsequence $\omega_\e$, $\Omega$, and $\Gamma_D$.

The proof parallels that of <ref> with appropriate changes. We give a fairly detailed outline of it, except in a few places where we refer back to the proof of <ref>.
Let $r_\e$ denote the remainder $r_\e := u_\e - u_0$, which is now the unique $H^1(\Omega)$ solution to the following problem

\begin{equation}\label{eq.repsneu}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla r_\e) = 0 & \text{in } \Omega~, \\
r_\e = 0 & \text{in } \Gamma_D \setminus \overline{\omega_\e}~, \\
\gamma \frac{\partial r_\e}{\partial n} = -\gamma \frac{\partial u_0}{\partial n} & \text{on } \omega_\e~, \\
\gamma \frac{\partial r_\e}{\partial n} = 0 & \text{on } \Gamma_N~. \\
\end{array}
\right.
\end{equation}

Let $x$ be a fixed point inside  $\Omega$. From the definition <ref> of the fundamental solution $N(x,y)$ to the background equation, we obtain after integration by parts

$$  \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
r_\e(x) &=& \int_\Omega{r_\e(y) (-\dv_y(\gamma(y) \nabla_y N(x,y))) \: \d y}\\
&=&  -\int_{\omega_\e }{r_\e(y) \gamma (y) \frac{\partial N}{\partial n_y}(x,y) \: \d s(y)} + \int_\Omega{\gamma (y) \nabla r_\e(y) \cdot \nabla_y N(x,y) \: \d y}~.\\
\end{array}

Another integration by parts of the second term in the above right-hand side reveals that the latter actually vanishes, so in conclusion

\begin{equation}\label{firsteq}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
r_\e(x) &=&  -\int_{\omega_\e }{r_\e(y) \gamma(y) \frac{\partial N}{\partial n_y}(x,y) \: \d s(y)}\\
&= & -\int_{\omega_\e }{r_\e(y) \eta(y)  \gamma(y) \frac{\partial N}{\partial n_y}(x,y) \: \d s(y)} ~.
\end{array}
\end{equation}

Following the proof of <ref>, we now proceed to calculate, for any given function $\phi \in {\mathcal C}^1(\partial\Omega)$ vanishing on $\{ y \in \partial \Omega,~\dist(y,\Gamma_N)<\dmin/3\}$,  the limit of the quantity

$$ -\int_{\omega_\e}{r_\e(y) \phi(y) \: \d s(y)}~.$$

For this purpose we introduce an extension $\psi \in {\mathcal C}^1(\overline\Omega)$ of $\phi$ satisfying the properties (see <ref>)

$$\psi = \phi \text{ on } \partial \Omega, \text{ and } || \psi ||_{{\mathcal C}^1(\overline{\Omega})} \leq C || \phi ||_{{\mathcal C}^1(\partial \Omega)}~,$$

and we consider the unique $H^1(\Omega)$ solution $\zeta_\e$ to the boundary value problem <ref>.

We calculate

$$   \begin{array}{>{\displaystyle}cc>{\displaystyle}l}
-\int_{\omega_\e}{r_\e(y) \phi(y) \: \d s(y)} &=& -\int_{\partial\Omega}{  \frac{\partial \zeta_\e}{\partial n} r_\e \psi \: \d s } \\
&=& -\int_{\Omega}{( \nabla \zeta_\e \cdot \nabla r_\e) \psi \: \d y} -\int_{\Omega}{( \nabla \zeta_\e \cdot  \nabla\psi) r_\e \: \d y} \\
&=& -\int_{\Omega}{(\nabla \zeta_\e \cdot \gamma \nabla r_\e) \frac{\psi}{\gamma} \: \d y}  \: + \: \O(e(\omega_\e)^{\frac54})\Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~,
\end{array}

where the last identity follows from the improved $L^2$ estimate (applied to $r_\e$) and the $H^1$ estimate (applied to $\zeta_\e$) from <ref>;
see the proof of <ref> for details. A repeated use of the same estimates (with the roles of $r_\e$ and $\zeta_\e$ interchanged) followed by an integration by parts yields

-\int_{\omega_\e}{r_\e(y) \phi(y) \: \d s(y)} &=&  -\int_{\Omega}{\gamma \nabla r_\e \cdot \nabla \left(\frac{\psi \zeta_\e}{\gamma}\right)  \: \d x}  + \O(e(\omega_\e)^{\frac54})\Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)} \\
&=& - \int_{\partial \Omega}{ \frac{\partial r_\e}{\partial n} \phi \zeta_\e \: \d s} + \O(e(\omega_\e)^{\frac54})\Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)};
\end{array}$$

see the proof of <ref>.  Using the boundary conditions satisfied by $r_\e$ and $\zeta_\e$ we finally end up with

\begin{equation}
\label{secondeq}
-\int_{\omega_\e}{r_\e(y) \phi(y) \: \d s(y)}=  \int_{\partial\Omega}{\left( \frac{\partial \zeta_\e}{\partial n} \zeta_\e \right) \frac{\partial u_0}{\partial n} \phi \: \d s} + \O(e(\omega_\e)^{\frac54})\Vert f\Vert_{H^m(\Omega)} \Vert \phi \Vert_{C^1(\partial \Omega)}~.
\end{equation}

From <ref>, we infer that the sequence $\frac{1}{e(\omega_\e)}\left(\frac{\partial \zeta_\e}{\partial n} \zeta_\e \right)$
has bounded norm in the dual space of ${\mathcal C}^1(\partial \Omega)$; see more precisely <ref> in the proof of <ref>. From the Banach-Alaoglu theorem, it now follows, after extraction of a subsequence (still labeled by $\e$), that there exists a bounded linear functional $\mu $ on ${\mathcal C}^1(\partial \Omega)$ such that

\begin{equation}
\label{thirdeq}
\forall \varphi \in {\mathcal C}^1(\partial \Omega) , \:\: \int_{\partial \Omega}{\frac{1}{e(\omega_\e)}\left( \frac{\partial \zeta_\e}{\partial n} \zeta_\e \right) \varphi \: \d s} \xrightarrow{\e\to 0} \mu(\varphi)~.
\end{equation}

Also, due to <ref>, it follows that $\mu(1)>0$, thus revealing that $\mu$ is non trivial.
Insertion of $\phi(y) = \eta(y)\gamma(y) \frac{\partial N}{\partial n_y}(x,y)$ into <ref> and application of <ref> with $\varphi(y)=\frac{\partial u_0}{\partial n}(y)\eta(y)\gamma(y) \frac{\partial N}{\partial n_y}(x,y)$ now gives

-\int_{\omega_\e }{r_\e(y) \eta(y)  \gamma(y) \frac{\partial N}{\partial n_y}(x,y) \: \d s(y)} = e(\omega_\e) \mu_y \left[  \eta(y) \frac{\partial u_0}{\partial n}(y) \gamma(y) \frac{\partial N}{\partial n_y}(x,y)\right] + \o(e(\omega_\e))~,

which in combination with <ref> leads to the desired representation formula <ref>. The uniformity of the convergence of the remainder $\o(e(\omega_\e))$, when $x$ is confined to a fixed compact subset of $\Omega$, follows as in the proof of <ref>.

Just as in <ref> we may show that the distribution $\mu$ is a non negative Radon measure compactly supported “near" the sets  $\omega_\e$; in other words, the following analogue of <ref> holds in the present context, whose nearly identical proof is left to the reader.

The limiting distribution $\mu$ introduced in <ref> is a non negative Radon measure on $\partial \Omega$.
Moreover, the support of $\mu$ is contained in any compact subset $K$ of $\partial \Omega$ such that $\omega_\e \subset K$ for $\e >0$ sufficiently small.

This proposition immediately leads to the following corollary to <ref>.

Suppose $d=2$ or $d=3$ and suppose $\omega_\e$ is a sequence of non-empty, open Lipschitz subsets of $\partial \Omega$, which are all contained in  $\Gamma_D$ and are well-separated from $\Gamma_N$, in the sense that <ref> holds; let $u_\e$ denote the solution to <ref>.
Assume that  the quantity $e(\omega_\e)$, defined by <ref>, goes to $0$ as $\e \rightarrow 0$.
Then there exists  a subsequence, still labeled by $\e$, and a non-trivial, non negative Radon measure $\mu$ on $\partial \Omega$, whose support is included in any compact subset $K \subset \partial \Omega$ containing the $\omega_\e$ for $\e >0$ small enough, such that the following asymptotic expansion

u_\e(x) = u_0(x) +e(\omega_\e) \int_{\partial \Omega} \frac{\partial u_0}{\partial n}(y) \gamma(y) \frac{\partial N}{\partial n_y} (x,y) \:\d \mu(y)  + \o(e(\omega_\e))~,

holds at any fixed point $x \in \Omega$.
The term $\o(e(\omega_\e))$ goes to zero faster than $e(\omega_\e)$ uniformly (in $x$) on compact subsets of $\Omega$. The measure $\mu$ depends only on the subsequence $\omega_\e$, $\Omega$, and $\Gamma_D$.

From the physical viewpoint, the second term in the representation formula of <ref> accounts for the potential created at $x$ by a distribution of dipoles located at the “limiting position” of the sets $\omega_\e$. We notice the sign change, when compared to the second term of the expansion in <ref>. A calculation similar to that found in <ref> (and under the same assumptions regarding the source term $f$) now leads to a non negative first term in the perturbation of the compliance, reflecting the intuitive fact that the compliance of $\Omega$ necessarily (asymptotically) increases when the homogeneous Dirichlet boundary condition on $\omega_\e$  is turned into a homogeneous Neumann condition.

§ AN EXPLICIT ASYMPTOTIC FORMULA FOR THE CASE OF SUBSTITUTING DIRICHLET CONDITIONS

In this section, we investigate a particular instance of the general situation of <ref>, where
the homogeneous Neumann boundary condition satisfied by the background potential $u_0$ on the whole region $\Gamma_N$ is modified to a Dirichlet boundary condition
on a subset $\omega_\e \subset \Gamma_N$ taking the form of a vanishing “surfacic ball”.

Without loss of generality, we assume that the origin $0$ belongs to $\Gamma_N$,
and that the normal vector $n(0)$ at $0$ coincides with the last coordinate vector $e_d$.
We select a smooth bounded domain ${\mathcal O} \subset \R^d$, and construct a smooth diffeomorphism $T: \R^d \to \R^d$ such that $\Omega = T({\mathcal O})$, and

\begin{minipage}{0.93\textwidth}
\begin{enumerate}[(i)]
\item The domain ${\mathcal O}$ lies inside the lower half-space $H$, and it coincides with $H$ in a fixed open neighborhood $U$ of $0$:
$${\mathcal O} \subset H, \text{ and } {\mathcal O}\cap U= H \cap U, \text{ where } H := \left\{ x=(x_1,\ldots, x_{d}) \in \R^d , \:\: x_d < 0\right\}.$$
\item $T(0) = 0$ and $\nabla T(0) = \Id$.
\end{enumerate}
\end{minipage}

Given such $T$ and ${\mathcal O}$, the subset $\omega_\e \subset \Gamma_N$ is now defined as follows:

\begin{equation}\label{eq.GNflat}
\omega_\e = T (\D_\e), \text{ where } \D_\e := \left\{ x = (x_1,\ldots, x_{d-1}, 0) \in \partial H, \:\: |x| < \e \right\},
\end{equation}

for $\e$ sufficiently small, see <ref> for an illustration.
We denote by $\widehat{\Gamma_N}\subset \partial {\mathcal O}$ the boundary set $\widehat{\Gamma_N} := T^{-1}(\Gamma_N)$, and purely for simplicity we also assume that ${\mathcal O}$ and $T$  are selected in such aa way that $T$ coincides with the identity mapping “far” from $0$, so that in particular $T^{-1}(\Gamma_D) = \Gamma_D$ (in terms of the original domain $\Omega$ this is achievable through the assumption that $\Omega$ lies below its tangent plane at $0$).

The setting in <ref>.

The “background” and perturbed potentials $u_0$ and $u_\e$ are the $H^1(\Omega)$ solutions to the following equations:

\begin{equation}\label{eq.ueu0si}
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla u_0) = f & \text{in } \Omega, \\
u_0 = 0 & \text{on } \Gamma_D, \\
\gamma\frac{\partial u_0}{\partial n} = 0 & \text{on } \Gamma_N,
\end{array}
\right.
\text{ and }
\left\{
\begin{array}{cl}
-\dv(\gamma \nabla u_\e) = f & \text{in } \Omega, \\
u_\e = 0 & \text{on } \Gamma_D \cup \omega_\e, \\
\gamma\frac{\partial u_\e}{\partial n} = 0 & \text{on } \Gamma_N \setminus \overline{\omega_\e},
\end{array}
\right.
\end{equation}

where the source term $f \in {\mathcal C}^\infty(\overline\Omega)$ is smooth.
Invoking classical elliptic regularity results, we observe that $u_0$ and $u_\e$ are smooth,
except in the vicinity of the points $x \in \partial \Omega$ where boundary conditions change type. More precisely, with $\Sigma= \overline{\Gamma_D} \cap \overline{\Gamma_N}$

* The function $u_0$ is of class ${\mathcal C}^\infty$ in a neighborhood of any point $x \in \overline\Omega \setminus \Sigma$;

* The function $u_\e$ is of class ${\mathcal C}^\infty$ in a neighborhood of any point $x \in \overline\Omega \setminus \left( \Sigma \cup \partial \omega_\e \right) $;

see for instance <cit.>, 9.6, or <cit.>.

We aim to derive a precise first order asymptotic expansion of $u_\e$ when $\e \to 0$, thus exemplifying the abstract structure of <ref>.
We start by providing the complete analysis for the two-dimensional case in <ref>.
The analysis for the three-dimensional case, which is quite similar in many aspects, is outlined in <ref>.

§.§ Asymptotic expansion of the perturbed potential $u_\e$ in 2d

This section deals with the case $d=2$, and our main result is

The following asymptotic expansion holds at any point $x\in \overline{\Omega}$, $x\notin \Sigma \cup \left\{ 0 \right\}$:

\begin{equation}\label{eq.expuedir2d}
u_\e(x) = u_0(x) - \frac{\pi}{| \log \e|} \gamma(0) u_0(0) N(x,0) + \o\left(\frac{1}{|\log \e|}\right).
\end{equation}

We proceed in four steps, relying on several intermediate, technical results, whose proofs are postponed to the end of the section
for the sake of clarity.

Step 1. We establish a representation formula for $u_\e$ which relates
its value at a point $x \in \partial \Omega$ “far” from the inclusion set $\omega_\e$ to its values inside $\omega_\e$ by means of the fundamental solution $N(x,y)$
to the background operator, defined by <ref>.

Considering a fixed point $x \in \Omega$, using the definition of $N(x,y)$ and integrating by parts twice, we obtain

u_\e(x) &=& - \int_\Omega{\dv_y(\gamma(y) \nabla_y N(x,y)) u_\e(y) \: \d y} \\
&=&  \int_{\Omega}{\gamma(y) \nabla_y N(x,y) \cdot \nabla u_\e(y) \: \d y} \\[0.7em]
&=& \int_{\partial \Omega}{\gamma(y)\frac{\partial u_\e}{\partial n_y}(y) N(x,y)\:\d s(y)} + \int_{\Omega}{f(y) N(x,y) \:\d y},
\end{array}$$

where the second line follows from the facts that $\gamma(y) \frac{\partial N}{\partial n_y}(x,y) = 0$ on $\Gamma_N$ and $u_\e = 0$ on $\Gamma_D$; see <ref>.
Using the background problem <ref> satisfied by $u_0$ and the boundary conditions for $u_\e$ and $N$, we
arrive at the following formula, for any point $x \in \Omega$

\begin{equation}\label{eq.repformulauesi0}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
u_\e(x) &=& u_0(x) + \int_{\omega_\e}{\gamma(y)\frac{\partial u_\e}{\partial n_y}(y) N(x,y)\:\d s(y)}\\
&=&u_0(x) + \int_{\omega_\e}{\gamma(y)\frac{\partial r_\e}{\partial n_y}(y) N(x,y)\:\d s(y)}~.
\end{array}
\end{equation}

Here we have taken advantage of the fact that $u_0$ satisfies homogeneous Neumann boundary conditions on $\Gamma_N$
to introduce the error $r_\e= u_\e - u_0$ in the last integral of the above right-hand side.
Note that

the identity <ref>
extends to the case of points $x \in \partial \Omega$, $x \notin \Sigma \cup \left\{0\right\}$ in the sense of traces, provided $\e$ is small enough, since all the quantities involved are smooth at such points.

Next, we introduce the mapped potentials $v_0 := u_0 \circ T$ and $v_\e := u_\e \circ T$.
A change of variables in the variational formulations of <ref> reveals that $v_0$ and $v_\e$ are the unique $H^1({\mathcal O})$ solutions to the problems

\begin{equation}\label{eq.vev0si}
\left\{
\begin{array}{cl}
-\dv(A \nabla v_0) = g & \text{in } {\mathcal O}, \\
v_0 = 0 & \text{on } \Gamma_D, \\
(A \nabla v_0) \cdot n = 0 & \text{on } \widehat{\Gamma_N},
\end{array}
\right.
\text{ and }
\left\{
\begin{array}{cl}
-\dv(A \nabla v_\e) =  g & \text{in } {\mathcal O}, \\
v_\e = 0 & \text{on } \Gamma_D \cup \D_\e, \\
(A\nabla v_\e)\cdot n = 0 & \text{on } \widehat{\Gamma_N} \setminus \overline{\D_\e},
\end{array}
\right.
\end{equation}

where $g \in {\mathcal C}^\infty(\R^2)$ and $A \in {\mathcal C}^\infty(\R^2 ,\R^{2\times 2})$ are the smooth function and the matrix field defined by

\begin{equation}\label{eq.defgA}
g(y) = |\det (\nabla T(y))| f(T(y))~, \text{ and } A(y) = |\det (\nabla T(y))| \gamma(T(y)) \nabla T(y)^{-1} (\nabla T(y)^t)^{-1}.
\end{equation}

Recalling the definition <ref> of $\omega_\e$, we now change variables in <ref> and then rescale the resulting integral to obtain

\begin{equation}\label{eq.repformulauesi}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
u_\e(x) &=& u_0(x) + \int_{\D_\e}{(A(y) \nabla s_\e(y))\cdot n(y) N(x,T(y))\:\d s(y)}\\
&=&u_0(x) + \int_{\D_1}{ \varphi_\e(z) N(x,T(\e z))\:\d s(z)}~,
\end{array}
\end{equation}

where we have introduced $s_\e := v_\e - v_0 = r_\e \circ T$, and the quantity

\begin{equation}\label{eq.vphie2ddir}
\varphi_\e(z) = \e (A\nabla s_\e \cdot n)(\e z) \in \widetilde{H}^{-1/2}(\D_1)~.
\end{equation}

The formula <ref> leads us to study the asymptotic behavior of $\varphi_\e$ as $\e \to 0$.
Step 2. We characterize $\varphi_\e$ as the solution to an integral equation.
To this end, we essentially repeat the derivation of Step 1,
except that we now use an approximate, explicit fundamental solution instead of the function $N(x,y)$.

For any symmetric, positive definite matrix $A$, and any $x\in H$, let $L_A(x,y)$ be a solution to the following equation posed on the lower half-space $H$:

\begin{equation}\label{eq.defLB}
\left\{
\begin{array}{cl}
-\dv_y(A \nabla_y L_A(x,y)) = \delta_{y=x}  & \text{in } H~,  \\
A\nabla_y L_A (x,y) \cdot n(y) = 0 & \text{on } \partial H~.
\end{array}
\right.
\end{equation}

The next lemma provides an explicit expression for such a function; its proof is postponed to the end of the present section.

Let $A$ be a symmetric, positive definite $2 \times 2$ matrix, and let $M := A^{-1/2}$.
Let $G(x,y)$ be the fundamental solution of the operator $-\Delta$ in the free space, defined in <ref>.
The function

\begin{equation}\label{eq.exprLB}
L_A(x,y) = |\det M| \left( G(Mx,My) + G\left(Mx, My - 2 y_2 \frac{M^{-1}e_2}{|M^{-1}e_2|^2}\right)\right), \quad x \neq y,
\end{equation}

satisfies the equation <ref>.

A straightforward calculation shows that,  for $x \in \partial H$, $y \in \R^2$, $y\neq x$,

$$ | Mx - My | = \left\lvert Mx - My + 2 y_2 \frac{M^{-1}e_2}{|M^{-1}e_2|^2} \right\lvert~.$$

For a given point $x\in H$, we now consider the function $y \mapsto L_{A(x)}(x,y)$ (by substituting $A(x)$ for $A$ in  <ref>) which satisfies

\begin{equation}\label{eq.defLAx}
\left\{
\begin{array}{cl}
-\dv_y(A(x) \nabla_y L_{A(x)}(x,y)) = \delta_{y=x}  & \text{in } H,  \\
A(x)\nabla_y L_{A(x)} (x,y) \cdot n(y) = 0 & \text{on } \partial H.
\end{array}
\right.
\end{equation}

For a point $x \in {\mathcal O}$, we obtain from <ref> and integration by parts that

v_\e(x) &=& - \int_{\mathcal O}{\dv_y(A(x) \nabla_y L_{A(x)}(x,y)) v_\e(y) \: \d y} \\[0.7em]
&=& - \int_{\partial {\mathcal O}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) v_\e(y)\:\d s(y) } + \int_{{\mathcal O}}{A(x) \nabla_y L_{A(x) }(x,y) \cdot \nabla v_\e(y) \: \d y} \\[0.7em]
&=& - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) v_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla v_\e(y) \: \d y} \\
&& + \int_{{\mathcal O}}{A(y) \nabla_y L_{A(x) }(x,y) \cdot \nabla v_\e(y) \: \d y} \\[0.7em]
&=& - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) v_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla v_\e(y) \: \d y} \\
&& + \int_{\partial {\mathcal O}}{(A\nabla v_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} + \int_{{\mathcal O}}{ g(y) L_{A(x) }(x,y) \: \d y}~,
\end{array}$$

where the third line follows from the equation <ref> satisfied by $L_{A(x)}(x,y)$, and the fact that $\partial {\mathcal O} \cap U = \partial H\cap U$. A similar calculation applied to $v_0$ instead of $v_\e$ yields

\begin{multline*}
v_0(x) = - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) v_0(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla v_0(y) \: \d y} \\
\hskip -50pt + \int_{\partial {\mathcal O}}{(A\nabla v_0 \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} + \int_{{\mathcal O}}{ g(y) L_{A(x) }(x,y) \: \d y}~.
\end{multline*}

Forming the difference of these identities, we get

\begin{multline*}
s_\e(x) = - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) s_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla s_\e(y) \: \d y} \\
\hskip -50pt + \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} + \int_{\D_\e}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)}~.
\end{multline*}

Letting $x$ tend to $\D_\e$, and invoking the boundary continuity of single layer potentials (as in the last term), we obtain for a.e. $x \in \D_\e$

\begin{eqnarray}\label{eq.v0De}
-v_0(x) = - \int_{\partial {\mathcal O} \setminus {\mathcal U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) s_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla s_\e(y) \: \d y}\nonumber \\
+ \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} + \int_{\D_\e}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)}~.
\end{eqnarray}

Rescaling the above equation, we finally obtain, for a.e. $x \in \D_1$

\begin{multline*}
-v_0(\e x) = - \int_{\partial {\mathcal O} \setminus {\mathcal U}}{ A(\e x) \nabla_y L_{A(\e x)}(\e x,y) \cdot n(y) s_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(\e x)-A(y)) \nabla_y L_{A(\e x) }(\e x,y) \cdot \nabla s_\e(y) \: \d y} \\
+ \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(\e x) }(\e x,y) \: \d s(y)} + \int_{\D_1}{ \varphi_\e(z) \: L_{A(\e x) }(\e x,\e z) \: \d s(z)},
\end{multline*}

where $\varphi_\e \in \widetilde{H}^{-1/2}(\D_1)$ is the function <ref> introduced in the course of Step 1.
We recast this equation in the form

\begin{equation}\label{eq.inteq}
T_\e \varphi_\e = -u_0(0) + \eta_\e~,
\end{equation}

where $T_\e: \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1) $ is the integral operator defined by

\begin{equation}\label{eq.defTe2ddir}
T_\e \varphi (x) =  \int_{\D_1}{ \varphi(z) \: L_{A(\e x) }(\e x,\e z) \: \d s(z)}~,
\end{equation}

and where the remainder $\eta_\e \in H^{1/2}(\D_1)$ is given by

\begin{multline}\label{eq.defetae}
\eta_\e(x) = \Big( u_0(0) - v_0(\e x) \Big)  +  \int_{\partial {\mathcal O} \setminus {\mathcal U}}{ A(\e x) \nabla_y L_{A(\e x)}(\e x,y) \cdot n(y) s_\e(y)\:\d s(y) }  \\
-  \int_{{\mathcal O}}{(A(\e x)-A(y)) \nabla_y L_{A(\e x) }(\e x,y) \cdot \nabla s_\e(y) \: \d y}  - \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(\e x) }(x,y) \: \d s(y)} ~.
\end{multline}

Step 3. We infer the asymptotic behavior of $\varphi_\e$ from the analysis of the integral equation <ref>.
The key ingredients in this direction are the next two lemmas; for clarity, their proofs are postponed to the end of this section.

The quantity $\eta_\e \in H^{1/2}(\D_1)$, defined in <ref>, satisfies

\begin{equation}\label{eq.cvetaeps}
\eta_\e \xrightarrow{\e \to 0} 0 \text{ strongly in } H^{1/2}(\D_1)~.
\end{equation}

The following asymptotic expansion holds

$$ \sup\limits_{\varphi \in \widetilde{H}^{-1/2}(\D_1) \atop || \varphi ||_{ \widetilde{H}^{-1/2}(\D_1)} \leq 1} \left\lvert\left\lvert \: T_\e \varphi - \frac{1}{\pi \gamma(0)} (  |\log \e|  + \alpha) \int_{\D_1}{\varphi \: \d s} - \frac{2}{\gamma(0)} S_1 \varphi \: \right\lvert\right\lvert_{H^{1/2}(\D_1)} \xrightarrow{\e \to 0}0~, $$

where $\alpha= \frac{1}{2} \log \gamma(0)$ and $S_1 : \widetilde{H}^{-1/2}( \D_1) \to H^{1/2}(\D_1)$ is the self-adjoint operator defined by

\begin{equation}\label{eq.S1}
S_1\varphi(x) = - \frac{1}{2\pi}\int_{\D_1}{\log|x-y| \varphi (y) \: \d s(y)}, \:\: x\in \D_1~.
\end{equation}

By use of these results, the integral equation <ref> may be rewritten

\begin{equation}\label{eq.inteqSe2}
(|\log \e | + \alpha) \langle \varphi_\e,1 \rangle + 2\pi S_1 \varphi_\e + R_\e \varphi_\e= - \pi \gamma(0) u_0(0) + \pi \gamma(0) \eta_\e~,
\end{equation}

where $\eta_\e$ converges to $0$ strongly in $H^{1/2}(\D_1)$ and $R_\e: \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1)$
is a sequence whose operator norm converges to $0$. The study of the approximate version <ref> of our integral equation <ref>
is based on yet another lemma, whose proof is also postponed.

* The operator $S_1 : \widetilde{H}^{-1/2}( \D_1) \to H^{1/2}(\D_1)$
is invertible.

* For $\e >0$ small enough, the operator $V_\e : \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1)$, defined by

$$ V_\e \varphi = (|\log \e |+  \alpha ) \langle \varphi, 1 \rangle + 2\pi S_1\varphi~,$$

is invertible with the uniformly bounded inverse

\begin{equation}\label{eq.Vem1}
V_\e^{-1} g = \frac{1}{2\pi}S^{-1}_1 g - \frac{(|\log \e| + \alpha) \: \langle S_1^{-1} g ,1 \rangle }{2\pi + ( |\log \e| +\alpha)  \:  \langle S_1^{-1} 1 ,1 \rangle} \frac{1}{2\pi}S_1^{-1} 1~,
\end{equation}

$\langle S_1^{-1} 1 ,1 \rangle = 2\pi/\log 2$. In particular,

\begin{equation}\label{eq.moyVe}
\langle V_\e^{-1} g, 1\rangle = \frac{ \langle S_1^{-1}g ,1 \rangle}{2\pi+(|\log \e| +\alpha) \langle S_1^{-1} 1, 1 \rangle}~.
\end{equation}

Since the operator norm of $R_\e: \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1)$ tends to $0$,
invoking <ref>, <ref>,
and a Neumann series for the solution of <ref>,
we see that the function $\varphi_\e \in \widetilde{H}^{-1/2}(\D_1)$ satisfies

$$ \varphi_\e = -\pi \gamma(0) u_0(0) V_\e^{-1} 1 + V_\e^{-1} \widetilde{\eta_\e}~,$$

for a sequence $\widetilde{\eta_\e}$ converging to $0$ strongly in $H^{1/2}(\D_1)$.
In particular, there exists a constant $C$ such that

\begin{equation}\label{eq.stabphie}
|| \varphi_\e ||_{\widetilde{H}^{-1/2}(\D_1)} \leq C~.
\end{equation}

Moreover, using <ref>, we calculate

\begin{equation}\label{eq.moyphie}
\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\langle \varphi_\e,1\rangle &=& -\pi \gamma(0) u_0(0)  \langle V_\e^{-1} 1, 1 \rangle + \langle V_\e^{-1} \eta_\e , 1 \rangle\\
&= & -\frac{ \pi}{|\log \e|} \gamma(0) u_0(0) +  \o\left( \frac{1}{|\log\e |} \right)~,
\end{array}
\end{equation}

which is the needed information about $\varphi_\e$ for the following Step 4.

Step 4. We pass to the limit in the initial representation formula <ref>.
Since $x$ does not belong to $\Sigma \cup \{ 0 \}$, we obtain from the estimate <ref> and a Taylor expansion of the smooth function $y \mapsto N(x,y)$ in a neighborhood of $0$

\begin{array}{>{\displaystyle}cc>{\displaystyle}l}
\left\lvert \int_{\D_1}{\varphi_\e(z) (N(x,T(\e z)) - N(x,0)) \d s(z)}  \right\lvert & \leq & || \varphi_\e||_{\widetilde{H}^{-1/2}(\D_1)}  || N(x,T(\e \cdot)) - N(x,0) ||_{H^{1/2}(\D_1)} \\
&\leq & C\e~,
\end{array}

and so

$$ u_\e(x) = u_0(x) + \left( \int_{ \D_1 }{ \varphi_\e (z) \d s(z)}\right) N(x,0) +  \O(\e)~.$$

Hence the desired expansion <ref> follows from <ref>.

We now provide the proofs of the missing links in the above analysis.

We seek a function $L_A(x,y)$ that satisfies, for any point $x \in H$, and any smooth function $\psi \in {\mathcal C}^\infty_c(\overline H)$,

\begin{equation}\label{eq.varfL2}
\psi(x) = \int_H{A \nabla_y L_A(x,y) \cdot \nabla \psi(y) \: \d y }~.
\end{equation}

Introducing the symmetric, positive definite matrix $M \in \R^{2\times 2}$ for which $M^{-2} = A$, we may write the latter requirement as follows

$$\forall \psi \in {\mathcal C}^\infty_c(\overline H), \:\: \psi(x) = \int_H (M^{-1} \nabla_y L_A(x,y)) \cdot (M^{-1} \nabla \psi(y)) \: \d y ~.$$

Changing variables and using test functions $\psi \in {\mathcal C}^\infty_c(\overline H)$ of the form $\psi(y) = \widetilde{\psi}(My)$, $\widetilde{\psi} \in {\mathcal C}^\infty_c(M\overline H)$, we arrive at

$$\forall x \in MH, \:\: \forall \widetilde{\psi} \in {\mathcal C}^\infty_c(M \overline H), \:\: \widetilde{\psi}(x) = \int_{MH}{ |\det M^{-1}| \: \nabla_z (L_A(M^{-1} x,M^{-1}z)) \cdot \nabla \widetilde{\psi}(z) \: \d z }~.$$

Therefore, it suffices that the function $(x,y) \mapsto \frac{1}{|\det M|} L_A(M^{-1}x, M^{-1}y)$ be a Neumann function for the Laplacian  on the rotated half-space $MH$. Such a function can easily be constructed  by reflection – more precisely

\begin{equation}\label{eq.LAconst}
\frac{1}{|\det M|}  L_A(M^{-1}x, M^{-1}y) = G(x,y) + G(x,s_M(y)),
\end{equation}

$$s_M(y) := y - 2\left(y \cdot  \frac{M^{-1}e_2}{|M^{-1}e_2|}\right)  \frac{M^{-1}e_2}{|M^{-1}e_2|} $$

is the symmetric image of a point $y \in MH$ with respect to the hyperplane $\partial (MH)$ (whose unit normal vector equals $\frac{M^{-1} e_2}{| M^{-1}e_2|}$). The desired expression <ref> for $L_A(x,y)$ follows immediately.

We next turn to the proof of <ref> concerning the remainder $\eta_\e$.

The definition of $\eta_\e$ as the right-hand side of <ref> features four terms, which we denote by $I^i_\e(x)$, $i=1,\ldots,4$, respectively.
We prove that each of these converges to $0$ strongly in $H^{1/2}(\D_1)$.

First, using the smoothness of $v_0$ near the point $0$ together with the fact that $v_0(0) = u_0(0)$, we get

\begin{equation}\label{eq.u0v0e}
I_\e^1(x) := u_0(0) - v_0(\e x) \xrightarrow{\e \to 0} 0 \text{ strongly in } H^{1/2}(\D_1)~.
\end{equation}

Secondly, the term

\begin{equation}\label{eq.I2rep}
I_\e^2(x) :=  \int_{\partial {\mathcal O} \setminus {U}}{ A(\e x) \nabla_y L_{A(\e x)}(\e x,y) \cdot n(y) s_\e(y)\:\d s(y) }
\end{equation}

is an integral over the set $\partial {\mathcal O} \setminus {U}$, which lies “far” from  $\D_\e$.
Since the function $(x,y) \mapsto A(\e x) \nabla_y L_{A(\e x)}(\e x,y)$ is smooth for $x \in \D_1$ and $y \in \partial {\mathcal O} \setminus {U}$ (uniformly with respect to $\e$), and since
$s_\e \to 0$ strongly in $H^1(\O)$ by virtue of <ref> ( <ref>) and <ref>, it follows easily that $I_\e^2(x) \to 0$ strongly in $H^{1/2}(\D_1)$.

For the very same reason, the term

\begin{equation}\label{eq.I3rep}
I_\e^4(x) := - \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(\e x) }(x,y) \: \d s(y)}
\end{equation}

also converges to $0$ strongly in $H^{1/2}(\D_1)$. Finally, we consider the term

\begin{equation}\label{eq.Kepsrep}
I^3_\e(x) :=  - \int_{{\mathcal O}}{(A(\e x)-A(y)) \nabla_y L_{A(\e x) }(\e x,y) \cdot \nabla s_\e(y) \: \d y}~.
\end{equation}

Using <ref> and the subsequent <ref>, we see that, for $x\in \D_1$ and $y \in {\mathcal O}$,

$$ \nabla_y L_{A(\e x)}(\e x , y) =
\frac{-1}{\pi \sqrt{\det(A(\e x))}} \frac{M^2(\e x) (y-\e x)}{|M(\e x)(y-\e x)|^2}.

As the matrix field $A(y)$ is smooth, there exists a constant $C>0$ such that:

$$\forall x \in \D_1, \: y \in {\mathcal O}, \quad ||A(\e x) - A(y)|| \leq C|\e x - y|~,$$

where $|| \cdot  ||$ denotes any matrix norm.  We then estimate

\begin{equation}\label{eq.estI4edir}
|I^3_\e(x)| \leq C \int_{{\mathcal O}}{ |\nabla s_\e (y)|\: \d y} \leq C || s_\e ||_{H^1({\mathcal O})}~.
\end{equation}

Invoking again <ref> (<ref>) and <ref>, we conclude that

$$ |I_\e^3(x)| \xrightarrow{\e \to 0} 0 \text{ uniformly in } x \in \D_1~,$$

which implies, in particular, the strong $L^2(\D_1)$ convergence of $I_\e^3$ to $0$. It remains to prove that $I_\e^3$ converges to $0$ strongly in $H^{1/2}(\D_1)$.
To this end, we return to the formula <ref>, which reads

\begin{multline*}
I_\e^3\left(\frac{x}{\e}\right) = -s_\e(x) - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) s_\e(y)\:\d s(y) }   +
\int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} \\
+ \int_{\widehat{\Gamma_N}}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)}, \quad x \in \D_\e~,
\end{multline*}

where we have taken advantage of the fact that $A\nabla s_\e \cdot n$ belongs to $\widetilde{H}^{-1/2}(\D_\e)$ (and vanishes in $\widehat{\Gamma_N}\setminus \D_\e$) to express the last integral in the above right-hand side as an integral on the whole  set $\widehat{\Gamma_N}$.
Using the mapping properties of the integral operator with kernel $L_{A(x) }(x,y)$ (see <ref>), we obtain

$$ \left\lvert I^3_\e\left(\frac{\cdot }{\e}\right) \right\lvert_{H^{1/2}(\D_\e)} \leq C || s_\e ||_{H^1(\mathcal O)}~,$$

where we recall the definition <ref> of the semi-norm $|\cdot|_{H^{1/2}(\D_\e)}$.
Changing variables in the definition of this semi-norm to rescale the above left-hand side, we now get

$$ \left\lvert I^3_\e \right\lvert_{H^{1/2}(\D_1)} \leq C || s_\e ||_{H^1(\mathcal O)}~.$$

We conclude from <ref> ( <ref>) and <ref>, that $\left\lvert I_\e^3 \right\lvert_{H^{1/2}(\D_1)} \to 0$.
Finally, as we already know that $I_\e^3$ converges to $0$ strongly in $L^2(\D_1)$, it follows that $I_\e^3$ converges to $0$ strongly in  $H^{1/2}(\D_1)$, which completes the proof of the lemma.

We next turn to the proof of the approximation <ref>.

Let $D \subset \R^2$ be a smooth bounded domain,
whose boundary $\partial D$ is a closed curve containing $\D_1$ as a subset.
Since $\widetilde{H}^{-1/2}(\D_1)$ is the space of distributions in $\D_1$ whose extension by $0$ to $\partial D$
belongs to $H^{-1/2}(\partial D)$, and since $H^{1/2}(\D_1)$ is the space of restrictions to $\D_1$ of elements from $H^{1/2}(\partial D)$ (see <ref>), it is enough to prove that the asymptotic formula in the statement of <ref> holds when all the operators at play are seen as operators from $H^{-1/2}(\partial D)$ into $H^{1/2}(\partial D)$.

To this end, let us first simplify the expression <ref> for the function $L_{A(\e x)}(\e x , \e y)$ featured in the definition <ref> of the operator $T_\e$

$$ \forall x,y \in \D_1, \: x \neq y, \quad L_{A(\e x)}(\e x,\e y) = \frac{-1}{\pi \sqrt{ |\det A(\e x)|}} \log \left\lvert M(\e x) ( \e x-\e y)\right\lvert~ .$$

The matrix field $A$ is given by <ref>, and its definition readily implies that $A(\e x)$ tends to $\gamma(0) \I$ in ${\mathcal C}^k(V)$ for any integer $k \geq 0$ and any relatively compact open neighborhood $V$ of $0$ in $\R^2$.
Hence, $T_\e$ may be decomposed as:

\begin{multline}\label{eq.Tedecomp2d}
T_\e \varphi =  \frac{1}{\pi \gamma(0)}  (|\log \e| + \alpha) \int_{\D_1}{\varphi \: \d s} + \frac{2}{\gamma(0)} S_1 \varphi \\
+ \left(  \frac{1}{\pi \sqrt{ |\det A(\e x)|}} -  \frac{1}{\pi \gamma(0)}  \right) ( |\log \e| + \alpha)  \int_{\D_1}{\varphi \: \d s} + T_{K_\e} \varphi~,
\end{multline}

where $T_{K_\e}$ is the integral operator with kernel $K_\e(x,x-y)$, and $K_\e$ is given by

$$ K_\e(x,z) := \frac{1}{\pi \gamma(0)} \log|z| - \frac{1}{\pi \sqrt{ |\det A(\e x)|}} \log \left\lvert \sqrt{\gamma(0)} M(\e x) z \right\lvert~.$$

The first two terms in the right-hand side of <ref> correspond to the desired limiting behavior for $T_\e$, and the third term is easily seen to converge to $0$ as an operator from $H^{-1/2}(\partial D)$ into $H^{1/2}(\partial D)$.
We then focus on the operator $T_{K_\e}$. It is easy to verify that $K_\e$ is a homogeneous kernel of class $-1$ in the sense of <ref>.
Hence, <ref> implies that $T_{K_\e}$ maps $H^{-1/2}(\partial D)$ into $H^{1/2}(\partial D)$. Note that we may modify $K_\e$, in such a way that it vanishes outside a sufficently large compact set (since the definition of $T_{K_\e}$ only involves values $K_\e(x,x-y)$ for $x,y \in \partial D$). With this modification we have

$$\sup\limits_{|\alpha| \leq k \atop |\beta| \leq k}\sup \limits_{x\in \R^d} \sup\limits_{|z|=1} \left\lvert \frac{\partial^\alpha}{\partial x^\alpha} \frac{\partial^\beta}{\partial z^\beta} K_\e(x,z) \right\lvert \xrightarrow{\e \to 0} 0~,$$

for any integer $k$.
In view of <ref>, $T_{K_\e}$ converges to $0$ in the operator norm

$$ \sup\limits_{\varphi \in H^{-1/2}(\partial D) \atop || \varphi ||_{H^{-1/2}(\partial D)} \leq 1} || T_{K_\e} \varphi ||_{H^{1/2}(\partial D)} \xrightarrow{\e \to 0} 0~,$$

which finishes the proof.

Proof of (i). Let $D \subset \R^2$ be a smooth bounded domain,
whose boundary $\partial D$ is a closed curve containing ${\D}_1$ as a subset.
We also introduce another bounded Lipschitz domain $V \subset \R^2$ such that $D \Subset V$, and a smooth cut-off function $\chi \in {\mathcal C}^\infty_c(\R^2)$
such that $\chi \equiv 1$ on a neighborhood of $\overline{D}$ and $\chi \equiv 0$ on $\R^2 \setminus \overline V$.

The proof follows an idea in <cit.>; it relies on the connection between $S_1$ and the single layer potential
${\mathcal S}_D : H^{-1/2}(\partial D) \to H^{1}_{\text{loc}}(\R^2) $ associated with  $D$, as defined in <ref>. More precisely

$$\forall \varphi \in \widetilde{H}^{-1/2}({\D}_1), \:\: S_1 \varphi = ({\mathcal S}_D \varphi) \lvert_{{\D}_1}  ~,$$

where the density $\varphi$ in the right hand side is extended by $0$ outside $\D_1$. We first show that $S_1: \widetilde{H}^{-1/2}({\D}_1) \to H^{1/2}({\D}_1) $
is a Fredholm operator with index $0$ by adapting the argument of the proof of Th. 7.6 in <cit.>.
The classical mapping properties of the single layer potential ${\mathcal S}_D$ imply that there exists a constant $C >0$ such that
for any density $\varphi \in  \widetilde{H}^{-1/2}({\D}_1) $, the associated potential $u  = {\mathcal S}_D\varphi \in H^1_{\text{loc}}(\R^2)$ satisfies

$$ || \chi u ||_{H^1(\R^2)} \leq C || \varphi ||_{H^{-1/2}(\partial D)}~. $$

Conversely, we infer from the jump relations <ref> of the single layer potential that

\begin{equation}\label{eq.estvphichiu}
|| \varphi ||_{H^{-1/2}(\partial D)} = \left\lvert \left\lvert  \frac{\partial (\chi u)^+}{\partial n} -  \frac{\partial (\chi u)^-}{\partial n}  \right\lvert\right\lvert_{H^{-1/2}(\partial D)} \leq C || \chi u||_{H^1(\R^2)}~.
\end{equation}

Now, using again <ref> together with integration by parts, we obtain that, for $\varphi \in \widetilde H^{-1/2}(\D_1)$,

\begin{eqnarray*}
\langle S_1\varphi, \varphi \rangle = \int_{\partial D}{{\mathcal S}_D \varphi \: \varphi \: \d s} &=& \int_{\partial D}{ \chi u \left(\frac{\partial (\chi u)^+}{\partial n} - \frac{\partial (\chi u)^-}{\partial n}\right) \: \d s} \\
&=& - \int_{\R^2}{|\nabla (\chi u ) |^2 \:\d x} - \int_{\R^2 \setminus \overline{D}}{\Delta (\chi u) \chi u \:\d x}\\
&=& - \int_{\R^2}{|\nabla (\chi u ) |^2 \:\d x} - \int_{\R^2 \setminus \overline{D}}{\Big(\Delta \chi u + 2\nabla \chi \cdot \nabla u \Big) \chi u \:\d x}\\
&=& - \int_{\R^2}{|\nabla (\chi u ) |^2 \:\d x} - \int_{\R^2 \setminus \overline{D}}{\Big(\chi (\Delta \chi) u^2 + 2u \nabla \chi \cdot \nabla (\chi u) - 2 u^2 |\nabla \chi|^2 \Big) \:\d x}~,\\
\end{eqnarray*}

which we rewrite as

\begin{equation}\label{eq.estchiu}
\int_{\R^2}{|\nabla (\chi u ) |^2 \:\d x}  = -\langle S_1\varphi, \varphi \rangle - \int_{\R^2 \setminus \overline{D}}{\Big(\chi (\Delta \chi) u^2 + 2u \nabla \chi \cdot \nabla (\chi u) - 2 u^2 |\nabla \chi|^2 \Big) \:\d x}~.
\end{equation}

The Cauchy-Schwarz inequality (and rearrangement) now implies the existence of a constant $C$ such that

$$|| \nabla (\chi u ) ||_{L^2(\R^2)^2}^2  \leq C\Big(|| S_1 \varphi ||_{H^{1/2}(\D_1)} || \varphi ||_{\widetilde{H}^{-1/2}(\D_1)} +  || u ||_{L^2(V)}^2  \Big)~.$$

A combination with <ref>, and insertion of $u= {\mathcal S}_D\varphi$, yields the existence of a constant $C$ such that, for arbitrary $\varphi \in  \widetilde{H}^{-1/2}({\D_1})$ (extended by $0$ outside $\D_1$)

\begin{equation}\label{eq.estpeetrevphi}
||\varphi ||_{\widetilde{H}^{-1/2}(\D_1)}  = ||\varphi ||_{H^{-1/2}(\partial D)} \leq C \Big( ||S_1 \varphi ||_{H^{1/2}(\D_1)} + || {\mathcal S}_D\varphi ||_{L^2(V)} \Big)~.
\end{equation}

Since the mapping $H^{-1/2}(\partial D) \ni \varphi \mapsto {\mathcal S}_D \varphi \in H^1(V)$ is continuous
and the injection $H^1(V) \to L^2(V)$ is compact, an application of  Peetre's <ref> to <ref> reveals that $S_1$ has finite dimensional kernel $\Ker(S_1) \subset \widetilde{H}^{-1/2}(\D_1)$,
and closed range $\Ran(S_1) \subset H^{1/2}(\D_1)$. Finally, since $S_1$ is self-adjoint, it follows that

$$ \Ker(S_1) = \Ran(S_1)^\perp, \text{ and so } \Ran(S_1) = \Ker(S_1)^\perp~.$$

In summary $S_1$ is a Fredholm operator with index $0$.

In order to prove that $S_1$ is invertible, it thus suffices to prove that it is injective on $ \widetilde{H}^{-1/2}({\D}_1)$.
To this end, let $\varphi \in  \widetilde{H}^{-1/2}({\D}_1)$ be such that $S_1 \varphi = 0$ on ${\D}_1$. We assume first that $\varphi$ has mean $0$, that is $\langle \varphi , 1\rangle = \int_{\partial D}{\varphi \: \d s} = 0$.
Then, the associated single layer potential ${\mathcal S}_D \varphi : \R^2 \to \R$ satisfies the decay property

\begin{equation}\label{eq.decaySDvphiS1}
| {\mathcal S}_D\varphi (x) | = {\mathcal O}\left( \frac{1}{|x|}\right) \text{ as } |x|\to \infty;
\end{equation}

see <ref>.
From the same integration by parts which led to <ref> (and which can now be carried out without introducing a cut-off function $\chi$ because of the decay property <ref>),
we obtain

\begin{equation}\label{eq.ippSDmean0}
\langle S_1\varphi, \varphi \rangle = 0 = - \int_{\R^2}{|\nabla ({\mathcal S}_D\varphi )|^2 \:\d x},
\end{equation}

and so ${\mathcal S}_D\varphi = 0$ on $\R^2$.
As a result,

$$ \varphi = \frac{\partial ({\mathcal S}_D\varphi)^+}{\partial n} - \frac{\partial ({\mathcal S}_D\varphi)^-}{\partial n}  = 0 \text{ on } \partial D, $$

as desired. Finally, let us consider the general case where $S_1\varphi = 0$ but $\langle \varphi, 1 \rangle$ does not necessarily vanish.
From <ref>, the function $\varphi_c \in \widetilde{H}^{-1/2}({\D}_1)$ defined by:

$$ \varphi_c(x_1) = \frac{1}{\pi\sqrt{1-x_1^2}}$$

is such that:

$$\langle \varphi_c, 1\rangle = 1, \text{ and } S_1 \varphi_c =  \frac{\log 2}{2\pi} \text{ on } \D_1.$$

Hence, the element $\varphi_0 \in  \widetilde{H}^{-1/2}(\D_1)$ defined by:

$$ \varphi_0 = \varphi - \langle\varphi,1\rangle \varphi_c,$$

satisfies the following properties:

$$ \langle \varphi_0, 1 \rangle = 0 \text{ and } S_1\varphi_0 = -\langle \varphi, 1 \rangle S_1 \varphi_c = -\frac{ \log 2}{2\pi} \: \langle\varphi,1\rangle,$$

so that $\langle S_1 \varphi_0, \varphi_0 \rangle = 0$.
The same calculation as in <ref> reveals that $\varphi_0=0$, and so
$\varphi = \langle \varphi,1 \rangle \varphi_c$.
Eventually, since $S_1 \varphi = 0$, we obtain $\langle \varphi,1\rangle = 0$, so that $\varphi = 0$, as desired.

Proof of (ii). Both formulas <ref> follow from simple calculations.

A significantly simpler proof of <ref> can be given, under the additional assumption that the boundary $\partial \Omega$ is completely flat in a fixed neighborhood $U$ of the $\omega_\e$ (i.e. $\partial \Omega \cap U = \partial H \cap U$) and that the conductivity $\gamma$ is constant in such a neighborhood.

§.§ Adaptation to the three-dimensional case

We proceed with the three-dimensional version of the general problem
described at the beginning of this section: the background and perturbed potentials $u_0$ and $u_\e$ are still characterized by the equations <ref>,
and we look for the asymptotic expansion of $u_\e$ as the size $\e$ of the subset $\omega_\e \subset \partial \Omega$, defined by <ref>, vanishes.

The counterpart of <ref> is the following. Since the proof is quite similar in most aspects, we only elaborate on the differences.

The following asymptotic expansion holds at any point $x\in \overline \Omega$, $x\notin \Sigma\cup \left\{ 0 \right\}$:

$$ u_\e(x) = u_0(x) - 4 \e \gamma(0) u_0(0) N(x,0) + \o(\e)~.$$

As in the two-dimensional case, we introduce the transported functions $v_0 := u_0 \circ T$ and $v_\e := u_\e \circ T$.
These are characterized as the unique $H^1({\mathcal O})$ solutions to the problems in <ref>,
which feature the smooth matrix field $A \in {\mathcal C}^\infty(\R^3, \R^{3\times 3})$ and source term $g \in {\mathcal C}^\infty(\mathbb{R}^3)$ defined as in <ref>.
We also introduce the error $r_\e := u_\e - u_0 \in H^1(\Omega)$ and its transformed version $s_\e := v_\e - v_0 \in H^1({\mathcal O})$. The proof of the theorem again proceeds in four steps.

Step 1. We construct a representation formula for $u_\e$ in terms of the values of $r_\e$ inside $\omega_\e$.
Arguing as in the first step of the proof of <ref>, we prove that, for any point $x \in \Omega$

\begin{equation*}\label{eq.repformulauesi3d}
u_\e(x) =u_0(x) + \int_{\omega_\e}{\gamma(y)\frac{\partial r_\e}{\partial n_y}(y) N(x,y)\:\d s(y)}~,
\end{equation*}

an identity which also holds for $x \in \partial \Omega$ in the sense of traces in $H^{1/2}(\partial \Omega)$.
Performing a change of variables based on the diffeomorphism $T$  we arrive at

\begin{equation}\label{eq.repformulauesi3d}
u_\e(x) = u_0(x) + \int_{\D_1}{ \varphi_\e(z) N(x,T(\e z))\:\d s(z)}~,
\end{equation}

where the rescaled density $\varphi_\e$ is given by

$$\varphi_\e(z) = \e^2 (A\nabla s_\e \cdot n)(\e z) \in \widetilde{H}^{-1/2}(\D_1)~.$$

Step 2. We characterize $\varphi_\e$ as the solution to an integral equation.
To this end, again, we rely on a variant of the representation formula <ref> adapted to the function $v_\e$, and obtained
with the use of a special function $y \mapsto L_{A(x)}(x,y)$ which satisfies, for given $x \in H$

\begin{equation}\label{eq.defLAx3d}
\left\{
\begin{array}{cl}
-\dv_y(A(x) \nabla_y L_{A(x)}(x,y)) = \delta_{y=x}  & \text{in } H,  \\
A(x)\nabla_y L_{A(x)} (x,y) \cdot n(y) = 0 & \text{on } \partial H.
\end{array}
\right.
\end{equation}

The construction of such a function is accomplished exactly as in the two-dimensional case; see <ref> and <ref>.
The same calculations as in Step 2 of the proof of <ref> then yield, for a.e. $x \in \D_\e$

\begin{eqnarray}\label{eq.v0De3d}
-v_0(x) = - \int_{\partial {\mathcal O} \setminus {U}}{ A(x) \nabla_y L_{A(x)}(x,y) \cdot n(y) s_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(x)-A(y)) \nabla_y L_{A(x) }(x,y) \cdot \nabla s_\e(y) \: \d y} \nonumber \\
+ \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)} + \int_{\D_\e}{(A\nabla s_\e \cdot n)(y) \: L_{A(x) }(x,y) \: \d s(y)}~,
\end{eqnarray}

which, after rescaling, reads

\begin{multline*}
-v_0(\e x) = - \int_{\partial {\mathcal O} \setminus {U}}{ A(\e x) \nabla_y L_{A(\e x)}(\e x,y) \cdot n(y) s_\e(y)\:\d s(y) }   + \int_{{\mathcal O}}{(A(\e x)-A(y)) \nabla_y L_{A(\e x) }(\e x,y) \cdot \nabla s_\e(y) \: \d y} \\
+ \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(\e x) }(\e x,y) \: \d s(y)} + \int_{\D_1}{ \varphi_\e(z) \: L_{A(\e x) }(\e x,\e z) \: \d s(z)}~,
\end{multline*}

for a.e. $x \in \D_1$. This can be recast in the form of an integral equation

\begin{equation}\label{eq.inteq3d}
T_\e \varphi_\e = -u_0(0) + \eta_\e~,
\end{equation}

where the operator $T_\e: \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1) $ is defined by

$$ T_\e \varphi (x) =  \int_{\D_1}{ \varphi(z) \: L_{A(\e x) }(\e x,\e z) \: \d s(z)}~,  $$

and $\eta_\e \in H^{1/2}(\D_1)$ denotes the remainder

\begin{multline}\label{eq.defetae3d}
\eta_\e(x) = u_0(0) - v_0(\e x)  +  \int_{\partial {\mathcal O} \setminus {U}}{ A(\e x) \nabla_y L_{A(\e x)}(\e x,y) \cdot n(y) s_\e(y)\:\d s(y) }  \\
-  \int_{{\mathcal O}}{(A(\e x)-A(y)) \nabla_y L_{A(\e x) }(\e x,y) \cdot \nabla s_\e(y) \: \d y}  - \int_{\Gamma_D}{(A\nabla s_\e \cdot n)(y) \: L_{A(\e x) }(\e x,y) \: \d s(y)}~.
\end{multline}

Step 3. We analyze the integral equation <ref>
to obtain information about the asymptotic behavior of $\varphi_\e$.
To this end, we rely on the following two lemmata, which are the exact counterparts of <ref> and <ref> in the present 3d situation;
their proofs are outlined at the end of this section.

The remainder term $\eta_\e \in H^{1/2}(\D_1)$, defined in <ref>, satisfies

\begin{equation}\label{eq.cvetaeps}
\eta_\e \xrightarrow{\e \to 0} 0 \text{ weakly in } H^{1/2}(\D_1)~.
\end{equation}

The following asymptotic expansion holds

\begin{equation}\label{eq.asymL3d}
\sup\limits_{\varphi \in \widetilde{H}^{-1/2}(\D_1) \atop || \varphi ||_{\widetilde{H}^{-1/2}(\D_1) } \leq 1}  \e \left\lvert \left\lvert T_\e \varphi
- \frac{2}{\e\gamma(0)} S_1 \varphi \right\lvert  \right\lvert_{H^{1/2}(\D_1)}  \xrightarrow{\e \to 0} 0~,
\end{equation}

where the operator $S_1 : \widetilde{H}^{-1/2}({\D}_1) \to H^{1/2}(\D_1)$ is defined by

\begin{equation}\label{eq.defS13ddir}
S_1\varphi(x) = \frac{1}{4\pi}\int_{{\D}_1}{\frac{1}{|x-y|} \varphi(y) \: \d s(y) }, \:\: x\in{\D}_1~.
\end{equation}

Using this result in combination with the integral equation <ref>, we see that the function $\varphi_\e \in  \widetilde{H}^{-1/2}(\D_1)$
satisfies the integral equation

\begin{equation}\label{eq.inteqS1dir3d}
S_1 \varphi_\e +  R_\e \varphi_\e = - \frac{\e}{2} \gamma(0) u_0(0) + \frac{\e\gamma(0)}{2} \eta_\e~,
\end{equation}

where $R_\e:  \widetilde{H}^{-1/2}(\D_1) \to H^{1/2}(\D_1)$ is a sequence of operators whose norms converge to $0$ and the sequence $\eta_\e$ converges to $0$ weakly in $H^{1/2}(\D_1)$. The study of this approximate version of our integral equation <ref>