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\begin{document}
\title[Boundary value problems on Klein surfaces]{Boundary value problems on
Klein surfaces}
\author[V.D. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address[V.D. R\u{a}dulescu]{Faculty of Applied Mathematics, AGH University
of Science and Technology, al. Mickiewicza 30, 30-059 Krak\'ow, Poland \&
Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, P.O. Box
1-764, 014700 Bucharest, Romania}
\email{\texttt{vicentiu.radulescu@imar.ro}}
\author[M. Ro\c{s}iu]{Monica Ro\c{s}iu}
\address[M. Ro\c{s}iu]{Department of Mathematics, University of Craiova,
Street A.I. Cuza No 13, 200585 Craiova, Romania}
\email{\texttt{monica\_rosiu@yahoo.com}}
\keywords{Klein surface, complex double, harmonic function, Radon-Nikodym
derivative, Dirichlet boundary condition, Neumann boundary condition.\\
\phantom{aa} 2010 AMS Subject Classification: 30F50, 35J25, 58J10}

\begin{abstract}
This survey paper deals with the unitary treatment of some classes of linear
partial differential equations on Klein surfaces. We are mainly concerned
with the study of harmonic functions with Dirichlet or Neumann boundary
condition. In such a way, the present paper extends several classical
results to the abstract setting of dianalytic manifolds of complex dimension
1. The analysis developed in this paper offers perspectives to the
qualitative analysis of other classes of linear or nonlinear elliptic
equations on Klein surfaces.
\end{abstract}

\maketitle

%\hfill\today\bigskip

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\section{Introduction}

\textcolor{red}{In the preface to the first edition of Courant-Hilbert's ``Methoden der mathematischen Physik" (see \cite{10}), R. Courant noted the danger that mathematical research would lose the initial link between the problems and methods of analysis and the physical and geometric intuition, the tendencies being to refine the methods and to extreme generalize the existing concepts.}

\textcolor{red}{Over the years, these trends led to an increasing distinction between pure and
applied mathematicians, who severely criticized each other. This constructive
criticism gave rise to the theory of real numbers and to many topological
concepts including non-orientable surfaces. It is obvious that some areas
that use mathematical methods but their object is derived from physical and
geometric intuition are disadvantaged in such a discussion.}

\textcolor{red}{The present paper  is a piece of the bridge between the theoretical
approach of the pure mathematician and the practical interest of the
engineer, physicist and applied mathematician. The main purpose is to bring
together various geometrical and physical concepts relating to surfaces
that have motivated the development of the theory of Klein surfaces.}

\textcolor{red}{Riemann surfaces, in the form of domains spread out over the complex plane
were introduced in Riemann's dissertation whose methods were developed much
further in the first edition of Riemann's paper on Abelian functions
``Theorie der Abel'schen Functionen" (see \cite{23b}), in 1857. Riemann's works
provided the basic tools to classify all compact orientable surfaces and,
more generally, to study the topology of manifolds. They are equally
important for the development of algebraic geometry and the geometric
treatment of complex analysis. As for the importance that was attached to
this topic, it suffices to say that Albert Einstein's ``general theory of
relativity" is wholly based on Riemann's ideas.}

\textcolor{red}{In his ``Extremale quasikonforme Abbildungen und quadratische Differentiale" (see \cite{30}) Teichm\"{u}ller considered the cases of oriented bordered Riemann surfaces and non-orientable Riemann surfaces. He defined the double of a oriented
bordered Riemann surfaces or of a non-orientable Riemann surfaces. These are
closed Riemann surfaces with genus depending on the original surfaces. He
introduced the notions of meromorphic functions, n-differentials and
divisors on bordered non-orientable Riemann surfaces.}

\textcolor{red}{Teichm\"{u}ller defined, through two examples, the notion of conformal
invariant for non-orientable regions.\ It is important to note that according
to Teichm\"{u}ller's definition, a conformal mapping (even in the orientable
case) preserves only the angles, but not necessarily the orientation. Thus,
such a mapping is also defined in the non-orientable case. Teichm\"{u}ller
exhibited complete systems of conformal invariants for some special
surfaces. For example, a simply connected domain with two distinguished
points in the interior has one conformal invariant, namely the Green function.}

\textcolor{red}{Teichm\"{u}ller considered special cases of the fact that one deals with
non-orientable surfaces by passing to the orientation double cover. To treat
the most general surfaces, he considered a symmetrization process on the
corresponding doubles. This idea was used to solve the main problem of
extremality, known as the Teichm\"{u}ller theorem. By using the two-sheeted
covering (an annulus) of the M\"{o}bius strip, he gets the Teichm\"{u}ller
distance. He showed that problems on the projective plane with two
distinguished points can be reduced to similar problems on the sphere with
four distinguished points. In two examples of non-orientable surfaces, M\"{o}bius strip and the projective plane, the problem of finding the conformal
invariants is lifted to the oriented double cover. Teichmuller considered
problems on the Klein bottle and lifted them to problems concerning the case
of the torus. The torus is the two-sheeted orientation covering of the Klein
bottle.}

\textcolor{red}{The genesis and development of the idea of symmetry are related to Lie's and
Klein's research which were inspired by their deep interest in the theory of
groups and in various aspects of the notion of symmetry. According to
Klein's Erlangen program, a geometry is determined by a ``domain of action"
(the plane, space, etc.) and a ``group of automorphisms" (or a symmetry
group) acting on the domain. When we change the symmetry group we change the
geometric scheme under consideration, namely we obtain a new ``geometry".}

\textcolor{red}{Thus, the main difference between, say, Euclidean and hyperbolic geometry is
not the possibility of constructing one or more lines passing through a
point and not intersecting a given line, but the difference in the structure
of the respective groups of symmetries of Euclidean and hyperbolic geometry.
Therefore the object of the geometry is the study of those properties of a
domain which are preserved by the transformations in a symmetry group. The
description of all possible geometries is an open problem.}

\textcolor{red}{In the same way, classifying non-classical topological compact surfaces is the same
thing as classifying all orientation reversing involutions of a classical
compact surface.}

\textcolor{red}{This brings us to the interesting question of the possible global forms of
various (say, two-dimensional) geometric systems (Euclidean, hyperbolic,
elliptic) first stated (in connection with Euclidean geometry) by the
outstanding geometer W. K. Clifford. Today this question is known as the
Clifford-Klein problem and the possible global forms of geometries are
called Clifford-Klein forms.}

\textcolor{red}{It is known that there are only two spatial forms of two-dimensional
elliptic geometry (the sphere and the elliptic plane), but there are as many
as five forms of two-dimensional Euclidean geometry (the ordinary Euclidean
plane, the infinite M\"{o}bius strip, the infinite cylinder, the torus and
the so-called Klein bottle). Finally, there are infinitely many forms of
two-dimensional hyperbolic geometry.}

\textcolor{red}{In this context, non-orientable surfaces are a possible geometric system
capable of ``modelling" the real shape of the universe surrounding us. For
more details, see Munzner's video \cite{16a}, about the different topological structures
that a three-dimensional universe could have and Weeks's book \cite{31}, which fills the gap between the simplest examples, such as the Mobius
strip and the Klein bottle, and the sophisticated mathematics found in
upper-level college courses.}

\textcolor{red}{Spencer and Schiffer in their advanced monograph ``Functionals of Finite
Riemann Surfaces", extended the investigation of finite Riemann surfaces
from the point of view of functional analysis, that is, the study of the
various Abelian differentials of the surface in their dependence on the
surface itself.}

\textcolor{red}{The methodology that Schiffer and Spencer employed is characterized by Ahlfors'
next comment: ``such a surface has a double, obtained by
reflection across the boundary, and one of the main features of the book is
the systematic use of this symmetrization process".}

The notion of \textit{Klein surface} goes back to Felix Klein due to his
closing remarks in \cite{12}, even though one does not find a definition of
a Klein surface there. Klein surfaces generalize Riemann surfaces and they
are dianalytic manifolds of complex dimension 1. Roughly speaking, a Klein
surface is a surface on which the notion of angle between two tangent
vectors at a given point is well-defined, and so is the angle between two
intersecting curves on the surface.

\textcolor{red}{Basic function theory on Klein surfaces and the relation between compact
Klein surfaces and real algebraic function fields were developed in the
monograph ``Foundations of the Theory of Klein surfaces" (see \cite{3}) by N.~Alling and N.~Greenleaf. They showed that every Klein surface can be represented as the
quotient of a Riemann surface by a conjugate analytic involution. Thus, it is
natural to extend on Klein surfaces the most fundamental problems in
engineering, physics and other sciences. Alling and Greenleaf were the ones
who introduced the name ``Klein surface".}

\textcolor{red}{Our approach is an alternative theory to the standard theory given by Alling
and Greenleaf and aims at the natural imbedded of calculus on Klein surfaces
in the well known Cartan's model of calculus on manifolds. We have developed
this theory because of the unusual behavior from the analytical point of
view of the Alling and Greenleaf's results. For instance, functions are not
usual functions but equivalence classes of families of meromorphic functions
relative to dianalytic atlases. Such a family defines an usual function if
and only if all its members are the same real constant. The meromorphic
differentials are also equivalence classes of families of functions
satisfying some compatibility conditions that lead to the impossibility of
defining a consistent integral on Klein surfaces (see \cite[Theorem 1.10.4]{3}).}

\textcolor{red}{We follow Schiffer and Spencer's method to study the objects on Klein
surfaces by means of the complex double, whose existence and uniqueness are
demonstrated in \cite{3}.}

\textcolor{red}{We are enabled to bring together systematically and concisely the concepts
of the Green and Neumann functions, the harmonic kernel function and the
harmonic measure and to build from them an elegant generalization for the
basic ideas of boundary value problems on Klein surfaces.}

The main objectives of study in this paper are the Dirichlet problem and the
Neumann problem for harmonic functions on Klein surfaces. The technique is
based on the fact that according to a classical result due to Klein, the
boundary value problems on a Klein surface can be reduced to similar
problems on its complex double. This process has many advantages, starting
from the fact that complex double is a symmetric Riemann surface, that is, a
Riemann surface endowed with a fixed point free antianalytic involution.
Consequently, we obtain harmonic functions on a Klein surface by adding
together a pair of harmonic functions on the symmetric Riemann surface,
whose singularities lye at symmetric points. In our study, we use methods
that have wide applicability in function theory and partial differential
equations.

The symmetric conditions on the boundary determine symmetric solutions on
the complex double, which lead to solutions for the similar problems on the
Klein surface. Specifically, in the case of Klein surfaces, the formula for
the solution of the Dirichlet problem is expressed in terms of an analogue
of the Green function, which has the symmetry in argument and parameter. In
these terms, we extend the use of the Green function to the study of the
harmonic measure on a Klein surface. That is why we distinguish the method
to solve the Dirichlet problem for harmonic functions on a Klein surface,
once the harmonic measure on a symmetric Riemann surface is known. This
procedure generates an explicit formula for the solution of the Dirichlet
problem on a Klein surface, which is similar to the Poisson integral. At the
same time, we rewrite the Radon--Nikodym derivative of harmonic measure
against the symmetric arc length.

The corresponding solution of the Neumann problem for harmonic functions on
a Klein surface is expressed in terms of an analogue of a Neumann function.

The harmonic kernel function is related to the classical domain functions,
such as the Green function and the Neumann function on a Klein surface. In
such a way it is possible to solve both boundary value problems of potential
theory on a Klein surface, once the harmonic kernel function on a symmetric
Riemann surface is known.

We refer to Krantz \cite{15a,15b,14,14b} for an excellent exposition of
various topics at the interplay between complex analysis and partial
differential equations.

\textcolor{red}{The study of objects on Klein surfaces is an important part of surface
topology due to the applications of these surfaces in several fields of
science such as quantum physics, chemistry and biology. Indeed, for a
physicist, M\"{o}bius's band and Klein's bottle are essential elements in
the so-called annulment of divergences (see \cite{11b}). In chemistry, the recent
synthesis and the ``half-cutting" of a molecular M\"{o}bius strip (see \cite{32})
was considered as a spectacular event, described as ``the most topologically
stimulating molecular structure synthesized to date" it catalyzed the birth
of extrinsic graph theory, dealing with topological chirality, a field now
burgeoning in mathematics.}

\textcolor{red}{The study of liquid crystals is another field where Klein surfaces have
surprisingly materialized themselves. In the so-called nematic liquids, the
molecules form ribbons which may or may not be orientable (see \cite{9b}). A
systematic topological analysis highlighted the double topological character
of distortions in liquid crystals differentiated for ``energetic reasons" (see \cite{9a}). If we consider the potential function of some form of internal energy
of the ribbons, then the normal derivative on the border characterizes the
flow of energy across the border. It may be necessary to determine this
potential knowing the respective flow or the values of the potential on the
border. These are boundary value problems which will be solved in this paper.}

\textcolor{red}{The natural tendency of some macromolecules to store energy through
distortions, a fact well known to chemists, might be the cause itself for
the formation of non-orientable strings, thus making obvious the practical
need of dealing with boundary value problems related to them.}

\textcolor{red}{A unified principle for science that works with dualism is presented in
terms of torsion fields and the non-orientable surfaces, namely the Klein
Bottle, the M\"{o}bius strip and the projective plane, in (see \cite{23a}). This
principle is applied to the complex numbers and cosmology, to non-linear
systems integrating the issue of hyperbolic divergences with the change of
orientability, to the biomechanics of vision and the mammal heart, to the
morphogenesis of crustal shapes on Earth in connection to the wavefronts of
gravitation, elasticity and electromagnetism, to pattern recognition of
artificial images and visual recognition, to neurology and the topographic
maps of the sensorium, to perception, in particular of music.}

\textcolor{red}{As it is noticed in (see \cite{11a}), these are the types
of problems which contribute to a unifying treatment of orientable and
non-orientable surfaces, not only from the topological point of
view (see \cite{8} and \cite{29}) but also from an analytical point of view (see \cite{28} and \cite{3}).}

\section{Klein surfaces and symmetric Riemann surfaces}

Klein surfaces are the most general two-manifolds that support harmonic
functions. In order to be able to extend results about boundary value
problems for harmonic functions on Riemann surfaces to Klein surfaces, we
have to review some results on the topology of surfaces, Klein surfaces, and
the uniformization of Riemann surfaces. The history of Klein surfaces is
going back to Klein (see \cite{12}) who considered the group of conformal
maps of the Klein bottle and other non-orientable surfaces. In their
monograph, Schiffer and Spencer (see \cite{28}) did the first modern study
of the surfaces endowed with dianalytic structures. Much of the material of
this Section will be presented without proofs and will be completed with
references to proofs. The main reference to topology of surfaces is the
monograph of Ahlfors and Sario (see \cite{2}).

A connected topological Hausdorff space $\mathcal{X}$ is a \textit{surface
with boundary} if every point $\widetilde{P}\in \mathcal{X}$ has an open
neighborhood $\widetilde{U},$ which is homeomorphic to a relatively open
subset of the closed upper half-plane. A homeomorphism $h:\widetilde{U}%
\rightarrow h(\widetilde{U})$ is called a \textit{local parameter} at the
point $\widetilde{P}\in \widetilde{U}$. The boundary $\partial \mathcal{X}$
of $\mathcal{X}$ consists of those points $\widetilde{P}\in \mathcal{X},$
such that $h(\widetilde{P})\in \mathbb{R,}$ for all the local parameters $h$
at the point $\widetilde{P}.$ The pair $(\widetilde{U},h)$ is called a
\textit{chart}. Let $h_{i}:\widetilde{U}_{i}\rightarrow h_{i}(\widetilde{U}%
_{i})$ and $h_{j}:\widetilde{U}_{j}\rightarrow h_{j}(\widetilde{U}_{j})$ be
two local parameters, such that $\widetilde{U}_{i}\cap \widetilde{U}_{j}\neq
\emptyset $, $i,j\in I.$ The mapping $h_{i}\circ h_{j}^{-1}$ $:$ $h_{j}(%
\widetilde{U}_{i}\cap \widetilde{U}_{j})$ $\rightarrow $ $h_{i}(\widetilde{U}%
_{i}\cap \widetilde{U}_{j})$ is called a \textit{transition function}.

Let $A$ and $B$ be nonempty open sets in the closed upper half-plane. A
continuous map of $A$ into $B$ is analytic on $A$ (resp., antianalytic on $A$%
) if it extends to an analytic (resp., antianalytic) function on some
neighborhood of $A$ in $\mathbb{C}$ into $\mathbb{C}.$ If $f$ or the complex
conjugate of $f$ is analytic on each connected component of the set $A$,
then $f$ is called \textit{dianalytic} on $A.$

An\textit{\ atlas} of the surface $\mathcal{X}$\textit{\ }is a family $%
\mathcal{A}=\{(\widetilde{U}_{i},h_{i})\left\vert i\in I\right. \}$ of
charts, where $(\widetilde{U}_{i})_{i\in I}$ is an open cover of $\mathcal{X}
$. The atlas $\mathcal{A}$ is \textit{dianalytic }if all of its transition
functions are dianalytic. Two dianalytic atlases $\mathcal{A}$ and $\mathcal{%
B}$ are called \textit{equivalent} if $\mathcal{A}$ $\cup \mathcal{B}$ is a
dianalytic atlas as well. An equivalence class $A$ of dianalytic atlases of $%
\mathcal{X}$ is called a \textit{dianalytic structure} on $X$.

A \textit{Klein surface} is a surface $\mathcal{X}$ with boundary endowed
with a dianalytic structure $A$\textit{\ }and will be denoted by $X.$
Observe that a classical Riemann surface is an orientable Klein surface with
empty boundary.

The main tool in our study is the complex double of a Klein surface. Details
about the history of this concept may be found in \cite{28} and for some of
its applications see \cite{3}, \cite{8} and \cite{29}.

Let $X$ be a\ Klein surface endowed with the maximal atlas $A=\{(\widetilde{U%
}_{i},h_{i})\left\vert i\in I\right. \}.$ We recall the construction of the
complex double of $X,$ which we shall use in order to apply results about
Riemann surfaces to Klein surfaces. We consider the disjoint union $S=%
\underset{i\in I}{\cup }\widetilde{U}_{i}.$ Let $(\widetilde{U}_{i},h_{i})$
and $(\widetilde{U}_{j},h_{j})$ be two charts, such that $\widetilde{U}%
_{i}\cap \widetilde{U}_{j}\neq \emptyset $. For a point $\widetilde{P}\in
\widetilde{U}_{i}\cap \widetilde{U}_{j}$, the set $S$ has two points which
both correspond to the point $\widetilde{P}$, namely the point $\widetilde{P}%
\in \widetilde{U}_{i}$ and the same point $\widetilde{P}\in \widetilde{U}%
_{j}.$We denote the latter one with $\widetilde{P}^{\ast }$. Next, we
identify the points $\widetilde{P}$ and $\widetilde{P}^{\ast },$ if the
corresponding transition function is analytic. If $\partial X\neq \emptyset
, $ then there are two points lying over each boundary point of $X $.
Identifying these two points, we obtain a surface $O_{2}$ which is called the%
\textit{\ complex double} of the surface $X$. The two points of $O_{2}$
which lie over the same point of $X$ are called \textit{symmetric points} of
$O_{2}.$ For more details, see Alling and Greenleaf \cite{3}.

Similar to the orientable case (see \cite{16}), it is obtained that the
cover group of the double cover $\pi :O_{2}\rightarrow X$ is generated by an
orientation reversing involution. For details, see Seppala and Sorvali \cite%
{29}.

The next theorem relates a Klein surface to its complex double. We refer to
\cite{3} for the proof and more details.

\begin{theorem}
\label{t1.1.1} Given a Klein surface $X$, there exist a double cover $\pi
:O_{2}\rightarrow X$ of the Klein surface $X$ by a Riemann surface $O_{2}$
and an antianalytic involution $k:O_{2}\rightarrow O_{2},$ with $\pi \circ
k=\pi ,$ such that $X$ is dianalytically equivalent with $O_{2}/\left\langle
k\right\rangle $, where $\left\langle k\right\rangle $ is the group
generated by $k$. Conversely, given a pair $(O_{2},k)$ consisting of a
Riemann surface $X$ and an antianalytic involution $k$, the orbit space $%
O_{2}/\left\langle k\right\rangle $ admits a unique structure of Klein
surface, such that $f:$ $O_{2}\rightarrow O_{2}/\left\langle k\right\rangle $
is a morphism of Klein surfaces, provided that one regards $O_{2}$ as a
Klein surface.
\end{theorem}

The mapping $\pi $ is a local homeomorphism at all points $P\in O_{2}$, for
which $\pi (P)\notin \partial X.$ At points lying over the boundary of $X$,
the mapping $\pi $ is a folding map similar to the mapping $x+iy\rightarrow
x+i\left\vert y\right\vert $ at the real axes. For more details about the
folding map and the morphisms of Klein surfaces, see \cite{3} and \cite{4}.

By Poincar\'{e}'s uniformization theorem, each compact Riemann surface of
algebraic genus $g\geq 2$ can be represented as an orbit space $H/\Gamma
^{+} $ of the upper half complex plane $H$. Next, $H$ is endowed with the
conformal structure induced by the group $M$ of the M\"{o}bius
transformations and the acting group $\Gamma ^{+}$ is a Fuchsian group, that
is, a discrete subgroup of $M.$ The group $\Gamma ^{+}$ can be chosen with
no elements of finite order. For details, we refer to Poincar\'{e} \cite{20}.

In his unpublished thesis, Preston proved the real counterpart of Poincar%
\'{e}'s uniformization theorem: for a Klein surface $X$ of algebraic genus $%
g\geq 2$, there exists a non-euclidean crystallographic ($NEC$ in short)
group $\Gamma ,$ that is, a discrete subgroup of the extended modular group,
such that $X$ and $H/\Gamma $ are isomorphic as Klein surfaces. This $NEC$
group can be assumed having no orientation preserving mapping of finite
order. For details, see \cite{21} and \cite{27}.

By Klein's definition, a \textit{symmetric Riemann surface, }$(O_{2},k),$ is
a Riemann surface $O_{2},$ together with an orientation reversing involution
$k:O_{2}\rightarrow O_{2}$. The involution $k$ is called a \textit{symmetry}
of $O_{2}.$ For more details about symmetries of a topologic surface, see
\cite{29}.

A set $D$ of $O_{2}$ is called \textit{symmetric} if $k(D)=D$. Thus, given $%
\Omega $ a subset of $X$, then $\pi ^{-1}(\Omega )=D$ is a symmetric subset
of $O_{2}.$

A function $f$ defined on a symmetric set is called a \textit{symmetric
function} if it is $k$-invariant, that is, $f=f\circ k$.

Next, we identify $X$ with the orbit space $O_{2}/\left\langle
k\right\rangle $ obtained by identifying $P$ with $k(P)$, for all $P\in
O_{2}.$ If $\widetilde{U}$ is a parametric disk on $X$, then $\pi ^{-1}(%
\widetilde{U})=U\cup k(U)$ is a pair of symmetric disks of $O_{2}$, hence it
is natural to consider restrictions on $U\cup k(U)$ for the local study of
the objects on $O_{2}$. Since $k$ is an involution without fixed points, one
can suppose that $U\cap k(U)=\emptyset .$

We identify the points of $O_{2}$, respectively $X$, with their images on $%
\mathbb{C}$ from the corresponding local parameters, with respect to the
relation between the dianalytic atlas on $X$ and the analytic atlases on $%
O_{2}$. Let $z$ be the local parameter on $U.$ Then $k(z)$ is the local
parameter on $k(U)$ and $\widetilde{z}=\widetilde{k(z)}=\pi (z)=\pi (k(z))=$
$\{z,k(z)\}$ is the local parameter on $\widetilde{U}$.

Let $\mathcal{F}(X)$ be the vector space of the complex functions on the
Klein surface $X$ and $\mathcal{F}_{s}(O_{2})$ the vector space of the
symmetric functions on $O_{2}.$ By Theorem \ref{t1.1.1}, we conclude that
there exists an isomorphism $\pi ^{\ast }:$ $F(X)\rightarrow F_{s}(O_{2}),$
between the vector spaces $F(X)$ and $F_{s}(O_{2}).$ Indeed, let $%
F:X\rightarrow \overline{\mathbb{C}}$ be a complex function on $X$, that can
take the value $\infty $ only on finite sets. Its lifting $f$ to $O_{2}$ is
given by
\begin{equation}
f(z)=f(k(z))=F(\widetilde{z}),\ z\in O_{2},\ \widetilde{z}=\pi (z).
\label{e1.1.1}
\end{equation}%
Then, it is easy to see that the function $\pi ^{\ast }$, defined by $\pi
^{\ast }(F)=f$ is an isomorphism.

Also, to any function $g:O_{2}\rightarrow \overline{\mathbb{C}}$, we can
associate a function $f=g+g\circ k$ which is a symmetric function on $O_{2}.$
Thus, \eqref{e1.1.1} defines a function $F$ on $X.$

Let $\widetilde{\gamma }$ be a piecewise smooth Jordan curve on a parametric
disk $\widetilde{U}.$ The curve $\widetilde{\gamma }$ has exactly two lifts
from $\pi ^{-1}(\widetilde{U}).$ If $\widetilde{\gamma }(0)=\widetilde{z}%
_{0}=\{z_{0},k(z_{0})\}$ and if $\gamma $ is the lift of $\widetilde{\gamma }
$ on $O_{2}$ from $z_{0}$, then $k\circ \gamma $ is the lift of $\widetilde{%
\gamma }$ on $O_{2}$ from $k(z_{0})$. We refer to \cite{1} for details about
covering surfaces. By definition of $\gamma $, we obtain $\pi \circ \gamma
=\pi \circ k\circ \gamma ,$ hence for any continuous real-valued function $F$
defined on $\widetilde{\gamma }$, the function $f=F\circ \pi $ is a
continuous real-valued symmetric function on $\gamma \cup k(\gamma ).$

The Euclidean lengths of the two curves $\gamma $ and its symmetric $k\circ
\gamma $, that is their lengths with respect to the metric $ds=\left\vert
dz\right\vert $, may be different. We modify this metric and get a new
metric $d\sigma $ on $O_{2},$ such that the lengths of $\gamma $ and $k\circ
\gamma ,$ with respect to the metric $d\sigma $, will be the same. We define
a symmetric metric on $O_{2}$ by
\begin{equation*}
d\sigma =\frac{1}{2}\left( ds+ds\circ k\right) .
\end{equation*}%
Then the $d\sigma $-lengths of $\gamma $ and $k\circ \gamma $ are equal. By
definition, the length of $\widetilde{\gamma }$ is the common $d\sigma $%
-length of $\gamma $ and $k\circ \gamma $. Then
\begin{equation*}
d\Sigma (\widetilde{z})=d\sigma (z)=d\sigma (k(z)),\widetilde{z}=\pi (z)\in X
\end{equation*}%
is a metric on $X$. The metric $d\Sigma $ is invariant with respect to the
group of conformal or anticonformal transition functions of $X$.

By definition,
\begin{equation*}
\underset{\widetilde{\gamma }}{\int }Fd\Sigma =\underset{\gamma }{\int }%
fd\sigma =\underset{k\circ \gamma }{\int }fd\sigma .
\end{equation*}

For more details about measure and integration on Klein surfaces, see \cite%
{6}.

Any Riemann surface $O_{2}$ of class $C^{1}$ is endowed with a \textit{%
Riemannian metric }determined by the line element
\begin{equation*}
ds=\nu \left\vert dz+\mu d\overline{z}\right\vert\,,
\end{equation*}%
where $\nu $ is a positive function. If $\mu $ is identically zero, then the
metric
\begin{equation*}
ds(z)=\nu (z)\left\vert dz\right\vert
\end{equation*}%
and the local parameter $z$ are called \textit{isothermal}.

It is known that the isothermal metric $ds$ defines a natural analytic
structure on $O_{2}$. Similar to the orientable case, the isothermal metric $%
d\sigma $ defines a dianalytic structure on the Klein surface $X$. See \cite%
{2} and \cite{16}, for details.

Next, we give an example of a symmetric isothermal metric (see Schiffer and
Spencer \cite{28}).

\begin{example}
\label{ex1.1.2} The simplest example of a Klein surface is provided by the M\"{o}bius strip. Consider $R>1$ and the annulus
\begin{equation*}
A_{R}=\left\{ z\in \mathbb{C}\left\vert \frac{1}{R}<\left\vert z\right\vert
<R\right. \right\}
\end{equation*}%
of the $z$-plane. The M\"{o}bius strip, denoted by $M,$ is obtained from $%
A_{R}$ by identifying the points $z$ and $-1/\overline{z}$. Let $%
k:A_{R}\rightarrow A_{R}$ defined by $k(z)=-1/\overline{z}.$ Then $(A_{R},k)$
is a \textit{symmetric Riemann surface and the quotient space }$%
A_{R}/\left\langle k\right\rangle $ is a M\"{o}bius strip. The M\"{o}bius
strip is obtained by cutting the ring along the real axis in the $z$-plane
and joining the two halves together along corresponding boundaries. Thus,
the annulus $A_{R}$ with points $z$ and $-1/\overline{z}$ identified is a
canonical form for the M\"{o}bius strip. The Euclidean metric%
\begin{equation*}
ds=\left\vert dz\right\vert
\end{equation*}%
is not symmetric. We define a symmetric isothermal metric on $A_{R}$ by%
\begin{equation*}
d\sigma =\frac{1}{2}\left( \left\vert dz\right\vert +\left\vert d\left( -%
\frac{1}{\overline{z}}\right) \right\vert \right)
\end{equation*}%
\begin{equation*}
=\frac{1}{2}\left( 1+\frac{1}{\left\vert z\right\vert ^{2}}\right)
\left\vert dz\right\vert .
\end{equation*}%
By definition, the metric on the M\"{o}bius strip is
\begin{equation*}
d\Sigma (\widetilde{z})=d\sigma (z)=d\sigma (k(z)
\end{equation*}%
thus,%
\begin{equation*}
d\Sigma (\widetilde{z})=\frac{1}{2}\left( 1+\frac{1}{\left\vert z\right\vert
^{2}}\right) \left\vert dz\right\vert .
\end{equation*}%
The area element $da$ on $A_{R}$ is
\begin{equation*}
da(z)=\frac{1}{4}\left( 1+\frac{1}{\left\vert z\right\vert ^{2}}\right)
^{2}dm(x,y),
\end{equation*}%
where $m$ is the Lebesgue measure in the complex plane. Then the area
element $d\mathcal{A}$ on the M\"{o}bius strip is
\begin{equation*}
d\mathcal{A}(\widetilde{z})=da(z)=da(k(z)),
\end{equation*}%
hence
\begin{equation*}
d\mathcal{A}(\widetilde{z})=\frac{1}{4}\left( 1+\frac{1}{\left\vert
z\right\vert ^{2}}\right) ^{2}dm(x,y).
\end{equation*}%
Let $\gamma :[a,b]\rightarrow A_{R}$ be a piecewise continuously
differentiable curve and let $f:\gamma ([a,b])\rightarrow \mathbb{C}$ be a
continuous function. The integral of $f$ on the curve $\gamma ,$ denoted by $%
\underset{\gamma }{\int }fd\sigma ,$ is defined by
\begin{equation*}
\underset{\gamma }{\int }fd\sigma =\frac{1}{2}\underset{a}{\overset{b}{\int }%
}f(\gamma (t))\left( 1+\frac{1}{\left\vert \gamma (t)\right\vert ^{2}}%
\right) \left\vert \gamma ^{\prime }(t)\right\vert dt
\end{equation*}%
and
\begin{equation*}
\underset{M}{\int \int }Fd\mathcal{A}=\frac{1}{8}\underset{A_{R}}{\int \int }%
\text{ }f(z)\left( 1+\frac{1}{\left\vert z\right\vert ^{2}}\right)
^{2}dm(x,y).
\end{equation*}
\end{example}

Let $\gamma $ be a $\sigma $-rectifiable Jordan arc $\gamma $, parametrized
in terms of the arc $\sigma $-length. Therefore, $\gamma :$ $%
z=z(s)=x(s)+iy(s)$, $s\in \lbrack 0,l]$, where $l$ is the $\sigma $-length
of $\gamma .$ Then the unit inward normal vector to $\gamma $ at $z(s)$ is $%
n_{\sigma }=\left( -\frac{dy}{d\sigma },\frac{dx}{d\sigma }\right) $ and we
denote by $\frac{\partial }{\partial n_{\sigma }}$ the inward normal
derivative, with respect to the symmetric metric $d\sigma .$ In this way,
our approach is consistent with Nevanlinna \cite{17}, Bergman \cite{9} and
Schiffer and Spencer \cite{28}. For more details about the normal derivative
and Green's identities in terms of $d\sigma ,$ see \cite{5}.

\section{The Dirichlet problem for harmonic functions}

This section is devoted to the study of harmonic functions with Dirichlet
boundary condition on a Klein surface. The similar analysis in the complex
plane has been developed in Krantz \cite[Section 1.2]{15a}.

The notion of harmonic function, as being a solution of the Laplace
equation, makes sense on a Klein surface. Moreover, a Klein surface is the
most general two-manifold in which this notion of harmonic function makes
sense. For details, see \cite{3}. We notice that the notion of analytic
function is meaningless on a Klein surface.

The Dirichlet problem on an arbitrary Riemann surface can be solved because
the property that a function that is harmonic remains invariant under
bi-holomorphic mappings. For the existence of a harmonic function which
vanishes on the boundary and has a finite number of isolated singularities
with given singular parts in a relatively compact region, which is contained
in a chart of a Riemann surface, we refer to Ahlfors and Sario \cite{2}.

Any Klein surface $X$ can be regularly imbedded in a border free surface
using a duplication process (see \cite{2}). Therefore, for the boundary
problems involving a part of $\partial X$ we can consider it as a part of
the boundary of a region on a border free surface.

Let $O_{2}$ be a region in the complex plane, bounded by a finite number of
analytic Jordan curves. Then $\overline{O_{2}}=O_{2}\cup \partial O_{2}$ can
be conceived as a bordered Riemann surface (see \cite{1}, \cite{28}).
Because the Klein surfaces $X$ and $O_{2}/\left\langle k\right\rangle $ are
dianalytically equivalent, a boundary value problem on a region $\Omega $ of
the Klein surface $X$, can be replaced by a similar problem on a symmetric
region $D$ of its double $O_{2}$, as follows.

Consider the Dirichlet problem on $X$ for harmonic functions%
\begin{equation}  \label{e1.2.1}
\left\{
\begin{array}{c}
\Delta U=0\text{ on }\Omega \\
U=F\text{ on }\partial \Omega.%
\end{array}%
\right.
\end{equation}%
where $\Omega $ is a region of $X$ bounded by a finite number of $\sigma $-
rectifiable Jordan curves and $F$ is a continuous real-valued function on $%
\partial \Omega .$

We define $D=\pi ^{-1}(\Omega )$ and $f=F\circ \pi $ on $\partial D.$ Then $%
D $ is a symmetric region of $O_{2},$ bounded by a finite number of $\sigma $%
-rectifiable Jordan curves on $O_{2}$, some of which may contain part of $%
\partial O_{2}.$ Since $\pi \circ k=\pi $, we obtain $f=f\circ k$ on $%
\partial D$, hence $f$ is a symmetric, continuous real-valued function on $%
\partial D$. The Dirichlet problem \eqref{e1.2.1} on $X$ is equivalent with
the following Dirichlet problem for harmonic functions on $O_{2}$
\begin{equation}  \label{e1.2.2}
\left\{
\begin{array}{c}
\Delta u=0\ \text{on }D \\
u= f\ \text{on }\partial D.%
\end{array}%
\right.
\end{equation}

For details about the Dirichlet problem on bordered Riemann surfaces, see
Ahlfors and Sario \cite{2}.

The Dirichlet problem turned out to be fundamental in many areas of
mathematics and physics. For example, if $D$ is a thin, heat-conducting
metal plate and $f$ is a continuous temperature distribution on $\partial D$%
, then the solution $u$ of problem \eqref{e1.2.2} represents the resulting
steady-state heat distribution on $D$ (see \cite{10}, \cite{15}).

Using the maximum principle for harmonic functions, it follows that the
Dirichlet problem \eqref{e1.2.2} with continuous boundary values has a
unique solution for any region $D$ with only regular points. For some basic
monotonicity, analytic and variational methods of the theory of partial
differential equations of elliptic type, we refer to \cite{22}.

The symmetric conditions on the boundary imply symmetric solutions for the
problem \eqref{e1.2.2}. For more details, see Schiffer and Spencer \cite{28}.

\begin{proposition}
\label{p1.2.1} A solution $u$ of the problem \eqref{e1.2.2} is a symmetric
function on $D$.
\end{proposition}

\begin{proof}
Let $u$ be a solution of the problem \eqref{e1.2.2}. We define $\widetilde{u}%
:\overline{D}\rightarrow \mathbb{R}$ by $\widetilde{u}=\frac{1}{2}\left(
u+u\circ k\right) .$ Then $\Delta \widetilde{u}$ $=0$ on $D.$ By hypothesis,
$f=f\circ k$ on $\partial D$, hence
\begin{equation*}
\widetilde{u}=\frac{1}{2}\left( f+f\circ k\right) =f\quad \mbox{on $\partial
D$}.
\end{equation*}
Thus, $\widetilde{u}$ is also a solution of the problem \eqref{e1.2.2}. The
uniqueness of the solution yields $\widetilde{u}=u$ on $D$, therefore $%
u=u\circ k$ on $D.$
\end{proof}

\subsection{The symmetric Green function}

Let $D$ be a symmetric region bounded by a finite number of $\sigma $%
-rectifiable Jordan curves on the symmetric Riemann surface $O_{2}.$

Fix a point $\zeta \in D.\ $The function $\upsilon \left( z,\zeta \right)
=-\ln \left\vert z-\zeta \right\vert $ is harmonic at all points $z\neq
\zeta .$ Let $w$ be the solution of the Dirichlet problem on $D,$ with the
boundary condition $w(z)=\upsilon (z,\zeta )$ on $\partial D.$ The unique
function $G_{D}(z,\zeta )=\upsilon \left( z,\zeta \right) -w(z)$ defined on $%
\overline{D}\backslash \left\{ \zeta \right\} $ is called the \textit{Green}
\textit{function} of the region $D$, with singularity at $\zeta $ (see \cite%
{2}).

We assume that $u$ and $v$ are continuously twice differentiable in $D$ and
once on the boundary $\partial D$. We will use the following Green formula:
\begin{equation*}
\underset{\partial D}{\int }\left( u\frac{\partial v}{\partial n_{\sigma }}-v%
\frac{\partial u}{\partial n_{\sigma }}\right) d\sigma =-\underset{D}{\int
\int }\left( u\Delta v-v\text{ }\Delta u\right) dxdy,
\end{equation*}%
where $d\sigma $ is the arc $\sigma $-length element on $\partial D$ and the
derivatives on the left are taken with respect to the inward normal on $%
\partial D$. For more details, see \cite{17}.

The next theorem is similar to the Cauchy integral formula for harmonic
functions in terms of the metric $d\sigma $.

\begin{proposition}
\label{p1.2.2} (Green representation formula) Let $D$ be a symmetric region
bounded by a finite number of $\sigma $-rectifiable Jordan curves and let $u$
be a harmonic function in $D$ and continuously differentiable on its
boundary $\partial D.$ Then, for all $\zeta $ in $D,$
\begin{equation}  \label{e1.2.3}
u(\zeta )=\frac{1}{2\pi }\underset{\partial D}{\int }\left( u(z)\frac{%
\partial \upsilon (z,\zeta )}{\partial n_{\sigma }}-\upsilon (z,\zeta )\frac{%
\partial u(z)}{\partial n_{\sigma }}\right) d\sigma ,
\end{equation}%
where the derivatives are taken with respect to the inward normal on $%
\partial D$.
\end{proposition}

\begin{proof}
Fix a point $\zeta \in D$ and a positive number $\varepsilon $ that is less
than the Euclidean distance of $\zeta $ to $\partial D.$ Define $%
D_{\varepsilon }=D\setminus \overline{D}(\zeta ,\varepsilon ).$ Let $%
C_{\varepsilon }$ be the negatively oriented circle of radius $\varepsilon $%
, centered at $\zeta .$ We apply the Green formula for $D_{\varepsilon },$
with the harmonic functions $u$ and $\upsilon .$ It follows that
\begin{equation}  \label{e1.2.4}
\underset{\partial D}{\int }\left( u\frac{\partial \upsilon }{\partial
n_{\sigma }}-\upsilon \frac{\partial u}{\partial n_{\sigma }}\right) d\sigma
=-\underset{C_{\varepsilon }}{\int }\left( u\frac{\partial \upsilon }{%
\partial n_{\sigma }}-\upsilon \frac{\partial u}{\partial n_{\sigma }}%
\right) d\sigma .
\end{equation}%
The curve $-C_{\varepsilon }$ is parameterized by $z$ $=z(\theta )$ $=\zeta
+\varepsilon e^{i\theta },$ $0\leq \theta \leq 2\pi $. We deduce that
\begin{equation*}
-\underset{C_{\varepsilon }}{\int }\upsilon \frac{\partial u}{\partial
n_{\sigma }}d\sigma =\underset{-C_{\varepsilon }}{\int }\upsilon \frac{%
\partial u}{\partial n_{\sigma }}d\sigma =-\varepsilon \underset{0}{\overset{%
2\pi }{\int }}\upsilon (z(\theta ),\zeta )\frac{\partial u(z(\theta ))}{%
\partial \rho }d\theta .
\end{equation*}%
As the function $u$ has continuous partial derivatives in $D$, there is a
constant $C$ such that $\left\vert \frac{\partial u}{\partial \rho }%
\right\vert \leq C$ on $C_{\varepsilon }.$ Then, on $C_{\varepsilon },$ we
obtain
\begin{equation*}
\left\vert \underset{-C_{\varepsilon }}{\int }\upsilon \frac{\partial u}{%
\partial n_{\sigma }}d\sigma \right\vert \leq 2\pi C\varepsilon \left\vert
\ln \varepsilon \right\vert.
\end{equation*}%
We observe that the right-hand side of the last inequality tends to zero as $%
\varepsilon $ tends to zero. Therefore
\begin{equation*}
\underset{\varepsilon \rightarrow 0}{\lim }\underset{-C_{\varepsilon }}{\int
}\upsilon \frac{\partial u}{\partial n_{\sigma }}d\sigma =0.
\end{equation*}
Using the mean value property, we have
\begin{equation*}
\underset{C_{\varepsilon }}{\int }u\frac{\partial \upsilon }{\partial
n_{\sigma }}d\sigma =-\underset{0}{\overset{2\pi }{\int }}u(z(\theta
))d\theta =-2\pi u(\zeta ).
\end{equation*}%
Then relation \eqref{e1.2.4} becomes
\begin{equation*}
\underset{\partial D}{\int }\left( u\frac{\partial \upsilon }{\partial
n_{\sigma }}-\upsilon \frac{\partial u}{\partial n_{\sigma }}\right) d\sigma
=2\pi u(\zeta ).
\end{equation*}
The proof is now complete.
\end{proof}

Following Nevannlina (see \cite{17}), we obtain that the values of $u$
inside $D$ are determined from its values and the values of the normal
derivative of the Green function on the boundary $\partial D.$

\begin{theorem}
\label{t1.2.3} Let $D$ be a symmetric region, whose boundary $\partial D$
consists of a finite number of $\sigma $-rectifiable Jordan curves. If $u$
is harmonic on $D$ and continuously differentiable on $\partial D,$ then\
for all $\zeta $ in $D$,
\begin{equation}  \label{e1.2.5}
u(\zeta )=\frac{1}{2\pi }\underset{\partial D}{\int }u(z)\frac{\partial
G_{D}(z,\zeta )}{\partial n_{\sigma }}d\sigma .
\end{equation}
\end{theorem}

\begin{proof}
Applying Green's formula for $D$ with the harmonic functions $u$ and $w,$ we
obtain%
\begin{equation}  \label{e1.2.6}
\underset{\partial D}{\int }\left( \upsilon \frac{\partial u}{\partial
n_{\sigma }}-u\frac{\partial w}{\partial n_{\sigma }}\right) d\sigma =0.
\end{equation}%
Dividing \eqref{e1.2.6} by $2\pi ,$ and adding this identity to the Green
representation formula, we obtain \eqref{e1.2.5}.
\end{proof}

The function
\begin{equation*}
P_{\zeta }(z)=\frac{1}{2\pi }\frac{\partial G_{D}(z,\zeta )}{\partial
n_{\sigma }}
\end{equation*}%
is called the \textit{Poisson kernel} of the Laplace operator and the
Dirichlet problem on the region $D$.

We define $G_{D}^{(k)}(z,\widetilde{\zeta })$ as
\begin{equation*}
G_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ G_{D}(z,\zeta
)+G_{D}(z,k(\zeta ))\right]
\end{equation*}%
on $\overline{D}\backslash \left\{ \zeta ,k(\zeta )\right\} .$

Let $w_{s}$ be the solution of the Dirichlet problem on $D,$ with the
boundary condition $w_{s}(z)=\frac{1}{2}\left[ \upsilon (z,\zeta )+\upsilon
(z,k(\zeta )\right] $ on $\partial D.$ Then
\begin{equation*}
G_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ \upsilon (z,\zeta
)+\upsilon (z,k(\zeta )\right] -w_{s}(z).
\end{equation*}%
Therefore, $G_{D}^{(k)}(z,\widetilde{\zeta })$ is a harmonic function of $z$
in $D\backslash \left\{ \zeta ,k(\zeta )\right\} ,$ with singularities $-%
\frac{1}{2}\ln \left\vert z-\zeta \right\vert $ and $-\frac{1}{2}\ln
\left\vert z-k(\zeta )\right\vert $ at $\zeta $ and $k(\zeta ),$
respectively. Also, $G_{D}^{(k)}(z,\widetilde{\zeta })=0$ for all $z$ on $%
\partial D.$

We can derive the following result (see \cite{5}):

\begin{proposition}
\label{p1.2.4} For every symmetric region $D$, the function $%
G_{D}^{(k)}(\cdot ,\widetilde{\zeta })$ is symmetric on $\overline{D},$ that
is, for all $z\in \overline{D},$
\begin{equation*}
G_{D}^{(k)}(z,\widetilde{\zeta })=G_{D}^{(k)}(k(z),\widetilde{\zeta }).
\end{equation*}
\end{proposition}

Consequently, the function $G_{D}^{(k)}(z,\widetilde{\zeta })$ is called the%
\textit{\ symmetric Green function} of the region $D$, with singularities at
$\zeta $ and $k(\zeta ).$

An explicit form for the symmetric Green function of the annulus is obtained
in \cite{5}. For additional information on this topic we refer to \cite{28}.

\subsection{The symmetric harmonic measure}

Let $D$ be a symmetric region bounded by a finite number of $\sigma $%
-rectifiable Jordan curves on $O_{2}$ and $\mathcal{B}(\partial D)$ the $%
\sigma $-algebra of Borel sets of $\partial D.$ The $\sigma $-algebra of
symmetric Borel sets of $\partial D$ is denoted by $\mathcal{B}_{s}(\partial
D)$ and $\mathcal{B}_{s}(\partial D)=\left\{ U\cup k(U)\left\vert U\in
\mathcal{B}(\partial D)\right. \right\} .$

The \textit{harmonic measure} for $D$ is a function $\omega _{D}:D\times
\mathcal{B}_{s}(\partial D)\rightarrow \lbrack 0,1]$ such that:

\begin{enumerate}
\item for each $\zeta \in D,$ the map $B\mapsto \omega _{D}(\zeta ,B)$ is a
Borel probability measure on $\partial D;$

\item if $f:\partial D\rightarrow
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$ is a continuous function, then the solution of the Dirichlet problem, for $%
D$ and the boundary function $f$,\ is the generalized Poisson integral of $f$
on $D$ given by
\begin{equation}  \label{e1.2.7}
P_{D}f(\zeta )=\underset{\partial D}{\int }f(z)d\omega _{D}(\zeta ,z),\zeta
\in D.
\end{equation}%
For details, see \cite{23}.
\end{enumerate}

\begin{remark}
\label{r1.2.5} The uniqueness of $\omega _{D}$ is a consequence of the Riesz
representation theorem.
\end{remark}

An extensive study of the harmonic measure is developed in \cite{11}.

A method of determining the harmonic measure is given by the following
characterization (see \cite{23}):

\begin{proposition}
\label{p1.2.6} The function $\omega _{D}(\cdot ,B),$ is the solution of the
generalized Dirichlet problem with boundary function $f=1_{B}$.
\end{proposition}

The harmonic measure for $D$ is related to another conformal invariant, the
Green function for the symmetric region $D$.

Using Theorem \ref{t1.2.3} and the fact that Borel measures are determined
by their actions on continuous functions, we obtain a representation of the
harmonic measure in terms of the inward normal derivative of the Green
function with respect to $d\sigma .$

\begin{proposition}
\label{p1.2.7} Let $D$ be a symmetric region, whose boundary $\partial D$
consists of a finite number of $\sigma $-rectifiable Jordan curves. If $%
\zeta \in D$, then for any $z\in \partial D,$
\begin{equation*}
d\omega _{D}(\zeta ,z)=\frac{\partial G_{D}(z,\zeta )}{\partial n_{\sigma }}%
\cdot \frac{d\sigma (z)}{2\pi }.\text{ }
\end{equation*}
\end{proposition}

Thus, the harmonic measure for $\zeta \in D$ is absolutely continuous to arc
$\sigma $-length on $\partial D$ and on $\partial D,$ the density being
\begin{equation*}
\frac{d\omega _{D}}{d\sigma }=\frac{1}{2\pi }\frac{\partial G_{D}(z,\zeta )}{%
\partial n_{\sigma }}=P_{\zeta }(z).
\end{equation*}

Let $\omega _{D}^{(k)}:D\times \mathcal{B}_{s}(\partial D)\rightarrow
\lbrack 0,1]$ be the function defined by
\begin{equation*}
\omega _{D}^{(k)}(\widetilde{\zeta },B)=\frac{1}{2}\left[ \omega _{D}(\zeta
,B)+\omega _{D}(k(\zeta ),B)\right] ,\
\end{equation*}%
where $\widetilde{\zeta }=\{\zeta ,k(\zeta )\}$, $\zeta \in D$, $B\in
\mathcal{B}_{s}(\partial D)$; see \cite{24}.

\begin{remark}
\label{r1.2.8} The symmetry of the region $D$ implies that the function $%
\omega _{D}^{(k)}(\widetilde{\zeta },B)$ is symmetric with respect to $B$ on
$\mathcal{B}_{s}(\partial D)$, that is, for any $B\in $ $\mathcal{B}%
_{s}(\partial D),$
\begin{equation*}
\omega _{D}^{(k)}(\widetilde{\zeta },B)=\omega _{D}^{(k)}(\widetilde{\zeta }%
,k(B)).
\end{equation*}
\end{remark}

The function $\omega _{D}^{(k)}(\widetilde{\zeta },B)$ is called the\textit{%
\ symmetric harmonic measure} for $D.$

The function
\begin{equation*}
P_{\widetilde{\zeta }}^{(k)}(z)=\frac{1}{2\pi }\frac{\partial G_{D}^{(k)}(z,%
\widetilde{\zeta })}{\partial n_{\sigma }},\ z\in D
\end{equation*}%
is called the\textit{\ symmetric Poisson kernel} for the region $D$.

\subsection{The Dirichlet problem on the complex double}

The following Poisson integral formula both reproduces and creates harmonic
functions on the complex double. Roughly speaking, the next theorem yields
the formula for the solution of the Dirichlet problem \eqref{e1.2.2} on a
symmetric region $D$, in terms of the symmetric Green function.

\begin{theorem}
\label{t1.2.9} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves and let $f$ be a symmetric, continuous
function on $\partial D.$ There is a unique symmetric function $u$ on $%
\overline{D}$, which is harmonic in $D$, continuous on $\overline{D},$ such
that $u=f$ on $\partial D.$ Moreover, for all $\zeta $ in $D$,
\begin{equation}  \label{e1.2.8}
u(\zeta )=\frac{1}{2\pi }\underset{\partial D}{\int }f(z)\frac{\partial
G_{D}^{(k)}(z,\zeta )}{\partial n_{\sigma }}d\sigma ,\ \zeta \in D.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t1.2.3}, for all $\zeta \in D,$
\begin{equation*}
u(\zeta )=\frac{1}{2\pi }\underset{\partial D}{\int }u(z)\frac{\partial
G_{D}(z,\zeta )}{\partial n_{\sigma }}d\sigma .
\end{equation*}%
Replacing $\zeta $ with $k(\zeta )$ we obtain
\begin{equation*}
u(k(\zeta ))=\frac{1}{2\pi }\underset{\partial D}{\int }u(z)\frac{\partial
G_{D}(z,k(\zeta ))}{\partial n_{\sigma }}d\sigma .
\end{equation*}%
Adding the last two equations and dividing by 2, we obtain
\begin{equation*}
\frac{u(\zeta )+u(k(\zeta ))}{2}=\frac{1}{4\pi }\underset{\partial D}{\int }%
u(z)\left[ \frac{\partial G_{D}(z,\zeta )}{\partial n_{\sigma }}+\frac{%
\partial G_{D}(z,k(\zeta ))}{\partial n_{\sigma }}\right] d\sigma ,
\end{equation*}%
for all $\zeta \in D.$ By Proposition \ref{p1.2.1}, $u$ is a symmetric
function on $D$, then the left-hand side of the last equality is $u(\zeta ).$
We conclude that for all $\zeta $ in $D$,%
\begin{equation*}
u(\zeta )=\frac{1}{4\pi }\underset{\partial D}{\int }u(z)\left[ \frac{%
\partial G_{D}(z,\zeta )}{\partial n_{\sigma }}+\frac{\partial
G_{D}(z,k(\zeta ))}{\partial n_{\sigma }}\right] d\sigma .
\end{equation*}%
The uniqueness of the solution of the Dirichlet problem for harmonic
functions implies relation \eqref{e1.2.8}.
\end{proof}

Theorem \ref{t1.2.9} is the equivalent of the Poisson formula for the
solution of the Dirichlet problem on the disc in the complex plane, see
Krantz \cite[Section 7.3]{14b}. In such a way, Theorem \ref{t1.2.9} creates
a function that agrees with $f$ on the boundary of the domain $D$ and is
harmonic inside.

The formula for the solution to the Dirichlet problem on the annulus is
obtained in \cite{7}.

In a similar way, we obtain the following representation of the solution of
the problem \eqref{e1.2.2} on a symmetric region $D,$ in terms of the
symmetric harmonic measure.

\begin{theorem}
\label{t1.2.10} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves and let $f$ be a symmetric, continuous
function on $\partial D.$ There exists a unique symmetric function $u$ on $%
\overline{D}$, which is harmonic on $D$, continuous on $\overline{D},$ such
that $u=f$ on $\partial D.$ For all $\zeta $ in $D,$ we have
\begin{equation}  \label{e1.2.9}
u(\zeta )=\underset{\partial D}{\int }f(z)d\omega _{D}^{(k)}(\widetilde{%
\zeta },z).
\end{equation}
\end{theorem}

\begin{proof}
Let $\zeta $ be a point in $D$. By \eqref{e1.2.7}, for all $\zeta \in D,$
\begin{equation*}
u(\zeta )=\underset{\partial D}{\int }f(z)d\omega (\zeta ,z)d\sigma .
\end{equation*}%
Replacing $\zeta $ with $k(\zeta )$ we get
\begin{equation*}
u(k(\zeta ))=\underset{\partial D}{\int }u(z)d\omega (k(\zeta ),z)d\sigma .
\end{equation*}%
Adding the last two equations and dividing by 2, we obtain
\begin{equation*}
\frac{u(\zeta )+u(k(\zeta ))}{2}=\frac{1}{2}\underset{\partial D}{\int }f(z)%
\left[ d\omega (\zeta ,z)+d\omega (k(\zeta ),z)\right] ,
\end{equation*}%
for all $\zeta $ in $D.$ By Proposition \ref{p1.2.1}, $u$ is a symmetric
function on $D$, then the left-hand side of the last equality is $u(\zeta )$
and we conclude that for all $\zeta $ in $D,$
\begin{equation*}
u(\zeta )=\underset{\partial D}{\int }f(z)d\omega _{D}^{(k)}(\widetilde{%
\zeta },z).
\end{equation*}
The proof is now complete.
\end{proof}

By Proposition \ref{p1.2.7}, we obtain the Radon-Nikodym derivative of
symmetric harmonic measure for $D$ against $\sigma $-arc length.

\begin{proposition}
\label{p1.2.11} Let $D$ be a symmetric region whose boundary $\partial D$
consists of a finite number of $\sigma $-rectifiable Jordan curves. If $%
\zeta \in D$, then for any $z\in \partial D,$
\begin{equation*}
d\omega _{D}^{(k)}(\widetilde{\zeta },z)=\frac{\partial G_{D}^{(k)}(z,%
\widetilde{\zeta })}{\partial n_{\sigma }}\cdot \frac{d\sigma (z)}{2\pi }.
\end{equation*}
\end{proposition}

This result shows that the symmetric harmonic measure for $D$ is absolutely
continuous to arc $\sigma $-length on $\partial D$ and on $\partial D,$ the
density being
\begin{equation*}
\frac{d\omega _{D}^{(k)}}{d\sigma }=\frac{1}{2\pi }\frac{\partial
G_{D}^{(k)}(z,\widetilde{\zeta })}{\partial n_{\sigma }}=P_{\widetilde{\zeta
}}^{(k)}(z).
\end{equation*}

\subsection{The Dirichlet problem on the Klein surface}

Let $X$ be a Klein surface and let $\Omega $ be a region bounded by a finite
number of $\sigma $-rectifiable Jordan curves. The Klein surface $X$ is the
factor manifold of the symmetric Riemann surface $O_{2}$ with respect to the
group $\left\langle k\right\rangle $. Then, $\Omega $ is obtained from the
symmetric region $D$ by identifying the corresponding symmetric points.

The Green function of $\Omega $ with singularity at $\widetilde{\zeta }$ is
defined by
\begin{equation*}
G_{\Omega }(\widetilde{z},\widetilde{\zeta })=G_{D}^{(k)}(z,\widetilde{\zeta
})=G_{D}^{(k)}(k(z),\widetilde{\zeta }),
\end{equation*}%
where $\widetilde{z}=\pi (z).$

By definition, the function $G_{\Omega }(\widetilde{z},\widetilde{\zeta })$
is continuous on $\overline{\Omega }$, harmonic on $\Omega \backslash
\left\{ \widetilde{\zeta }\right\} $ and\ has the singularity at $\widetilde{%
\zeta }=\pi (\zeta ).$

\begin{remark}
\label{r1.2.12} By Proposition \ref{p1.2.4}, it follows that $G_{\Omega }(%
\widetilde{z},\widetilde{\zeta })$ is well-defined on $\Omega .$
\end{remark}

An explicit form for the Green function of the M\"{o}bius strip is obtained
in \cite{5}.

The harmonic measure for $\Omega ,$ $\omega _{\Omega }:\Omega \times
\mathcal{B}(\partial \Omega )\rightarrow \lbrack 0,1],$ is defined by%
\begin{equation*}
\omega _{\Omega }(\widetilde{\zeta },\widetilde{B})=\omega _{D}^{(k)}(%
\widetilde{\zeta },B)=\omega _{D}^{(k)}(\widetilde{\zeta },k(B)).
\end{equation*}%
for all $\widetilde{\zeta }\in \Omega $ and $\widetilde{B}=\pi (B)\in
\mathcal{B}(\partial \Omega ).$

The function
\begin{equation*}
P_{\widetilde{\zeta }}(\widetilde{z})=P_{\widetilde{\zeta }}^{(k)}(z)=P_{%
\widetilde{\zeta }}^{(k)}(k(z)),\ z\in D
\end{equation*}%
is called the\textit{\ Poisson kernel }for the region\textit{\ }$\Omega $.

\begin{remark}
\label{r1.2.13} By Remark \ref{r1.2.8}, it follows that the function $\omega
_{\Omega }$ is well-defined. By Proposition \ref{p1.2.4}, it follows that
the function $P_{\widetilde{\zeta }}$ is well-defined, too.
\end{remark}

The symmetric solutions on $O_{2}$ determine the solutions of the similar
problems on the Klein surface $X$.

Consequently, we obtain the solution of the Dirichlet problem on the region $%
\Omega $, with respect to the Green function of $\Omega .$

\begin{theorem}
\label{t1.2.14} Let $F$ be a continuous real-valued function on the border $%
\partial \Omega . $ The solution of the Dirichlet problem \eqref{e1.2.1}
with the boundary function $F$ is the function $U$ defined on $\overline{%
\Omega },$ by the relation $u=U\circ \pi ,$ where $\pi $ is the canonical
projection of $O_{2}$ on $X$ and $u$ is the solution \eqref{e1.2.8} of the
Dirichlet problem \eqref{e1.2.2} on the symmetric region $D$, with the
boundary function $f,$ given by $f=F\circ \pi .$
\end{theorem}

\begin{proof}
By definition, $\Delta U(\widetilde{\zeta })=\Delta u(\zeta )$ $=0,$ for all
$\widetilde{\zeta }\in \Omega $, where $\widetilde{\zeta }=\pi (\zeta ).$
Thus, $U$ is a harmonic function on $\Omega $. The symmetry of the function $%
f $ on $\partial D$ implies
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=f(\zeta )=f(k(\zeta ))=F(\widetilde{\zeta }%
),\quad
\end{equation*}%
for all $\widetilde{\zeta }\in \partial \Omega .$ Due to the uniqueness of
the solution, the function $U$ defined on $\overline{\Omega }$ by
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=u(k(\zeta )),
\end{equation*}%
for all $\widetilde{\zeta }$ in $\overline{\Omega },$ where $\widetilde{%
\zeta }=\pi (\zeta ),$ is the solution of the Dirichlet problem %
\eqref{e1.2.1} on $\Omega $.
\end{proof}

In a similar way, we obtain the solution of the problem \eqref{e1.2.1} on
the region $\Omega $, with respect to the harmonic measure for the region $%
\Omega .$

\begin{theorem}
\label{t1.2.15} Let $F$ be a continuous real-valued function on the border $%
\partial \Omega . $ The solution of the problem \eqref{e1.2.1} with the
boundary function $F$ is the function $U$ defined on $\overline{\Omega },$
by the relation $u=U\circ \pi ,$ where $\pi $ is the canonical projection of
$O_{2}$ on $X$ and $u$ is the solution \eqref{e1.2.9} of the problem %
\eqref{e1.2.2} on the symmetric region $D$, with the boundary function $f,$
given by $f=F\circ \pi .$
\end{theorem}

By Proposition \ref{p1.2.11}, we obtain the Radon-Nikodym derivative of
harmonic measure for $\Omega $ against $\Sigma $-arc length.

\begin{proposition}
\label{p1.2.16} Let $\Omega $ be a region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. If $\widetilde{\zeta }\in \Omega $, then
for all $\widetilde{z}\in \partial \Omega ,$
\begin{equation*}
d\omega _{\Omega }(\widetilde{\zeta },\widetilde{z})=d\omega _{D}^{(k)}(%
\widetilde{\zeta },z)=d\omega _{D}^{(k)}(\widetilde{\zeta },k(z)).
\end{equation*}
\end{proposition}

This result implies that the harmonic measure for $\Omega $ is absolutely
continuous to arc $\Sigma $-length on $\partial \Omega $ and on $\partial
\Omega ,$ the density being
\begin{equation*}
\frac{d\omega _{\Omega }}{d\Sigma }=P_{\widetilde{\zeta }}(\widetilde{z}).
\end{equation*}

\section{The Neumann problem for harmonic functions}

This section is devoted to the study of harmonic functions with Neumann
boundary condition on a Klein surface. The similar analysis in the complex
place has been developed by Schiffer and Spencer \cite{28}.

Consider the Neumann problem for harmonic functions%
\begin{equation}  \label{e1.3.1}
\left\{
\begin{array}{c}
\Delta U=0\text{ on }\Omega \\
\displaystyle\frac{\partial U}{\partial n_{\Sigma }}=G\text{ on }\partial
\Omega,%
\end{array}%
\right.
\end{equation}%
where $\Omega $ is a region of $X$ bounded by a finite number of $\sigma $%
-rectifiable Jordan curves and $G$ is a continuous real-valued function on $%
\partial \Omega .$

We define $D=\pi ^{-1}(\Omega )$ and $g=G\circ \pi $ on $\partial D.$ Since $%
\pi \circ k=\pi $, we obtain that $D$ is a symmetric region bounded by a
finite number of $\sigma $-rectifiable Jordan curves on $O_{2}$, some of
which may contain part of $\partial O_{2}$ and $g$ is a symmetric,
continuous real-valued function on the boundary $\partial D.$

The Neumann problem on $X$ is equivalent with the following Neumann problem
on $O_{2}$%
\begin{equation}  \label{e1.3.2}
\left\{
\begin{array}{c}
\Delta u=0\text{ on }D \\
\displaystyle\frac{\partial u}{\partial n_{\sigma }}=g\text{ on }\partial D.%
\end{array}%
\right.
\end{equation}

Since $k$ is an antianalytic involution, the symmetry of $D$ and the
symmetry of $g$ on $\partial D,$ imply that the prescribed values of the
normal derivative satisfy the compatibility condition
\begin{equation*}
\underset{\partial D}{\int }gd\sigma =0.
\end{equation*}
Therefore the Neumann problem on $O_{2}$ for the region $D$ and the boundary
function $g$ has solutions. For details, see \cite{18}.

\begin{proposition}
\label{p1.3.1} If the problem \eqref{e1.3.2} admits a solution, then it is
unique up to an additive constant.
\end{proposition}

\begin{proof}
Let $u_{1}$ and $u_{2}$ be solutions of the problem \eqref{e1.3.2}. If $%
u=u_{1}-u_{2}$, then $u$ is harmonic on $D$ and $\frac{\partial u}{\partial
n_{\sigma }}=0$ on $\partial D$. Applying Green's first identity, we get
\begin{equation*}
\underset{D}{\int \int }\left( u_{x}^{2}+u_{y}^{2}\right) dxdy=0.
\end{equation*}
Therefore $u$ is constant on $D$.
\end{proof}

\begin{proposition}
\label{p1.3.2} The solution of the problem \eqref{e1.3.2} is a symmetric
function on $D$.
\end{proposition}

\begin{proof}
Let $u$ be a solution of the problem \eqref{e1.3.2}. We define $\widetilde{u}%
:\overline{D}\rightarrow \mathbb{R}$ by $\widetilde{u}=\frac{1}{2}\left(
u+u\circ k\right) $. By hypothesis $g=g\circ k$ on $\partial D$, then $\frac{%
\partial \widetilde{u}}{\partial n_{\sigma }}=\frac{\partial u}{\partial
n_{\sigma }}=g$ on $\partial D$ and $\Delta \widetilde{u}$ $=0$ on $D.$
Thus, $\widetilde{u}$ is also a solution of the problem \eqref{e1.3.2}. By
Proposition \ref{p1.3.1}, there is a constant $c$ such that $\widetilde{u}%
=u+c$ on $D$. Thus $u\circ k=u+2c$ on $D$ and using the symmetry of the
region $D$, we obtain $u=u\circ k+2c$ on $D$. Hence $c=0$, that is, $u\circ
k=u$ on $D$.
\end{proof}

\subsection{The symmetric Neumann function}

Let $\zeta $ be a point inside $D$. A \textit{Neumann function} $%
N_{D}(z,\zeta )$ for the region $D$, with singularity at $\zeta $, in terms
of the metric $d\sigma $, is the function
\begin{equation*}
N_{D}(z,\zeta )=\upsilon (z,\zeta )-h(z,\zeta ),\text{ }z\in D,\ z\neq \zeta,
\end{equation*}%
where $h(z,\zeta )$ is a solution of the following Neumann problem in terms
of the metric $d\sigma $:
\begin{equation*}
\left\{
\begin{array}{cc}
\Delta h(z,\zeta )=0\text{ }, & z\in D \\
\displaystyle\frac{\partial h}{\partial n_{\sigma }}(z,\zeta )=\frac{%
\partial \upsilon }{\partial n_{\sigma }}\left( z,\zeta \right) -\frac{2\pi
}{l}, & \text{ }z\in \partial D,%
\end{array}%
\right.
\end{equation*}%
where $l=\underset{\partial D}{\int }d\sigma $ is the $\sigma $-length of $%
\partial D$.

\begin{remark}
\label{r1.3.3} The boundary value of the inward normal derivative of the
Neumann function is a constant equal to $\frac{2\pi }{l}.$
\end{remark}

\begin{theorem}
\label{t1.3.4} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. If $u$ is harmonic in $D$ and
continuously differentiable on $\partial D$ then, up to an additive constant,%
\begin{equation*}
u(\zeta )=-\frac{1}{2\pi }\underset{\partial D}{\int }\frac{\partial u(z)}{%
\partial n_{\sigma }}N_{D}(z,\zeta )d\sigma ,\ \zeta \in D.
\end{equation*}
\end{theorem}

\begin{proof}
Fix a point $\zeta \in D$ and a positive number $\varepsilon $ that is less
than the Euclidean distance of $\zeta $ to $\partial D.$ Define $%
D_{\varepsilon }=D\setminus \overline{D}(\zeta ,\varepsilon ).$ Let $%
C_{\varepsilon }$ be the negatively oriented circle of radius $\varepsilon $%
, centered at $\zeta $. Applying Green formula for $D_{\varepsilon }$ with
the harmonic functions $h$ and $u,$ we obtain
\begin{equation}
\underset{\partial D}{\int }\left( h\frac{\partial u}{\partial n_{\sigma }}-u%
\frac{\partial h}{\partial n_{\sigma }}\right) d\sigma =0.  \label{e1.3.3}
\end{equation}%
Dividing \eqref{e1.3.3} by $2\pi $ and adding this identity to the Green
representation formula, it follows that
\begin{equation*}
u(\zeta )=-\frac{1}{2\pi }\underset{\partial D}{\int }N(z,\zeta )\frac{%
\partial u}{\partial n_{\sigma }}d\sigma +\frac{1}{l}\underset{\partial D}{%
\int }ud\sigma .
\end{equation*}%
Thus, $u$ is determined up to the additive constant $\frac{1}{l}\underset{%
\partial D}{\int }u(z)d\sigma .$
\end{proof}

Let $N_{D}^{(k)}(z,\widetilde{\zeta })$ be the function defined by
\begin{equation*}
N_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ N_{D}(z,\zeta
)+N_{D}(z,k(\zeta ))\right] ,\ z\in D\setminus \left\{ \zeta ,k(\zeta
)\right\},
\end{equation*}%
where $N_{D}(z,k(\zeta ))$ is a Neumann function for the region $D$, with
singularity at $k(\zeta )$ and $\widetilde{\zeta }=\left\{ \zeta ,k(\zeta
)\right\} .$ Therefore
\begin{equation*}
N_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ \upsilon (z,\zeta
)+\upsilon (z,k(\zeta ))\right] -h_{s}(z,\widetilde{\zeta }),\ z\neq \zeta
,\ z\neq k(\zeta ),
\end{equation*}%
where $h_{s}$ is a harmonic function on $D$ that satisfies
\begin{equation*}
\frac{\partial h_{s}}{\partial n_{\sigma }}(z,\widetilde{\zeta })=\frac{1}{2}%
\left[ \frac{\partial \upsilon }{\partial n_{\sigma }}\left( z,\zeta \right)
+\frac{\partial \upsilon }{\partial n_{\sigma }}\left( z,k(\zeta \right) )%
\right] -\frac{2\pi }{l}\,.
\end{equation*}

Therefore, $N_{D}^{(k)}(z,\widetilde{\zeta })$ is a harmonic function of $z$
in $D\setminus \left\{ \zeta ,k(\zeta )\right\} $, with singularities at $%
\zeta $ and $k(\zeta )$ and $\frac{\partial N_{D}^{(k)}}{\partial n_{\sigma }%
}(z,\widetilde{\zeta })=\frac{2\pi }{l}$, for all $z\in \partial D.$

An explicit form for the function $N_{D}^{(k)}(z,\widetilde{\zeta })$ of the
annulus and of the M\"{o}bius strip are obtained in \cite{25}.

\begin{proposition}
\label{p1.3.5} If $D$ is a symmetric region, then the function $%
N_{D}^{(k)}(z,\widetilde{\zeta })$ is symmetric with respect to $z$ on $D$,
that is, for any $z\in D$,
\begin{equation*}
N_{D}^{(k)}(z,\widetilde{\zeta })=N_{D}^{(k)}(k(z),\widetilde{\zeta }).
\end{equation*}
\end{proposition}

\begin{proof}
Let $h^{\ast }(\cdot ,\zeta )$ be a harmonic function in $D$, such that
\begin{equation*}
\frac{\partial h^{\ast }}{\partial n_{\sigma }}(z,\zeta )=\frac{1}{2}\left(
\frac{\partial \upsilon }{\partial n_{\sigma }}(z,\zeta )+\frac{\partial
\upsilon }{\partial n_{\sigma }}(k(z),\zeta )\right) -\frac{2\pi }{l},\ z\in
\partial D.
\end{equation*}

Therefore
\begin{equation*}
\frac{\partial h^{\ast }}{\partial n_{\sigma }}(z,\zeta )=\frac{\partial
h^{\ast }}{\partial n_{\sigma }}(k(z),\zeta ),\quad \mbox{for all}\ z\in
\partial D.
\end{equation*}
By Proposition \ref{p1.3.2}, $h^{\ast }(\cdot ,\zeta )$ is a symmetric
function. Hence the function
\begin{equation*}
M_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ \upsilon (z,\zeta
)+\upsilon (k(z),\zeta )\right] -h^{\ast }(z;\zeta )
\end{equation*}%
is a symmetric function, harmonic in $D\setminus \left\{ \widetilde{\zeta }%
\right\} $ and $\frac{\partial M_{D}^{(k)}}{\partial n}(z,\widetilde{\zeta }%
)=\frac{2\pi }{l}$. So, $N_{D}^{(k)}(z,\widetilde{\zeta })$ and $%
M_{D}^{(k)}(z,\widetilde{\zeta })$ are solutions of the same Neumann
problem. Thus, by Proposition \ref{p1.3.1}, there is a constant $c$ such
that $N_{D}^{(k)}(z,\widetilde{\zeta })=M_{D}^{(k)}(z,\widetilde{\zeta })+c.$
Since $M_{D}^{(k)}(z,\widetilde{\zeta })$ is a symmetric function, we obtain
that $N_{D}^{(k)}(z,\widetilde{\zeta })$ is also a symmetric function.
\end{proof}

Let $\zeta _{0}$ be a point of $D$. A Neumann function $N_{D}(z,\zeta )$ is
not a conformal invariant, but the difference $N_{D}(z,\zeta )-N_{D}(z,\zeta
_{0})$ is a Neumann function and has a vanishing normal derivative on $%
\partial D$, hence it is a conformal invariant. We redefine the difference
\begin{equation*}
N_{D}(z,\zeta ,\zeta _{0})=N_{D}(z,\zeta )-N_{D}(z,\zeta _{0})
\end{equation*}%
to be a \textit{Neumann function} for the region $D$ on the Riemann surface $%
O_{2},$ see \cite{28}.

The function $N_{D}^{(k)}(z,\widetilde{\zeta },\widetilde{\zeta _{0}})$
defined by
\begin{equation*}
N_{D}^{(k)}(z,\widetilde{\zeta },\widetilde{\zeta _{0}})=\frac{1}{2}\left[
N_{D}(z,\zeta ,\zeta _{0})+N_{D}(z,k(\zeta ),k\left( \zeta _{0}\right) )%
\right] ,
\end{equation*}%
for all $z\in \overline{D}\backslash \{\widetilde{\zeta },\widetilde{\zeta
_{0}}\}$, is called \textit{a} \textit{symmetric Neumann function} for the
region $D.$

\subsection{The symmetric harmonic kernel function}

Let $D$ be a symmetric region in the complex plane, bounded by a finite
number of $\sigma $-rectifiable Jordan curves. In this section, we introduce
closed systems $(\varphi _{i})_{i\text{ }}$ of harmonic functions in $D,$
which are orthonormal with respect to the Dirichlet integral%
\begin{equation*}
D\{\varphi _{i},\varphi _{j}\}=\underset{D}{\int \int }\left( \frac{\partial
\varphi _{i}}{\partial x}\frac{\partial \varphi _{j}}{\partial y}+\frac{%
\partial \varphi _{i}}{\partial y}\frac{\partial \varphi _{j}}{\partial x}%
\right) dxdy.
\end{equation*}

We recall some notions and results about orthogonal harmonic functions. For
more details, see \cite{9}.

Let $\Lambda ^{2}(D)$ be the set of harmonic functions $\varphi $ in $D$
with a finite Dirichlet integral
\begin{equation*}
D\{\varphi \}=D\{\varphi ,\varphi \}<\infty
\end{equation*}%
such that
\begin{equation*}
D\{N_{D}(z,\zeta ),\varphi (\zeta )\}=2\pi \varphi (\zeta ),
\end{equation*}%
where $N_{D}(z,\zeta )$ is the Neumann function of $D$ with its singularity
at the fixed point $\zeta $, $\zeta \in D.$

\begin{remark}
\label{r1.3.6} The second condition is imposed to normalize $D\{\varphi
,\varphi \}$ to be zero if and only if $\varphi $ vanishes identically.
\end{remark}

\begin{proposition}
\label{p1.3.7} There exists a closed system $(\varphi _{i})_{i}$ for the
class $\Lambda ^{2}(D)$, which is orthonormal with respect to the Dirichlet
integral, that is,
\begin{equation*}
D\{\varphi _{i},\varphi _{j}\}=\delta _{ij},\ \delta _{ii}=1,\ \delta
_{ij}=0,\ i\neq j.
\end{equation*}
\end{proposition}

Let $\zeta $ be a point inside $D$. The \textit{harmonic kernel function} $%
K_{D}(z,\zeta )$ of the closed orthonormal system $(\varphi _{i})_{i}$ for
the region $D$, with respect to the point $\zeta ,$ is the function defined
by
\begin{equation*}
K_{D}(z,\zeta )=\underset{i=1}{\overset{\infty }{\sum }}\varphi
_{i}(z)\varphi _{i}(\zeta ),\ z\in \overline{D}.
\end{equation*}

The harmonic kernel function is uniquely characterized by the following
properties:%
\begin{equation*}
K_{D}(z,\zeta )=K_{D}(\zeta ,z)
\end{equation*}%
and%
\begin{equation*}
D\{K_{D}(z,\zeta ),\varphi (\zeta )\}=\varphi (\zeta ),\ \varphi \in \Lambda
^{2}(D).
\end{equation*}

An extensive study of the harmonic kernel function is due to \cite{9}.

The representation of the harmonic kernel function in terms of a closed
orthonormal system gives the opportunity to solve numerically the Dirichlet
problem for arbitrarily multiply connected regions. This is an important
tool in physics, in particular in fluid mechanics, elasticity and
electricity.

It is known that the harmonic kernel function $K_{D}(z,\zeta )$, the Green
function $G_{D}(z,\zeta )$ and the Neumann function $N_{D}(z,\zeta )$
satisfy the relation%
\begin{equation}  \label{e1.3.4}
K_{D}(z,\zeta )=\frac{1}{2\pi }\left[ N_{D}(z,\zeta )-G_{D}(z,\zeta )\right]
,\ z\in \overline{D}.
\end{equation}

We first derive a formula that solves the Dirichlet problem \eqref{e1.2.2}.
We prove that if $u$ is harmonic inside a region $D$ and continuous on $%
\partial D$, then we can determine the values of $u$ inside of $D$ by
integrating on $\partial D$ the product of $u$ times the inward normal
derivative\ of the harmonic kernel function for the region $D$, which is a
fixed function that depends only on $D.$

\begin{theorem}
\label{t1.3.8} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. If $u$ is harmonic in $D$ and continuous
on $\overline{D},$ then, up to an additive constant,
\begin{equation}
u(\zeta )=-\underset{\partial D}{\int }u(z)\frac{\partial K_{D}(z,\zeta )}{%
\partial n_{\sigma }}d\sigma ,\ \zeta \in D.
\end{equation}
\end{theorem}

\begin{proof}
From \eqref{e1.2.5}, the solution of the Dirichlet problem \eqref{e1.2.2} is
\begin{equation}  \label{e1.3.6}
u(\zeta )=\frac{1}{2\pi }\underset{\partial D}{\int }u(z)\frac{\partial
G_{D}(z,\zeta )}{\partial n_{\sigma }}d\sigma ,\ \zeta \in D.
\end{equation}%
Using \eqref{e1.3.4}, we obtain
\begin{equation*}
\frac{\partial K_{D}(z,\zeta )}{\partial n_{\sigma }}=\frac{1}{2\pi }\frac{%
\partial N_{D}(z,\zeta )}{\partial n_{\sigma }}-\frac{1}{2\pi }\frac{%
\partial g_{D}(z,\zeta )}{\partial n_{\sigma }}
\end{equation*}
\begin{equation*}
=\frac{1}{l}-\frac{1}{2\pi }\frac{\partial G_{D}(z,\zeta )}{\partial
n_{\sigma }},
\end{equation*}
for any $z\in \partial D,$ where $l$ is the length of $\partial D$ (see \cite%
{18}). Combining this relation with \eqref{e1.3.6}, we find
\begin{equation*}
u(\zeta )=-\underset{\partial D}{\int }u(z)\frac{\partial K_{D}(z,\zeta )}{%
\partial n_{\sigma }}d\sigma +\frac{1}{l}\underset{\partial D}{\int }%
u(z)d\sigma .
\end{equation*}%
Thus, $u$ is determined up to the additive constant $\frac{1}{l}\underset{%
\partial D}{\int }u(z)d\sigma .$
\end{proof}

Next, we derive a formula that solves the Neumann problem \eqref{e1.3.2}.

\begin{theorem}
\label{t1.3.9} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. If $u$ is harmonic in $D$ and
continuously differentiable on $\partial D$ then, up to an additive
constant,
\begin{equation}  \label{e1.3.7}
u(\zeta )=-\underset{\partial D}{\int }\frac{\partial u(z)}{\partial
n_{\sigma }}K_{D}(z,\zeta )d\sigma ,\ \zeta \in D.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t1.3.4}, using Green formula, it follows that, up to an
additive constant, a solution of the Neumann problem is given by%
\begin{equation}  \label{e1.3.8}
u(\zeta )=-\frac{1}{2\pi }\underset{\partial D}{\int }\frac{\partial u(z)}{%
\partial n_{\sigma }}N_{D}(z,\zeta )d\sigma ,\ \zeta \in D.
\end{equation}

The constant is chosen such that $u(z)$ is in $\Lambda ^{2}(D)$.

By \eqref{e1.3.4}, for $\zeta \in \partial D,$ we have
\begin{equation*}
K_{D}(z,\zeta )=\frac{1}{2\pi }N_{D}(z,\zeta ).
\end{equation*}%
Substituting this in \eqref{e1.3.8}, we obtain \eqref{e1.3.7}.
\end{proof}

Let $K_{D}^{(k)}(z,\widetilde{\zeta })$ be the function defined by
\begin{equation*}
K_{D}^{(k)}(z,\widetilde{\zeta })=\frac{1}{2}\left[ K_{D}(z,\zeta
)+K_{D}(z,k(\zeta ))\right] ,\ z\in \overline{D},
\end{equation*}%
where $K_{D}(z,k(\zeta ))$ is the harmonic kernel function of the closed
orthonormal system $(\varphi _{i})_{i},$ for the region $D$, with respect to
the point $k(\zeta ).$ The function $K_{D}^{(k)}(z,\widetilde{\zeta })$ is
in $\Lambda ^{2}(D)$ (see \cite{9}, \cite{26}).

\begin{proposition}
\label{p1.3.10} If $D$ is a symmetric region, then the function $%
K_{D}^{(k)}(z,\widetilde{\zeta })$ is symmetric with respect to $z$ on $%
\overline{D}$, that is, for every $z\in \overline{D}$,
\begin{equation*}
K_{D}^{(k)}(z,\widetilde{\zeta })=K_{D}^{(k)}(k(z),\widetilde{\zeta }).
\end{equation*}
\end{proposition}

\begin{proof}
We use \eqref{e1.3.4} and the symmetric properties of the symmetric Green
function and symmetric Neumann function.
\end{proof}

Let $\zeta _{0}$ be a point of $D$. Let $\Lambda _{0}^{2}(D)$ be the class
of harmonic functions $\varphi $ that satisfy the conditions:%
\begin{equation*}
D\{\varphi ,\varphi \}<\infty
\end{equation*}%
and
\begin{equation*}
\varphi (\zeta _{0})=0.
\end{equation*}
The harmonic kernel function $K_{D}(z,\zeta ,\zeta _{0})$ of the class $%
\Lambda _{0}^{2}(D)$ is related to the harmonic kernel function $%
K_{D}(z,\zeta )$ of the class $\Lambda _{0}^{2}(D)$ by the following
identity:%
\begin{equation*}
K_{D}(z,\zeta ,\zeta _{0})=K_{D}(z,\zeta )-K_{D}(\zeta ,\zeta _{0}).
\end{equation*}

The harmonic kernel function $K_{D}(z,\zeta )$ for the region $D$, with
respect to the point $\zeta $ is not a conformal invariant but the harmonic
kernel function $K_{D}(z,\zeta ,\zeta _{0})$ is invariant under conformal
mapping (see \cite{9}), therefore $K_{D}(z,\zeta ,\zeta _{0})$ is
well-defined on the Riemann surface $O_{2}.$

The function $K_{D}^{(k)}(z,\widetilde{\zeta },\widetilde{\zeta _{0}})$
defined by
\begin{equation*}
K_{D}^{(k)}(z,\widetilde{\zeta },\widetilde{\zeta _{0}})=\frac{1}{2}\left[
K_{D}(z,\zeta ,\zeta _{0})+K_{D}(z,k(\zeta ),k\left( \zeta _{0}\right) )%
\right] ,
\end{equation*}%
for all $z\in \overline{D}\backslash \{\widetilde{\zeta },\widetilde{\zeta
_{0}}\}$, is called the \textit{symmetric harmonic kernel function} for the
region $D.$

\subsection{Integral representations on the double cover}

We first express the solution of the Neumann problem \eqref{e1.3.2} for
harmonic functions in terms of $d\sigma $ as a line integral involving the
boundary function and a symmetric Neumann function.

\begin{theorem}
\label{t1.3.11} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves and let $g$ be a symmetric, continuous
function on $\partial D$. If $u$ is harmonic in $D$ and $g$ is its inward
normal derivative on $\partial D,$ then up to an additive constant
\begin{equation}  \label{e1.3.9}
u(\zeta )=-\frac{1}{2\pi }\underset{\partial D}{\int }g(z)N_{D}^{(k)}(z,%
\widetilde{\zeta })d\sigma ,\ \zeta \in D.
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t1.3.4}, up to the additive constant $\frac{1}{l}\underset{%
\partial D}{\int }u(z)d\sigma $, we have for all $\zeta \in D,$%
\begin{equation*}
u(\zeta )=-\frac{1}{2\pi }\underset{\partial D}{\int }g(z)N_{D}(z,\zeta
)d\sigma\,.
\end{equation*}%
Replacing $\zeta $ with $k(\zeta )$ we get
\begin{equation*}
u(k(\zeta ))=-\frac{1}{2\pi }\underset{\partial D}{\int }g(z)N_{D}(z,k(\zeta
))d\sigma .
\end{equation*}%
Adding the last two equations and dividing by 2, we obtain, up to the
additive constant $\frac{1}{l}\underset{\partial D}{\int }u(z)d\sigma $,
\begin{equation*}
\frac{u(\zeta )+u(k(\zeta ))}{2}=-\frac{1}{2\pi }\underset{\partial D}{\int }%
g(z)\frac{N_{D}(z,\zeta )+N_{D}(z,k(\zeta ))}{2}d\sigma .
\end{equation*}%
By Proposition \ref{p1.3.2}, $u$ is a symmetric function on $D$, then the
left-hand side of the last equality is $u(\zeta ).$ Therefore
\begin{equation*}
u(\zeta )=-\frac{1}{4\pi }\underset{\partial D}{\int }g(z)\left[
N_{D}(z,\zeta )+N_{D}(z,k(\zeta ))\right] d\sigma ,
\end{equation*}%
up to the additive constant $\frac{1}{l}\underset{\partial D}{\int }%
u(z)d\sigma .$
\end{proof}

Similarly, we obtain a formula for the symmetric solution of the Neumann
problem \eqref{e1.3.2} on a symmetric region $D,$ in terms of the symmetric
harmonic kernel function.

\begin{theorem}
\label{t1.3.12} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. Let $g$ be a symmetric, continuous
function on $\partial D.$ If $u$ is harmonic in $D$ and $g$ is its inward
normal derivative on $\partial D,$ then up to an additive constant,%
\begin{equation}  \label{e1.3.10}
u(\zeta )=-\underset{\partial D}{\int }g(z)K_{D}^{(k)}(z,\widetilde{\zeta }%
)d\sigma ,\ \zeta \in D.
\end{equation}
\end{theorem}

\begin{proof}
It is similar with the proof of the Theorem \ref{t1.3.11}. Here we use
Theorem \ref{t1.3.9}.
\end{proof}

The next theorem yields a formula for the symmetric solution of the
Dirichlet problem \eqref{e1.2.2} on a symmetric region $D,$ in terms of the
symmetric harmonic kernel function.

\begin{theorem}
\label{t1.3.13} Let $D$ be a symmetric region bounded by a finite number of $%
\sigma $-rectifiable Jordan curves. Let $f$ be a symmetric, continuous
function on $\partial D.$ There is a unique symmetric function $u$ on $%
\overline{D}$, which is harmonic on $D$, continuous on $\overline{D},$ such
that $u=f$ on $\partial D.$ For all $\zeta $ in $D$,%
\begin{equation}  \label{e1.3.11}
u(\zeta )=-\underset{\partial D}{\int }f(z)\frac{\partial K_{D}^{(k)}(z,%
\widetilde{\zeta })}{\partial n_{\sigma }}d\sigma .
\end{equation}
\end{theorem}

\begin{proof}
By Theorem \ref{t1.3.8}, for all $\zeta \in D,$
\begin{equation*}
u(\zeta )=-\underset{\partial D}{\int }u(z)\frac{\partial K_{D}(z,\zeta )}{%
\partial n_{\sigma }}d\sigma .
\end{equation*}%
Replacing $\zeta $ with $k(\zeta )$ we get, for all $\zeta \in D,$
\begin{equation*}
u(k(\zeta ))=-\underset{\partial D}{\int }u(z)\frac{\partial K_{D}(z,k(\zeta
))}{\partial n_{\sigma }}d\sigma .
\end{equation*}%
Adding the last two equations and dividing by 2, it follows that
\begin{equation*}
\frac{u(\zeta )+u(k(\zeta ))}{2}=-\frac{1}{2}\underset{\partial D}{\int }u(z)%
\left[ \frac{\partial K_{D}(z,\zeta )}{\partial n_{\sigma }}+\frac{\partial
K_{D}(z,k(\zeta ))}{\partial n_{\sigma }}\right] d\sigma ,
\end{equation*}%
for all $\zeta \in D.$

By Proposition \ref{p1.2.1}, $u$ is a symmetric function on $D$, then the
left-hand side of the last equality is $u(\zeta )$ and we conclude that for
all $\zeta $ in $D,$
\begin{equation*}
u(\zeta )=-\frac{1}{2}\underset{\partial D}{\int }u(z)\left[ \frac{\partial
K_{D}(z,\zeta )}{\partial n_{\sigma }}+\frac{\partial K_{D}(z,k(\zeta ))}{%
\partial n_{\sigma }}\right] d\sigma .
\end{equation*}%
The uniqueness of the solution of the Dirichlet problem for harmonic
functions implies \eqref{e1.3.11}.
\end{proof}

\subsection{Integral representations on the Klein surface}

Let $\widetilde{\zeta }$ be a point inside $\Omega .$ A \textit{Neumann
function} $N_{\Omega }(\widetilde{z},\widetilde{\zeta })$ for the region $%
\Omega ,$ with singularity at $\widetilde{\zeta }$ is defined by
\begin{equation}  \label{e1.3.12}
N_{\Omega }(\widetilde{z},\widetilde{\zeta })=N_{D}^{(k)}(z,\widetilde{\zeta
})=N_{D}^{(k)}(k(z),\widetilde{\zeta }),
\end{equation}%
where $\widetilde{z}=\pi (z).$

\begin{remark}
\label{r1.3.14} By Proposition \ref{p1.3.5}, it follows that $N_{\Omega }(%
\widetilde{z},\widetilde{\zeta })$ is well-defined on $\Omega $.
\end{remark}

Therefore $N_{\Omega }(\widetilde{z},\widetilde{\zeta })$ is a harmonic
function on $\Omega \setminus \left\{ \widetilde{\zeta }\right\} $, which
has a constant normal derivative $\frac{\partial N_{\Omega }}{\partial
n_{\Sigma }}$ on the boundary $\partial \Omega $ and has a logarithmic pole
at the point $\widetilde{\zeta }=\pi (\zeta ).$

Next, we derive the solution of the Neumann problem \eqref{e1.3.1} on the
region $\Omega .$

\begin{theorem}
\label{t1.3.15} Let $G$ be a continuous real-valued function on $\partial
\Omega .$ Then, up to an additive constant, the solution of problem %
\eqref{e1.3.1} is the function $U$ defined by the relation $u=U\circ \pi $,
where $\pi $ is the canonical projection of $O_{2}$ on $X$ and $u$ is the
solution \eqref{e1.3.9} of the problem \eqref{e1.3.2} on the symmetric
region $D,$ with the inward normal derivative $g$ given by $g=G\circ \pi $
on $\partial D.$
\end{theorem}

\begin{proof}
The symmetry of the function $u$ on $D$, yields
\begin{equation*}
\Delta U(\widetilde{\zeta })=\Delta u(\zeta )=\Delta u(k(\zeta ))=0\quad
\mbox{for all $\widetilde{\zeta }\in
\Omega $},
\end{equation*}
where $\widetilde{\zeta }=\pi (\zeta )$.

Using the symmetry of the function $g$ on $\partial D$, we obtain
\begin{equation*}
\frac{\partial U}{\partial n_{\Sigma }}(\widetilde{\zeta })=\frac{\partial U%
}{\partial n_{\sigma }}(\zeta ) =g(\zeta )=g(k(\zeta ))=G(\widetilde{\zeta }%
),
\end{equation*}
for all $\widetilde{\zeta }\in \partial \Omega $. Then, up to an additive
constant, the function $U$ defined on $\Omega $ by
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=u(k(\zeta )),
\end{equation*}%
for all $\widetilde{\zeta }$ in $\Omega $, is the solution of problem %
\eqref{e1.3.1}.
\end{proof}

Let $\widetilde{\zeta }$ be a point inside $\Omega $. The harmonic kernel
function $K_{\Omega }(\widetilde{z},\widetilde{\zeta })$ of the closed
orthonormal system $(\varphi _{i})_{i},$ for the region $\Omega $, with
respect to the point $\widetilde{\zeta }=\{\zeta ,k(\zeta )\}$ is defined by%
\begin{equation*}
K_{\Omega }(\widetilde{z},\widetilde{\zeta })=K_{D}^{(k)}(z,\widetilde{\zeta
})=K_{D}^{(k)}(k(z),\ \widetilde{\zeta }),\widetilde{z}=\pi (z)\in \Omega .
\end{equation*}

\begin{remark}
\label{r1.3.16} By Proposition \ref{p1.3.10}, it follows that $K_{\Omega }(%
\widetilde{z},\widetilde{\zeta })$, is well-defined on $\Omega $.
\end{remark}

The symmetric solutions on $O_{2}$ determine the solutions of the similar
problems on the Klein surface $X$. Thus, we obtain the solution of the
Dirichlet problem \eqref{e1.2.1} on the region $\Omega $, with respect to
the harmonic kernel function, for the region $\Omega .$

\begin{theorem}
\label{t1.3.17} Let $F$ be a continuous real-valued function on the border $%
\partial \Omega . $ The solution of the problem \eqref{e1.2.1} with the
boundary function $F$ is the function $U$ defined on $\overline{\Omega },$
by the relation $u=U\circ \pi ,$ where $\pi $ is the canonical projection of
$O_{2}$ on $X$ and $u$ is the solution \eqref{e1.3.11} of the problem %
\eqref{e1.2.2} on the symmetric region $D$, with the boundary function $f$
given by $f=F\circ \pi .$
\end{theorem}

\begin{proof}
By definition,
\begin{equation*}
\Delta U(\widetilde{\zeta })=\Delta u(\zeta )=0\quad \mbox{for all}\
\widetilde{\zeta }\in \Omega ,
\end{equation*}
where $\widetilde{\zeta }=\pi (\zeta ),$ thus $U$ is a harmonic function.
The symmetry of the function $f$ on $\partial D$, implies
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=f(\zeta ) =f(k(\zeta ))=F(\widetilde{\zeta }%
)\quad\mbox{for all $\widetilde{\zeta }\in
\partial \Omega .$}
\end{equation*}
Due to the uniqueness of the solution, the function $U$ defined on $%
\overline{\Omega }$ by
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=u(k(\zeta )),
\end{equation*}%
for all $\widetilde{\zeta }$ in $\overline{\Omega },$ where $\widetilde{%
\zeta }=\pi (\zeta ),$ is the solution of problem \eqref{e1.2.1} on $\Omega $%
.
\end{proof}

The next theorem gives the solution of the Neumann problem \eqref{e1.3.1} on
the region $\Omega $, with respect to the harmonic kernel function, for the
region $\Omega .$

\begin{theorem}
\label{t1.3.18} Let $G$ be a continuous real-valued function on the border $%
\partial \Omega . $ Then, up to an additive constant, the solution of the
problem \eqref{e1.3.1} with the normal derivative $G$ on $\partial \Omega $
is the function $U$ defined on $\overline{\Omega },$ by the relation $%
u=U\circ \pi ,$ where $\pi $ is the canonical projection of $O_{2}$ on $X$
and $u$ is the solution \eqref{e1.3.10} of the problem \eqref{e1.3.2} on the
symmetric region $D$, with the normal derivative function $g$ given by $%
g=G\circ \pi $ on $\partial D.$
\end{theorem}

\begin{proof}
By definition,
\begin{equation*}
\Delta U(\widetilde{\zeta })=\Delta u(\zeta )=0 \quad%
\mbox{for all
$\widetilde{\zeta }\in \Omega $},
\end{equation*}
where $\widetilde{\zeta }=\pi (\zeta ),$ thus $U$ is a harmonic function.
The symmetry of the function $g$ on $\partial D,$ implies
\begin{equation*}
\frac{\partial U(\widetilde{\zeta })}{\partial n_{\Sigma }}=\frac{\partial
u(\zeta )}{\partial n_{\sigma }}=g(\zeta )=g(k(\zeta ))=G(\widetilde{\zeta }%
),
\end{equation*}
for all $\widetilde{\zeta }\in \partial \Omega .$ Thus, up to an additive
constant, the function $U$ defined on $\overline{\Omega }$ by
\begin{equation*}
U(\widetilde{\zeta })=u(\zeta )=u(k(\zeta )),
\end{equation*}%
is the solution of the problem \eqref{e1.3.1} on $\Omega $.
\end{proof}

\subsection*{Concluding remarks}

The methods developed in this paper remain valid in the case of all
differential operators associated to \textit{conformal invariant metrics}.
Such an example corresponds to the \textit{invariant Laplacian} (or
sometimes the \textit{Laplace-Beltrami operator} for the Poincar\'e-Bergman
metric), see Krantz \cite[Section 6.5]{15a}. We also refer to the \textit{%
pseudo-hyperbolic metric}, which is conformally invariant, but it does not
arise from integrating an infinitesimal metric (that is, lengths of tangent
vectors at a point). A comprehensive analysis of the pseudo-hyperbolic
metric on the disc may be found in Krantz \cite{15b}.

To the best of our knowledge, there are not further results involving either
\textit{linear} or \textit{nonlinear} elliptic equations on Klein surfaces.
This study can include qualitative and quantitative properties of solutions
but also related singular or degenerate phenomena. We consider that the
mathematical analysis of these classes of PDEs on \textit{Klein surfaces} is
a very rich and attractive research field at the interplay between complex
analysis and nonlinear analysis.

\begin{thebibliography}{KLE1882}
\bibitem[AHL]{1} L.V. Ahlfors, \textit{Conformal Invariants: Topics in
Geometric Function Theory}, McGraw-Hill Series in Higher Mathematics,
McGraw-Hill Book Co., New York-D\"usseldorf-Johannesburg, 1973.

\bibitem[AHS]{2} L.V. Ahlfors and L. Sario, \textit{Riemann Surfaces},
Princeton Mathematical Series, No. 26, Princeton University Press,
Princeton, N.J., 1960.

\bibitem[ALG]{3} N.L. Alling and N. Greenleaf, \textit{Foundations of the
Theory of Klein Surfaces}, Lecture Notes in Math., vol. 219,
Springer-Verlag, New York, 1971.

\bibitem[ANC]{4} C. Andreian Cazacu, On the morphisms of Klein surfaces,
\textit{Rev. Roumaine Math. Pures Appl. } \textbf{31} (1986), 461-470.

\bibitem[BAR]{6} I. B\^{a}rz\u{a}, Integration on non-orientable Riemann
surfaces, In: \textit{Almost} \textit{Complex Structures}, World Scientific,
Singapore-New Jersey-London-Hong Kong, 1995, 63-97.

\bibitem[BAG1]{5} I. B\^{a}rz\u{a} and D. Ghi\c{s}a, Explicit formulas for
Green's functions on the annulus and on the M\"{o}bius strip, \textit{Acta
Applicandae Mathematicae} \textbf{54} (1998), 289-302.

\bibitem[BAG2]{7} I. B\^{a}rz\u{a} and D. Ghi\c{s}a, Boundary value problems
on non-orientable surfaces, \textit{Rev. Roumaine Math. Pures Appl. } \textbf{%
43} (1998), 67-79.

\bibitem[BER]{9} S. Bergman, \textit{The Kernel Function and Conformal
Mapping}, Mathematical Surveys, No. 5, American Mathematical Society, New
York, 1950.

\bibitem[BDPPT]{9a} \textcolor{red} {Y. Bouligand, B. Derrida, V. Poenaru,Y. Pomeau and G. Toulouse, Distortions with double topological character - case of cholesterics, \textit{Journal de Physique}, Vol.39, Issue:8, 863-867, 1978.}

\bibitem[BOU]{9b} \textcolor{red} {Y. Bouligand, Geometry and topology of defects in liquid crystals, In R. Balin et al. (Editors), \textit{Physics of Defects}, North Holland Publ. Co., 1981.}

\bibitem[BEGG]{8} E. Bujalance, J.J. Etayo, J.M. Gamboa and G. Gromadzki,
\textit{Automorphism Groups of Compact Bordered Klein Surfaces. A
Combinatorial Approach}, Lecture Notes in Math., vol. 1439, Springer-Verlag,
Berlin, 1990.

\bibitem[COH]{10} \textcolor{red} {R. Courant and D. Hilbert, \textit{Methoden der mathematischen
Physik}, Springer-Verlag, 1924. First English edition \textit{Methods of Mathematical Physics},Vol.1, Wiley-Interscience, New York, 1953.}

\bibitem[GAM]{11} J.B. Garnett and D.E. Marshall, \textit{Harmonic Measure},
New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge,
2005.

\bibitem[JSS]{11a}  \textcolor{red} {M. Jurchescu, K. Spallek, and F. Succi, DC-Spaces. In: S. Dimiev and K. Sekigawa (Editors), \textit{Topics of Complex Analysis, Differential
Geometry and Mathematical Physics}, World Scientific, Singapore-New
Jersey-London-Hong Kong, 1997.}

\bibitem[KAT] {11b} \textcolor{red} {M. Kaku and J. Trainer, \textit{\ Beyond Einstein: The Cosmic Quest for the Theory of Universe}, Bantam Books, 1987.}

\bibitem[KLE1882]{12} F. Klein, \textit{Ueber Riemanns Theorie der
algebraischen Funktionen und ihrer integrale}, Teubner, Leipzig, 1882.

\bibitem[KLE2]{13} F. Klein, \textit{Riemann'sche Fl\"{a}che I und II},
Vorlesungen G\"{o}ttingen, Wintersemester 1891-1892 und Sommersemester 1892.

\bibitem[KRA1]{15a} S.G. Krantz, \textit{Partial Differential Equations and
Complex Analysis}, Studies in Advanced Mathematics, CRC Press, Boca Raton,
FL, 1992.

\bibitem[KRA2]{15} S.G. Krantz, \textit{A} \textit{Panorama of Harmonic
Analysis}, Mathematical Association of America, Washington, DC, 1999.

\bibitem[KRA3]{15b} S.G. Krantz, \textit{Complex Analysis: The Geometric
Viewpoint}, 2nd ed., Mathematical Association of America, Washington, DC,
2004.

\bibitem[KRA4]{14} S.G. Krantz, \textit{Cornerstones of Geometric Function
Theory:} \textit{Explorations in Complex Analysis}, Birkh\"{a}user, Boston,
2006.

\bibitem[KRA5]{14b} S.G. Krantz, \textit{A Guide to Complex Variables}, The
Dolciani Mathematical Expositions, vol. 32, Mathematical Association of
America, Washington, DC, 2008.

\bibitem[LEH]{16} O. Lehto, \textit{Univalent Functions and Teichm\"{u}ller
spaces}, Graduate Texts in Math., vol. 109, Springer-Verlag, New York, 1987.

\bibitem [MUN] {16a} \textcolor{red} {T. Munzner, {\tt https://www.youtube.com/watch?v=-gLNlC\_hQ3M}}

\bibitem[NEV1]{18} R. Nevanlinna, \textit{Uniformisierung}, Grundlehren
Math. Wiss., Bd. 64., Springer-Verlag, Berlin-G\"{o}ttingen-Heidelberg, 1953.

\bibitem[NEV2]{17} R. Nevanlinna, \textit{Analytic Functions},
Springer-Verlag, Berlin, 1970.

\bibitem[PIR]{19} Y. Pinchover and J. Rubinstein, \textit{An Introduction to
Partial Differential Equations}, Cambridge University Press, Cambridge, 2005.

\bibitem[POI]{20} H. Poincar\'{e}, \textit{Sur l'uniformisation des
fonctions analytiques}, Acta Math. \textbf{31} (1907), 1-63.

\bibitem[PRE]{21} R. Preston, \textit{Projective structures and fundamental
domains on compact Klein surfaces}, Ph.D. thesis, Univ. of Texas, 1975.

\bibitem[RAD]{22} V. R\u{a}dulescu, \textit{Qualitative Analysis of
Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic,
and Variational Methods}, Contemporary Mathematics and Its Applications,
vol. 6, Hindawi Publ. Corp., 2008.

\bibitem[RAN]{23} T. Ransford, \textit{Potential Theory in the Complex Plane}%
, Cambridge University Press, Cambridge, 2003.

\bibitem[RAP]{23a} \textcolor{red} {D. L. Rapoport, Klein bottle logophysics: a unified principle for
nonlinear systems, cosmology, geophysics, biology, biomechanics and
perception, \textit{J. Phys.: Conf. Ser.} 437 (2013) 012024.}

\bibitem[RIE]{23b} \textcolor{red}{B. Riemann, Theorie der Abel'schen Functionen,
\textit{J. Reine Angew. Math.} {\bf 54} (1857), 115-155.}

\bibitem[ROS1]{27} M. Ro\c{s}iu, Associating divisors with quadratic
differentials on Klein surfaces, \textit{Complex} \textit{Variables and
Elliptic Equations}, 60 (2) (2015), 181-190.

\bibitem[ROS2]{24} M. Ro\c{s}iu, Harmonic measures and Poisson kernels on
Klein surfaces, \textit{Electronic} \textit{Journal of Differential Equations%
}, Vol. 2017 (2017), No. 269, 1-7.

\bibitem[ROS3]{25} M. Ro\c{s}iu, On a class of Neumann problems on Klein
surfaces, \textit{Proceedings of the} \textit{Romanian Academy, Series A},
Vol. 18, No. 2/2017, 116-123.

\bibitem[ROS4]{26} M. Ro\c{s}iu, Kernel function and integral
representations on Klein surfaces, \textit{Electronic} \textit{Journal of
Differential Equations}, Vol. 2017 (2017), No. 132, 1-8.

\bibitem[SCS]{28} M. Schiffer and D.C. Spencer, \textit{Functionals of
Finite Riemann Surfaces}, Princeton University Press, Princeton, NJ, 1954.

\bibitem[SES]{29} M. Seppala and T. Sorvali, \textit{Geometry of Riemann
Surfaces and Teichm\"{u}ller Spaces}, North-Holland Mathematics Studies,
vol. 169, North-Holland, Amsterdam, 1992.

\bibitem[TEI]{30} \textcolor{red}{O. Teichm\"{u}ller, \textit{Extremale quasikonforme Abbildungen und quadratische Differentiale}, Abh. Preuss. Akad. Wiss., Math.-Naturw. Kl.
1940, No. 22, 1-197, 1940. English translation by G. Th\'{e}ret, Extremal
quasiconformal mappings and quadratic differentials. In \textit{Handbook of
Teichm\"{u}ller theory} (A. Papadopoulos, ed.), Volume V, EMS Publishing
House, 321-484, 2015.}

\bibitem[WHRH]{32} \textcolor{red} {D.M. Walba, T.C. Homan, R.M. Richards, and R.C. Haltiwanger, Topological stereochemistry .9. synthesis and cutting in
half of a molecular M\"{o}bius strip, \textit{New J.  Chemistry} {\bf 17} (1993), 661-681.}

\bibitem[WEE]{31} \textcolor{red} {J. Weeks, \textit{The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds}, Pure and Applied Mathematics, Vol. 96, Marcel
Dekker, New York, 1985.}
\end{thebibliography}

\end{document}
