text:
\section{Introduction}
Motivated by questions related to some conjecture of Bishop \cite{Bis}, we
continue
 the analysis of spectral minimal partitions developed for plane
 domains
 in  \cite{HHOT}, \cite{HHO}, \cite{BHV:2007}, \cite{BHHO} and analyze
 the case of the two-dimensional sphere $\mathbb S^2$.\\

In the whole  paper the Laplacian is the Laplace-Beltrami operator on
$\mathbb S^2$.
We describe as usual $\mathbb S^2$ in $\mathbb R^3_{x,y,z}$ by the spherical  coordinates,
 \begin{equation}
x= \cos \phi\sin \theta , y =\sin \phi \sin \theta, z= \cos
\theta\,,\,
 \mbox{ with } \phi\in [-\pi,\pi[\,, \, \theta \in ]0,\pi[\,,
\end{equation}
and we add the two
 poles ``North'' and ``South'', corresponding to the two points
  $(0,0,1)$ and $(0,0,-1)$. \\
If $\Omega$ is a regular bounded open set
 with piecewise $C^{1,+}$ boundary\footnote{ i.e. with piecewise
$C^{1,\alpha}$ boundary, for some $\alpha >0$},
 we consider the Dirichlet Laplacian $H=H(\Omega)$ and we would like to analyze
 the question of the existence and of the properties of minimal
 partitions
 for open sets $\Omega$ on $\mathbb S^2$. When $\Omega$ is strictly
 contained
 in $\mathbb S^2$, the question is not fundamentally
different of the case of planar domains, hence
 we will focus on the case of the whole sphere $\mathbb S^2$ and
 on the search for  possible candidates for minimal $k$-partitions
 of the sphere for $k$ small.\\

To be more precise, let us now recall a few definitions that the reader
can for example find in \cite{HHOT}.
For  $1\le k\in \mathbb N$ and $\Omega\subset \mathbb S^2$, we call  a
{\bf spectral}
$ k$-\textbf{partition}\footnote{We say more shortly ``$k$-partition''.} of ${\Omega}$   a family $\mathcal D=\{D_i\}_{i=1}^k$ of pairwise  disjoint
open regular domains  such that
\begin{equation}\label{opar}
\cup_{i=1}^kD_i\subset {\Omega}.
\end{equation}
It is called  \textbf{strong} if
\begin{equation}\label{regcov}
{\rm Int\,}(\overline{\cup_{i=1}^k D_i})\setminus \pa {\Omega}={\Omega}\,.
\end{equation}
We denote by $\mathfrak O_k$ the set of such partitions.
 For $\mathcal D\in \mathfrak O_k$ we introduce
\begin{equation}\label{LA}
\Lambda(\mathcal D)=\max_{i}{\lambda}(D_i)\,,
\end{equation}
where $\lambda(D_i)$ is the ground state energy of $H(D_i)\,$,
and
\begin{equation}\label{Lfrak}
\mathfrak L_k(\Omega) =\inf_{\mathcal D\in \mathfrak O_k}\Lambda(\mathcal D)\,.
\end{equation}
We call  a spectral minimal  $k$-partition, a $k$-partition
$\mathcal D\in \mathfrak O_k$
such that  $$  \mathfrak L_k(\Omega)=\Lambda(\mathcal
 D)\,.
$$
More generally we can consider (see in \cite{HHOT}) for $p\in [1,+\infty[$
\begin{equation}\label{LAp}
\Lambda^p(\mathcal D)=(\frac 1k \sum_i {\lambda}(D_i)^p)^{\frac 1p}\,,
\end{equation}
and
\begin{equation}\label{Lfrakp}
\mathfrak L_{k,p}(\Omega)=\inf_{\mathcal D\in \mathfrak O_k}\Lambda^p(\mathcal D)\,.
\end{equation}
We write  $\mathfrak L_{k,\infty}(\Omega)=\mathfrak L_k(\Omega)\,$ and
recall the monotonicity property
\begin{equation}\label{monoto}
\mathfrak L_{k,p} (\Omega) \leq \mathfrak L_{k,q}(\Omega)\,\mbox{ if }
p\leq q\,.
\end{equation}
The notion of $p$-minimal $k$-partition can be extended accordingly,
by minimizing $\Lambda^p(\mathcal D)$.\\
We would like to give in this article the proof of the following theorem.
\begin{theorem}\label{conj1}~\\
Any minimal  $3$-partition of $\mathbb S^2$ is up to a fixed rotation
obtained by the so called {\bf Y}-partition whose boundary is given by the intersection
of $\mathbb S^2$ with  the three half-planes  defined respectively by
$\phi =0, \frac{2\pi}{3}, \frac{-2 \pi}{3}\,$. Hence
\begin{equation}\label{valeur}
\mathfrak L_3(\mathbb S^2)=\frac{15}{4}\,.
\end{equation}
\end{theorem}
This theorem is immediately related (actually a consequence of) to
a conjecture of Bishop  (Conjecture 6)  proposed in \cite{Bis} stating that~:
\begin{conjecture}[Bishop 1992]\label{conj2}~\\
The minimal $3$-partition for $\frac 13 (\sum_{i=1}^3 \lambda(D_i))$
 corresponds to the {\bf Y}-partition.
\end{conjecture}
We can indeed observe that if for some $(k,p)$ there exists a  $p$-minimal
$k$-partition $\mathcal D_{k,p}$ such that $\lambda(D_i) =\lambda(D_j)$ for all $i,j$,
then by the monotonicity property $\mathcal D_{k,p}$ is a $q$-minimal
partition for any $q\geq p$.\\
\begin{remark}~\\
At the origin, Bishop's Conjecture was motivated by the analysis
 of the properties of Harmonic functions in conic sets. The whole
 paper by Friedland-Hayman \cite{FrHa} (see also references therein) which inspires our Section \ref{s5} is written
 in this context. The link between our problem of minimal partitions
 and the problem in harmonic analysis can be summarized in this way.
If we consider a homogeneous Lipschitzian function of the form
 $u(x)=r^\alpha g(\theta,\phi)$ in $\mathbb R^3$,
 which is harmonic outside its nodal set
 and such that the complementary of the nodal set divides
 the sphere in three parts, then
 $$ \alpha (\alpha +1)\geq \mathfrak L_3(\mathbb S^{2}) \,.$$
Hence Theorem \ref{conj1} (and more
specifically
 \eqref{valeur}) implies  $\alpha\geq 3/2 $.
This kind of property can be
 useful to improve some statements in \cite{CL1,CL2} (see inside the
 proofs of Lemmas~2 in
 \cite{CL1} and 4.1 in \cite{CL2}).
\end{remark}
A similar question was analyzed (with partial success) when looking in
\cite{HHO} at
candidates of minimal $3$-partitions  of the unit disk $D(0,1)$ in
$\mathbb R^2$. The most natural
candidate
 was indeed  the Mercedes Star, which is  the $3$-partition
given   by three disjoint sectors with opening
angle $2\pi/3$, i.e.
 \begin{equation}\label{sector}
D_1=\{x\in {\Omega}\:|\:\omega\in ]0,2\pi/3[\}
\end{equation}
and $D_2, D_3$ are obtained by rotating $D_1$ by $2\pi/3\,$,
respectively by  $4\pi/3\,$.
Hence the Mercedes star in \cite{HHO}  is replaced here by the ${\bf
  Y}$-partition in Theorem \ref{conj1}.
We observe that  {\bf Y}-partition  can also be described  the inverse image of the mercedes-star
 partition  by the map
$\mathbb S^2\ni (x,y,z)\mapsto (x,y)\in D(0,1)$.\\

Here let us mention the two main statements giving the proof of Theorem~\ref{conj1}.
\begin{proposition}\label{antipodal}~\\
If  $\mathcal D=(D_1,D_2,D_3)$ is a $3$-minimal partition, then its
boundary
 contains two antipodal points.
\end{proposition}
The proof of Proposition \ref{antipodal} will be achieved in Section \ref{s6} and involves Euler's
formula
 and the thorem of  Lyusternik and Shnirelman.
\begin{proposition}\label{centera}~\\
If there exists a  minimal $3$-partition $\mathcal D=(D_1,D_2,D_3)$
 of $\, \mathbb S^2$ with two antipodal points in
$\cup_{i} \pa D_i$,  then it is (after possibly a rotation)
 the  ${\bf Y}$-partition.
\end{proposition}
The proof of Proposition \ref{centera}  will be done in  Section \ref{s8} by lifting this  $3$-partition
  on the double covering $\mathbb S^2_{\mathcal C}$ of $\ddot{\mathbb S^2}$, where
  $\ddot{\mathbb S^2}$ is the
  sphere
 minus two antipodal points.\\

More precisely, following what has been done in the approach of the Mercedes-star
conjecture in \cite{HHO}, the steps for the proof
 of Theorem \ref{conj1} (or towards Conjecture \ref{conj2} if we were
 able to
 show that the minimal partition for $\Lambda^1$ has all the $\lambda(D_j)$  equal)
are the following~:
\begin{enumerate}
\item
One has to prove that minimal partitions on $\mathbb S^2$ exist
 and share the same properties as for planar domains : regularity and
 equal  angle meeting  property. This will be done in Section~\ref{s2}.
\item One can observe that  the
 minimal $3$-partition cannot be a nodal partition. This is a
 consequence of Theorem \ref{L=L} in Section \ref{s2} and
  of the fact that the multiplicity of the second
  eigenvalue (i.e. the first non zero one) is more than $2$ actually
   $3$.
\item The Euler formula implies that there exists only one
 possible  type of minimal $3$-partitions. Its boundary
 consists of  two points $x_1$ and $x_2$ and three arcs
 joining  these two points. This will be deduced in Subsection~\ref{s3}.
\item The next point is  to show a  minimal partition
  has in its boundary two antipodal points.
\item The next point is that any minimal $3$-partition  which contains
  two antipodal  points in its boundary can be lifted in a symmetric $6$-partition
  on the double covering $\mathbb S^2_{\mathcal C}$. More precisely,
if $\mathcal D
 =(D_1,D_2,D_3)$
 and $\Pi$ is the canonical projection of $\mathbb S^2_{\mathcal C}$
 onto $\ddot{\mathbb S^2}$ , we get the $6$-partition $\mathcal
   D_{\mathcal C}$
 by considering
$$
 \mathcal
   D_{\mathcal C} = (D_1^+,D_2^+,D_3^+, D_1^{-}, D_2^{-}, D_3^-)\,,
$$
where for $j=1,2,3$ $D_j^{+}$ and $D_{j}^{-}$ denote the two
components
 of $\Pi^{-1} (D_j)$. If $\mathcal I$ denotes the map on $\mathbb
 S^2_{\mathcal C}$
 defined, for $m\in \mathbb S^2_{\mathcal C}$  by
\begin{equation}\label{defIrev}
\Pi (\mathcal I(m))= \Pi(m) \mbox{ with }\mathcal I(m)\neq m \,,
\end{equation}
we observe that
$$
\mathcal I (D_j^+)=D_j^-\;.
$$

\item The last point is to show that on this  double covering
 a minimal  symmetric $6$-partition is necessarily the double {\bf
 Y}-partition, which is the inverse image in $\mathbb
 S^2_{\mathcal C}$
 of the {\bf Y}-partition.
\end{enumerate}
All these points will be detailed in the following sections
 together with analogous questions in the case of minimal
 $4$-partitions.\\

In the last section, we will describe what can be said towards
 the proof of Bishop's conjecture and about the large $k$ behavior
 of $\mathfrak L_k$ using mainly the tricky estimates of
 Friedland-Hayman
 \cite{FrHa}.

\section{Definitions, notations and extension of  previous results to
  the sphere.}\label{s2}
We first recall more  notation, definitions and results essentially
  extracted
 of \cite{HHOT} but which have to be extended from the case of planar
 domains  to the case of domains  in $\mathbb S^2$.
\begin{definition}\label{def:regopenset}~\\
We say that an open  domain  $D\subset  \mathbb S^2$ is
 \textbf{regular}
 if it satisfies an interior cone condition and if
$\pa D$ is the union a finite number of simple regular arcs
$\gamma_i(\overline{I_i})$, with $\gamma_i\in\mathcal C^{1,+}(\overline I_i)$ with no mutual
nor self intersections, except possibly at the endpoints.
\end{definition}

For a given set $\Omega\subset \mathbb S^2$, we  are
interested in the eigenvalue problem for $H({\Omega})$, the Dirichlet
realization of the Laplace Beltrami operator in $\Omega$.  We shall denote for any open domain
$\Omega$
 by $\lambda(\Omega)$  the lowest
eigenvalue of $H(\Omega)$.
 We define for any eigenfunction $u$ of $H(\Omega)$
\begin{equation}
N(u)=\overline{\{x\in {\Omega}\:\big|\: u(x)=0\}}
\end{equation}
and call the components of ${\Omega}\setminus N(u)$ the nodal domains of $u$. The number of
nodal domains of such a function will be called $\mu(u)$.

If $\mathcal D$ is a strong partition,
we say $D_i,D_j\in \mathcal D$ are \textbf{neighbors}
if
\begin{equation}\label{DsimD}
{\rm Int\,}(\overline{D_i\cup D_j})\setminus \pa {\Omega}\text{ is connected}
\end{equation}
and write in this case  $D_i\sim D_j$.
We  then define   a {\bf graph}  $G(\mathcal D)$
by associating to
 each $D_i\in \mathcal D$  a vertex $v_i$ and to  each pair $D_i\sim D_j$ we associate
an edge $e_{i,j}$.

Attached to a regular partition $\mathcal D$ we can associate its
boundary  $N=N(\mathcal D)$ which is the closed set in $\overline{\Omega}$
defined by
\begin{equation}
N(\mathcal D)=\overline{\bigcup_i(\pa D_i\cap {\Omega})}.
\end{equation}
This leads us to introduce the  set $\mathcal M({\Omega})$ of the regular closed sets.

\begin{definition}\label{AMS}~\\
A closed set $N\subset \overline{\Omega}$ belongs to $\mathcal M({\Omega})$ if $N$ meets the following requirements:
\\
\textbf{(i)}
There are finitely many distinct critical points $x_i\in {\Omega}\cap N$ and associated positive
integers $\nu(x_i)$  with $\nu(x_i)\ge 3$
such that, in a sufficiently small neighborhood of each of the $x_i$,
$N$ is the union of $\nu(x_i)$ disjoint
 (away from $x_i$ non self-crossing) smooth arcs  with one end at $x_i$ (and each pair
defining at $x_i$ a positive angle in $]0,2\pi[$)
and such that in the complement of these points in ${\Omega}$, $N$
is locally diffeomorphic to a smooth arc. We denote by $X(N)$ the set of these critical points.  \\
\textbf{(ii)}
$\pa{\Omega}\cap N$ consists of a (possibly empty) finite set of points $z_i$, such that at each $z_i$,
$\rho(z_i), \text{with } \rho(z_i)\ge 1$  arcs hit the boundary. Moreover for each $z_i\in \pa {\Omega}$,
then $N$ is near $z_i$ the union of $\rho_i$ distinct smooth arcs  which hit
$z_i$ with strictly positive distinct angles.  We denote by $Y(N)$ the set of these critical points. \end{definition}

Conversely, if $N$ is a regular closed set, then the family $\mathcal D(N)$ of connected
components of ${\Omega}\setminus N$ belongs (by definition) to $\mathcal R({\Omega})$, hence
regular and strong.

\begin{definition}\label{eacp}~\\
We will say that a closed set has the \textbf{equal angle meeting
  property} ({\bf eamp}), if the arcs meet
with equal angles at each critical point  $x_i\in N\cap {\Omega}$ and also with equal angles
at the $z_i\in N\cap  \pa {\Omega}$. For the boundary points $z_i$ we mean that the two arcs in the
boundary are included.
\end{definition}

We will say that the partition is {\bf eamp}-regular if it is regular and
satisfies the equal angle meeting property.\\

It has been proved by Conti-Terracini-Verzini \cite{CTV0, CTV2, CTV:2005} that
\begin{theorem}\label{thstrreg}~\\
For any $k$ there exists a minimal {\bf eamp}-regular strong  $k$-partition.
\end{theorem}
It has been proved in \cite{HHOT} the
\begin{theorem}\label{thanyreg}~\\
Any  minimal spectral $k$-partition admits a representative which is
{\bf eamp}-regular and strong.
\end{theorem}

A basic result concerns  the regularity (up to the boundary if any) of the nodal partition associated to an eigenfunction.

We first observe that the  results about minimal partitions for
 plane domains can be transfered to the sphere $\mathbb S^2$.
It is indeed enough
 to use the stereographic projection on the plane which gives  an
 elliptic operator on the plane with analytic coefficients.This map is
 a conformal map, hence respecting the angles. The
 regularity questions being local
 there are no particular problem for
 recovering the equal angle meeting  property.

A natural question is whether a minimal partition is the nodal partition
induced by an eigenfunction.
The next  theorem gives a simple criterion for a partition to be
 associated to a nodal set.  For this we need some additional
 definitions.

 We  recall  that the graph $G(\mathcal D)$ is \textbf{bipartite} if its vertices
can be colored by two colors (two neighbours having  different
colors).
In this case, we say that the partition is \textbf{admissible}.  We recall that  a collection of nodal domains of an eigenfunction is always
 admissible.

We have now the following converse theorem \cite{HHOT}~:
\begin{theorem}\label{partnod}~\\
An admissible  minimal  $k$-partition is nodal, i.e. associated to the nodal set of an
eigenfunction of $H(\Omega)$ corresponding to an eigenvalue
 equal to $\mathfrak L_k(\Omega)$.
\end{theorem}

This theorem was already obtained for planar domains in \cite{HH:2005a} by adding a strong
a priori regularity
and the assumption that $\Omega$ is simply
connected. Any subpartition of cardinality $2$ corresponds indeed
 to a second eigenvalue and the criterion of pair compatibility (see
 \cite{HH:2005a}) can be applied.\\

A natural question is now to determine how general is  the situation
described in Theorem \ref{partnod}. As for partitions in planar
domains,
 this can only occur in very particular cases when $k>2$.
If $\lambda_k(\Omega)$ denotes the $k$-th eigenvalue of the Dirichlet
realization of the Laplacian in an open set $\Omega$ of $\mathbb S^2$,
the Courant Theorem says~:
\begin{theorem}\label{ThC}~\\
 The number of nodal domains $\mu(u)$ of
 an eigenfunction $u$ associated with $\lambda_k(\Omega)$
satisfies
 $\mu(u)\leq k\,.
$
\end{theorem}
 Then we say, as in
\cite{HHOT}, that  $u$ is \textbf{Courant-sharp} if $\mu(u)=k$.
For any integer $k\ge 1$, we denote by $L_k(\Omega)$ the smallest eigenvalue whose eigenspace
contains an eigenfunction with $k$ nodal domains. In general we have
\begin{equation}
\lambda_k(\Omega)\leq\mathfrak L_k(\Omega) \leq L_k(\Omega) \,.
\end{equation}

The next result of \cite{HHOT} gives  the full picture of the equality cases~:

\begin{theorem}\label{L=L}~\\
Suppose that ${\Omega}\subset \mathbb S^2$ is regular.
If $\mathfrak L_k(\Omega) =L_k(\Omega)$ or $\lambda_k(\Omega) =\mathfrak L_k(\Omega)$, then $${\lambda}_k(\Omega)=\mathfrak L_k(\Omega)=L_k(\Omega)\,,$$
and  any minimal $k$-partition is nodal and admits a representative
  which is the family of nodal domains of some eigenfunction $u$
associated  to ${\lambda}_k(\Omega)$.
\end{theorem}

This theorem will be quite useful for showing for example that, for $k=3$ and
$k=4$, a $k$-minimal partition of $\mathbb S^2$ cannot be nodal. This
will be further discussed in Section \ref{s4} (see Theorem \ref{Unter}).

\section{Courant's nodal Theorem with inversion symmetry.}\label{s4}
We collect here some easy useful observations  for the
analysis of the sphere. These considerations already appear in
\cite{Ley1} but the application to minimal partitions is new.
We consider Courant's Nodal Theorem for $
H({\Omega})$
where ${\Omega}$ is an open connected set in $\mathbb S^2$.
Let
\begin{equation}
\mathbb S^2\ni (x,y,z)\mapsto I(x,y,z)=(-x,- y,-z)
\end{equation}
 denotes the  inversion map
  and
assume that
\begin{equation}\label{OVinv}
I{\Omega} ={\Omega} \,.
\end{equation}
Note that $\Omega =\mathbb S^2$ satisfies the condition.
These assumptions imply that we can write $H({\Omega})$
as a direct sum
\begin{equation}\label{HSHA}
H({\Omega})=H_S({\Omega})\bigoplus H_A({\Omega})\,,
\end{equation}
where $H_S({\Omega})$ and $ H_A({\Omega})$ are respectively
the restrictions of $ H({\Omega})$ to the $I$-symmetric
(resp. antisymmetric)
 $L^2$-functions in $\Omega$ in $D( H({\Omega}))$.

For simplicity we just write $H_S,H_A$.
For the spectrum of $H({\Omega})$, $\sigma$, we have
\begin{equation*}
\sigma=\sigma_S\cup \sigma_A
\end{equation*}
so that  $\sigma_S=\{{\lambda}^S_k\}_{k=1}^\infty$ and analogously $\sigma_A=\{{\lambda}_k^A\}_{k=1}^\infty$. It is of independent
interest to investigate how $\sigma_S$ and $\sigma_A$ are related. Obviously we have
$$
{\lambda}_1^S<{\lambda}_1^A\le {\lambda}_2^A\mbox{ and } {\lambda}_1^S<{\lambda}_2^S\,,
$$ by standard spectral  theory.

In the present situation we can ask the question
of a theorem \`a la Courant separately for the eigenfunctions
of $H_S$ and $H_A$.

First we  note the following easy properties~:
\begin{enumerate}
\item
Suppose $u$ is an eigenfunction and that $u$ is either symmetric or
anti-symmetric. Then $I\, N(u)=N(u)$, i.e. the
nodal set is symmetric with respect to inversion.
\item
If  $u^A$ is an eigenfunction of $H_A$, then for each nodal domain
$D_i$ of $u^A$, $ID_i$ is a distinct  nodal domain of $u^A$.
Hence the nodal domains come in pairs and $\mu(u^A)$
 is even.
\item
If $u^S$ is a symmetric eigenfunction, then there are two classes of
nodal domains~:
\begin{itemize}
\item  the symmetric domains,
\begin{equation}\label{DSS}
D_{i,S}=ID_{i,S}\,,
\end{equation}
\item the symmetric pairs of domains $D^1_{i,S}, D^2_{i,S}$ so that
\begin{equation}\label{DSA}
ID^1_{i,S}=D^2_{i,S}\,.
\end{equation}
\end{itemize}
\end{enumerate}
\begin{theorem}\label{CourantInv}~\\
Suppose that $\Omega$ satisfies the symmetry assumption \eqref{OVinv}.
Then, if $(\lambda^A_k,u^A)$ is a spectral pair for $H_A(\Omega)$, we have
\begin{equation}\label{mua}
\mu(u^A)\le 2k\,.
\end{equation}
If $(\lambda^S_k, u^S)$ is a spectral pair for $H_S(\Omega)$ and if we
denote by $\ell(k)$  the number of  pairs of  nodal domains of
$u^S$ satisfying \eqref{DSA}
and $m(k)$ the number of domains satisfying \eqref{DSS}, then we have
\begin{equation}\label{muS}
\ell(k)+m(k)\le k\,,
\end{equation}
and
\begin{equation}\label{muSkl}
\mu(u^S)\le k+\ell(k)\,.
\end{equation}
\end{theorem}
\begin{remark}\label{Icourant}~\\
 Of course the original  Courant Theorem holds, but the above result gives additional informations.
\end{remark}

\textbf{Proof.}~\\
We just have to mimick the proof of Courant's original theorem. Let us
first show \eqref{mua}. We can of course add the condition that~:
$$
\lambda^A_{k-1} < \lambda^A_k\;.
$$
Assume for contradiction that, for some
 $u^A_k$ we have $\mu(u^A_k)>2k$. To each pair $(D_i,\: ID_i)$ of nodal
domains of $u^A_k$,  we associate the corresponding
ground states, so that $$H(D_i)\phi_i={\lambda}_k^A\phi_i,\:\phi_i\in
W_0^{1,2}(D_i)\,,$$ and, with $I\phi_i=-\phi_i\circ I\,$,
 $$H(ID_i,V)I\phi_i={\lambda}_k^AI\phi_i\,.$$
We use the variational principle in the form domain of $H_A\,$.
We have
\begin{equation}\label{lkas}
{\lambda}_k^A=\inf_{\varphi^A\bot\mathcal
  Q_A^{k-1}}\frac{\int_{\Omega}\big(|\nabla
  \varphi^A|^2\big)\,d\mu_{\mathbb S^2}}
{\int_{\Omega}|\varphi^A|^2\,d\mu_{\mathbb S^2}}
\end{equation}
where $I\varphi^A=-\varphi^A$ and $\varphi^A\in W_0^{1,2}({\Omega})$.  Here
$\mathcal Q_A^{k-1}$
 is just the space spanned by the
first $(k-1)$  eigenfunctions of $H_A$.
We  proceed now as  in the proof of Courant's nodal Theorem.  Hence, in
other words, we have just replaced in this proof
the single domains by pairs of domains.

The proof of \eqref{muSkl} is similar.
\finbox

We can also find some immediate consequences concerning the relation between the $\sigma_A$ and $\sigma_S$.
Take for instance a spectral pair  $(u^A,{\lambda}^A_j)$ and assume that $\mu(u^A)=2k$. Then we can
construct from the $2k$  ground states of each connected component
$k$ symmetric ones, each one being supported in a symmetric pair of components. By the variational principle, this time for $H_S$ we obtain
\begin{equation}\label{lksym}
{\lambda}_k^S=\inf_{\varphi^S\bot\mathcal
  Q_S^{k-1}}\frac{\int_{\Omega}|\nabla\varphi^S|^2\,d\mu_{\mathbb
    S^2}}
{\int_{\Omega}|\varphi^S|^2d\mu_{\mathbb S^2}}.
\end{equation}

This implies~:
\begin{proposition}~\\
 Any eigenvalue ${\lambda}^A$ of $H_A$,  whose
corresponding eigenspace contains an eigenfunction with $2k$ nodal
domains, satisfies ${\lambda}^A\ge{\lambda}_k^S$.
\end{proposition}
A  similar argument can be also made for the symmetric case if $\ell(k)>0$.
This gives us new versions of Courant-sharp properties.\\
If we call {\bf pair symmetric partition}  a partition  which is invariant
by
the symmetry but such that no element of the
 partition is invariant,  we   have the following Courant-sharp
 analog~:\\
\begin{theorem}\label{CourantA}~\\
If, for some eigenvalue $\lambda_k^A$ of $H_A(\Omega)$, there exists an eigenfunction $u^A$ such
that
 $\mu(u^A) =2 k$, then the corresponding family of nodal domains
 is a minimal pair symmetric partition.
\end{theorem}
Note that, if the labelling  of the eigenvalue (counted as eigenvalue of
$H(\Omega)$)
 is $>2k$),  then it is not a minimal $(2k)$-partition of $\Omega$.
\begin{remark}\label{rembesse}~\\
Let us finally mention as connected result (see for example
\cite{Besse}),
 that if $\Omega$ satisfies \eqref{OVinv}, then $\lambda_2(\Omega)=
 \lambda_1^A(\Omega)$.
\end{remark}

\paragraph{Application}~\\
It is known that the eigenfunctions are the restriction to $\mathbb S^2$ of
the homogeneous harmonic polynomials. Moreover, the eigenvalues are
$\ell (\ell +1)$ ($\ell \geq 0$) with multiplicity $(2\ell +1)$.
 Then,
the  Courant nodal  Theorem says that for a spherical harmonic
$u_\ell$
 corresponding to $\ell (\ell +1)$, one should have
\begin{equation}
\mu (u_\ell) \leq \ell^2 +1\,.
\end{equation}
As observed in \cite{Ley1}, one can, using the fact that
\begin{equation}
u_\ell (-x) =(-1)^{\ell} u_\ell (x)\,,
\end{equation}
improve this result by using a variant of  Courant's nodal Theorem with symmetry (see
Theorem \ref{CourantInv})
 and this leads to the improvement
\begin{equation}\label{impCo}
\mu (u_\ell) \leq \ell (\ell -1) +2\,.
\end{equation}
Let us briefly sketch the proof of \eqref{impCo}. If $\ell$ is odd,
any eigenfunction is odd with respect to inversion. Hence the
number of nodal domains is even $\mu(
u_\ell) = 2 n_\ell$
 and there are no nodal domains invariant by inversion. Using Courant's
 nodal Theorem  for $H_A$,
we get with $\ell = 2p+1$ that
$$
\begin{array}{ll}
n_\ell &\leq \sum_{q=0}^{p-1} (2 (2q+1) +1)\;\; +1 \\
& = 2 p (p-1) + 3 p +1\\
& = p (2p +1) +1\\
&= \frac 12 \ell (\ell -1)
 +1\,.
\end{array}
$$
If $\ell$ is even, we can only write
$$
\mu(u_\ell) = 2 n_\ell + p_\ell\,,
$$
where $p_\ell$ is the cardinality of the nodal domains which are
invariant by inversion.\\
 Using Courant's nodal  Theorem for $H_S$,
we get with $\ell = 2p$, that
$$\begin{array}{ll}
n_\ell + p_\ell &\leq \left(\sum_{q=0}^{p-1} (4q +1)\right) +1\\&
 = 2 p(p-1) + p +1\\& = p (2p -1) +1\\& = \frac 12 \ell (\ell -1)
 +1\,.
\end{array}
$$

Using this improved estimate, we immediately obtain ~:
\begin{proposition}~\\
 The only cases where $u_\ell$ can be Courant-sharp are for $\ell =0$
 and $\ell =1$.\end{proposition}
This proposition has the following consequence~:
\begin{theorem}\label{Unter}~\\
A minimal $k$-partition of $\mathbb S^2$ is nodal if and only if
 $k\leq 2$.
\end{theorem}

Note that in  \cite{Ley1, Ley2} the more sophisticated conjecture
 (verified for $\ell \leq 6$)  is proposed~:
\begin{conjecture}~\\
$$
\mu (u_\ell) \leq \left\{\begin{array}{lr} \frac 12 (\ell+1)^2& \mbox{
  if }
\ell \mbox{ is odd}\,, \\
 \frac 12 \ell (\ell+2)& \mbox{ if }
\ell \mbox{ is even}\,. \end{array}\right.
$$
\end{conjecture}
\begin{remark}~\\
As indicated by D.~Jakobson to one of us, there is also a
probabilistic version of  this conjecture \cite{NaSo}.
V. N. Karpushkin \cite{Kar} has also  the following bound for the number of
components~:\\
$$
\mu (u_\ell) \leq \left\{\begin{array}{lr} (\ell-1)^2 +2  & \mbox{
  if }
\ell \mbox{ is odd}\,, \\
  (\ell-1)^2 +1 & \mbox{ if }
\ell \mbox{ is even} \,.\end{array}\right.
$$
This is for $\ell$ large slightly better than what we obtained
 with the refined Courant-sharp Theorem.
Let us also mention the recent paper \cite{ErJaNa} and references therein.
\end{remark}

\begin{remark}\label{remcov}~\\
Considering the Laplacian on the double covering $\mathbb S_{\mathcal C}^2$
of $\ddot {\mathbb S}^2:= \mathbb S^2\setminus \{North,South\}$,
Theorems \ref{CourantInv} and \ref{CourantA}
 hold true where  $I$ is replaced by $\mathcal I$ (introduced in \eqref{defIrev})  corresponding to the map $\phi
\mapsto \phi + 2\pi$. The $\mathcal I$-symmetric eigenfunctions can be identified
 to the eigenfunctions of $H(\mathbb S^2)$ by
 $u_S(x)=u(\pi(x))$  and the restriction
 $H_{\mathcal A}(\mathbb S^2_{\mathcal C})$ of $H(\mathbb
 S^2_{\mathcal C})$ to the $\mathcal
 I$-antisymmetric space leads to a new spectrum, which will be
 analyzed in Section \ref{s8}.
 \end{remark}

\section{On topological properties of minimal $3$-partitions of the
  sphere}
As in the case of planar domains, a classification of the possible
types
 of minimal partitions could simplify the analysis. The case of the
 whole sphere $\mathbb S^2$ shows some difference with for example the
 case of the disk.
\subsection{Around Euler's formula}\label{s3}
As in the case of domains in the plane \cite{HHO},
 we will use the following  result.
\begin{proposition}\label{Euler}~\\
Let ${\Omega}$ an open set in $\mathbb S^2$ with piecewise $C^{1,+}$ boundary
 and let  $N\in\mathcal M({\Omega})$ such that the associate
 $\mathcal D$  consists of $\mu$ domains $D_1,\dots, D_\mu$. Let $b_0$ be the number of components of
$\pa {\Omega}$ and $b_1$ be the number of components of $N\cup\pa {\Omega}$. Denote by $\nu(x_i)$ and $\rho(z_i)$
the numbers associated to the $x_i\in X(N)$, respectively  $z_i\in Y(N)$. Then
\begin{equation}\label{Emu}
\mu=b_1-b_0+\sum_{x_i\in X(N)}(\frac{\nu(x_i)}{2}-1)+
\frac{1}{2}\sum_{z_i\in Y(N)}\rho(z_i)+1\,.
\end{equation}
\end{proposition}
\begin{remark}~\\
In the case when $\Omega =\mathbb S^2$, the statement simply reads
\begin{equation}\label{Emuprime}
\mu=b_1 +\sum_{x_i\in X(N)}(\frac{\nu(x_i)}{2}-1)\;
+1\,,
\end{equation}
where $b_1$ is the number of components of $N$.

\end{remark}
\subsection{Application to $3$- and $4$-partitions.}
\paragraph{The case of $3$-partitions}~\\
Let us analyze in this spirit the topology of minimal $3$-partitions
of $\mathbb S^2$. \\
First we recall that a minimal $3$-partition cannot be nodal. The
multiplicity of the second eigenvalue of $-\Delta_{\mathbb S^2}$ is indeed
$3$.
Hence, our minimal $3$-partition cannot be admissible.\\ Let us look now
to the information given by Euler's formula. We argue like in \cite{HHO}
for the case of the disk.
We recall that at any critical point $x_c$ of $N$,
\begin{equation}
\nu(x_c)\geq 3\,.
\end{equation}
Hence \eqref{Emuprime} implies that $b_1\leq 2$ and  we have
\begin{equation}
1\leq b_1\leq 2\,.
\end{equation}

When $b_1=1$, we get
as unique solution $\# X(N) =2$. So $X(N)$ consists of two points
$x_1$ and $x_2$ such that  $\nu(x_i)=3$ for $i=1$ and
$2$. The other case when $\# X(N) =1$ leads indeed to an even number
of half lines arriving to the unique critical point and to an
admissible (hence excluded) partition.\\
When  $b_1=2$, we obtain that $X(N)$ is empty and the partition
should be admissible which is excluded. Hence we have shown
\begin{proposition}\label{propotop}~\\
If $\mathcal D$ is a regular non admissible  strong $3$-partition of $\mathbb S^2$, then
 $X(N)$ consists of two points $x_1$ and $x_2$ such that $\nu(x_i)=3$,
 and
 $N$  consists of
 three non crossing (except at their ends) arcs joining  the two points
 $x_1$ and $x_2$.
\end{proposition}
In particular this can be applied to minimal $3$-partitions of
$\mathbb S^2$.
\paragraph{The case of $4$-partitions}~\\
We can analyze in the same way the case of non admissible
 $4$-partitions.  Euler's formula leads to the following classification.
\begin{proposition}\label{propotop4}~\\
If $\mathcal D$ is a regular non admissible  strong $4$-partition of
$\mathbb S^2$, then we are in one of the following cases~:
\begin{itemize}
\item
 $X(N)$ consists of four  points $x_i$ ($i=1,\dots, 4$)  such that $\nu(x_i)=3$,
 and
 $N$  consists of
 six non crossing (except at their ends) segments, each one  joining  two points
 $x_i$ and $x_j$ ($i\neq j$).
\item $X(N)$ consists of three points  $x_i$ ($i=1,2, 3$)
 such that $\nu(x_1)=\nu(x_2)=3$, $\nu(x_3)=4$ and $N$ consists of five
 non crossing (except at their ends)  segments joining two critical points.
\item $X(N)$ consists of two  points $x_i$ ($i=1, 2$)  such that $\nu(x_i)=3$, and
 $N$  consists of
 three non crossing (except at their ends) segments joining  the two points
 $x_1$ and $x_2$ and of one closed line.
\item $X(N)$ consists of two  points $x_i$ ($i=1, 2$) such that
 $\nu(x_1)=3$, $\nu(x_2)=5$ and $N$ consists of four
 non crossing (except at their ends)  segments joining the two critical points
 and of one non crossing (except at his ends) segment starting from one critical point and coming back
 to the same one.
\end{itemize}
\end{proposition}
Note that the spherical tetrahedron corresponds to the first type and
we recall from Theorem \ref{Unter} that minimal $4$-partitions
 are not admissible.
\section{Lyustenik-Shnirelman Theorem and proof of Proposition \ref{antipodal}}\label{s6}

As  we have shown in the previous section $N(\mathcal D)$ consists of two points $x_1\neq x_2$ and $3$ mutually non-crossing arcs $\gamma_1, \gamma_2,\gamma_3$
connecting $x_1$ and $x_2$.
This means that each $D_i$ has a boundary which is a closed curve
which is away from $x_1,x_2$ smooth.\\

We first recall the well known theorem of Lyusternik and Shnirelman from 1930, that can be found for instance in \cite{M}
on page 23. It states the following.
\begin{theorem}\label{LS}~\\
 Suppose $S_1,S_2,\dots, S_d$ are closed subsets of $\mathbb S^{d-1}$
 such
that $\cup_{i=1}^dS_i=\mathbb S^{d-1}$. Then there is at least one $S_i$ that  contains a pair of antipodal points.
\end{theorem}

 We will use this theorem in the case $d=3$  and apply it with  $S_1,S_2,S_3$ defined by
\begin{equation}\label{SD}
S_i=\overline D_i.
\end{equation}
In order to prove Proposition \ref{antipodal} it suffices to show that
\begin{equation}\label{IN}
 N(\mathcal D)\cap  I\, N(\mathcal D)\neq \emptyset\,,
\end{equation}
where we recall that $I$ is the antipodal map.\\

By Theorem \ref{LS} we know that there is an $S_i=\overline D_i$ which contains a pair of
antipodal points. After relabelling the $D_i$'s, we can  assume that
\begin{equation}\label{LS1}
 I \, \overline{ D_1} \cap \overline{ D_1 } \neq \emptyset\;,
\end{equation}
and the goal is to show that
\begin{equation}
 I \, \pa D_1 \cap \pa D_1 \neq \emptyset\;.
\end{equation}
Our $D_i$'s have the properties of $3$-minimal partitions established
in the previous section. In
particular $\pa D_1$ has one component and is also the boundary of
$\pa D_{13}$, where $D_{13}={\rm Int\,} ( \overline{D_2} \cup \overline{D_3})$.

The proof is by contradiction. Let us assume by contradiction that
\begin{equation}\label{contradiction}
 I \, \pa D_1 \cap \pa D_1 = \emptyset\;.
\end{equation}
Then there are two possible cases
\begin{enumerate}
\item[Case a:] $ I\, \pa D_1 \subset D_1$
\item[Case b:]   $ I\,\pa D_1 \cap \overline{ D_1}=\emptyset $
\end{enumerate}

{\bf Let us start with Case a.} Again there are two possibilities.
\begin{enumerate}
\item[Case a1:] $I\, D_1 \subset\subset D_1$ ($\subset\subset$ means
 compactly included).
\item[Case a2:] $I\, D_2 \subset\subset D_1$
\end{enumerate}
But Case $a1$ is in contradiction with the fact that $I$ is an
isometry and
Case~$a2$ is in contradiction with $\lambda (D_2)=\lambda(D_1)$.
Of course we could have taken $D_3$ instead of  $D_2$.

{\bf We now look at Case b.} Then $ I \, \pa D_1$ is delimiting in $\mathbb S^2$ two components and
$D_1$ is compactly supported in one component. One of the component is
 $I\, D_1$. But \eqref{LS1} implies that $D_1$ is in this last component~:
$$
D_1 \subset\subset I \, D_1\,.
$$
This can not be true for two isometric domains. Hence we have a
contradiction  with \eqref{contradiction} in all the cases. This
achives the proof of the proposition.

\section{The Laplacian on $\mathbb S^2_{\mathcal C}$} \label{s8}
\subsection{Spherical harmonics with half integers}
These spherical harmonics appear from the beginning
 of Quantum mechanics in connection with  the representation theory \cite{Pau}. We
 refer to \cite{Fl} (Problem 56 (NB2) in the first volume together
 with Problem 133 in the second volume). We are looking for
 eigenfunctions
 of the Friedrichs extension of
\begin{equation}
{\bf L}^2= -\frac{1}{\sin^2\theta} \frac{\pa^2}{\pa \phi^2} - \frac{1}{\sin
 \theta} \frac{\pa}{\pa \theta} \, \sin \theta \, \frac{\pa}{\pa \theta}
\end{equation}
in $L^2(\sin \theta d\theta \,d\phi)$,
satisfying
\begin{equation}
{\bf L}^2 Y_{\ell m}  = \ell (\ell +1)  Y_{\ell m}\,.
\end{equation}
The standard spherical harmonics, corresponding to $\ell\geq 0$
 are defined,  for an integer $m \in \{-\ell,\dots,\ell\}$, by
\begin{equation}\label{YLM}
Y_{\ell m}(\theta,\phi) = c_{\ell,m} \exp i m \phi \frac{1}{\sin^m\theta}
 (-\frac{1}{\sin \theta} \frac{d}{d\theta})^{\ell -m} \sin^{2\ell} \theta \,,
\end{equation}
where $c_{\ell,m}$ is an explicit normalization constant.\\
For future extensions, we prefer to take this as a definition for
 $m\geq 0$ and then to observe that
\begin{equation}\label{YLMa}
Y_{\ell, -m} = \hat c_{\ell ,m}\overline{ Y_{\ell,m}}\,.
\end{equation}
For $\ell = 0$, we get $m=0$ and the constant. For $\ell =1$, we
 obtain, for $m=1$, the function  $(\theta,\phi)\mapsto \sin \theta
 \exp i \phi$ and for $m=-1$, the function
 $\sin \theta \exp -i \phi$ and for $m=0$ the function  $\cos \theta$, which shows
 that the multiplicity is $3$ for the eigenvalue $2$.\\
Of course concerning nodal sets, we look at the real valued functions\break
  $(\theta,\phi)\mapsto \sin \theta
 \cos \phi$ and  $(\theta,\phi)\mapsto \sin \theta
 \sin \phi$ for $|m|=1$.

As  observed a long time ago, these formulas still define
eigenfunctions
 for pairs $(\ell, m)$ with $\ell$ a positive half-integer (and not integer),
  $m \in  \{-\ell,\dots,\ell\}$ and $m-\ell$ integer.\\
For definiteness, we prefer (in the half-integer case) to only consider the pairs with $\ell >0$ and
$m>0$
 and to complete the set of eigenfunctions by introducing
\begin{equation}\label{YLMb}
\widehat Y_{\ell,m} = \overline{Y_{-\ell,m}}
\end{equation}
These functions are only defined on the double covering $\mathbb
S^{2}_\mathcal C$ of\break $\ddot{ \mathbb
S}^2:=\mathbb S^2 \setminus
\{\{\theta=0\}
 \cup \{\theta =\pi\}\}$, which can be  defined by extending $\phi$ to
 the interval $]-2\pi,2\pi]$.\\
When restricted to $\phi \in ]-\pi,\pi]$, they correspond to the
antiperiodic problem with respect to period $2\pi$ in the $\phi$
variable.\\

To show the completeness it is enough to show that, for given $m>0$,
 the orthogonal family (indexed by $\ell \in \{m+\mathbb N\}$) of functions  $\theta\mapsto \psi_{\ell,m}(\theta):=\frac{1}{\sin^m \theta} (-\frac{1}{\sin
 \theta} \frac{d}{d\theta})^{\ell -m} \sin^{2\ell}\theta$
 span all $L^2(]0,\pi[,\sin \theta \,d\theta)$.\\
For this, we consider $\chi \in C_0^\infty(]0,\pi[)$
 and assume that
$$
\int_{0}^\pi \chi (\theta) \psi_{\ell,m}(\theta) \sin \theta d\theta
 =0\,,\, \forall \ell \in\{ m+ \mathbb N\}\;.
$$
We would like to deduce that this implies $\chi=0$. After a change of
variable
 $t=\cos \theta$ and an integration by parts, we obtain that this problem is equivalent to the
problem to show that, if
$$
\int_{-1}^1 \psi (t ) \;[(1-t^2)^\ell]^{(\ell -m)} \; dt
 =0\,,\, \forall \ell\in \{m+\mathbb N\}\;,
$$
then $\psi =0$.\\
Observing that the space spanned  by the functions
 $(1-t^2)^{-m}  ((1-t^2)^\ell)^{(\ell -m)}$ (which are actually
polynomials
 of exact order $\ell$)
 is the space of all polynomials we can conclude the completeness.\\
 Hence we have obtained the
\begin{theorem}~\\
The spectrum of
 the  Laplace Beltrami operator on $\mathbb S^2_{\mathcal C}$ can be
 described by the eigenvalues $\mu_\ell=\ell (\ell+1)$ ($\ell \in \mathbb N/2$),
 each eigenvalue being of multiplicity $(2\ell +1)$. Moreover the
 $Y_{\ell,m}$, as introduced in \eqref{YLM}, \eqref{YLMa} and
 \eqref{YLMb},  define an orthonormal basis of
 the eigenspace $E_{\mu_\ell}$.
\end{theorem}

In particular, for $\ell =\frac 12$, we get a basis of two orthogonal real
 eigenfunctions $\sin \frac{\phi}{2}
(\sin \theta)^{\frac 12}$  and $\cos \frac \phi 2 (\sin \theta)
 ^{\frac 12}$
 of the eigenspace associated with $\frac 34$.
 For $\ell =\frac 32$, the multiplicity is  $4$
 and the functions
 $\sin \frac{3 \phi}{2}(\sin \theta)^{\frac 32}$,
 $\cos \frac{3 \phi}{2}(\sin \theta)^{\frac 32}$, $\sin \frac \phi 2
(\sin\theta)^{\frac 12}  \cos \theta$ and  $\cos \frac \phi 2 (\sin
 \theta)^{\frac 12} \cos \theta$ form  a basis of the eigenspace
 associated with the eigenvalue $\frac{15}{4}$.
\subsection{Covering argument and minimal partition}
Here we give one part of the proof of Proposition~\ref{centera}.
\begin{lemma}\label{lemmacentera}~\\
Let us assume that there exists a $3$-minimal partition $\mathcal D=(D_1,D_2,D_3)$ of $\mathbb S^2$ containing two antipodal
 points in its boundary.  Then,  considering the associated punctured
 $\ddot{\mathbb S}^2$ and the corresponding double covering
  $\mathbb S^2_{\mathcal C}$ and  denoting  by $\Pi$ the projection
of $\mathbb S^2_{\mathcal C}$ on $\ddot{\mathbb S}^2$, $\Pi^{-1}(D_i)$
 consists of two components and $\pi^{-1} (\mathcal D)$ defines a
 $6$-partition of $\mathbb S^2_{\mathcal C}$ which is pairwise symmetric.
\end{lemma}

The only point to observe is that, according to the property of a
minimal partition established in Proposition \ref{propotop},  the boundary of the partition
necessarily contains a ``broken'' line joining the two antipodal
points.\\
Using the minimax principle, one immediately gets that, {\bf under the
assumption of the lemma},
\begin{equation}\label{inega}
 \mathfrak L_3(\mathbb S^2) \geq \lambda^3_{AS}(\mathbb S^2_{\mathcal
  C})\,,
\end{equation}
where $
\lambda^3_{AS}$ is the third eigenvalue of the Laplace-Beltrami
operator on $\mathbb S^2_{\mathcal
  C}$ restricted to the antisymmetric spectrum.
$\lambda^3_{AS}(\mathbb S^2_{\mathcal
  C})$ will be computed in the next subsection.

\subsection{Covering argument and Courant-sharp eigenvalues}
In the case of the double covering $\mathbb S^2_{\mathcal C}$ of $\ddot{\mathbb S}^2$,
 we have seen that we have to add the antisymmetric (or antiperiodic)
 spectrum  (corresponding to the map $\Pi$, which writes  in spherical
 coordinates~:
       $\phi \mapsto \phi+2\pi$). This  adds
 the eigenvalue $\frac 34 = \frac 12 (1 +\frac 12)$ with multiplicity
       $2$
 and the eigenvalue $\frac {15}{4} = \frac 32 (1+\frac 32)$
with multiplicity $4$.
Hence $\frac {15}{4}$ is the $7$-th eigenvalue of the Laplacian on
 $\mathbb S^2_{\mathcal C}$
 hence not Courant-sharp, but it is the third antisymmetric eigenvalue
$$\lambda^3_{AS} = \frac {15}{4}\,.$$
Hence, observing that the nodal set of an eigenfunction associated
 to $\lambda^3_{AS}$ has six nodal domains which are pairwise
 symmetric
 and giving by projection the $Y$-partition, we immediately obtain
 that under the assumption of the lemma
$$
\mathfrak L_3(\mathbb S^2) = \frac{15}{4}\;.
$$

But Proposition \ref{centera} says more. For getting this result, we
have to prove the following proposition~:
\begin{proposition}\label{last}~\\
Let $\mathfrak L_{2\ell}^{AS}(\mathbb S^2_{\mathcal C})$ the infimum
 obtained over the pairwise symmetric (by $\Pi$) $(2 \ell)$- partitions
 of $\mathbb S^2_{\mathcal C}$. Then, if
 $$\mathfrak L_{2\ell}^{AS}(\mathbb S^2_{\mathcal C})=
 \lambda^\ell_{AS}\,,
$$
then $ \lambda^\ell_{AS}$ is Courant-sharp in the sense of the
antisymmetric spectrum and any minimal pairwise symmetric
$(2\ell)-$partition  is nodal.
\end{proposition}
The proof is the same as for Theorem 1.17  in \cite{HHOT} and Theorem
2.6 in \cite{HHO}, with the
difference that we consider everywhere antisymmetric states.\\

Applying this proposition for $\ell =3$, we have the proof of
Proposition \ref{centera}.

\section{On Bishop's approach for   minimal $2$-partitions and
  extensions
 to strong $k$-partitions}\label{s5}
\subsection{Main result for $k=2$}
For $2$-partitions, it is
 immediate to show that  the minimal  $2$-partitions realizing
$\mathfrak L_{2}(\mathbb S^2)$  are  given by the two
 hemispheres. One is indeed in the Courant-sharp situation.
The case of $\mathfrak L_{2,p}(\mathbb S^2)$ for $p<\infty$
 is more difficult. Bishop has described in \cite{Bis} how one can
 show
 that the
 minimal $2$-partitions realizing  $\mathfrak L_{2,1}(\mathbb S^2)$
 are  also
given by the two
 hemispheres.  It is then easy to see that it implies the property
 for any $p\in [1,+\infty[$. Hence we obtain~:
\begin{theorem}\label{ThBishop}~\\
For any $p\in[1,+\infty]$, $\mathfrak L_{2,p}(\mathbb S^2)$
 is realized by the partition of $\mathbb S^2$ by two hemispheres.
\end{theorem}
The proof is based on two theorems due respectively to Sperner \cite{Sper}
 and Friedland-Hayman \cite{FrHa}. We will discuss their proof because it  will
 have some consequences for the analysis of
 minimal
$3$ and $4$-partitions.\\
\subsection{The lower bounds of Sperner and Friedland-Hayman}
For a given domain $D$  on the unit sphere $S^{m-1}$ in $\mathbb R^m$,
 Sperner shows
 the following theorem, which plays on the sphere the same role
as the Faber-Krahn Inequality  plays in $\mathbb R^m$~:
\begin{theorem}\label{Th5.1}~\\
Among all sets $E\subset \mathbb S^{m-1}$ with given $(m-1)$-dimensional surface area
$\sigma_m S$ on  (with $\sigma_m$
being the area
 of  $\mathbb S^{m-1}$), a spherical cap has the smallest characteristic constant.
\end{theorem}
Here the characteristic constant for a domain $D$ is related to the
ground state energy by
\begin{equation}
\lambda (D) = \alpha (D) (\alpha(D) +m-2)\,,
\end{equation}
with $\alpha (D)\geq 0\,$.\\
We introduce for short
\begin{equation}
\alpha (S,m)=\alpha (\mathcal{SC}(\sigma_mS))\,,
\end{equation}
where $\mathcal{SC}(\sigma_mS)$ is a spherical cap of surface area
$\sigma_m S$.

This theorem is not sufficient in itself for the problem. The second
 ingredient\footnote{See for example \cite{AltCaf} p.~441 or C.~Bishop \cite{Bis}}
 is a lower bound of $\alpha (S,m)$ by various convex decreasing functions. It is
 indeed proven\footnote{We only write the result for $m=3$ but
   \eqref{prem}
 holds for any $m\geq 3$.}
 in \cite{FrHa} that~:
\begin{theorem}\label{Th4}~\\
We have the  following lower bound ~:
\begin{equation}\label{prem}
\alpha (S,3)\geq \Phi_3(S)\,,\;
\end{equation}
where $\Phi_3$ is the convex decreasing function
 defined by
\begin{equation}
 \Phi_3 = \max ( {\widehat \Phi_3}, \Phi_\infty) \,,
\end{equation}
\begin{equation}
\Phi_\infty (S)=\left\{
\begin{array}{ll}
 \frac 12\log (\frac{1}{4S})+ \frac 32\,, &\mbox{ if } 0<S\leq \frac 14\,,\\
2 (1-S)\,,&\mbox{ if } \frac 14 \leq S<1\,.\end{array}\right.
\end{equation}
\begin{equation}
\widehat \Phi_3(S) = \left\{
\begin{array}{ll}
2 (1-S)\,,&\mbox{ if } \frac 12 \leq S <1\,,\\
 \frac 12 j_0\left( \frac 1S -
 \frac 12\right)^{\frac 12}-\frac 12 \,,&\mbox{ if }  S <\frac 12\,,
\end{array}\right.
\end{equation}
and  $j_0$ being the first zero of Bessel's function of order $0$~:
\begin{equation}
j_0\sim  2.4048\,.
\end{equation}
\end{theorem}
\subsection{Bishop's proof for $2$-partitions}
With these two ingredients, we observe (following a remark of
C.~Bishop) that, for a $2$-partition, we have
necessarily
\begin{equation}
\alpha (D_1)+ \alpha (D_2) \geq 2\,,
\end{equation}
the equality being obtained for two hemispheres.\\
The minimization for the sum corresponds to
\begin{equation}
\inf \left(\alpha (D_1) (\alpha(D_1) +1) +  \alpha (D_2) (\alpha(D_2)
+1)\right)\,.
\end{equation}
This infimum is surely larger or equal to
$$
\inf_{\alpha_1+\alpha_2\geq 2\,,\; \alpha_1\geq 0,\alpha_2\geq 0}
\left(\alpha_1(\alpha_1+1) +\alpha_2 (\alpha_2 +1)\right)\,.
$$
It is then easy to see that the infimum is obtained for
 $\alpha_1=\alpha_2 =1$,
\begin{equation}\label{cas2}
\inf_{\alpha_1+\alpha_2\geq 2\,,\; \alpha_1\geq 0,\alpha_2\geq 0}
\left(\alpha_1(\alpha_1+1) +\alpha_2 (\alpha_2 +1)\right)=4\,.
\end{equation}
 This gives a lower bound for $\mathfrak L_{2,1} (\mathbb S^2)$
 which is equal to the upper bound of  $\mathfrak
 L_2(\mathbb S^2)$ and which is attained for the two hemispheres. This
 achieves
 the proof of Theorem \ref{ThBishop}.
\begin{remark}\label{remsusanna}~\\
A natural question is to determine under which condition the infimum
 of $\Lambda^1(\mathcal D)$ for $\mathcal D \in \mathfrak O_2$
 is realized for a pair $(D_1,D_2)$ such that $\lambda(D_1)=\lambda(D_2)\,$. Let us illustrate the
 question by a simple example.
 If we consider two disks
 $C_1$ and $C_2$ such that $\lambda(C_1)<\lambda(C_2)\leq
 \lambda_2(C_1)\,$,
 it is not too difficult to see that if we take $\Omega$ as the union
 of these two disks and of a thin channel joining the two disks, then
 $\mathfrak L_2(\Omega)=\lambda_2(\Omega)$
 will be very close to $\lambda(C_2)$
 and the infimum of  $\Lambda^1(\mathcal D)$ will be less than $\frac 12
 (\lambda(C_1)+\lambda(C_2))\,$. Hence we will have strict inequality
 if the channel is small enough. We refer to \cite{BHM1,BHM2,Ann,JM}
 for the spectral analysis of this type of situation. These authors
 are actually more interested in the symmetric situation where
 tunneling plays an important role.
\end{remark}
\subsection{Application to general $k$-partitions}\label{ss53}
One can also discuss what can be obtained in the same spirit
for  $k$-partitions ($k\geq 3$). This will not lead to the proof
 of Bishop's conjecture but give rather accurate lower bounds
 corresponding in a slightly different context to the ones proposed in
 Friedland-Hayman \cite{FrHa} for harmonic functions in cones of
 $\mathbb R^m$.\\

Let us first mention the easy result extending \eqref{cas2}.
\begin{lemma}\label{lemmeeasy}~\\
Let $k\in \mathbb N^*$ and $\rho >0$.
If $$
T^{k,\rho}:=\{\alpha \in \overline{\mathbb R}_{+}^{\;k}\;\big |\;\sum_{j=1}^k
\alpha_j\geq \rho\},
$$
then
$$
\frac 1k \inf_{\alpha\in T^{k,\rho}} \sum_{j=1}^k \alpha_j (\alpha_j +1)
 \geq \frac\rho k (\frac \rho k +1)\,.
$$
\end{lemma}

For a $k$-partition $\mathfrak D=(D_1,\dots,D_k)$, the corresponding characteristic numbers satisfy~:
\begin{equation}
\sum_j\alpha (D_j)
 \geq \sum_j \Phi_3 (S_j)
\end{equation}
with
\begin{equation}
 \sigma_3\, S_j = \mbox{ Area } (D_j)\; (j=1,\,\dots,\,k) \,,
\end{equation}
and
\begin{equation}
\sum_j S_j =1\,.
\end{equation}
Using the convexity of $\Phi_3$, we obtain
\begin{proposition}~\\
If $\mathfrak D= (D_j)_{j=1,\dots,k}$ is a strong $k$-partition
 of $\mathbb S^2$, then
\begin{equation}\label{lbalpha}
\frac 1k\, \sum_{j=1}^k \alpha(D_j)\geq  \Phi_3(\frac 1k)\,.
\end{equation}
\end{proposition}

Applying Lemma \ref{lemmeeasy} with  $\rho = k\Phi_3(\frac 1k) $, this leads
 together with \eqref{lbalpha} and \eqref{monoto} to the lower bound
 of $\mathfrak L_{k,1}(\mathbb S^2)$~:
\begin{proposition}\label{prop4.6}
\begin{equation}
\mathfrak L_k (\mathbb S^2) \geq \mathfrak L_{k,1}(\mathbb S^2)
\geq \Phi_3(\frac 1k)\,( 1 + \Phi_3(\frac 1k)) \,.
\end{equation}
\end{proposition}
Let us see what it gives coming back to the definition of $\Phi_3$.
\begin{corollary}
\begin{equation}
\mathfrak L_k (\mathbb S^2) \geq \mathfrak L_{k,1}(\mathbb S^2)
\geq \gamma_k  \,,
\end{equation}
with
\begin{equation}
\gamma_k:= \Phi_\infty(\frac 1k)\,( 1 + \Phi_\infty(\frac 1k))\,,
\end{equation}
and
\begin{equation}
 \Phi_\infty(\frac 1k):=\left\{
\begin{array}{ll}  \frac{2(k-1)}{k} &\mbox{ if } k \leq  4\,,\\
  2 \log (\frac k4)  + \frac 32 & \mbox{ if } k>4\,.
\end{array}\right.
\end{equation}
In particular
\begin{equation}
\gamma_2=2\,,\, \gamma_3 = \frac{28}{9}\,,\, \gamma_4= \frac{15}{4}\;.
\end{equation}
\end{corollary}
We note that $\gamma_2$ is optimal and  that $\gamma_3 <
\frac{15}{4}$. Hence for $k=3$, the lower bound is not optimal and
does not lead to a proof of Bishop's Conjecture.
Let us now consider the estimates associated with $\Phi_3$.
\begin{corollary}
\begin{equation}
\mathfrak L_k (\mathbb S^2) \geq \mathfrak L_{k,1}(\mathbb S^2)
\geq \delta_k  \,,
\end{equation}
with
\begin{equation}
\delta_k:= \widehat \Phi_3(\frac 1k)\,( 1 + \widehat \Phi_3(\frac 1k))\,,
\end{equation}
In particular
\begin{equation}
 \delta_3 = \frac 5 8  j_0^2  -\frac 14\,,\,
\delta_4= \frac 78 j_0^2 -\frac 14 \;.
\end{equation}
\end{corollary}
\subsection{Discussion for the cases $k=3$ and $k=4$.}
We observe that $\delta_3 >\gamma_3$ and $\delta_4 >\gamma_4$.
Small computations\footnote{already done in \cite{FrHa}} show indeed  that
\begin{equation}\label{valimpa}
\widehat \Phi_3(\frac 13) \sim 1.401\,,
\end{equation}
which is higher than $\Phi_\infty (\frac 13)=\frac 43$,
and
\begin{equation}
\widehat \Phi_3(\frac 14)\sim 1.748\,,
\end{equation}
which is higher than $\Phi_\infty(\frac 14)= \frac 32\,$.
This
leads to  the lower bound~:
\begin{proposition}
\begin{equation}\label{minoL4b}
\mathfrak L_4 (\mathbb S^2) \geq \mathfrak L_{4,1}(\mathbb S^2) >
15/4 =\mathfrak L_3 (\mathbb S^2)\,.
\end{equation}
\end{proposition}
In particular the best lower bound of $\mathfrak L_{4,1}(\mathbb S^2) $
 is approximately
\begin{equation}\label{minoL4app}
\delta_4 \sim 4.8035\,.
\end{equation}
Note that by a third method, one can find in \cite{FrHa} (Theorem 5 and
table~1, p.~155, computed by J.G. Wendel) another convex function
$\tilde \Phi$ such that $\alpha(D)\geq \tilde \Phi(S)$ and
\begin{equation}\label{valimpabis}
\tilde \Phi (\frac 13)\sim 1.41167\,.
\end{equation}
Note that $\tilde \Phi(\frac 14) < \widehat \Phi_3(\frac14)$ so this improvment
occurs only for $3$-partitions.\\

In the case of $\mathbb S^2$, unlike the case of the square or the
disk, the minimal $4$-partition is not nodal (as proven in Theorem \ref{Unter}).  Note that this implies that the $4$-minimal partition
realizing $\mathfrak L_{4,p}(\Omega)$
 for  $p\in [1,+\infty]$ is
 neither nodal.
\\
 As already mentioned in \cite{FrHa}, there is at least a natural candidate which is the
spherical regular tetrahedron. Numerical
computations\footnote{transmitted to us by M. Costabel}  give, for
the corresponding $4$-partition $\mathcal D_4^{Tetra}$,
\begin{equation}
\Lambda (\mathcal D_4^{Tetra}) \sim 5.13\,.
\end{equation}
Hence we obtain   that
\begin{equation}\label{Tetra}
\frac{15}{4} < \mathfrak L_4(\mathbb S^2) \leq \Lambda (\mathcal D_4^{Tetra})
< 6 =L_4(\mathbb S^2)\,.
\end{equation}
It is interesting to compare it with \eqref{minoL4app}.\\
According to a personal communication of M. Dauge, one can also
observe
 that the largest circle inside a face of the tetrahedron is actually
 a nodal line corresponding to an eigenfunction with eigenvalue
 $6$ (up to a rotation, this is the (restriction to $\mathbb S^2$ of the)
 harmonic polynomial  $\mathbb S^2\ni (x,y,z)\mapsto x^2+y^2 - 2 z^2\,$. This
 gives  directly
 the comparison $\Lambda (\mathcal D_4^{Tetra})
< 6\,$.

\subsection{Large $k$ lower bounds}

We can push the argument by looking at the asymptotic as $k\ar +\infty$
of $\delta_k$.
This gives~:
\begin{equation}\label{lbk}
\mathfrak L_{1,k}(\mathbb S^2) \geq \frac 14 j_0^2 k  - \frac 18\,
j_0^2
-\frac 14 \;.
\end{equation}
We note that at least for $k$ large, this is much better than
 the trivial lower bound
\begin{equation}
\mathfrak L_{1,k}(\mathbb S^2) \geq
 \frac{1}{k} \mathfrak L_{k}(\mathbb S^2)\;.
\end{equation}
We discuss below in Remark \ref{remfermio} an independent improvment.\\
Multiplying \eqref{lbk} by $4\pi$, the area of $\mathbb S^2$
and dividing by $k$, we obtain
\begin{equation}
\mbox{Area}(\mathbb S^2)\;\liminf_{k\ar +\infty} \frac{\mathfrak L_{1,k}(\mathbb S^2)}{k}
 \geq \pi j_0^2\,.
\end{equation}
But $\pi j_0^2$ is the groundstate energy $\lambda(D^1)$
of the Laplacian on the  disk $D^1$ in $\mathbb R^2$
 of area $1$. Although not
written explicitly\footnote{The authors mention only the lower bound
  for $\mathfrak L_k(\Omega)$} in \cite{HHOT, BHV:2007}, the Faber-Krahn Inequality gives
for planar
 domains
\begin{equation}
|\Omega| \frac{\mathfrak L_{k,1}(\Omega)}{k} \geq \lambda(D^1)=\pi j_0^2\;.
\end{equation}
We have not verified the details, but we think that as in
\cite{BHV:2007}
 for planar domains, we will have
\begin{equation}
\mbox{Area}(\mathbb S^2)\;\limsup_{k\ar +\infty} \frac{\mathfrak
  L_{k}(\mathbb S^2)}{k}\leq \lambda(Hexa^1)\,,
\end{equation}
where $Hexa^1$ denotes the regular hexagon of area $1$.\\

As for the case of plane domains, it is natural to conjecture
(see for example \cite{BHV:2007, CL1} but we first heard of this
question from
M. Van den Berg five years ago
)
 that~:
\begin{conjecture}~\\
$$
\lim_{k\ar + \infty}  \frac{\mathfrak
  L_{k}(\mathbb S^2)}{k} = \lim_{k\ar + \infty}  \frac{\mathfrak
  L_{k,1}(\mathbb S^2)}{k}=\lambda(Hexa^1)\;.
$$
\end{conjecture}
The first equality in the conjecture corresponds to the idea, which is
well illustrated in the recent paper by Bourdin-Bucur-Oudet \cite{BBO}
 that, asymptotically as $k\ar +\infty$, a minimal $k$-partition for $\Lambda^p$ will
 correspond  to $D_j$'s such that the $\lambda(D_j)$ are equal.

\begin{remark}\label{remfermio}~\\
If $\Omega$ is a regular bounded open set in $\mathbb R^2$ or
 in $\mathbb S^2$, then
\begin{equation}\label{fermionic}
\frac 1 k \sum_{j=1}^k \lambda_j(\Omega) \leq \mathfrak L_{k,1}(\Omega)\,.
\end{equation}
The proof is "fermionic". It is enough to apply the minimax characterization
 for the groundstate energy $\lambda^{Fermi,k}$
 of the Dirichlet realization of the  Laplacian on $\Omega^k$ (in $(\mathbb R^2)^k$ or in $(\mathbb S^2)^k$)
 restricted to the Fermionic space $\wedge^k L^2(\Omega)$
 which is $$
 \lambda^{Fermi,k}=\sum_{j=1}^k \lambda_j(\Omega)\,.
$$ For any $k$-partition $\mathcal D$
 of $\Omega$, we can
 indeed
 consider the Slater determinant of the normalized groundstates $\phi_j$  of each
$D_j$ and observe that the corresponding energy is
 $k\Lambda^1(\mathcal D)$.\\
This suggests the following conjecture (which is proven for $p=1$
 and $p=+\infty$)~:
\begin{equation}\label{fermionicp}
\left(\frac 1 k \sum_{j=1}^k \lambda_j(\Omega)^p\right)^{\frac 1 p}
 \leq \mathfrak L_{k,p}(\Omega)\,,\,\forall k\geq 1\,,\, \forall p \in [1,+\infty]\,.
\end{equation}
The case when $\Omega$ is the union of  two disks considered in Remark \ref{remsusanna} gives an example
 for $k=2$ where \eqref{fermionic} becomes an equality. In this case,
 we have indeed $\lambda_1(\Omega)=\lambda(C_1)$ and $\lambda_2(\Omega)=\lambda(C_2)$.
  \end{remark}

\newpage

{\bf Acknowledgements.}~\\
This work was motivated by questions of  A.~Lemenant
 about Bishop's Conjecture.
Discussions with (and numerical computations of) M.~Costabel and
 M.~Dauge were also quite helpful. Many thanks also to A.~Ancona and
  D.~Jakobson for
 indicating to us useful references.

\footnotesize
\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Q: nodejs formidable changing uploadDir i have a proyect that im trying to finish i wanna upload images
i can upload the images but not in the dir that supose to upload
ok lets code:
 app.post('/register',function(req,res)
    {
var form = new formidable.IncomingForm();
form.keepExtensions = true;
form.uploadDir ='./uploaded/';
form
    .on('error', function(err) {
        throw err;
    })

    .on('field', function(field, value) {
        //receive form fields here
    })

    /* this is where the renaming happens */
    .on ('fileBegin', function(name, file){
            //rename the incoming file to the file's name
            file.path = form.uploadDir + "/" + file.name;
    })

    .on('file', function(field, file) {
        //On file received
    })

    .on('progress', function(bytesReceived, bytesExpected) {
        //self.emit('progess', bytesReceived, bytesExpected)

        var percent = (bytesReceived / bytesExpected * 100) | 0;
        process.stdout.write('Uploading: %' + percent + '\r');
    })

    .on('end', function() {

    });

form.parse(req);

the problem is that when i post the image... the image still saving on process.env.TMP
my proyect has this dir arq
proyect:
    node_modules
    uploaded
    public
    routes
    views
    app.js
    package.json

EDIT 2: PROBLEM IS BODYPARSER
ok from the last code... the problem was the app.js when i configure the app has a bodyParser so that bodyParser method use formidable from the express
the last code is right and functionally, if anybody wanna use the code... just comment the line from express.configure
app.use(express.bodyParser());

to turn off the express bodyparser but express use formidable too so is same thing...
there is another problem
i configure express.bodyParser like this
  app.use(express.bodyParser(
    {
      uploadDir: './uploads',
      keepExtensions: true
    }));

i change the first code like this
app.post('/register',function(req,res)
{
  var oldDir=req.files.img.path;
  var newDir='./uploads/'+req.body.email+'/perfil/';

if(req.files)
{
  mkdirp(newDir,0777,function(err){
    if(err) throw err;
  })
}
if(req.files)
{
  fs.rename(oldDir,newDir+req.files.img.name,function(err){
    if(err) throw err;
  });

}
res.send('pow!');

it is functionally and when this code receive a file put the file in the uploads folder next create folder with the name of the email and last move the file into the email folder and rename it like the origin file with the extension
its perfect"!
nooo there is no perfection... i wanna use this methods
.on ('fileBegin', function(name, file){
            //rename the incoming file to the file's name
            file.path = form.uploadDir + "/" + file.name;
    })
    .on('progress', function(bytesReceived, bytesExpected) {
        //self.emit('progess', bytesReceived, bytesExpected)

        var percent = (bytesReceived / bytesExpected * 100) | 0;
        process.stdout.write('Uploading: %' + percent + '\r');
    })

because from the first code i can configure where put the file from the beginning
in the second code i move the file after the server receive the file
i think the method is in express because express use formidable like middleware
how can i configure the method on??
where is the method...?'?
how can i know the methods contained in req.files from the second code??
(because req.files using bodyParser() is same like form.file using formidable in first code)
tnx all

A: The IncomingForm constructor takes an options object.  You'll probably need to use the __dirname global in your path as well.
var form = new formidable.IncomingForm({ uploadDir: __dirname + '/uploaded' });
You can see how the option is applied in the formidable source.

A: You can keep the renaming code when 'end' is triggered so that it renames after the file is completely uploaded.

redpajama_set_name: RedPajamaStackExchange


5 Introduction to Public Spaces A lot of geographers are noting that the idea of 20th century planned city is getting transformed into an understanding of city as a self-organising system (Portugali, 2000). Planners no longer deal only with resource planning and ecological benefits but work around a complex interrelationship between different components of environmental, social, economic and technological interventions (Iqbal, 20). Many factors like technological developments, modification of the production process, cluster of public activities, etc. effect the dynamics of user s needs and their preferences (Amando et al, 20; Scott, 20). With this new discourse, people are now realising that cities are constantly shaped itself by its occupants; in a never-ending process of transformation, growth and adaptation (Ernsten, Broekmans, Feenstra & Urhahn, 20). Planning, in all cases, works as a juxtaposition between a present condition and a desirable future (Healey, 200), both of which keeps changing over the course of time. Figure : Three dimensions of sustainability conservation, etc. is the main objective of sustainable urban development (Iqbal, 20). The essence lies in establishing a stable relationship between human activities, their habitation and the natural world. An intercultural city should be able to provide a combination of these different aspects (Amando et al, 20): Ethnical, cultural and natural diversity as potential to urban life Pluralism in community participation Socio-cultural integration in policies To inculcate these, a good inclusive city should provide spaces for social engagement, to foster social cohesion and tackle poverty and inequality. One of the ways to do this is through provision of inclusive, safe and accessible public spaces (UN Habitat, 205). Public Spaces Public spaces in economic terms is a public good; it should be free for all (non-excludable) and should not create conflict between the users (nonrivalrous) (UN Habitat, 208). Rival Environmental (Care for nature) Non-Rival Excludable Private good Natural Monopoly Non-excludable Common resource Public Good Sustainability Social (Social Integrity and Quality of life) Image 2 Abandoned warehouse being used an exhibition space in Milano. They are a key element of (PPS, 20): Individual and social well-being Community s collective life Expressions of diversity (common, natural and cultural) A foundation of their identity Economic (Prosperity to people) Maintaining a balance between the development of urban areas and the interests of the stakeholders with an eye to equity in employment, protection of the environment, social infrastructure, basic services, ecological Image Restaurants extending their seating space in the adjoining street. Every community has some public spaces; they may or may not be apparent. They can be in the form of a plaza around buildings, a park with benches and walkways, a boulevard with sidewalks, a library, museum, community halls, even the spaces found between private spaces (PPS, 20). Spaces like these are mostly undervalued and underutilized assets. Public spaces are also an extension to the communities. They are the banners of urban civility. Civility, here, attributes to both citizen s behaviour as well as the collective behaviour of the community (UN Habitat, 206). Figure 2 Components of a Self-organising sustainable public space Multi-agent Supports Open Communication Generative Constantly Learning Evolves incrementally Sustainable Urban System Collaborative India being signatory to Sustainable Development Goals (SDG) 200, is accountable for creation of Sustainable Cities and Communities. Navi Mumbai Smart City plan intends to solve dual purpose of environmental sustainability and enhancing quality of life of residents through smart solutions. It has taken several initiatives such Based on simple dynamic rules Source - Tan, Bekkering and Corp, 20 8 Public spaces facilitate social capital, economic development, impact physical and mental health and community revitalisation (PPS, 20). In urban areas, they play a vital role in improving the quality of urban life in three ways: environmentally, socially and towards one s health (Hashim, Akmaniza, & Nasir, 205). They help restore the environment, allow community to meet up and make urban dwellers be able to engage in physical activities. Therefore, planning and designing public spaces is critical for sustainable urban development. as Infrastructure innovation: LED lighting, Grey Water Pipeline, Water Leakage Prevention System; Development of natural open spaces: Nature Park, Mangrove Park, Regional Park; Environmental monitoring, publishing Environmental Status Report, etc. Need For Development of Public Spaces a meaningful role (Abbott, 6). It plays a decisive part in developing a relation between the community, other stakeholders and the design of the space (Amando et al, 200). A balanced involvement of public in decisionmaking process provides a long-term support to policy implementations, investments or proposed concepts of land use plan (Agenda 2, ; Nawrath et al., n.d.). This exchange and confrontation between the worlds is more productive when there is public participation (active or passive) involved during space transformation and management. In any public space, multiple stakeholders with their individual interests and interpretations are present. To achieve long-term benefits and to preserve the stakeholders interests and rights, effective community participation plays Along with public participation, a public space works around specific parameters or principles around which a development model is produced. Following are some of the parameters that a development model should inculcate (UN Habitat, 205): Public Spaces has a positive impact on one s health and quality of life They enhance environmental sustainability They promote income, investment and wealth creation They promote equity and social inclusion They are tools for gender and age-friendly cities They enhance public safety and safety perception This issue will further discuss these parameters in detail. It will take one or more of the parameters discussed above and explain with different case studies of development of public spaces in India and around the world. Cities, along with the spatial conditions for diverse economic activities and lifestyles, should also have enough opportunities for these worlds to interact with each other (Bertolini, 2006). Public open spaces act as lungs of a city and have the potential to address all dimensions of sustainability (mentioned in figure ). Therefore, city always needs a sufficiently developed public realm to accommodate spaces for exchange and confrontation between the different worlds (Bertolini, 2006; Crawford, 5).
7 SMART CITY CORNER BEST PRACTICES Designing Gender Sensitive Public Spaces Case Study of Public Spaces in Vienna Introduction Gender equality is not only a fundamental human right, but a necessary foundation for a prosperous and sustainable world (United Nations, 208). UN Women defines gender equality as equal rights, responsibilities and opportunities for women and men (UN WOMEN, 208). Providing equal access to education, public spaces, health care, decent work, and representation in decision-making processes ensures sustainable development (United Nations, 208). Perceptions of gender equality differ between men and women, societies and countries of different developmental status. Globally, many countries have achieved important milestones towards gender parity, however developing countries like India still face women safety as the basic issue in gender equality. Gender-based violence in urban areas can be attributed to factors such as poverty, discrimination, exclusion and lack of gender mainstreaming in urban development leading to public spaces and structures not catering to all genders equally (Jagori, 205). We have previously looked into gender mainstreaming in housing sector and women safety audits in India. This article talks about the importance of gender equality in planning and design of public spaces. The article focuses on case studies from Vienna describing the implementation of gender sensitive practises in their public spaces. Gender Equality in Public Spaces Public spaces enable women, girls, elderly and other marginalised groups (transgenders, migrants, etc.) to participate in public life (UCLG, 206). Though they are meant for everyone to use regardless of their gender or age, women use public parks and streets lesser than men (Harth, 208). In India it is noticed that women tend to limit their participation in public sphere to day time in markets or parks in urban areas (Shukla, 207). Reported cases of physical and psychological harassment in parks, streets and public transports have raised the levels of fear or vulnerability among them (Phadke, 20). Studies show that women prefer active public spaces with characteristics of safer perimeter, cleanliness and safety (Gholamhosseini et al., 208). They perceive lack of proper lighting, deserted roads, absence of street vendors and stores as unsafe situations. Public spaces that ensures comfort, accessibility and safety through features like clean toilets, proper lighting, etc. are preferred by women, elders and children (PUKAR, 20). Image : Paved path, clear visibility and seating in Einsiedler Park Source: wien.gv.at Gender equality in public spaces can be achieved by accommodating features that improve women s safety (UNIFEM, 200). Planning and designing should put special focus on (UCLG, 206; UNIFEM, 200): Proper lighting Landscaping Visibility Clean toilets Motorized and Pedestrian traffic Signages Security personnel Proximity to other public spaces and emergency services Access to public transportation Mixed-land use Women s participation in decision making Case study - Gender Equality in Public Spaces of Vienna Vienna, the capital city of Austria functions as its economic, cultural and political centre. It has been focusing on gender mainstreaming while designing its public spaces, housing, mobility and infrastructure since 0. The gender mainstreaming concept is being incorporated mainly in the design of streetscapes, public squares and public parks (Damyanovic, Reinwald and Weikmann, 20). Image 2: Platforms to sit and chat in St. Johann Park Gender-sensitive Public Parks Design: Re-design of Einsiedler Park and St. Johann Park A need to redesign Einsiedler Park and St. Johann Park was perceived by the City of Vienna when Source: wien.gv.at girls aged between 0 and were found using parks lesser. By focusing on their interests, gender sensitive solutions were implemented to make them feel safer and better in these spaces (UCLG, 206). The main objectives of the project were to (Policytransfer.metropolis.org, 208): motivate girls and young women to use the parks more often improve safety perception in the parks improve elements to attract elderly and parents with little children, and have intense professional exchange of ideas during the planning phase. The city of Vienna selected the design proposals of Tilia planning office and Koselika planning office for Einsiedler Park and St. Johann Park respectively through a design challenge. By 200, detailed planning for re-structuring and re-designing the parks was done and renovation works were completed (Policytransfer.metropolis.org, 208). Gender-sensitive planning measures Image : Hammocks, quick attraction elements at Einsiedler Park Source: WPS Prague.com Source: wien.gv.at Image 5: Seating arrangements in Chrisian Broda Platz The participating consultancies conducted meetings and workshops with residents, mothers, representatives of schools and kindergartens in the district, etc. to identify joint goals for the project. They paid attention to girls interests specifically Image : Platform at Einsiedler Park Source: WPS Prague.com to develop strategies for encouraging their involvement in public activities (Policytransfer. metropolis.org, 208). Several gender sensitive design elements were introduced in these parks, such as (Harth, 208): Football cages were converted for activities that accommodates both genders; in this case, badminton and volleyball courts Hollows in the meadowland were converted to be used as arenas, for ball-games, gymnastics and sitting together Multifunctional play areas Efficient lighting was provided on the main paths Park keepers ensured that the rules are followed Good visibility and clear-cut organisation of footpaths Well-maintained public toilets playing equipment, barrier free toilets, drinking fountains, etc. The pilot project resulted in a generous use of the public square by all genders among youth, children and senior citizens (Damyanovic, Reinwald and Weikmann, 20). Similar measures were adopted in Liesinger Platz of the 2rd district also to achieve a gendersensitive design. Impacts Conclusion The projects witnessed considerable physical and social impacts over time. Physical transformations such as open common areas, gender-neutral activity field, places for group chatting, etc. motivated women and girls to spend more time in the park. Features like visibility in main avenues and proper lighting improved the safety aspects also (Policytransfer.metropolis.org, 208). Noticeable presence of women of all age groups was found in St. Johann s park (Harth, 208). Looking at the response, City of Vienna implemented pilot projects of gender sensitive re-design in other parks of the city. On similar concepts, gender sensitive design elements such as structured footpath network, efficient illumination, multifunctional plazas, multifunctional lawns, etc. were incorporated in Rudolph-Bednar Park (Damyanovic, Reinwald and Weikmann, 20). Gender-sensitive public square design: redesign of Christian Broda Platz Public squares are another focus area for gender mainstreaming in the planning of public spaces in Vienna (Chalaby, 207). On submitting the winning entry for a gender-sensitive architectural competition, architects Beitl and Wallmann redesigned the Christian-Broda-Platz in the 6th district of Vienna. The team designed the square by paying attention to direct walking routes, In addition to these projects, gender mainstreaming is also incorporated in designing walkways. A survey conducted by City of Vienna in identified that females use public transit and pedestrian paths more than males. As a result, city planners adopted steps to improve pedestrian mobility and access to public transit (Foran, 20). This includes 26 street lighting projects, widening of sidewalks and barrier free designs by the City of Vienna Women s Office (Chalaby, 207). Over the years, re-designing several parks and public squares in Vienna has resulted in an inclusive city planning model. Certain design elements such as multifunctional play areas, raised platforms to sit and chat, etc. are easily transferable and can be installed in other places. Assuring safety through efficient lighting and multiple activities in any public space is an important factor in gender-sensitive planning. From the cases of gender mainstreaming in public spaces explained here, it is evident that through effective planning measures, public spaces can have equal utility and benefits for everyone. In India s diverse social setting, women s safety and factors for comfort are often neglected while designing public spaces like parks, streets, markets, public transit, institutions, etc. However, several positive initiatives to improve the safety of public spaces are being taken by many Indian cities. Apps such as SafetiPin are useful for women safety audits. The data acquired is used by the police and PWD to augment facilities such as lighting in public spaces. Government missions like JNNURM seeks to promote planned urban development and equitable cities as an opportunity to build gender-fair and inclusive cities (Khosla, 200). In patriarchal economies like India where women s interests are conventionally under-represented, there is still a lot to achieve.


This ornate ring features a rich cherry amber oval set in oxidized .925 sterling silver, blackened to add definition and give the ring an antiqued look. Surrounded by curls and accented on either side by a small silver leaf, this ring captures the classic beauty of amber.
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In January 2009, she pleaded no contest to misdemeanor reckless driving, avoiding a more severe charge, and was ordered to attend a 12-hour drug education course, fined 0 and sentenced to three years' informal probation.In 2011 Locklear was arrested on a hit-and-run charge when she took off after hitting a parking sign with her vehicle, but was never prosecuted.
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Locklear was arrested and booked shortly after 1 a.m. "She grew up in Thousand Oaks and really did well in Hollywood, so she is the sweetheart of the family and they all rally around her. When he arrived, she and her boyfriend were fighting.
this morning on a charge of felony domestic violence Sunday night, as well as three counts of misdemeanor battery on a peace officer."She was uncooperative with deputies and battered our deputies," a Ventura County Sheriff's Office spokesperson told E! "We conducted an investigation into the original call of the domestic disturbance and determined she had battered her boyfriend... It was a terrible scene."Just two days ago Locklear had posted a photo of herself and her beloved dog, Mister, on a plane headed to Boston.
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A.)But the story behind their split would become even more convoluted shortly.You always have the choice to avoid personalized advertising based on your web browsing activity by visiting the DAA's Consumer Choice page, the NAI's website, and/or the EU online choices page, from each of your browsers or devices.To avoid personalized advertising based on your mobile app activity, you can install the DAA's App Choices app."I made a big mistake," he acknowledged to Despite how her marriage ended, Locklear was not deterred from dating rock stars.(And apparently it's all just water under the bridge, because in 2016 she posted a throwback pic with Lee on what would have been their 30th wedding anniversary.)A year after her divorce from Lee was finalized, she married Bon Jovi guitarist Richie Sambora on Dec. They welcomed daughter Ava Elizabeth Sambora on Oct. Locklear and Sambora carried on as one of Hollywood's most enduring couples, a rare rocker-actress union that worked...until it stopped working.Publicly, they were still going strong in November.But apparently there were major issues behind closed doors. Before the authorities arrived, "she was freaking out and texting her brother nonstop and wasn't making sense," a source told , noting that Locklear has the full support of her family, who only want her to be healthy.Her parents were very glad when the doctors decided she was well enough to be discharged."While in hindsight that seems like a lot of on-the-record information from a hospital, at the time Locklear's family wanted to immediately clear up any rumors that she was suicidal or had tried to harm herself. Marc Mani, and she couldn't help but feel a little déjà vu when she was making headlines for her bikini bod once again in 2015.It was soon reported that she had mixed her prescription medication with alcohol and her sister had worried she might do something to herself. She and Sambora were spotted together more frequently and were vacationing as a family with Ava."Time does heal old wounds. Toward the end of 2016 Locklear decided it was time for a timeout to focus on her health."I am feeling great and am taking steps to enrich and better my life," she said in a statement in early January 2017, responding to rumors that she was in rehab again."Unfortunately our friendship had to dissolve and I'm sad about that.The last thing I would want to do is create a media frenzy like this, especially as I'm trying to get through a divorce."The actress—who after , but in 2008 she was forced to take a break and focus on herself.
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The Challenges of Crypto Sponsorship
With blockchain sponsorship predicted to reach $5 billion by 2026 according to Nielson, it is clear that the sector will form a key component of a sports organisation's sponsorship revenue stream. However, like any industry that experiences hypergrowth, greater regulation and due diligence procedures must be in place to prevent some of the disastrous partnerships we have seen over the past couple of months.
Few industries have had such a polarising effect as crypto amongst both sports fans and the general population, with good reason. These companies are agreeing vast sponsorship deals with rights holders, bringing an often much-needed injection of funds into the sport. In contrast, many fans feel these partnerships are inauthentic and untrustworthy, the very opposite of what a successful sponsorship should look like. We have seen the likes of Man City and Barcelona enter into now terminated partnerships with 3Key and Ownix after it occurred that the former's company executives couldn't be found in an online search and the latter's company consultant was arrested on fraud charges. Now, this begs the question were there any basic background checks completed? It seems it is just another case of a sports organisation being more than happy for a company to become a sponsor simply because enough money has been offered.
Elsewhere, a further negative crypto trend we have seen over the past year that has led to deteriorating fan trust is the increasing number of football clubs launching their own fan tokens via namely Socios, the trading platform run by the cryptocurrency Chilliz. Socios allows each partnered club to release a certain number of tokens for their fan base to trade. On the surface, this may seem like the modern way to engage fans, upon further inspection, as with anything if there is money to be made people will take advantage. The idea is that the tokens that are traded by fans lead to higher levels of engagement, yet many will purchase tokens purely for trading purposes and thus to make money. Once more, Iqoniq, another token trading platform recently went into liquidation which left thousands of sports fans at clubs such as Wigan Warriors and Crystal Palace with tokens that were virtually worthless, furthering calls for regulation of the industry.
Whether fans like it or not crypto sponsorship in sport is a trend to stay and will continue to grow over the coming years to become one of the biggest spenders in sport. Last year, Crypto.com announced a string of sponsorship deals across various sports across the globe accumulating to a commitment well into the billions of dollars over the next decade. Elsewhere FTX and Tezos have also committed millions to F1 and football in the form of the Mercedes-AMG F1 team and Manchester United. The dominance of these crypto brands can also be evidenced by the numerous ads that were shown during halftime at the Superbowl, with these brands spending over $50 million on the ads.
On the whole, it will be important to remain critical of the crypto sponsorship landscape, yet we cannot always lay the blame on the rights holder for permitting these sponsorship deals, with many operating on a financial knife edge. Many partnerships we have seen have also been hugely impactful for the rights holder and once more robust regulation is applied, we will potentially see the transformative effect these deals can have on sport over the coming years.
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