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\begin{document}
\title{\bf {\Large On the best constants in Sobolev inequalities on the solid torus
in the limit case $p=1$ }}
\author{  Nikos Labropoulos$\,^1$ and Vicen\c tiu D. R\u adulescu$\,^{2,3}$ \\
{\small $^1\,$Department of Mathematics, University of Patras, Patras 26110, Greece}\\
 {\small    $^2\,$Department of Mathematics, Faculty of Sciences, King Abdulaziz University,}\\
      {\small P.O. Box 80203, Jeddah 21589, Saudi Arabia}\\
      {\small    $^3\,$Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy,}\\
      {\small P.O. Box 1-764,  014700 Bucharest, Romania}\\
\footnotesize{E-mail addresses: \texttt{nal@upatras.gr}; {\tt vicentiu.radulescu@imar.ro}}}
\date{}
\maketitle
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\begin{abstract} In this paper, we analyze the problem
of determining  the best constants for the Sobolev inequalities in
the limiting case where $p=1$. Firstly, the special case of the
solid torus  is studied , whenever it is proved that the solid
torus is an extremal domain with respect to the second best
constant and totally optimal with respect to the  best constants
in the trace Sobolev inequality. Secondly, in the spirit of
Andreu, Mazon and Rossi in \cite{And-Maz-Ros}, a Neumann problem
involving the 1-Laplace operator in the solid  torus is solved.
Finally, the existence of both  best constants in the case of a
manifold with boundary is studied, when they exist. Further
examples are provided where there are none. The impact of
symmetries which appears in the manifold, is also discussed.
\end{abstract}
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\smallbreak {\noindent}{\textbf{Keywords:} Sobolev inequalities;
best constants; limit case; manifolds with boundary; symmetries;
solid torus.}
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\smallbreak\noindent{\textbf{Mathematics Subject Classification
(2010):} 46E35;  41A44; 35B33. }
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\tableofcontents
\section{Introduction}
\setcounter{equation}{0} $\,\,\,\,\,$ Let $(M,g)$ be a smooth,
compact $n$-dimensional Riemannian manifold, $n\geq 3$, with
boundary. We define  the Sobolev space $H^p_1(M)$ as the
completion of $C^\infty(M)$ with respect to the norm:
$$
\left\| u\right\|_{H_1^p \left( M \right)}=\left(\int_M \left|
\nabla u \right|^p d\upsilon _g\right)^{1/p} +\left(\int_M \left|
u \right|^p d\upsilon _g \right)^{1/p},
$$
and $\stackrel{\circ}{H}\!^p_1(M)$ as the closure of
$C^\infty_0(M)$ in $H^p_1(M)$.\\
For any $p \in[1, n)$ we denote:
$$
p^*=\textstyle\frac{\textstyle
np}{\textstyle n - p}\quad \mathrm{and }\quad \tilde p^ * =
\frac{\textstyle(n - 1)p}{\textstyle n - p}\,.
$$

According to Sobolev's theorem (see Aubin \cite{Aub}) the
embeddings $ H_1^p (M) \hookrightarrow L^q (M)$ and $H_1^p (M)
\hookrightarrow L^{\tilde q} (\partial M)$ are compact for any $q
\in \left[ {1,p^*  } \right)$ and $\tilde q \in \left[ {1,\tilde
p^ * } \right)$, respectively, but the embeddings $ H_1^p (M)
\hookrightarrow L^{p^ *  } (M) $ and $ H_1^p (M) \hookrightarrow
L^{\tilde p^ * } (\partial M) $ are only continuous. So, there
exist constants $A,B$ and $\tilde A, \tilde B$ such that for all $
u \in H_1^p (M)$ the following inequalities hold:
\begin{equation}\label{E1}
\left( {\int_M {\left| u \right|^{p^* } d\upsilon _g } }
\right)^{1/p^* }  \le A\left( {\int_M {\left| {\nabla u} \right|^p
d\upsilon _g } } \right)^{1/p}  + B\left( {\int_M {\left| u
\right|^p d\upsilon _g } } \right)^{1/p}
\end{equation}
and
\begin{equation}\label{E2}
\left( {\int_{\partial M} {\left| u \right|^{\tilde p^* } ds_g } }
\right)^{1/\tilde p^* }  \le \tilde A\left( {\int_M {\left|
{\nabla u} \right|^p d\upsilon _g } } \right)^{1/p}  + \tilde
B\left( {\int_{ M} {\left| u \right|^p d\upsilon_g } }
\right)^{1/p}
\end{equation}

In these two inequalities, we are interested in the values of the
best possible constants. Define the sets:
$$
\mathcal{A}_p{(M)} =\inf \{A \in\mathbb{R}: \; \exists \; B \in
\mathbb{R }\;\;\mathrm{such}\;\;
\mathrm{that}\;\;\mathrm{inequality}\; (\ref{E1}) \;
\mathrm{holds}\;\mathrm{for}\;\mathrm{all} \; u\in H^p_1(M)\};
$$
$$
\mathcal{B}_p{(M)} =\inf \{B \in\mathbb{R}: \; \exists \; A \in
\mathbb{R }\;\;\mathrm{such}\;\;
\mathrm{that}\;\;\mathrm{inequality}\; (\ref{E1}) \;
\mathrm{holds}\;\mathrm{for}\;\mathrm{all} \; u\in H^p_1(M)\}.
$$
One can define in the same way the sets $\mathcal{\tilde
A}_p{(M)}$ and $\mathcal{\tilde B}_p{(M )}$.

\smallbreak In the case  $ 1<p<n $, the best constants are known
in almost all cases both on Euclidean space (here only appears the
first constant) and on Riemannian manifolds (with boundary or
without boundary), and a host of literature exists for all these
cases; see e.g. \cite{Aub2, Aub3, Bie, Che3, Cot-Lab1, Cot-Lab2,
Cot-Lab3, Dru, Dru-Heb, Fag1, Fag2, Heb, Heb-Vau1, Heb-Vau2,
Heb-Vau,  Lio,  Tal} and the references therein. We point out that
Druet and Hebey \cite{Dru-Heb} are working exclusively on the
Euclidean space and the manifolds without boundary, provide a
complete study on the area. In addition, they answered questions
like when and under what conditions optimal Sobolev inequalities
are true or not. Some more cases towards this direction were
studied by Druet \cite{Dru} and Faget \cite{Fag2} on manifolds
without boundary, while Biezuner in \cite{Bie1} answered  the same
questions on manifolds with boundary. When $p=1$ is also known
quite remarkable and interesting results. (For example, see
\cite{Maz}, \cite{Lio}, \cite{Mot}, \cite{Par},
\cite{And-Maz-Ros}) and the references therein. Complete and
thorough study on the best constants in Sobolev inequalities is
presented by the books \cite{Aub}, \cite{Dru-Heb},
\cite{Heb} and \cite{Maz1}.\\

\smallbreak This paper  is organized as follows: In Section 2, a
short survey on known results about the best constants is
presented; this is done for the sake of completeness so that the
paper is self-contained. Section 3 is devoted to the qualitative
presentation of the results established in this paper. In Section
4, the case of the solid torus is studied in detail. A Neumann
problem involving the $\Delta_1$-Laplace operator in the solid
torus is considered in Section 5. Finally,  some results are given
for smooth, compact, Riemannian manifolds with boundary. In
particular, the general case of a manifold with boundary is
treated in subsection $\S\,6.1$, and  in $\S\, 6.2$ some results
are presented  for manifolds in the presence of symmetries.

\section{A short survey on the best constants}
\subsection{The case $1<p<n$}

In this paragraph, we recall certain important results concerning
the best constants for some Sobolev inequalities, which are well
known and they are directly related with the context of the
present paper.\\
We now give  a short survey of known results about the best
constants for the classical as much as for the trace Sobolev
inequalities in the case $1<p<n$. Cf. Aubin \cite{Aub}, if $M$ is
a compact Riemannian manifold with boundary, $H_1^p(M)\neq
\stackrel{\circ}{H}\!\!^p_1(M)$. Therefore, as far as the first
best constant of (\ref{E1}) is concerned, we have to consider two
distinct Sobolev spaces:\\
$\textbf{(i)}\;$ When we consider (\ref{E1}) on
$\stackrel{\circ}{H}\!\!^p_1(M)$, the same results for best
constants on compact Riemannian manifolds without boundary remain
true. Recall that for compact Riemannian manifolds  without
boundary the best constant in front of the gradient term in
inequality (\ref{E1}) is the same as the best constant for the
Sobolev embedding for $M=\mathbb{R}^n$ under the Euclidean metric
(see Aubin \cite{Aub3}), that is,
\[
\frac{1} {{K\left( {n,p} \right)}} = \mathop {\inf
}\limits_{\mathop {u \in L^{p*} \left( {\mathbb{R}^n }\right)
\setminus \{ 0 \}}\limits_{\nabla u \in L^{p} \left( {\mathbb{R}^n
} \right)} } \displaystyle\frac{{\int_{\mathbb{R}^n } {\left| {\nabla u}
\right|} ^p dx}} {{\left( {\int_{\mathbb{R}^n } {\left| u \right|}
^{p^* } dx} \right)^{p/p^* } }}\,.
\]
It has been proven that
\begin{eqnarray*}
K\left( {n,1} \right) &=& \frac{1} {n}\left( {\frac{n} {{\omega
_{n -1}}}} \right)^{1/n},\\
K\left( {n,p} \right) &=& \frac{1} {n}\left( {\frac{{n\left( {p -
1} \right)}} {{n - p}}} \right)^{1 - 1/p} \left( {\frac{{\Gamma
\left( {n + 1} \right)}} {{\Gamma \left( {n/p} \right)\Gamma
\left( {n + 1 - n/p} \right)\omega _{n - 1} }}} \right)^{1/p} ,
\end{eqnarray*}
where $\omega_{n-1}$ is the area of the unit sphere in
$\mathbb{R}^n$ and $\Gamma$ is the gamma function.\\
In particular, when $p=1$ and $M=\mathbb{R}^n$, (\ref{E1}) is the
usual isoperimetric inequality, (see \cite{Fed-Fle},
\cite{Fle-Ris} and \cite{Fed}). The exact value of $K(n,1)$ was
computed by Federer and Fleming   \cite{Fed-Fle} and by Maz'ya
\cite{Maz}; the extremum functions in this case are the
characteristic functions of the balls of $ \mathbb{R}^n$. The
value of $K(n,p)$ was explicitly computed independently by Aubin
\cite{Aub2} and Talenti \cite{Tal} and is attained by the
functions $\varphi(x)=(\lambda+|x|^{p/(p-1)})^{1-{n/p}}$, where
$\lambda$ is any positive real number. In addition, Aubin
\cite{Aub} proved that the constant $K(n,1)$ is obtained as a
limit of $K(n,p)$ as $p$ tends to $1^+$.


 \smallbreak
\noindent$\textbf{(ii)}\;$ When we consider inequality (\ref{E1})
on $H_1^p(M)$, Cherrier \cite{Che3} showed that the first best
constant is equal to $2^{1/n}K(n, p)$.
 \smallbreak
\noindent Contrary to the first best constant $K(n,p)$ of
(\ref{E1}), which depends only on the dimension $n$ of the
manifold and $p$, the second best constant $\mathcal{B}_p(M)$
depends on the geometry of the manifold and was computed to be
$\mathcal{B}_p(M)=|M|^{-1/n}$, (see Hebey \cite{Heb}).

  \smallbreak Concerning  the inequality (\ref{E2}), Lions \cite{Lio} proved
that the best constant in front of its gradient term in the case
of the Euclidean half-space $\mathbb{R}^n_+=\left\{x=(x',y)\in
\mathbb{R}^n:x'\in \mathbb{R}^{n-1}, y\geq 0\right\}$ is
$$
\frac{1}{\tilde K\left(n,p\right)} ={\mathop {\inf
}\limits_{\begin{subarray}{l}
  u \in L^{\tilde p * } (\partial R_ + ^n ) \setminus \{ 0\}\\
  \nabla u \in L^p (R_ + ^n )
\end{subarray} } }\displaystyle\frac{{\int_{R_ + ^n } {\left| {\nabla u} \right|} ^p dx}}
{{\left( {\int_{\partial R_ + ^n } {\left| u \right|^{\tilde p * }
} dx'} \right)^{p/{\tilde p * }} }}\,.
$$
Biezuner \cite{Bie} used inequality (\ref{E2}) and a standard
contradiction argument in order to show  that there exists a
constant $C>0$ such that
\begin{equation}\label{E3}
\int_M \left| u \right|^p d\upsilon_g   \le C\left(  \int_M |
\nabla u |^p d\upsilon_g+ B \int_{\partial M}|u|^p ds_g \right),
\end{equation}
for all $u\in H_1^p(M)$.\\
>From (\ref{E2}) and (\ref{E3}) we conclude  that there
are positive constants $A,\, B$, which may depend on $M$ and $g$,
such that:
\begin{equation}\label{E4}
\left( {\int_{\partial M} {\left| u \right|^{\tilde p^* } ds_g } }
\right)^{1/\tilde p^* }  \le \tilde A\left( {\int_M {\left|
{\nabla u} \right|^p d\upsilon _g } } \right)^{1/p}  + \tilde
B\left( {\int_{\partial M} {\left| u \right|^p ds_g } }
\right)^{1/p}\,.
\end{equation}

Inequality (\ref{E3}) is a `bridge' to move from (\ref{E2}) to
(\ref{E4}) and to our knowledge has not presented any interest on
its best constants.

\smallbreak Regarding the first best constant in inequality
(\ref{E4}), Biezuner showed that Lions' conclusion still remains
valid for any smooth, compact $n$-dimensional Riemannian manifold
with boundary and for all $p\in(1, n)$. The explicit value of
$\tilde{K}(n, p)$ was computed independently by Escobar \cite{Esc}
and Beckner \cite{Bec}, only in the case where $p=2$:
$$
\tilde{K}(n,2)={\frac{2}{n-2}}\, \omega_{n-1} ^{-1/(n-1)},
$$
Unfortunately, their proofs both deeply used the conformal
invariance of the associated variational problem, and thus cannot
be generalized to other values of $p$, and the problem was still
open. However, Nazaret \cite{Naz} studying optimal Sobolev trace
inequalities on the half-space proved a conjecture made by Escobar
\cite{Esc} about the minimizers in (\ref{E4}) and founded that the
functions $f(x) = \left((\lambda+y)^2+|x'|^2 \right)^{(p -n)/2(p -
1)},\,\,\forall \,x = \left( {x', y} \right)\in
\mathbb{R}^n_+\,\,\mathrm{and}\,\,\lambda
> 0 $, are optimal for this inequality.\\ The second best constant
of (\ref{E4}), for $1<p<n$,  was computed as $ \mathcal{\tilde
B}_p(M)=|\partial M|^{-1/n}$, see Biezuner \cite{Bie}.

\smallbreak In order to make the paper  more self-contained  we
introduce at this point some background material from the
geometry. (More details see in \cite{Bre} or \cite{Kob}).\\
Consider a group $G$ acting on a set $X$. The \emph{orbit} of a
point $x$ in $X$ is the set of elements of $X$ to which $x$ can be
moved by the elements of $G$. (Just as gravity moves a planet
around in its orbit, the group action moves an element around in
its orbit.) The $G-orbit$ of $x$ is denoted by $O_G(x)=\left\{
\tau(x), \tau \in G \right\}$. If $Y\subseteq X$, we write
$G(Y)=\{\tau(y): y\in Y \,\,\mathrm{and}\,\, \tau\in G\}$. We call
the subset $Y$ \emph{invariant} under the action of $G$ if $G(Y) =
Y$ and denoted by $Y_G$. For every $x\in X$, we define the
\emph{stabilizer subgroup} of $G$ with respect to $x$ (also called
the \emph{isotropy group}) as the set of all elements in $G$ that
fix $x$: $S_G(x)=\left\{ \tau \in G: \tau(x)=x \right\}$.
Moreover, if the set $X$ is equipped with a metric, then the
\emph{isometry group} of this metric space is the set of all
isometries (i.e. distance-preserving maps) from the metric space
onto itself, with the function composition as group operation. Its
identity element is the identity function, (i.e. the isometry
group of a two-dimensional sphere is the orthogonal
group $O(3)$). \\
Given $(M,g)$ a Riemannian manifold (complete or not, but
connected), we define by $I(M,g)$ its group of isometries. It is
well known (see for instance \cite{Kob}) that $I(M,g)$ is a Lie
group with respect to the compact open topology, and that $I(M,g)$
acts differentiably on $M$. Since (this is actually due to
E.Cartan) any closed subgroup of a compact Lie group is a Lie
group, we get that any compact subgroup of $I(M,g)$ is a sub-Lie
group of $I(M,g)$. It is now classical (see \cite{Bre} and
\cite{Die}), that for any $x\in M$, $O_G(x)$ is a smooth compact
submanifold of $M$. Denote by $|O_G(x)|$ the volume of $O_G(x)$
for the Riemannian metric induced on $O_G(x)$. (In the special
case where $O_G(x)$ has finite cardinal, then
$|O_G(x)|=\mathrm{card}O_G(x)$.) Let $G$ be a closed subgroup of
$I(M,g)$. Assume that for any $x\in M$,
$\mathrm{card}O_G(x)=+\infty$, and set $k = \mathrm{min}_{x\in
M}\mathrm{dim}O_G(x)$. Then $k\geq 1$ (see Hebey-Vaugon
\cite{Heb-Vau}), and is called \emph{minimum orbit dimension}.

\smallbreak If $G$ denotes some subgroup of $I(M,g)$, we set
$$
C_G^\infty(M) =\left\{u\in C^\infty(M): u\circ \tau=u\, ,\,
\forall \: \tau \in G\right\},
$$
and
$$
C_{0,G}^\infty(M) =\left\{u\in C_0^\infty(M): u\circ \tau=u\, ,\,
\forall \: \tau \in G\right\},
$$
where $C^\infty(M)$ denotes the space of smooth functions on $M$
and where $C_0^\infty(M)$  denotes the space of smooth functions
with compact support $M$.\\
Similarly, for $p\geq 1$, we set
$$
H_{1,G}^p(M)=\left\{u\in H_1^p(M): u\circ \tau=u\, ,\, \forall \:
\tau \in G\right\},
$$
and
$$
\stackrel{\circ}{H}\!\!_{1,G}^p(M)=\left\{u\in
\stackrel{\circ}{H}\!\!_1^p(M): u\circ \tau=u\, ,\, \forall \:
\tau \in G\right\},
$$
where $\stackrel{\circ}{H}\!\!_{1,G}^p(M)$ is the completion  of
$C_{0, G}^\infty(M)$ with respect to the norm
$\left\|u\right\|_{H_1^p(M)}$.\\
If $(M,g)$ is complete and $G$ is compact, by the existence of the
a Haar measure on any Lie group, one gets that $H_{1,G}^p(M)=
\stackrel{\circ}{H}\!\!_{1,G}^p(M)$ for any $p$.

\smallbreak If $k$ denotes the minimum orbit dimension of $G$, it
is known from Hebey \cite {Heb} that for a $G$-invariant
Riemannian manifold $(M,g)$ without boundary, the embeddings
$H_{1,G}^p \left( M \right) \hookrightarrow L^q \left( M \right)$
are continuous for any $ p \in \left[ {1,n - k} \right)$ and $ q
\in \left[1,\frac{\textstyle (n - k)p}{ \textstyle n - k - p}
\right] $ and compact if $q \in \left[ 1,\frac{\textstyle (n -
k)p}{\textstyle n - k - p} \right)$. The best constant $K_G $ in
front of the gradient term in inequality (\ref{E1})  in this case
was computed by Faget \cite{Fag1}:
\[
  K_G  = \frac{{K\left( {n - k,p} \right)}}
{{V^{1/\left( {n - k} \right)} }},
\]
where $V$ is the minimum volume of orbits of dimension $k$.

Consider now a compact smooth $n$-dimensional Riemannian manifold
$(M,g)$, $n\geq 3$,  with boundary invariant under the action of a
subgroup $G$ of the isometry group $I(M,g)$. Let $k$ denote  the
minimum orbit dimension of $G$ and let $V$ be  the minimum of the
volume of the $k$-dimensional orbits. Denote, also in this case:
$$
p^*= \frac{\textstyle (n-k)p}{\textstyle
n-k-p}\quad \mathrm{and} \quad\tilde p^* =\frac{\textstyle \left(
{n - k - 1} \right)p}{\textstyle n - k - p}\,.
$$

Cotsiolis and Labropoulos \cite{Cot-Lab3}  showed   that for any
$p \in \left( {1,n - k} \right)$ and for all  $u \in {H}^p_{1,G}
\left( M \right)$ the first best constants in the inequalities:
\begin{equation}\label{E5}
\left( {\int_M {\left| u \right|^{p^* } d\upsilon _g } }
\right)^{p/p^* }  \le A\int_M {\left| {\nabla u} \right|^p
d\upsilon _g }  + B\int_M {\left| u \right|^p d\upsilon _g }
\end{equation}
 and
\begin{equation}\label{E6}
\left( {\int_{\partial M} {\left| u \right|^{\tilde p^* } ds_g } }
\right)^{p/\tilde p^* }  \le \tilde A\int_M {\left| {\nabla u}
\right|^p d\upsilon _g }  + \tilde B\int_{\partial M} {\left| u
\right|^p ds_g }
\end{equation}
are $2^{p/(n-k)} K_G^p $ and $\tilde K_G^p$  respectively,  where:
$$
K_G  = \frac{{K\left( {n - k,p} \right)}} {{V^{1/\left( {n - k}
\right)} }}\quad \mathrm{and } \quad \tilde{K}_G =\frac{{\tilde
K\left( {n - k,p} \right)}} {{V^{\left( {p -1} \right)/\left( {n -
k - 1} \right)p} }}.
$$
Since  inequalities (\ref{E5}) and (\ref{E6}) are stronger than
(\ref{E1}) and (\ref{E4}), it  follows that  the first best
constants in inequalities (\ref{E1}) and (\ref{E4}) in this case
are $2^{1/(n-k)} K_G$ and $\tilde K_G$. In addition, Cotsiolis and
Labropoulos \cite{Cot-Lab1, Cot-Lab2} computed the first best
constants in the inequalities (\ref{E1}) and (\ref{E4}) in the
special case where the manifold is the $3$-dimensional solid torus
$T$, which is invariant under the action of a subgroup $G=O\left(
2 \right) \times I$ of the isometry group $O(3)$.

\subsection{The case $p=1$}
The case where  $p=1$ can be said \emph{limiting}  because it can
be seen as a limit case as $p$ tends to 1. This case is more
complicated due to the  lack of compactness of the embedding
$H^1_1(M)\hookrightarrow L^1(\partial M)$. However, in the
direction
of interest in this article, there are some important results.

The limit of the Sobolev trace inequality in the Euclidean upper
half-space $\mathbb{R}^n_+$, is given by:
\begin{equation}\label{E7}
\int_{\partial \mathbb{R}^n_+} {\left| u (x')\right| dx' }  \le
\tilde C {\int_{\mathbb{R}^n_+}\left| {\nabla u}(x) \right| dx }.
\end{equation}
This problem was studied by Motron  \cite{Mot} and   Park
\cite{Par}. The best constant $\tilde K\left(n,1\right)$ defined
by:
$$
\frac{1}{\tilde K\left(n,1\right)} ={\mathop {\inf
}\limits_{\begin{subarray}{l}
  u \in L^1(\partial R_ + ^n ) \setminus \{ 0\}\\
  \nabla u \in L^1 (R_ + ^n )
\end{subarray} } }\displaystyle\frac{{\int_{R_ + ^n } {\left| {\nabla u}(x) \right|} dx}}
{{\left( {\int_{\partial R_ + ^n } {\left| u (x')\right| } dx'}
\right)  }}
$$
was computed for this inequality  to be equal to $1$. Moreover,
Park proved that if $n=1$, then  the same result is obtained from
recognizing the Euler-Lagrange equation for the inequality as the
main curvature formula of plane curves. Motron \cite{Mot} also
proved that if $\Omega $ is a connected bounded open set of
$\mathbb{R}^n, n\geq 2$, whose boundary is piecewise $C^1$, then
the second best constant in the inequality:
\begin{equation}\label{E8}
\int_{\partial \Omega} {|u|} dS \leq \tilde A\int_\Omega {|\nabla
u|} dV+\tilde B\int_\Omega | u| dV
\end{equation}
is equal to $|\partial \Omega|/|\Omega|$.

Andreu, Mazon and  Rossi  \cite{And-Maz-Ros} studied the best
constant $\lambda _1(\Omega)$ for the trace map from
$W^{1,1}(\Omega)$ into $L^1(\partial\Omega)$,  that is, the best
constant in the inequality:
\begin{equation}\label{E9}
\lambda\int_{\partial \Omega} {\left| u \right| dx' }  \le
{\int_\Omega \left| {\nabla u} \right| dx}+ {\int_\Omega \left|
u\right| dx },
\end{equation}
where $\Omega $ is a bounded set in $\mathbb{R}^n$ with Lipschitz
continuous boundary $\partial\Omega$. Obviously, if we set
$1/\lambda=\tilde A=\tilde B$, the best constant in (\ref{E9}) can
be obtained as  a special case of (\ref{E8}). The authors showed
that if $\lambda_1(\Omega)<1$ then this best constant is attained
in $BV(\Omega)$, the space of functions of bounded variation on
$\Omega$, which  is defined as the space of functions in
$L^1(\Omega)$ whose derivatives in the sense of distributions are
bounded measures on $\Omega$. Moreover, the authors proved that
this constant can be obtained as limit when $p\to 1^+$ of the best
constant of the compact embedding $W^{1,p}(\Omega)\hookrightarrow
L^p(\partial\Omega)$ with $p>1$, that is, of the best constant in
the inequality:
\begin{equation}\label{E10}
\lambda\int_{\partial \Omega} {\left| u \right|^p dx' }  \le
{\int_\Omega \left| {\nabla u} \right|^p dx}+{\int_\Omega \left|
u\right|^p dx }.
\end{equation}
In aim to perform the proofs, they looked at Neumann problems
involving the 1-Laplace operator defined by $\Delta_1(u)=\mathrm{div}(Du/|Du|)$
in the context of bounded variation functions (the natural context
for this type of problems).

Consider now the case where  $p=1$ is a smooth, compact
$n$-dimensional Riemannian manifold, $n\geq 3$, with boundary.
Since for $p=1$, $p^*=\frac{\textstyle n}{\textstyle n-1}$ and
$\tilde p^*=1$, inequality (\ref{E4}) has no interest for $p=1$.
Thus, we are interested in the following two inequalities, arising
from (\ref{E1}) and (\ref{E2}) respectively:
\begin{equation}\label{E11}
\left( {\int_M {\left| u \right|^{n/(n-1) } d\upsilon _g } }
\right)^{(n-1)/n }  \le A\int_M {\left| {\nabla u} \right|
d\upsilon _g }  + B\int_M {\left| u \right| d\upsilon _g }
\end{equation}
 and
\begin{equation}\label{E12}
\int_{\partial M} {\left| u \right| ds_g }  \le \tilde A {\int_M
\left| {\nabla u} \right| d\upsilon _g }+\tilde B {\int_M \left|
u\right| d\upsilon _g },
\end{equation}\\
where $A, B$ and $ \tilde A , \tilde B$ are positive constants.

Traditionally for the study of best constants  it is used the
space $W^{1,1}(\Omega)$ of functions in $L^1(\Omega)$ whose
gradient in the distributional sense is in $L^1(\Omega)$:
\[
W^{1,1}(\Omega)=\left\{u\in L^1(\Omega):\nabla u\in
L^1(\Omega)\right\}\,.
\]
Although the Sobolev space $W^{1,1}(\Omega)$ is a proper subset of
$BV(\Omega)$,  from density results (see in \cite{Dem-Tem, Giu,
Str-Tem}), we can derive that if $A, B$  (or $\tilde A, \tilde B$)
are such that (\ref{E11}) (or (\ref{E12})) is valid for all $u\in
W^{1,1}(\Omega)$, then (\ref{E11}) (or (\ref{E12})) may be
expanded to functions in $BV(\Omega)$ with the same constants $A,
B$ (or $\tilde A, \tilde B$). Concerning  the inequality
(\ref{E11}), it was proved by Andreu, Mazon and Rossi
\cite{And-Maz-Ros} that the best constant is the same in both the
$W^{1,1}(\Omega)$ and $BV(\Omega)$. Thus, when we have to solve
problems on best constants in the limiting case $p=1$ on an
arbitrary  bounded set $\Omega \in \mathbb{R}^n$  we can remain in
the space $W^{1,1}(\Omega)$. Furthermore, by Meyers-Serrin's
theorem (see \cite{Mey-Ser} or \cite[Theorem  3.17]{Ada-Fou}), the
equality $H_1^p(\Omega)=W^{1,p}(\Omega)$, $1\leq p<\infty$, is
known to hold on all open subsets $\Omega$ of the Euclidean space
$\mathbb{R}^n$. It seems to be unknown whether this extends to
arbitrary manifold $M$. However,  by definition of $H_1^p(M)$, we
have $H_1^p(M)\subseteq W^{1,p}(M)$. On the other hand, by
Hopf-Rinow's theorem we have that every compact Riemannian
manifold $(M,g)$ is geodesically complete. In addition, it is
known that if $M$ is geodesically complete then $C^\infty_0 (M)$
is dense in $W^{1,p}(M)$ (see G\"uneysu and Pallara
\cite[Proposition 2.10]{Gun-Pal}). Moreover, $C^\infty_0(M)$ is
dense in $H^{1,p}(M)$ (see Aubin
 \cite[Theorem 1]{Aub2}). In particular, one has $H^{1,p}(M)
=W^{1,p}(M)$ but we have been unable to find a direct reference for it.

In the rest of this paper we remain in the space $H_1^p(M)$,
and also in the space $BV(\Omega)$, where it is absolutely
necessary (i.e., when we have to solve equations).

\section{Qualitative presentation of results}

The analysis presented in this paper is divided into three parts, as described in the following.
\begin{enumerate}
    \item[(i)] We first study
    inequalities (\ref{E11}), (\ref{E12}), as
well as the inequality (\ref{E9}), and we compute
all the best constants in the case where the domain is the solid
torus:
$$
\overline{T}=\left\{(x,y,z)\in \mathbb{R}^3: \left(
\sqrt{x^2+y^2}-l\right)^2+z^2\leq r^2, \, \,l>r>0\right\}.
$$

One of our main interests  in this paper, is to study the
dependence of the best constant in theses inequalities and as well
as the existence of extremals (functions where the constant is
attained) in (\ref{E9})  on the geometry. The  related problem in
the general case was studied by Andreu, Mazon and  Rossi in
\cite{And-Maz-Ros} and the same problem in a more overall context
was studied by Demengel in \cite{Dem}. So, we compute both the
best constants in  inequalities (\ref{E11}) and (\ref{E12}), we
prove that the solid torus is an extremal domain with respect to
the second best constant in inequality (\ref{E12}), in the sense
that this constant cannot be lowered for all bounded
anti-symmetric domain $\Omega$ in $\mathbb{R}^3$, and we prove
that the solid torus is totally optimal with respect to the
constants. Moreover, we compute the best constant in the
inequality (\ref{E9}). The calculation of this best constant
allows us to study the corresponding of boundary value problem for
the 1-Laplace differential operator in  the solid torus.
    \item[(ii)]Secondly, the dependence of the existence of a solution to the
Neumann problem involving the $1$-Laplacian of geometry is also
considered. In particular, it is proved that this problem has a
solution only in the cases when we have ``big'' tori. For
``small'' tori the problem does not have solution.
    \item[(iii)]Finally, we give some answers to the same problems in the case
where the domain is a smooth, compact Riemannian manifold with
boundary both in the general case and in the presence of
symmetries. More precisely, we are concerned with the following problems:\\
\begin{enumerate}
\item[(a)] In the first part, we
study the case of a smooth, compact Riemannian manifold with
boundary. Concerning inequality (\ref{E11}), we prove that the
best constants are the same as those in the Euclidean case.
Regarding inequality (\ref{E12}) we prove that the first best
constant is equal to $1$, remaining the same as that of the
Euclidean space. The second best constant is $|\partial
M|/|M|$, where $|\partial M|$ denotes the $(n-1)$-dimensional
 measure of $\partial M $ and $| M|$ the $n$-dimensional measure of $ M $.\\
\item[(b)] In the second part, we study the impact of the
symmetries which appear in the manifold in the general case.
Specifically, we compute the best constants in  both inequalities
(\ref{E11}) and (\ref{E12})  and we give general theorems
concerning to the best constants on manifolds in the presence of
symmetries for $p=1$. The values of both best constants in
inequality (\ref{E11}) are strongly influenced by the geometry.
The first best constant depends on the dimension and the volume of
the orbit with the minimum volume. The second best constant
depends on the volume of the manifold and the dimension of the
orbit with minimum volume. Finally, surprising results occur on
the best constant in the inequality (\ref{E12}).  For instance,
the first best constant remains the same for all smooth, compact
Riemannian manifolds and is neither depending  on the dimension
nor on the geometry. Contrary to the first best constant, the
second best constant depends strongly on the geometry.
\end{enumerate}
\end{enumerate}




\section{Best constants on the solid torus in the case $p=1$}

Consider the solid torus represented by:
$$
\overline{T}=\left\{(x,y,z)\in \mathbb{R}^3: \left(
\sqrt{x^2+y^2}-l\right)^2+z^2\leq r^2, \, \,l>r>0\right\},
$$
its boundary:
$$
\partial T=\left\{(x,y,z)\in \mathbb{R}^3: \left(
\sqrt{x^2+y^2}-l\right)^2+z^2= r^2, \, \,l>r>0\right\},
$$
and the group $G=O(2)\times I$ of $O(3)$. Note that the solid
torus $\overline{T}\in\mathbb{R}^3$ is invariant under the action
of the group $G$.

We  now recall some background material and results from Cotsiolis
and Labropoulos \cite{Cot-Lab2}. Let
$\mathcal{A}=\{(\Omega_i,\xi_i):i=1,2\}$ be an atlas on
$T=\overline{T}\backslash \partial T$ defined by
\begin{gather*}
\Omega_1=\{(x,y,z)\in T :(x,y,z)\notin H^+_{XZ}\},\\
\Omega_2=\{(x,y,z)\in T :(x,y,z)\notin H^-_{XZ}\},
\end{gather*}
where:
\begin{gather*}
H_{XZ}^+=\{(x,y,z)\in \mathbb{R}^3 : x>0\, ,\, y=0 \}\\
H^-_{XZ}=\{(x,y,z)\in \mathbb{R}^3 : x<0\, ,\, y=0 \}
\end{gather*}
and $\;\;\xi_i:\Omega_i\to I_i\times D\;\;$,  $i=1,2,\;\;$ with $
\;\;I_1=(0,2\pi), \quad I_2=(-\pi,\pi), $
$$
D=\{(t,s)\in \mathbb{R}^2 :t^2+s^2<1\}, \quad \partial
D=\{(t,s)\in \mathbb{R}^2 :t^2+s^2=1\},
$$
$\xi_i(x,y,z)=(\omega_i,t,s)$, $i=1,2$  with $
\cos\omega_i=\frac{\textstyle x}{\textstyle\sqrt{x^2+y^2}},\;
\sin\omega_i=\frac{\textstyle y}{\textstyle\sqrt{x^2+y^2}},$ where
$$
\omega_1=  \begin{cases}
\mathop{\rm arctan}\frac{\textstyle y}{\textstyle x},&x\neq 0 \\
 \,\;\;\frac{\textstyle \pi}{\textstyle 2} ,&x=0,y>0\;\;,   \\
\;3\frac{\textstyle \pi}{\textstyle 2},&x=0\,,\,y<0
 \end{cases}\quad
\omega_2=\begin{cases}
\mathop{\rm arctan}\frac{\textstyle y}{\textstyle x}, &x\neq 0 \\
     \;\;\;\frac{\textstyle \pi}{\textstyle 2},&x=0,\;y>0 \\
    -\frac{\textstyle \pi}{\textstyle 2},&x=0\,,\,y<0
     \end{cases}
$$
and
$$
 t=\frac{\sqrt{x^2+y^2}-l}{r}\,,\quad  s=\frac{z}{r}\,,\quad  0\leq t,\,s \leq 1.
$$
Then the Euclidean metric $g$ on $(\Omega,\xi)\in \mathcal{A}$ and
the induced metric on the boundary $\bar g$ can be expressed,
respectively as
\begin{equation}\label{E13}
(\sqrt{g}\circ\xi^{-1})(\omega,t,s)=r^2(l+rt)
\end{equation}
and
\begin{equation}\label{E14}
(\sqrt{\bar g}\circ\xi^{-1})(\omega,t,s)=r(l+rt).
\end{equation}

If for any $G$-invariant function $u$ defined on $T$ we define the
function:
\begin{equation}\label{E15}
\phi(t,s)=(u\circ \xi^{-1})(\omega,t,s),
\end{equation}
then we obtain the following equalities:
\begin{equation}\label{E16}
\int_T {|u|^p } dV = 2\pi r^2 \int_D {\left| {\phi \left( {t,s}
\right)} \right|} ^p \left( {l + rt} \right)dtds
 \end{equation}
 \begin{equation}\label{E17}
\;\;\;\int_T {\left| {\nabla u} \right|^p} dV = 2\pi r^{2-p}\int_D
{\left| {\nabla \phi \left( {t,s} \right)} \right|^p} \left( {l +
rt} \right)dtds,
 \end{equation}
\begin{equation}\label{E18}
\int_{\partial T} {\left| u \right|^p} dS = 2\pi r\int_{\partial
D} {\left| {\phi \left( {t,0} \right)} \right|^p\left( {l + rt}
\right)} dt,\
\end{equation}
where by $\phi$ we denote the extension of  $\phi$ on $\partial
D$.

Let $K(2,1)=1/\left(2\sqrt{\pi}\right)$ be the best constant of
the Sobolev inequality:
\begin{equation}\label{E19}
\left\|\varphi\right\|_{L^2(\mathbb{R}^2)}\leq
K(2,1)\\\left\|\nabla\varphi\right\|_{L^1(\mathbb{R}^2)}
\end{equation}
for the Euclidean space  $\mathbb{R}^2$  (see Aubin  \cite{Aub})
and let $\tilde{K}(2,1)=1$ be the best constant in the Sobolev
trace embedding:
\begin{equation}\label{E20}
\left\|\varphi\right\|_{L^1(\partial\mathbb{R}^2_+)} \leq
\tilde{K}(2,1)\\\left\|\nabla\varphi\right\|_{L^1(\mathbb{R}^2_+)}
\end{equation}
for the Euclidean half-space $\mathbb{R}^2_+$  (see Motron
\cite{Mot} and Park \cite{Par}). \smallbreak It is known (see
Cotsiolis and Labropoulos \cite{Cot-Lab1, Cot-Lab2})) that the
solid torus $\bar{T}$ is invariant under the action of the group
$G=O(2)\times I$, all the orbits are of dimension $1$, and the
`classical' Sobolev inequality in this case states as follows: for
any real $p$ such that $1 \leq p<2$, $p^*=2p/(2-p)$ and for all
$u\in H_{1,G}^1(T) $ there exist two positive constants $A$ and
$B$ such that:
\begin{equation}\label{E20.1}
\left(\int_T {|u|^{p^* }} dV \right)^{1/{p^*}}  \leqslant A \left(
\int_T \left|\nabla u\right|^pdV \right)^{1/p}+ B \left(\int_T
\left| u \right|^pdV\right)^{1/p}.
\end{equation}
We recall here that the displayed exponent $p^*=2p/(2-p)$ in the
above inequality is the highest possible supercritical exponent
(critical of supercritical) because of symmetry presented by the
solid torus.\\
Aubin \cite{Aub2} proved that in a compact Riemannian manifold
$M$, the best constant in the embedding of the Sobolev space
$H^p_1(M)$ in $L{p^*}(M)$, where $p^*=np/(n-p)$ for $p\in(1,n)$ is
equal to $K(n,p)$, the norm of the inclusion $H^p_1\hookrightarrow
L^{p^*}$ on  $\mathbb{R}^n$. Thus for any $\varepsilon> 0$ there
exists a constant $B_p(\varepsilon)$ such that every $u\in
H^p_1(M)$ satisfies:
\begin{equation}\label{E20.2}
\left( {\int_M {\left| u \right|^{p^* } d\upsilon _g } }
\right)^{1/p^* }  \le \left(K(n,p)+\varepsilon\right)\left(
{\int_M {\left| {\nabla u} \right|^p d\upsilon _g } }
\right)^{1/p}  + B_p(\varepsilon)\left( {\int_M {\left| u
\right|^p d\upsilon _g } } \right)^{1/p}
\end{equation}
A natural question arises: Is the best constant achieved? i.e.,
does there exist $B_p=B_p(0)$? We can expect a positive answer.
Aubin made a conjecture \cite{Aub2} concerning the following
inequality of interest, among other very significant inequalities
\begin{equation}\label{E20.2}
\left( {\int_M {\left| u \right|^{p^* } d\upsilon _g } }
\right)^{1/p^* }  \le K(n,p)\left( {\int_M {\left| {\nabla u}
\right|^p d\upsilon _g } } \right)^{1/p}  + B_p\left( {\int_M
{\left| u \right|^p d\upsilon _g } } \right)^{1/p}\,\!,\,\, 1\leq
p\leq 2,
\end{equation}
This conjecture was first proved for $p = 2$ by Hebey and Vaugon
\cite{Heb-Vau2}, then for any $p$ by Druet \cite{Dru}.  On
Riemannian manifolds in the presence of symmetries a positive
answer is given by Faget \cite{Fag2}. In the case of the solid
torus a positive answer is, also, given by Cotsiolis and
Labropoulos \cite{Cot-Lab1}. However, a new question arises: What
happens in the case where $p = 1$. If $p=1$ then $p^*=1^*=2$  and
thus by (\ref{E20.1}) we obtain the following Sobolev inequality:
\begin{equation}\label{E21}
\left(\int_T {u^2} dV \right)^{1/2}  \leqslant A  \int_T
\left|\nabla u\right|dV + B \int_T \left| u \right|dV\,.
\end{equation}
The question most clearly now is: Is it possible to have an
optimal inequality from (\ref{E21}) without $\varepsilon$? The
answer is positive in the sense:  We can not find someone
$\varepsilon> 0$, arbitrarily small such that the inequality
(\ref{E21}) is valid for $A = K(2,1) + \varepsilon$ for any $B$
and for any $u\in H_{1,G}^1(T) $. In particular, we can state the
following theorem:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{T4.1}
Let $T$ be the $3$-dimensional solid torus. Then the following properties are true.\\
$\mathrm{(i)}$ There exists $B\in\mathbb{R}$ such that for any
$u\in H_{1,G}^1(T) $,
\begin{equation}\label{E22}
\left( {\int_T {u^2 } dV} \right)^{1/2}  \leqslant \frac{{K\left(
{2,1} \right)}} {{\sqrt {\pi \left( {l - r} \right)} }}\int_T
{\left| {\nabla u} \right|} dV +B\int_T {\left| u \right|} dV\,.
\end{equation}
$\mathrm{(ii)}$ There exists $A\in\mathbb{R}$ such that for  any
$u\in H_{1,G}^1(T) $,
\begin{equation}\label{E23}
\left( {\int_T {u^2 } dV} \right)^{1/2}  \leqslant A\int_T {\left|
{\nabla u} \right|} dV +|T|^{-1/2}\int_T {\left| u \right|} dV\,.
\end{equation}
In addition, $\frac{\textstyle
K(2,1)}{{\textstyle\sqrt{\pi(l-r)}}}=\frac{\textstyle
1}{\textstyle 2\pi\sqrt{\,l-r}}$ and
$\;\textstyle|T|^{-1/2}=\frac{\textstyle 1}{\textstyle\pi
r\sqrt{2l}}$ are the best constants for these inequalities.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{{\bf Proof of Theorem \ref{T4.1}.} We carry through the
proof of the theorem in two steps.\\
\noindent{\bf{Step 1.}} This first step is devoted to study the
first best constant. Cf. Cotsiolis and Labropoulos
\cite{Cot-Lab1},  if $p$ is a positive real number such that
$1<p<2$ then there exists $B=B(p)>0$ such that for any $u\in
H_{1,G}^ p(T) $,
\begin{equation}\label{E24}
\left( {\int_T {\left| u \right|^{2p/\left( {2 - p} \right)} } dV}
\right)^{\left( {2 - p} \right)/2p}  \le \left(
{\frac{{K(2,p)}}{{\sqrt {\pi (l - r)} }}} \right)^p \int_T {\left|
{\nabla u} \right|} ^p dV + B\int_T {\left| u \right|} ^p dV\,.
\end{equation}
The constant $\frac{\textstyle K(2,p)}{\textstyle\sqrt{\pi(l-r)}}$
is the best constant for which inequality (\ref{E24}) remains true
for any $u\in H_{1,G}^ p(T) $.

Our purpose here  is to prove that inequality (\ref{E24}) also holds
 in the case where $p=1$, namely that for any $u\in H_{1,G}^
1(T) $, there exists a positive real number $B>0$, such that:
\begin{equation}\label{E25}
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le
\frac{{K(2,1)}}{{\sqrt {\pi (l - r)} }}\int_T {\left| {\nabla u}
\right|} dV + B\int_T {\left| u \right|} dV\,.
\end{equation}
Assume  that inequality (\ref{E25}) is false. Then for any
$\beta>0$ there exists $u\in H_{1,G}^1(T)$ such that:
\begin{equation}\label{E26}
\left( {\int_T {u^2 } dV} \right)^{1/2} > \frac{{K(2,1)}}{{\sqrt
{\pi (l - r)} }}\int_T {\left| {\nabla u} \right|} dV +
\beta\int_T {\left| u \right|} dV\,.
\end{equation}
Thus by (\ref{E26}), we deduce that  for any $\beta>0$ there
exists $u\in H_{1,G}^1(T)$ such that:
\begin{equation}\label{E27}
 I(u)=\frac{\left(\int_T {\left| {\nabla u} \right|}
dV + \beta\int_T {\left| u \right|} dV\right)}{\left( {\int_T {u^2
} dV} \right)^{1/2}} <\left( \frac{{K(2,1)}}{{\sqrt {\pi (l - r)}
}}\right)^{-1}.
\end{equation}
By (\ref{E27}) we obtain:
\[
0<I(u)<\left( \frac{{K(2,1)}}{{\sqrt {\pi (l - r)} }}\right)^{-1}
\]
and so
\begin{equation}\label{E28}
0\leq\mathop {\inf}\limits_{u \in H_{1,G}^1 \left( T \right), u
\not\equiv 0}I(u)<\left( \frac{{K(2,1)}}{{\sqrt {\pi (l - r)}
}}\right)^{-1}.
\end{equation}
The inequality (\ref{E28}) allows us, using variation arguments,
to prove that for any $\beta>0$ the infimum $\lambda_\beta=
\mathop {\inf}\limits_{u \in H_{1,G}^1 \left( T \right), u
\not\equiv 0}I(u) $ is achieved by a function $u_\beta\geq0$.
Namely, it holds $I(u_\beta)=\lambda_\beta$.\\
Due to (\ref{E28}), $\lambda_\beta$ is bounded. In addition, we
conclude that for sufficiently large $\beta$, $u_{\beta}$ is not a
constant. Otherwise, we would have:
$$
\lim_{\beta \to  + \infty } \frac{{\,\int_T
{\left| {\nabla u_\beta  } \right|} dV + \beta \int_T {\left|
{u_\beta  } \right|dV} }} {{\left( {\int_T {\left( {u_\beta  }
\right)^2 dV} } \right)^{1/2} }} = \lim_{\beta
\to  + \infty } \frac{{\,\beta \int_T {\left| {u_\beta  }
\right|dV} }} {{\left( {\int_T {\left( {u_\beta  } \right)^2 dV} }
\right)^{1/2} }} =  + \infty,
$$
which is false because of (\ref{E28}). This implies that
for sufficiently large $\beta$, we have $ \int_T {\left| {\nabla
u_\beta } \right|} dV  \neq 0 $ and then $u_\beta\not\equiv 0$.

\smallbreak For any $p\in[1,2)$, we define:
\[
I_p^\beta \left( u \right)\equiv \frac{{\int_T {\left| {\nabla u}
\right|} ^p dV + \beta\int_T {\left| u \right|} ^p dV}} {{\left(
{\int_T {\left| u \right|^{2p/\left( {2 - p} \right)} } dV}
\right)^{\left( {2 - p} \right)/2p} }}\,.
\]
Since the function $f(p)=I_p^\beta(u)$ with $p\in[1,2)$ is
continuous on $p$ for any $u \in H_{1,G}^p \left( T \right)$, we
conclude that $ \mathop {\lim }\limits_{p \to 1} I_p ^\beta\left(
u \right) = I \left( u \right) $. On the other hand, cf. Aubin
\cite{Aub} we have
 $ \mathop {\lim }\limits_{p \to 1} K(n,p) = K(n,1)$. Thus by
(\ref{E24}), we obtain that for any $p\in[1,2)$
\begin{equation*}
\mathop {\inf}\limits_{u \in H_{1,G}^p \left( T \right), u
\not\equiv 0}\lim\limits_{p \to 1} I_p^\beta \left( u \right) \geq
\lim\limits_{p \to 1} {\left( {\frac{{K\left( {2,p} \right)}}
{{\sqrt {\pi \left( {l - r} \right)} }}} \right)^{-1} }
\end{equation*}
or
\begin{equation}\label{E29}
\mathop {\inf}\limits_{u \in H_{1,G}^1 \left( T \right), u
\not\equiv 0} I \left( u \right) \geq {\left( {\frac{{K\left(
{2,1} \right)}} {{\sqrt {\pi \left( {l - r} \right)} }}}
\right)^{-1} }.
\end{equation}
As it is impossible to apply (\ref{E28}) and (\ref{E29})
simultaneously, we have reached a contradiction.\\

It remains to prove that the constant $\frac{K(2,1)}{\sqrt{\pi(l -
r)}}$ is the best possible for the inequality (\ref{E25}). Since,
we proved above that for functions  the first best constant of the
Sobolev inequality (\ref{E20.1}) is greater or equal to
$\frac{K(2,1)}{\sqrt{\pi(l - r)}}$, to complete the proof we must
exclude the first case.

 We define the \emph{smallish}  torus $T_\delta   = \{
{P \in \mathbb{R}^3: d(P,\mathcal{O}) < \delta } \}$, where
$\mathcal{O}$ is the orbit of minimum length $2\pi(l-r)$ and
$d({\cdot,\mathcal{O}})$ is the distance to the orbit. Since, for
functions belonging to ${H}_{1,G}^1(T\cap T_\delta)$ the first
best constant of the Sobolev inequality (\ref{E20.1}) in $T$ has
the same value and in $T\cap T_\delta$, (see Theorem 3.1 in
\cite{Cot-Lab1}), in the sequel of this proof firstly we will stay
in $H_{1,G}^1(T\cap T_\delta)$.\\
Assume now by contradiction that for some arbitrarily small but
fix $\varepsilon$ and for all $\beta>0$ the inequality:
\begin{equation*}
\left( {\int_{T\cap T_\delta} {u^2 } dV} \right)^{1/2}
\leqslant\left( \frac{K(2,1)} {\sqrt {\pi(l -
r)}}+\varepsilon\right)\int_{T\cap T_\delta} \left|\nabla u\right|
dV +\beta\int_{T\cap T_\delta} \left| u \right| dV,
\end{equation*}
holds for all $u\in {H}_{1,G}^1(T\cap T_\delta)$, or,
equivalently:
\begin{equation}\label{E29.1 }
I^\beta(u)=\frac{\left(\int_{T\cap T_\delta} {\left| {\nabla u}
\right|} dV + \beta\int_{T\cap T_\delta} {\left| u \right|}
dV\right)}{\left( {\int_{T\cap T_\delta} {u^2 } dV} \right)^{1/2}}
<\left( \frac{{K(2,1)}}{{\sqrt {\pi (l - r)}
}}+\varepsilon\right)^{-1}
\end{equation}
If we define:
\begin{equation}\label{E29.01}
 \mathcal{I}^\beta=\mathop {\inf}\limits_{u \in
H_{1,G}^1 \left( T\cap T_\delta \right), u \not\equiv
0}I^\beta(u),
\end{equation}
it follows that for all $\beta>0$, there exists
$\theta(\varepsilon)>0$ such that:
\begin{equation}\label{E29.2}
\mathcal{I}^\beta<\left( \frac{{K(2,1)}}{{\sqrt {\pi (l - r)}
}}+\varepsilon\right)^{-1}=\frac{\sqrt { \pi (l -
r)}}{K(2,1)}-\theta(\varepsilon)
\end{equation}
Let now a minimizing sequence $(u_j)\in
 {H}_{1,G}^1(T\cap T_\delta)$ of $I^\beta(u)$. For
all $ u_j$ we define on $D$, the unit disk of $\mathbb{R}^2$, the
functions:
\begin{equation}\label{E29.3}
\phi _j (t,s) \equiv (u_j \circ \xi ^{ - 1} )(\omega ,t,s)
\end{equation}
and for any $\phi _j \in H^p_1(D)$ and any $\lambda\geq0$ we set
\begin{equation}\label{E29.4}
\phi _{j_\lambda  } (t,s) \equiv \phi _j (\lambda t,\lambda s)
\end{equation}
Consider now  the $\lambda-$parametric sequence $ \phi_{j_\lambda
}  $ defined by (\ref{E29.4}) and the $\lambda-$parametric
sequence $u_{j_\lambda }  $ defined by (\ref{E29.3}) to be $
u_{j_\lambda  }  = \phi _{j_\lambda  } \circ \xi $. By (\ref{E16})
and (\ref{E17}) we obtain respectively:
\begin{eqnarray}\label{E29.5}
\int_{T\cap T_\delta} {|{u_{j_\lambda  } }|^2 } dV = 2\pi \left(
{\frac{\delta } {\lambda }} \right)^2 \int\limits_D \left|\phi _{j
} \right|^{2 } \left( {l - r + \frac{\delta } {\lambda }t}
\right)dtds \\
                \label{E29.6}
\;\;\;\int_{T\cap T_\delta} {\left| {\nabla {u_{j_\lambda  } }}
\right|} dV = 2\pi {\frac{\delta } {\lambda }}\int_D {\left|
\nabla \phi_j  \right|} \left( {l - r + \frac{\delta } {\lambda
}t} \right)dtds,
\end{eqnarray}
Set:
\begin{eqnarray}\label{E29.7}
\mathrm{I}^\beta_{j_\lambda} \equiv =\frac{\left(\int_{T\cap
T_\delta} {\left| {\nabla u_{j_\lambda  }} \right|} dV +
\beta\int_{T\cap T_\delta} {\left| u_{j_\lambda  } \right|}
dV\right)}{\left( {\int_{T\cap T_\delta} {u^2_{j_\lambda  } } dV}
\right)^{1/2}}
\end{eqnarray}
By (\ref{E29.7}), because of (\ref{E29.5}) and (\ref{E29.6}) a
direct computation we obtain successively:
\begin{eqnarray}\label{E29.8}
\mathrm{I}^\beta_{j_\lambda}  =\frac{{2\pi \int_D {\left| {\nabla
\phi_{j_\lambda  } } \right|\left( {l - r + \delta \frac{t}
{\lambda }} \right)dtds} }} {{\left( {2\pi \int_D {\left|
\phi_{j_\lambda  } \right|^2 \left( {l - r + \delta \frac{t}
{\lambda }} \right)dtds} } \right)^{1/2} }} + \frac{\delta }
{\lambda }\frac{{\beta 2\pi \int_D {\left| \phi_{j_\lambda  }
\right|\left( {l - r + \delta \frac{t} {\lambda }} \right)dtds} }}
{{\left( {2\pi \int_D {\left| \phi_{j_\lambda  } \right|^2 \left(
{l - r + \delta \frac{t} {\lambda }} \right)dtds} } \right)^{1/2}
}}
\end{eqnarray}
Letting $\lambda\to\infty$ the inequality (\ref{E29.8}) yields:
\begin{eqnarray}\label{E29.8}
I_j^\beta   &\equiv& \mathop {\lim }\limits_{\lambda  \to \infty }
I_{j\lambda }^\beta =  \sqrt {2\pi \left( {l - r} \right)}
\frac{{\int_D {\left| {\nabla \phi_j } \right|dtds} }} {{\left(
{\int_D {\left| \phi_j  \right|^2 dtds} } \right)^{1/2} }}
\end{eqnarray}
Now by (\ref{E29.8}), because of  (\ref{E29.01}), letting
$j\rightarrow\infty$ we obtain the equality:
\begin{eqnarray}\label{E29.9}
\mathcal{I}^\beta   =\sqrt {2\pi \left( {l - r} \right)}
\frac{{\int_D {\left| {\nabla \phi } \right|dtds} }} {{\left(
{\int_D {\left| \phi  \right|^2 dtds} } \right)^{1/2} }},
\end{eqnarray}
where the function $\phi$ is defined in the same way as the
$\phi_j$'s by (\ref{E29.3}) to be $\phi(t,s)\equiv(u\circ \xi ^{ -
1})(\omega,t,s)$ and $u$ is the limit of $(u_j)_{j=1,2,...} $,
defined above.\\
Finally, it is known that the best constant of the Sobolev
inequality for  functions defined in $H^1_1(D)$   is equal to
$\sqrt{2}\,K(2,1)$, (see Aubin \cite[Lemma 2.31]{Aub}; see
Cherrier \cite{Che3} for a complete proof). Thus, by (\ref{E29.2})
and (\ref{E29.9}) we obtain:
$$
\sqrt {2\pi ( l - r)}  \frac{1} {\sqrt{2}\, K (2,1) } < \,
\frac{\sqrt{\pi(l - r)}}{K(2,1)}- \theta (\varepsilon),
$$
which is a contradiction.

\smallbreak \noindent{\bf{Step 2.}} In this second step, we
compute the second bast constant in inequality (\ref{E22}). By
taking $u=1$ in (\ref{E21}), we obtain that $B\geq |T|^{-1/2}$. In
particular,
\begin{equation}\label{E30}
\mathcal{B}_1(T)\geq |T|^{-1/2}.
\end{equation}
Let $u\in H_{1,G}^1(T)$ and $\bar u =\frac{1}{|T|}\int_T {u\,dV}
$. Since  $\bar u$ is a constant function and because  $\left( {u
- \bar u} \right) \circ \tau  = u \circ \tau  - \bar u \circ \tau
= u - \bar u $  for any $\tau\in G$, we conclude that $ \left( {u
- \bar u}\right) \in H_{1,G}^1 (T)$.
 Setting $\left( {u - \bar u} \right) \circ \xi ^{ - 1} =
\phi ^ * $ in (\ref{E16}) we obtain:
\begin{eqnarray}\label{E31}
\nonumber\left( {\int_T {\left( {u - \bar u} \right)^2 } dV}
\right)^{1/2} &=& \left( {2\pi r^2 \int_D {\left( {\phi ^ * }
\right)^2 } (l +rt)dtds} \right)^{1/2}\\
& \le & \left( {2\pi r^2 \left( {l + r} \right)} \right)^{1/2}
\left( {\int_D {\left( {\phi ^ *  } \right)^2 } dtds}
\right)^{1/2}.
\end{eqnarray}
Moreover, by the Sobolev-Poincar\'e inequality there exists some
positive real number $C$ such that for any $\phi ^
*   \in {\rm H}_1^1 (D)$, the following inequality holds:
\begin{eqnarray}\label{E32}
\left( {\int_D {\left( {\phi ^ *   - \bar \phi ^ *  } \right)^2 }
dtds} \right)^{1/2}  \le C\int_D {\left| {\nabla \phi ^ * }
\right|} dtds,
\end{eqnarray}
where $ \bar \phi ^ *   = \frac{1}{|T|}\int_D {\phi ^ *
dtds} $.

We may assume that  $\bar \phi ^ *   = 0$. Actually, if  $ \bar
\phi ^ *   = \eta  \ne 0$   instead of $\phi ^ *$  we could take
the function $\phi ^ *   - \eta$  and then:
\begin{eqnarray*}
 \frac{1}{{|T|}}\int _D {(\phi ^ *  }  - \eta )dtds
 &=&
 \frac{1}{|T|}\int _D {\phi ^ *  } dtds - \frac{1}{|T|}\int_D \eta
 dtds   = \eta  - \frac{\eta }{|T |}| T | = 0.
\end{eqnarray*}
Thus if $\bar\phi^*=0$, inequality (\ref{E32}) yields:
\begin{eqnarray}\label{E33}
\left( {\int_D {\left( {\phi ^ *  } \right)^2 } dtds}
\right)^{1/2}  \le C\int_D {\left| {\nabla \phi ^ *  } \right|}
dtds.
\end{eqnarray}
Relation (\ref{E17}) yields:
\begin{eqnarray}\label{E34}
\int_D {\left| {\nabla \phi ^ *  } \right|} dtds \le
\frac{1}{{2\pi r\left( {l - r} \right)}}\int_T {\left| {\nabla u}
\right|} dV.
\end{eqnarray}
Combining inequalities (\ref{E33}) and (\ref{E34}) we obtain:
\begin{eqnarray}\label{E35}
\left( {\int_D {\left( {\phi ^ *  } \right)^2 } dtds}
\right)^{1/2}  \le C_1 \int_T {\left| {\nabla u} \right|} dtds,
\end{eqnarray}
where $ C_1  =C/\left(2\pi r(l - r)\right)$.\\
Thus by (\ref{E35}), because of (\ref{E31}) arises:
\begin{eqnarray}\label{E36}
\left( {\int\limits_T {\left( {u - \bar u} \right)} ^2 dV}
\right)^{1/2}  \le C_2 \int\limits_T {\left| {\nabla u} \right|}
dV,
\end{eqnarray}
where $ C_2  = \sqrt{ 2\pi r^2 \left( {l + r} \right)} C_1$.

\smallbreak\noindent By (\ref{E36}), because of  $  \left\| u
\right\|_p  \le \left\| {\,u - \bar u} \right\|_p  +
\left\|{\,\bar u} \right\|_p$,  we obtain:
\[
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le \left( {\int_T
{\left( {u - \bar u} \right)} ^2 dV} \right)^{1/2} + \left(
{\int_T {\bar u^2 dV} } \right)^{1/2},
\]
or equivalently:
\[
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le \left( {\int_T
{\left( {u - \bar u} \right)} ^2 dV} \right)^{1/2} + \left| {\bar
u} \right|\left( {\int_T {dV} } \right)^{1/2}.
\]
>From the last inequality, since  $\bar u =\frac{1}{|T|}\int_T
{u\,dV} $, we deduce that:
\begin{eqnarray}\label{E37}
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le \left( {\int_T
{\left( {u - \bar u} \right)} ^2 dV} \right)^{1/2} + |T|^{ - 1/2}
\left| {\int_T {udV} } \right|\,.
\end{eqnarray}
By (\ref{E37}) and using (\ref{E36}) we find
\[
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le C\int_T {\left|
{\nabla u} \right|} dV + | T|^{ - 1/2} \left| {\int_T {udV} }
\right|,
\]
from which arises:
\begin{eqnarray}\label{E38}
\left( {\int_T {u^2 } dV} \right)^{1/2}  \le C\int_T {\left|
{\nabla u} \right|} dV +| T |^{ - 1/2} {\int_T { |u|dV} }\,.
\end{eqnarray}
Combining the inequality (\ref{E38}) with (\ref{E30}) we conclude
that $\mathcal{B}_1(T)= |T|^{-1/2}$. \mbox{} $\Box$

\smallskip
The following property is the natural extension
of Theorem \ref{T4.1} in the general case  where $1<p<2$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{C4.1}
Let $T$ be the solid torus  and let $p$ be a real number such that
$1<p<2$.\\
$\mathrm{(i)}$ There exists $B\in\mathbb{R}$ such that for all
$u\in {H}_{1,G}^p(T)$,
\begin{equation}\label{E39}
\left( {\int_T {\left| u \right|^{2p/\left( {2 - p} \right)} } dV}
\right)^{\left( {2 - p} \right)/2p}  \le
\frac{K(2,p)}{{\sqrt{\pi(l-r)}}}\left( {\int_T {\left| {\nabla u}
\right|} ^p dV} \right)^{1/p}  + B \left( {\int_T {\left| u
\right|} ^p dV} \right)^{1/p}.
\end{equation}
$\mathrm{(ii)}$ There exists $a\in\mathbb{R}$ such that for all
$u\in {H}_{1,G}^p(T)$,
\begin{equation}\label{E40}
\left( {\int_T {\left| u \right|^{2p/\left( {2 - p} \right)} } dV}
\right)^{\left( {2 - p} \right)/2p}  \le A\left( {\int_T {\left|
{\nabla u} \right|} ^p dV} \right)^{1/p}  + | T |^{ - 1/2} \left(
{\int_T {\left| u \right|} ^p dV} \right)^{1/p}.
\end{equation}
In addition, $\frac{\textstyle
K(2,p)}{{\textstyle\sqrt{\pi(l-r)}}}$ and $\;|T|^{-1/2}$ are the
best constants for these inequalities.
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent{{\bf Proof of Corollary \ref{C4.1}.} For the second best
constant in inequality (\ref{E40}) the calculation follows the
same ideas as in the proof of Theorem \ref{T4.1} and is completed
with the help of the inequality:
\[
\left| {\int_T {udV} } \right| \le \left| {\int_T {\left| u
\right|dV} } \right| \le |T|^{1 - (1/p)} \left( {\int_T {\left| u
\right|} ^p dV} \right)^{1/p},
\]
which arises  directly from the H\"older inequality.

Concerning the calculation of the first best constant in
inequality (\ref{E39}) see Theorem 3.1 in Cotsiolos and Labropoulos \cite{Cot-Lab1}.
\mbox{}$\Box$
\smallbreak The last result concerning the case of
the torus is as follows.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{T4.2}
Let $T$ be the solid torus. For all $u\in H_{1,G}^ 1(T)$, the
following inequality holds:
\begin{equation}\label{E41}
{\int_{\partial T }{\left| u \right| dS } }   \le \int_T {\left|
{\nabla u} \right| dV}+ \frac{2}{r}\int_T {\left| {u} \right| dV}.
\end{equation}
In particular, $1=\tilde K(2,1)$ and ${2}/{r}$ are the best
constants for this inequality and the function $u_0=(1/|\partial
T|)\mathcal{\chi}_{_{T}}$ is the only extremal function for this
inequality.\\
Furthermore, for all $\phi\in H_{1}^ 1(D_r)$, the following
inequality holds:
\begin{equation}\label{E42}
{\int_{\partial {D_r} }{\left| \phi \right| ds } }  \le \int_{D_r}
{\left| {\nabla \phi} \right| d\upsilon}+\frac{2}{r} \int_{D_r}
{\left| \phi \right| d\upsilon},
\end{equation}
where $ D_r = \left\{ {(t,s) \in { \mathbb{R}}^2 : t^2  + s^2 <
r^2} \right\} $  is the disc of radius $r$ which  rotating around
the $z$-axis and remaining coplanar with it  produces the solid
torus $T$.
\end{theorem}

The proof of this theorem makes use of the following auxiliary property.
\begin{lemma}\label{L4.1}
Let $T$ be the solid torus. Suppose  that there exist  real
numbers $\tilde A$ and $\tilde B$ such that for all $u\in
H^1_{1,G}(T)$ the following inequality holds:
\begin{equation}\label{E43}
\int_{\partial T} {|u|} dS \le \tilde A\int_T {|\nabla u|} dV
+\tilde B\int_T | u| dV.
\end{equation}
Then $ \tilde A\geq 1=\tilde{K}(2,1) $.
\end{lemma}
The proof of Lemma \ref{L4.1} is provided in Appendix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallbreak\noindent{{\bf Proof of Theorem \ref{T4.2}.}  In order
to prove inequality (\ref{E41}), it is equivalent to show firstly
that for all $u\in H_{1,G}^ 1(T)$, there exists a constant $\tilde
A\in\mathbb{R}$ such that
\begin{equation}\label{E44}
{\int_{\partial T }{\left| u \right| dS } }   \le\tilde  A\int_T
{\left| {\nabla u} \right| dV}+ \frac{2}{r}\int_T {\left| { u}
\right| dV},
\end{equation}
and secondly, that for all $u\in H_{1,G}^ 1(T)$, there exists a
constant $\tilde B\in\mathbb{R}$ such that:
\begin{equation}\label{E45}
{\int_{\partial T }{\left| u \right| dS } }   \le \int_T {\left|
{\nabla u} \right| dV}+\tilde  B\int_T {\left| { u} \right| dV}.
\end{equation}

\smallskip\emph{Proof of inequality $\mathrm{(\ref{E44})}$}.
Cf. Proposition 3.10  in Motron \cite{Mot}, the best second
constant for the inequality:
\begin{equation}\label{E46} \int_{\partial T}
{|u|} dS \leq \tilde A\int_T {|\nabla u|} dV+\tilde B\int_T | u|
dV\end{equation} is
\begin{equation}\label{E47}
\frac{{\left| {\partial T} \right|}} {{\left| T \right|}} =
\frac{{4\pi ^2 rl}} {{2\pi ^2 r^2 l}} = \frac{2} {r}\,.
\end{equation}
This fact means that for all $u\in H^1_{1,G}(T)$, there exists
$\tilde A>0$ such that the inequality (\ref{E44})  holds.

\smallskip\emph{Proof of inequality $\mathrm{(\ref{E45})}$}.
We first establish an auxiliary inequality, that is, for any
$\varepsilon>0$ and all $u\in H_{1,G}^ 1(T)$, there exists a
constant $\tilde B\in\mathbb{R}$ such that:
\begin{eqnarray}\label{E48}
\int_{\partial T} {\left| u \right|}  dS  &\leqslant&
(1+\varepsilon)\int_T {\left| {\nabla u} \right| dV +C'\int_T }
|u| dV\,.
\end{eqnarray}
The proof is rather classic  but we  give a brief
outline of the arguments. Let $P_j\in T$, $O_{P_j}$ be its orbit under the
action of subgroup $G=O(2)\times I$ and $l_j=\sqrt{x_j^2+y_j^2}$
be the distance of $O_{P_j}$ from the $z$-axis. For any
$\varepsilon_0>0$, consider $\delta_j=\varepsilon_0 l_j<1$, and
the set $ T_j  = \left\{ {Q \in \mathbb{R}^3 :d(Q,O_{P_j } ) <
\delta _j } \right\}$.  Choose a finite covering of $\overline{
T}$ by sets of the type $T_j$ such that:\\ $(a)$ if $P_j \in T $,
then the entire $T_j$ lies in $T$ and the entire coordinate of
$\xi_j(\Omega_j)$ on $ D$
lies in $D$;\\
$(b)$ if  $P_j\in \partial T$, then the coordinate of
$\xi_j(\Omega_j\cap \partial T)$ on $\bar D$ lies in $\partial D$.

Now, for  $T$ we build a partition of unity that covers $T$ in the
following way. For each $j$ we consider $h_j \in C_{0,G}^\infty
\left( D \right)$, $h_j\geq 0$. Then $h_j$ can be seen as a
function defined on $I\times D$ and depending only on $D$
variables.

Let $ \eta _j = \frac{\textstyle{h_j \circ \xi _j }}
{{\textstyle\sum\nolimits_{j = 1}^N {\left( {h_j  \circ \xi _j }
\right)} }}$. The $\eta_j$'s form a new partition of unity
relative to $T_j$, is $G$-invariant, and   $\eta _j \circ \xi _j^{
- 1}$ depends only on $D$ variables. Moreover, since the covering
of $T$ is finite, there exists a positive constant $H$ depending
on the chosen covering, namely such that $\left| {\nabla \eta _j }
\right|\leqslant H$ for all $ j\in\mathbb{N} $. Thus for any $u\in
H^1_{1,G}(T)$,  if we set $\phi_j=(\eta_ju)\circ \xi^{-1}_j$,
because of (\ref {E31})  we obtain:
\begin{eqnarray}\label{E49}
\int\limits_{\partial T} {\left| u \right|dS}   \le \sum\limits_{j
= 1}^N {\int\limits_{\partial T} {\left| {\eta _j u} \right|dS} }
\le \sum\limits_{j = 1}^N\left( {1 + \varepsilon _0 } \right)2\pi
\,\delta _j \,l_j \int_{\partial D} {\left| {\phi _j (t,0)}
\right|}dt.
\end{eqnarray}
By relation (\ref {E49}) and the Sobolev embedding theorem in
$\partial \mathbb{R}_ + ^2 $ we deduce that:
\begin{eqnarray*}
\int_{\partial T} {\left| { u} \right|} dS & \le & \frac{{1 +
\varepsilon _0 }}{{1 - \varepsilon _0 }}\sum\limits_{j =
1}^N\int_{T } {\left(\eta _j \left| {\nabla u} \right| + H\left| u
\right| \right)} dV\\
&\leqslant& \frac{1 + \varepsilon _0}{1 - \varepsilon _0}\int_T
{\left( {\left| {\nabla u} \right| +HN\left| u \right|} \right)}
dV,
\end{eqnarray*}
and if we set $\varepsilon=O(\varepsilon_0)$ and
$C'=HN(1+\varepsilon _0)/(1 - \varepsilon _0)$ we obtain the
inequality (\ref{E48}).\\
Combining  inequality (\ref{E48}) with Lemma \ref{L4.1} and taking
into account that we can choose the $\varepsilon_0$  arbitrarily
small, therefore the $\varepsilon$ deduce that $1=\mathcal{\tilde
A}_1(T)$.\\
Furthermore, it is easy to verify that the constant function
$u_0=(1/|\partial T|)\mathcal{\chi}_{_{T}}$ is an extremal
function for the inequality (\ref{E45}), and the first part of the
theorem is proved.

For the first best constat in the inequality (\ref{E42}) we need
to repeat the same steps as in the case of the inequality
(\ref{E41}).  Regarding the second best constant for this by the
same argument as in the inequality (\ref{E41}) we find that is
equal to:
\begin{eqnarray}\label{E50}
\frac{{\textstyle \left| {\partial D_r} \right|}}
{\textstyle{\left| D_r \right|}} = \frac{\textstyle{2\pi r}}
{\textstyle{\pi  r^2 }} = \frac{\textstyle 2} {\textstyle r}
\end{eqnarray}
Finally with a simple substitution we can prove that the constant
function $u_0=(1/|\partial T|)\mathcal{\chi}_{_{T}}$ is an
extremal function for this inequality.

The proof is now complete. \mbox{}$\Box$\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{R4.1}
\emph{Observing  equalities  (\ref{E47}) and (\ref{E50}) we see
that the second best constants in inequalities (\ref{E41}) and
(\ref{E42}) are the same. This geometrical aspect means that the
solid torus behaves exactly like the disc of which is produced by
rotation in the $yz$-plane about the $z$-axis far from $z$-axis.
This result confirms in some sense the fact that each
axi-symmetric object $E$ of the three-dimensional Euclidean space
is identified by its description in $\mathbb{R}^2$ and so, for
simplicity, we may view $E$ as a subset of $\mathbb{R}^2$.}
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\label{R4.2}
\emph{The solid torus is an extremal domain with respect to the
second best constant $2/r$ in the inequality of the Theorem
\ref{T4.2}, in the sense that this constant cannot be lowered for
all bounded anti-symmetric  domains $\Omega$ in $\mathbb{R}^3$,
since
\begin{equation}\label{E56}
\frac{{2}} {r} = \frac{{\left| {\partial T} \right|}} {{\left| T
\right|}} =\frac{{\left| {\partial D_r} \right|}} {{\left| D_r
\right|}}
\end{equation}
and because of the isoperimetric equality (see Federer
\cite{Fed})
\begin{equation}\label{E57}
\frac{{\left| {\partial D_r} \right|}} {{\left| D_r \right|}}=\inf
\limits_{\Omega\in\mathbb{R}^3}\left\{\frac{{\left| {\partial
\Omega} \right|}} {{\left| \Omega \right|}}\right\}.
\end{equation}}
\end{remark}
\begin{remark}\label{R4.3}
\emph{Since the first best constant of the second inequality in Theorem
\ref{T4.2} is  equal to 1 for all manifolds, we conclude that the
solid torus is totally optimal with respect to the
constants.}
\end{remark}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{A Neumann problem involving the $1$-Laplace operator in the solid torus}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Mathematical background. Auxiliary results}
 At this point we need some background material
concerning functions in the space $BV(\Omega)$ (see
\cite{And-Maz-Ros},  \cite{ Bel-Cas-Nov}), where $\Omega $ is a
bounded set in $\mathbb{R}^n$ with Lipschitz continuous boundary
$\partial\Omega$.

A function $u\in L^1(\Omega)$ whose gradient  in the sense of
distributions is a (vector valued) Radon measure with finite total
variation in $\Omega$ is called a \emph{function of bounded
variation}. Thus $u\in BV(\Omega)$ if and only if there are Radon
measures $\mu_1, \mu_2,...,\mu_n$ defined in $\Omega$ with finite
total mass in $\Omega$ and:
$$
\int_\Omega u\, D_i \varphi \,dx=-\int_\Omega\varphi\, d\mu_i\;\;
\forall\; \varphi \in C_0^\infty(\Omega),\;i=1,2,\ldots ,n.
$$

The gradient  of $u$ is the vector measure $\mu=(\mu_1,
\mu_2,...,\mu_n)$  denoted by $Du$ with finite total variation:
\[
\sup\left\{\int_\Omega u\, \mathrm{div}\psi dx:\;\psi\,\in
C_0^\infty(\Omega, \mathbb{R}^n),\,|\psi(x)|\leq 1 \;\;\mathrm{for
}\,x\in \Omega\right\}
\]
and will be denoted by $|Du|(\Omega)$ or by $\int_\Omega |Du|$.

The function space $BV(\Omega)$ is a Banach space when endowed
with the norm:
\[
 \|\nabla u\|_{BV}=\int_\Omega |u|+\int_\Omega |Du|\,.
\]

A measurable set $E \in \mathbb{R}^n$ is said to be of
\emph{finite perimeter} in $\Omega$ if $\mathcal{ \chi}_{_E}\in
BV(\Omega)$, and in this case the perimeter of $E$ in $\Omega$ is
defined as $P(E,\Omega)=|D\mathcal{\chi}_{_E}|$. We shall use the
notion  $P(E)=P(E,\mathbb{R}^n)$. If $E$ has a smooth boundary
then $P(E)$ and the classical measure $|\partial E|$ of the
boundary correspond. It is well known (see, e.g., Ambrosio, Fusco and
Pallara \cite{Amb-Fus-Pal}, Evans and Gariepy \cite{Eva-Gar} or
Ziemer \cite{Zie}) that for a given function $u \in BV(\Omega)$
there exists a sequence $u_j\in W^{1,1}(\Omega)$ such that $u_j$
strict converges to $u$, that is:
\[u_j\to u\;\; \mathrm{in}
\;\;L^1(\Omega)\;\;\;\mathrm{and}\;\;\;\int_\Omega|\nabla
u_j|dx\to \int_\Omega|D u|\,.
\]
Moreover, there exists a trace operator $\tau$ which sends
$BV(\Omega)$ into $L^1(\Omega)$, namely for all $u \in BV(\Omega)$
\[\|{{\tau}}(u)\|_{L^1(\partial\Omega)}\leq C
\|u\|_{BV(\Omega)}\,,
\]
for some constant $C$ depending only on $\Omega$. The trace
operator $\tau$ is continuous between $BV(\Omega)$ endowed with
the topology induced by the strict convergence and
$L^1(\partial\Omega)$. In the sequel we write $\tau(u) = u$.

For further information concerning functions of bounded variation
we refer to Evans and Gariepy \cite{Eva-Gar} and Ziemer \cite{Zie}
and for a short generalization on Riemannian manifolds a good
reference is Druet \cite{Dru1}.

We now recall several results  from Andreu, Mazon and Rossi
\cite{And-Maz-Ros} and Anzellotti \cite{Anz}.\\ Let:
\[X(\Omega)=\{z\in L^\infty(\Omega,\mathbb{R}^n):
\mathrm{div}(z)\in L^1(\Omega)\}.
\]
If $z\in X(\Omega)$ and $w \in BV(\Omega)\cap L^\infty(\Omega)$,
the functional $(z, Dw): C^\infty_0(\Omega)\rightarrow\mathbb{R}$
is defined by:
\[
\left\langle {\left( {z,Dw} \right),\varphi } \right\rangle  =  -
\int_\Omega  {\mathrm{div}\left( z \right)w\varphi \,}  -
\int_\Omega  {zw \cdot \nabla \varphi \,}
\,\,\,\mathrm{for}\,\,\,\mathrm{all}\,\,\,\varphi  \in C_0^\infty
\left( \Omega \right).
\]
Then $(z, w)$ is a Radon measure in $\Omega$,
\[
\int_\Omega  {\left( {z,Dw} \right)\,}  = \int_\Omega  {z \cdot
\nabla w\,} \,\,\,\mathrm{for}\,\,\,\mathrm{all}\,\,\,w \in H_1^1
\left( \Omega \right) \cap L^\infty  \left( \Omega  \right)
\]
and
\[
\left| {\int_B {\left( {z,Dw} \right)\,} } \right| \leqslant
\int_B {\left| {\left( {z,Dw} \right)} \right|\,}  \leqslant
\left\| z \right\|_\infty  \int_B {\left| {Dw} \right|\,}
\,\,\,\mathrm{for}\,\,\mathrm{any}\,\,\mathrm{Borel}\,\,\mathrm{set}\,\,B
\subseteq \Omega\,.
\]
In addition,  a week trace on $\partial \Omega$ of the normal
component of $z\in X(\Omega)$ is defined. More precisely, it is
proved that there exists a linear operator $\gamma: X(\Omega)\to
L^\infty(\partial\Omega)$ such that:
\[\|\gamma(z)\|_\infty\leq\|z\|_\infty\;\;\mathrm{and}\;\;\gamma(z)(x)=z(x)\cdot \nu(x)\;
\;\mathrm{for}\;\;x\in \partial\Omega\;\;\mathrm{if}\;\;z\in
C^1(\bar \Omega, \mathbb{R}^n),\] where $\nu$ denotes the outward
unit normal along $\partial\Omega$.

We shall denote $\gamma(z)(x)$ by $[z, \nu](x)$. Moreover, we have the
following Green formula relating the function $[z, \nu]$ and
the measure $(z,Dw)$: for $z \in X_1(\Omega)$ and $w \in
BV(\Omega)\cap L^\infty(\Omega)$
\begin{eqnarray}\label{E58}
\int_\Omega  {\mathrm{div}\left( z \right)wdx\,}  + \int_\Omega
{\left( {z,Dw} \right)\,}  = \int_{\partial \Omega } {\left[
{z,\nu } \right]wd\mathcal{H}^{n - 1} \,},
\end{eqnarray}
where $\mathcal{H}^{n - 1}$ is the $(n-1)$-dimensional Hausdorff
measure. For a proof of this result we refer to Anzellotti \cite{Anz}.

\subsection{Resolution of the problem}

Consider  the solid torus defined by:
$$ T=\left\{(x,y,z)\in \mathbb{R}^3: \left(
\sqrt{x^2+y^2}-l\right)^2+z^2< r^2, \, \,l>r>0\right\}.
$$
We are interested in the variation problem:
\begin{equation}\label{E59}
\lambda_1(T)=\inf\left\{\int_T|\nabla u|dV+\int_T| u|dV:\; u\in
H^1_1(T),\;\int_{\partial T}| u|dS=1\right\}.
\end{equation}
In view of the results presented in the previous results, this
problem is equivalent to:
\begin{equation}\label{E60}
\lambda_1(T)=\inf\left\{\int_T|D u|+\int_T| u|dV:\; u\in
BV(T),\;\int_{\partial T}| u|dS=1\right\}.
\end{equation}

Andreu, Mazon and Rossi \cite{And-Maz-Ros} studied the dependence
of the best constant $\lambda_1(\Omega)$ and its extremals on the
domain. Here, we are interested to study the dependence of the
existence of extremals on the best constant $\lambda _1(T) $, and
therefore the geometrical characteristics of the torus. We note
that since the variation method fails due the lack of compactness
of the embedding  $H_1^1 (T) \hookrightarrow L^{1} (\partial T)$,
the study of the problem is not trivial.

For $1<p<2$, let us consider the variation problem:
\begin{equation}\label{E61}
\lambda_p(T)=\inf\left\{\int_T|\nabla u|^pdV+\int_T| u|^p dV:\;
u\in H^p_1(T),\;\int_{\partial T}| u|^pdS=1\right\}\,.
\end{equation}
Due to the compactness of the embedding  $H_1^p(T) \hookrightarrow
L^p(\partial T)$, it is known (see Bonder and Rossi \cite{Fer-Ros}) that
problem (\ref{E65}) has a minimizer in $H^p_1(T)$ and the
extremals are week solutions of the following problem:
\begin{equation}\label{E62}
\left\{ {\begin{array}{*{20}c} {\Delta _p u = \left| u \right|^{p
- 2} u\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\,\,\,T ,}  \\
\displaystyle {\left| {\nabla u} \right|^{p - 2} \frac{{\textstyle \partial u}}
{{\textstyle\partial \nu}} = \lambda _p \left( T \right)\left| u
\right|^{p - 2} u\,\,
\,\,\,\mathrm{on}\,\,\,\,\,\partial T},  \\
 \end{array} } \right.
\end{equation}
where $ \Delta _p u =\mathrm{ div}\left( {\left| {\nabla u}
\right|^{p - 2} \nabla u} \right)$ is the $p$-Laplacian and
$\frac{{\textstyle
\partial u}} {{\textstyle\partial\nu}}$ is the outer unit normal
derivative (see Martinez and Rossi \cite{Mar-Ros}).

Therefore, it seems natural to search for an extremal for
$\lambda_1(T)$  as the limit of extremals for $\lambda_p(T)$ when
$p \to 1^+$. Unfortunately,   there is no hope to prove an
existence result in $H^1_1(T)$, since this space is not
reflexive. The convenient space in which one must look for
solutions is the space $BV(T)$. So, the extremals in this limit
case are solutions of the following Neumann problem involving the
$1$-Laplacian operator $ \Delta _1 u = \mathrm{div}\left(
{Du/\left| {Du} \right|} \right) $,
\begin{equation}\label{E63}
\left\{ {\begin{array}{*{20}c} {\Delta _1 u = \frac{\textstyle u}
{{\textstyle \left|u \right|}}\,\,\,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{in}\,\,\,\,\,\,T,}  \\
{\frac{{\textstyle Du}} {{\left| {\textstyle Du} \right|}} \cdot
\nu = \lambda _1 \left( T \right)\frac{\textstyle u}
{{\textstyle\left| u \right|}}\,\,\,\,\,\mathrm{on}\,\,\,\,\,\partial T},  \\
 \end{array} } \right.
\end{equation}
in the context of bounded variation functions, where  $Du$
denotes  the gradient of $u$ in this space.

We now recall the definition of the solution of the  problem
(\ref{E63}) (see \cite{And-Maz-Ros}).
\begin{definition}\label{D5.1}
A function $u\in BV(T)$ is said to be a solution of problem
$\mathrm{(\ref{E63})}$ if there exist $z\in X_1(T)$ with
$\|Z\|_\infty\leq1$, $\tau\in L^\infty(T)$ with
$\|\tau\|_\infty\leq1$ and $\theta\in L^\infty(\partial T)$  with
$\|\theta\|_\infty\leq1$ such that:
\begin{eqnarray}
\label{E64}\mathrm{div}\left(
z\right)&=&\tau\,\,\,\mathrm{in}\,\,\,\mathcal{D}'(T),
 \hfill \\
\label{E65}\tau u &=& \left| u
\right|\,\,\,\mathrm{a.e.}\,\,\,\mathrm{in}\,\,\,T\,\,\,\mathrm{and}
\,\,\,\left( {z,Du} \right) = \left| {Du} \right|\,\,\,\mathrm{as}\,\,\,\mathrm{measures}, \hfill \\
\label{E66}\left[ {z,\nu } \right]& =& \lambda _1 \left( T
\right)\theta \,\,\,\mathrm{and} \,\,\,\theta u = \left| u
\right|\mathcal{H}^{n - 1} \,\,\,\mathrm{a.e.}\,\,\,
\mathrm{on}\,\,\,\partial T.
\end{eqnarray}
\end{definition}

\begin{proposition}\label{P5.1}
The problems $\mathrm{(\ref{E60})}$ and $\mathrm{(\ref{E63})}$ are
equivalent in the sense that if $\upsilon$ is a solution of $\mathrm{(\ref{E63})}$  and $ \int_{\partial T} {\left| \upsilon \right|dS\,}
\ne 0 $ then $ w = \upsilon /\int_{\partial T} {\left| \upsilon
\right|dS\,}$ is a minimizer of $\mathrm{(\ref{E60})}$.
\end{proposition}

\noindent{{\bf Proof of Proposition \ref{P5.1}.}
(\cite{And-Maz-Ros}) Multiplying (\ref{E64}) by $\upsilon$ and
integrating by parts due to (\ref{E65}), (\ref{E58}) and
(\ref{E66}), we obtain:
\begin{eqnarray*}
\int_T {\left| \upsilon  \right|dV\,} & =& \int_T {\tau \upsilon
dV\,}  = \int_T {\mathrm{div}\left( z \right)\upsilon dV\,}  =  -
\int_T {\left( {z,D\upsilon }
 \right)\,}  + \int_{\partial T} {\left[ {z,\nu } \right]\upsilon dS \,}  \hfill \\
&=&  - \int_T {\left| {D\upsilon } \right|\,}  + \lambda _1 \left(
T \right)\int_{\partial T} {\left| \upsilon  \right|dS.\,}  \hfill \\
\end{eqnarray*}
Therefore:
\[
\lambda _1 \left( T \right) = \int_T {\left| {Dw} \right|\,}  +
\int_T {\left| w \right|dV.\,}
\]
This completes the proof.
\mbox{}$\Box$

The result of Proposition \ref{P5.1}} certifies the equivalence of
the above problems in the sense noticed above, namely if the
problem (\ref{E60}) has a solution $u$ then the problem
(\ref{E63}) has the solution $w = \upsilon /\int_{\partial T}
{\left| \upsilon \right|dS}$.  Also, in \cite{And-Maz-Ros}
necessary and sufficient conditions of the  existence of a
solution to the above problems are given. In this paper, as
already mentioned above, we are not interested in solving these
problems in the torus. We use the above results in order to study
the dependence of the existence of the solutions to the above
problems of the geometry of the torus. In the following theorem it
is confirmed that the above problems have a solution only in the
cases when we have ``small'' tori. For ``big'' tori the problems
have not solutions.  In addition, it is proved that in the case
when $r=2$, (where $r$ is the range of the circle which is rotated
in the $yz$-plane about the $z$-axis far from $z$-axis), the torus
behaves as the disk $D_2$ of radius 2 in $\mathbb{R}^2$, since in
this case $\lambda_1(T)=\lambda_1(D_2)$.

\begin{theorem}\label{T5.1}
If $\,r\leq 2$, there exists a nonnegative function of bounded
variation which is a solution of the problem
$\mathrm{(\ref{E60})}$. In particular for $\,r=2$, this problem is
equivalent to the same problem considered in the disk $D_2$ in
$\mathbb{R}^2$ in the sense that if $\upsilon$ is a solution to
the first one then the function $\phi=\upsilon\circ\xi^{-1}$,
defined by $\mathrm{(\ref{E15})}$, is a solution of the second problem. In
addition,   problem $\mathrm{(\ref{E64})}$ has a solution if
$\,r\leq 2$ and has not solution if $r >2$.
\end{theorem}

\noindent{{\bf Proof of Theorem \ref{T5.1}.} We study  three
cases concerning the range of $r$.\\
$\mathrm{\textbf{(i)}}\;$ Let $r<2$. Since $\frac{\textstyle
|T|}{\textstyle |\partial T|}=\frac{\textstyle r}{\textstyle 2}$,
 we obtain (see (\ref{E50}))  that $\frac{\textstyle
|T|}{\textstyle |\partial T|}<1$ for all $r<2$ and then by Theorem
1 in \cite {And-Maz-Ros} we conclude that $\lambda_1(T)<1$. Thus
by Theorem 2 in \cite {And-Maz-Ros}, we deduce that there exists a
nonnegative function of bounded variation which is a minimizer of
the variational problem (\ref{E60})  and a
solution of the problem (\ref{E63}) see Proposition \ref{P5.1}.\\
$\mathrm{\mathbf{(ii)}}\;$ For  $r=2$, because of the Theorem
\ref{T4.2}, $\lambda_1(T)=\lambda_1(D_2)$, and since
$\frac{\textstyle |D_2|}{\textstyle |\partial D_2|}=1$, the
variational problem (\ref{E64}) is equivalent to the  same problem
considered in the disk $D_2$, for which the function
$\phi_0=(1/|\partial D_2|)\mathcal{\chi}_{_{D_2}}$ is a minimizer
(see Example 1 in \cite{Mot}). Thus the function $u_0=(1/|\partial
T|)\mathcal{\chi}_{_{T}}$, ($u_0=\phi_0\circ \xi$ by definition
(\ref{E15})) is a minimizer to the problem (\ref{E64}), being the
only minimizer in the case $\frac{\textstyle |T|}{\textstyle
|\partial  T|}=1$  (see (\ref{E56})).\\
$\mathrm{\textbf{(iii)}}\;$ If $r>2$, then  $\frac{\textstyle
|T|}{\textstyle |\partial T|}>1$ and then the variational problem
(\ref{E60}) does not have any minimizer (see \cite{Mot}). \mbox{}$\Box$

\section{Best constants on Riemannian manifolds with boundary}

\subsection{The general case}

This part of the paper is devoted to the study of the classical
Sobolev inequality on manifolds with boundary, to the Sobolev
trace inequality, and also to the existence and calculation of
best constants, when they exist. The proofs of the  related
theorems are not difficult and probably are classical in the sense
that in general these have been used in the case of manifolds
without boundary and the presence of the boundary does not affect
the classical proofs. So, we do not give in detail the proofs of
these theorems  but we outline the arguments as much as briefly as
possible by making the necessary adjustments in the case of
manifolds with boundary. In addition, we note that the study of
these inequalities are necessary because they have never been
studied in the past and we do not know the values of the best
constants. Furthermore, we will give some counter-examples
demonstrating that in some cases there are no best constants for
the above Sobolev inequalities.

Concerning  the first best constant for the  classical Sobolev
inequality on manifolds with boundary in the case $p=1$,  the
following theorem holds.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{T6.1}
Let $(M,g)$ be a smooth, compact $n$-dimensional Riemannian
manifold with boundary, $n\geq 3$. For any $\varepsilon>0$, there
exists $B\in\mathbb{R}$ such that for any $u\in {H}_1^1(M)$,
\begin{equation}\label{E67}
\left( {\int_M {\left| u \right|^{n/(n-1) } d\upsilon _g } }
\right)^{(n-1)/n }  \le
\left(2^{1/n}K(n,1)+\varepsilon\right)\int_M {\left| {\nabla u}
\right| d\upsilon _g }  + B\int_M {\left| u \right| d\upsilon _g }.
\end{equation}
Moreover, $\;2^{1/n}K(n,1)$ is the best constant for this
inequality.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent {{\bf{Proof of Theorem \ref{T6.1}.}} Our first purpose
is to establish the first best constant in inequality (\ref{E67}).
The proof of this theorem uses some ideas from the proof  of
Theorem 4.5 in Hebey \cite{Heb}, which are adapted to our case on
manifolds with boundary. Let us sketch the proof. Fix
$\varepsilon>0$. For any $P$ in $M$ and any $\varepsilon_0>0$,
there exists some chart $(\Omega, \xi)$ on $P$ such that:
$$1-\varepsilon_0 \leq \sqrt{\mathrm{det}(g_{ij})}\leq
1+\varepsilon_0 $$ Choosing $\varepsilon_0$ small enough, by
Theorem $4.5$ in Hebey \cite{Heb}   we can assume that for any
smooth function $u$ with compact support in $\Omega$,
\begin{equation}\label{E68}
\left( {\int_M {\left| u \right|^{n/(n-1) } d\upsilon _g } }
\right)^{(n-1)/n }  \le \left(K(n,1)+\varepsilon_0\right)\int_M
{\left| {\nabla u} \right| d\upsilon _g }.
\end{equation}
Since $M$ is compact, it can be covered by a finite number of
charts  $(\Omega_k, \xi_k)$, ${k=1,...,N}$. Denote by $(\alpha_k)$
a smooth partition of unity subordinated to the covering
$(\Omega_k)$, and set:
$$
\eta_k=\frac{\textstyle\alpha_k^2}{\textstyle\Sigma_{m=1}^M
\alpha_m^2}\quad\mbox{for}\ k=1,...,N.
$$
Then  $\eta_k \in C^1(M)$, $\eta_i$ has compact support in
$\Omega_i$ for any $i$ and there exists $H\in\mathbb{R}$ such that
for any $k$, $\;|\nabla
\eta_k|\leq H$.\\
Furthermore, for any $u\in C^\infty_0(M)$, after rather standard
computations we can write:
\begin{eqnarray}\label{E69}
\left( {\int_M {\left| u \right|^{n/(n-1) } d\upsilon _g } }
\right)^{(n-1)/n }\leq  \sum\limits_{i=1}^N\left( {\int_M {\left|
\eta_k u \right|^{n/(n-1) } d\upsilon _g } } \right)^{(n-1)/n }.
\end{eqnarray}
Thus by  (\ref{E69}), because of (\ref{E68}) and  following standard
steps we obtain that for any $u\in C^\infty_0(M)$,
\begin{eqnarray}\label{E70}
\left( {\int\limits_M {\left| u \right|^{n/\left( {n - 1} \right)}
} d\upsilon _g } \right)^{\left( {n - 1} \right)/n} \leqslant
\left( {K\left( {n,1} \right) + \varepsilon } \right)\int_M
{\left( {\left| {\nabla u} \right| + NH\left| u \right|} \right)}
d\upsilon _g,
\end{eqnarray}
where $\varepsilon=O(\varepsilon_0)$.

On the other hand, using Proposition 4.2 in Hebey \cite{Heb}, we
know that if there are real numbers $A$, $B$ such that
 inequality (\ref{E7}) holds for all $u \in H_1^1 \left( M
\right)$, then $ A \geq K\left( {n,1} \right), $ which is the best
constant in the classical Sobolev inequality:
\begin{equation*}
\left( {\int_{\mathbb{R}^n } {\left| u \right|^{n/\left( {n - 1}
\right)} dx} } \right)^{\left( {n - 1} \right)/n}  \leqslant
K\left( {n,1} \right)\int_{\mathbb{R}^n } {\left| {\nabla u}
\right|dx},
\end{equation*}
that holds for all $u\in C^\infty_0(\mathbb{R}^n)$.

Since inequality (\ref{E70}) holds for all $u\in
C^\infty_0(M)$, we have $ \mathcal{A}_1(M)=K(n,1) $.

In order to complete the proof,  we need to prove that inequality
(\ref{E70}) holds for all $u\in H_1^1(M)$. Let $(\Omega_i,
\xi_i)$, ${i=1,...,N}$ be a finite atlas of $M$, each $\Omega_i$
being homeomorphic either to a ball of $\mathbb{R}^n$ or to a half
ball of $\mathbb{R}^n_+$. We choose the atlas so that in each
chart the metric tensor is bounded. Consider a $C^\infty$
partition of unity $\{\alpha_i\}$ subordinated to the covering
$\Omega_i$. Then for all $u\in H_1^1(M)$,  $\alpha_iu$ has support
in $\Omega_i$. When $\Omega_i$ is homeomorphic to a ball, the
proof is that of first part. When $\Omega_i$, is homeomorphic to a
half ball, the proof is similar, but in this case the best
constant is $2^{1/n}K(n,1)$ (see Aubin   \cite[Lemma 2.31]{Aub};
see Cherrier \cite{Che3} for a complete proof).  \mbox{} $\Box$

\smallskip As regards  the existence of the second best constant the
situation is  confusing in the sense that it exists in certain
cases of manifolds, while in others does not seem to be possible
to formulate a relevant theorem that clarifies the situation.  For
instance, we proved in Theorem \ref{T4.2} that in inequality
(\ref{E23}) the second best constant exists and is equal to
$|T|^{-1/2}$. Also, it is well known that for any smooth, compact
Riemannian $n$-manifold without boundary, $n\geq2$,  then (see
Hebey \cite[Theorem 4.1]{Heb})  for any $u\in {H}_1^1(M)$ there
exists $A\in\mathbb{R}$ such that:
\begin{equation}\label{E71} \left( {\int_M {\left| u
\right|^{n/(n-1) } d\upsilon _g } } \right)^{(n-1)/n }  \le
A\int_M {\left| {\nabla u} \right| d\upsilon _g }  +
|M|^{-1/n}\int_M {\left| u \right| d\upsilon _g },
\end{equation}
which means that  $\mathcal{B}_1(M)=|M|^{-1/n}$.\\
We can therefore conclude that on the second constant in the case
of the torus, (which is manifold with boundary), the same theorem
as in the case of manifolds without boundary is valid. \\
However, as demonstrated in the following example, we can not
formulate a theorem that relates to all the manifolds with
boundary and calculate the value of the second constant.

\begin{example}\label{ex6.1}
\emph{Let $M=M_1\cup M_2\cup M_3$ be the smooth manifold  with
boundary $\partial M$, where $M_1$ and $M_2$ are two smooth
disjoint bounded domains in $\mathbb{R}^n$ connected smoothly by a
small thin ``tube'' $M_3$.  Consider now a smooth function $u$
which is equal to $1$ on $M_1$ and $0$ on $M_2$. Then we
 can adjust the sizes of $M_1$ and  $M_2$ in such a
way that inequality (\ref{E71}) becomes false.}
\end{example}
\noindent\textbf{Proof of Example \ref{ex6.1}.} By definition of
$u$, there exists $H>0$ such that $\left|\nabla u \right|\leq H$
in $M=M_1\cup M_2\cup M_3$.  Furthermore, we can  choose the small
thin ``tube" such that $|M_3|=\varepsilon$, for any arbitrarily
small $\varepsilon>0$; (that is, in the 3-dimensional case $M_3$ is a
cylinder of radius $a$ and of length  $b$ and then $|M_3|=\pi a^2
b$. Thus, for any $\varepsilon>0$ and for any arbitrary $b$ we can
choose $a=\sqrt{\varepsilon/\pi b}$ and then
$|M_3|=\varepsilon$).\\
Suppose now that in our case the inequality (\ref{E71}) holds.
Therefore:
\begin{eqnarray*}
\left(\int_{M_1\cup M_3} d\upsilon _g  \right)^{(n-1)/n }& \le &
A\int_{M_3} {\left| {\nabla u} \right| d\upsilon _g }  + |M_1\cup
M_2\cup M_3|^{-1/n}\int_{M_1\cup M_3}  d\upsilon _g,\\
|M_1\cup M_3|^{(n-1)/n }& \le & AH|M_3|  + |M_1\cup M_2\cup
M_3|^{-1/n}|M_1\cup M_3|,\\ \left(|M_1|+\varepsilon|
\right)^{(n-1)/n} & \le & \varepsilon AH + \left( |M_1|+
|M_2|+\varepsilon\right)^{-1/n}(|M_1|+\varepsilon),
\\
\left(\frac{|M_1|+ |M_2|+\varepsilon}{|M_1|+\varepsilon}
\right)^{1/n} & \le & \varepsilon AH\frac{\left( |M_1|+
|M_2|+\varepsilon\right)^{1/n}}{|M_1|+\varepsilon}  + 1,
\\
1+\frac{ |M_2|}{|M_1|+\varepsilon} & \le &\left(
1+O(\varepsilon)\right)^n,
\end{eqnarray*}
where $O(\varepsilon)=\varepsilon AH\left( |M_1|+ |M_2|+\varepsilon\right)^{1/n}/(|M_1|+\varepsilon) $.

Obviously, since we can choose $M_2$ as we want huge,  the last
inequality does not always hold, and our assertion is
proved.\mbox{} $\Box$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallbreak\noindent As demonstrated by the counterexample
\ref{ex6.1}, the results related to the value of the second best
constant in the inequality (\ref{E71}) in some cases fail,
however in all cases the existence and the value of this depends
on the ``shape'' of the manifold $M$. Although, another case when
this constant exists, is presented in the following result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{P6.1}
If $M$ is not a connected manifold and $M=\bigcup\limits_{i = 1}^J
{M_j }$, where $J$ is a positive integer, $M_j$ is connected, and
such that the second best constant in inequality (\ref{E7}) exists
for all $M_j$, $j\in\{1,2,...,J\}$, then the second best constant
is given by $\mathcal{B}_1(M)=\mathop {\sup }\limits_{1 \leqslant
j \leqslant J} |M_j|^{-1/n}$.
\end{proposition}
\noindent\textbf{Proof of Proposition \ref{P6.1}. }  By the
definition of $\mathcal{B}_1(M)$ we have
$\mathcal{B}_1(M)\leq\mathop {\sup }\limits_{1 \leqslant j
\leqslant J} |M_j|^{-1/n}$.

For the reverse inequality, let $M_{j_0}$ the component of $M$ of
minimum $n$-dimensional measure $|M_{j_0}|$. Then  $
|M_{j_0}|^{-1/n}=\mathop {\sup }\limits_{1 \leqslant j \leqslant
J} |M_j|^{-1/n}$. Fix $u \in{H}_1^1(M)$ equal to $1$ in $M_{j_0}$
and equal to $0$ outside $M_{j_0}$. Then the inequality:
\begin{equation}\label{E72}
 \left( {\int_M {\left| u \right|^{n/(n-1) } d\upsilon
_g } } \right)^{(n-1)/n }  \le A\int_M {\left| {\nabla u} \right|
d\upsilon _g }  + B\int_M {\left| u \right| d\upsilon _g }
\end{equation}
implies that $B\geq |M_{j_0}|^{-1/n}$, and since it is true for
all $B>0$ such that (\ref{E72}) occurs, we deduce that:
$\mathcal{B}_1(M)\geq |M_{j_0}|^{-1/n}$.\\
Finally, we have $\mathcal{B}_1(M)=\mathop {\sup }\limits_{1 \leqslant j
\leqslant J} |M_j|^{-1/n}$. \mbox{} $\Box$

\smallbreak Our second result in this part is the following
theorem, which concerns the first best constant in Sobolev trace
inequality with $p=1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{T6.2}
Let $(M,g)$ be a smooth, compact $n$-dimensional Riemannian
manifold with boundary, $n\geq 3$. For any $\varepsilon>0$, there
exists $\tilde B\in\mathbb{R}$ such that for all $u\in
{H}_1^1(M)$,
\begin{equation}\label{E73}
{\int_{\partial M }{\left| u \right| ds _g } }  \le
\left(1+\varepsilon\right)\int_M {\left| {\nabla u} \right|
d\upsilon _g }+ \tilde B \int_M {\left| u \right| d\upsilon _g }.
\end{equation}
In particular,  $1=\tilde K\left(n,1\right)$ always is the best
constant for this inequality.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 The proof of this theorem makes use of the following auxiliary property.

\begin{lemma}\label{L6.1}
Let $(M,g)$ be a smooth, compact Riemannian $n$-dimensional
manifold with boundary, $n\geq 3$. Suppose that there exist real
numbers $\tilde A, \tilde B$  such that the inequality:
\begin{equation}\label{E74}
\int_{\partial M} {\left| u \right| ds_g }  \le \tilde A {\int_M
\left| {\nabla u} \right| d\upsilon _g }+ \tilde B {\int_M \left|
{  u} \right| d\upsilon _g }
\end{equation}
holds for any $u \in H_1^1 \left( M \right)$. \smallbreak
\noindent Then $ \tilde A \geq 1=\tilde K\left(n,1\right)$,  the
best constant in the classical Sobolev trace inequality:
\begin{equation}\label{E75}
\int_{\partial \mathbb{R}^n_+} {\left| u \right| dx' }  \le \tilde
K\left(n,1\right) {\int_{\mathbb{R}^n_+}\left| {\nabla u} \right|
dx },
\end{equation}
which holds for all $u \in H_1^1(\mathbb{R}^n_+)$.
\end{lemma}
The proof of Lemma \ref{L6.1} is provided in Appendix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallbreak\noindent{{\bf{Proof of Theorem \ref{T6.2}.}}   In
attempting to compute the first best constant in inequality
(\ref{E73}) we choose a finite covering of $M$ consisted by
geodesic balls $B_k=B_k(P_k)$, $k=1,...,N$ in the following way:\\
$(i)$ if the center $P_k$ of the ball lies in the interior of the
manifold, then the entire ball lies in its interior, and then
$B_k$ is a normal geodesic neighborhood with
normal geodesic coordinates $x_1,...,x_n$;\\
 $(ii)$ if the center
$P_k$ of the ball lies in the boundary of the manifold, then $B_k$
is a Fermi neighborhood with Fermi coordinates
$x_1,...,x_{n-1},y$.\\
 In all these neighborhoods, the following
hold: $$ 1-\varepsilon_0\leq \sqrt{\mathrm{det}(g_{ij})}\leq
1+\varepsilon_0, $$ where $\varepsilon_0$ can be as small as we
want, depending on the chosen covering.

Let $(\eta_k)_{k=1,2,...,N}$ be a partition of unity associated to
the covering $B_k$. Then for any $u\in H^1_1(M)$ we obtain:
\begin{equation}\label{E76}
{\int_{\partial M }{\left| u \right| ds _g } }   = {\int_{\partial
M } {\left|\sum\limits_{k=1}^N\left( \eta_k u\right) \right| ds_g
} } \leq \sum\limits_{k=1}^N\left( {\int_{\partial M } {\left|
\eta_k u \right| ds _g } } \right).
\end{equation}
Furthermore, by  (\ref{E76})  because of the Sobolev embedding
theorem on $\partial\mathbb{R}^n_+$, passing the integration in
the Euclidean space and returning to the manifold we deduce that:
\begin{equation}\label{E77}
\int_{\partial M} {\left| u \right|}  ds_g \leq \frac{1 +
\varepsilon _0}{1 - \varepsilon _0}\int_M {\left( {\left| {\nabla
u} \right| + \sum\limits_{k = 1}^N \left| {\nabla \eta _k }
\right|\left| u \right|} \right)} d\upsilon _g.
\end{equation}
Let  $C$ be a positive constant depending  on the chosen finite
covering of the compact manifold $M$ such that $|\nabla
\eta_k|\leq C$ for all $k$.  Then by (\ref{E77}), we have:
\begin{eqnarray*}
\int_{\partial M} {\left| u \right|}  ds_g  \leqslant \frac{1 +
\varepsilon _0}{1 - \varepsilon _0}\int_M {\left( {\left| {\nabla
u} \right| +CN\left| u \right|} \right)} d\upsilon _g
\end{eqnarray*}
or
\begin{equation}\label{E78}
\int_{\partial M} {\left| u \right|}  ds_g  \leqslant
(1+\varepsilon)\int_M {\left| {\nabla u} \right| d\upsilon _g
+C'\int_M } |u| d\upsilon _g,
\end{equation}
where $\varepsilon=O(\varepsilon_0)$ and $C'=CN(1 + \varepsilon
_0)/(1 - \varepsilon _0)$.\\
Since inequality (\ref{E78}) holds for all $u\in H^1_1(M)$ we deduce by
Lemma \ref{L6.1}   that $ \mathcal{\tilde A}_1(M)=1=\tilde
K(n,1) $.  \mbox{} $\Box$


\smallbreak Suppose now that we are interested in studying  the
existence of the second best constant in Sobolev trace inequality
with $p=1$ and  to calculate its value if it exists. This problem
is answered in the case of a connected bounded open set of
$\mathbb{R}^n$, see Motron \cite[Proposition 3.10]{Mot}. However,
in the following counterexample it is proved that this result is
not always true even if the manifold is connected.

\begin{example}\label{ex6.2}
\emph{On a huge sphere $\mathbb{S}^n$ consider a cap $M_1$ around
a point $P$, the complement $M_2$ of a bigger cap around the same
point $P$ and a thin ``tube'' $M_3$ connecting smoothly $M_1$ and
$M_2$. Let now consider the smooth  function $u$ on $M=M_1\cup M_2\cup
M_3$ which is equal to $1$ on $M_1$ and $0$ on $M_2$. Then we
 can adjust the sizes of $M_1$ and  $M_2$ in
such a way that inequality (\ref{E75}) becomes false.}
\end{example}
The proof of Example \ref{ex6.2} is omitted since it is similar to
that of Example \ref{ex6.1}.
\begin{remark}\label{R6.1}
\emph{If the manifold $M$ is not connected then the result of
Theorem \ref{T4.2} concerning the second best constant fails.
However, if $M$ is not a connected manifold and
$M=\bigcup\limits_{i = 1}^p {M_i }$, where $p$ is a positive
integer and $M_i$ is connected for all $i\in\{1,2,...,p\}$ then
the second best constant for inequality  (\ref{E12}) is given by
$\mathcal{\tilde B}_1(M)=\mathop {\sup }\limits_{1 \leqslant i
\leqslant p}  \frac{\textstyle|\partial M_i|}{\textstyle|M_i|}$
(see Proposition 3.12 and example on page 81  in Motron \cite{Mot}).}
\end{remark}

\subsection{Best constants on Riemannian manifolds in the presence
of symmetries  }

This part of the paper is devoted to manifolds which present symmetries.
The following two examples may be regarded as representatives of
these manifolds.
\begin{example}\label{ex6.3} \emph{ Consider the three dimensional solid
torus:
$$
T=\left\{(x,y,z)\in \mathbb{R}^3:  \left(
\sqrt{x^2+y^2}-l\right)^2+z^2< r^2,\, l>r>0\right\},
$$
with the metric induced by the $\mathbb{R}^3$ metric.\\ Let
$G=O(2)\times I$ be the the group of rotations around axis $z$.
Then, all the  $G$-orbits of the $T$ are circles, thus of
dimension $1$, the orbit of minimum volume is the circle of radius
$l-r$, and the  volume of it is equal to $2\pi(l-r)$.  Therefore,
$T$ is a compact $3$-dimensional manifold with boundary, invariant
under the action of  the subgroup $G$ of the isometry group
$O(3)$.}
\end{example}

\begin{example}\label{ex6.4}
\emph{($\cite{Cot}, \cite{Cot-Ili})$ Let
$\mathbb{R}^n=\mathbb{R}^k\times\mathbb{R}^m $, $k\geq 2$, $m\geq
1$ and $\overline{\Omega}\subset \left(\mathbb{R}^k \backslash
\{0\}\right)\times \mathbb{R}^m $. Denote by $G_{k,m}=O(k)\times
Id_m$ the subgroup of the isometry group $O(n)$ of the type
$$ \tau: (x_1, x_2)\longrightarrow (\sigma (x_1),x_2),\,\, \sigma\in
O(k),\,\, x_1 \in \mathbb{R}^k , \,\,x_2 \in \mathbb{R}^m ,$$ and,
suppose that $\overline{\Omega}$ is invariant under the action of
$G_{k,m}$ $\left(\tau (\overline{\Omega})=\overline{\Omega},
\forall \tau \in G_{k,m}\right)$. Then $\overline{\Omega}$ is a
compact $n$-dimensional manifold with boundary, invariant under
the action of  the subgroup $G_{k,m}$ of the isometry group
$O\left(n\right)$.}
\end{example}
Considering that we studied  the case of the solid torus, we need
some background material and results concerning the
``decomposition'' of a manifold with boundary which presents
symmetries.

In the following, we assume the notations and background material
from Hebey and Vaugon \cite {Heb-Vau} and Cotsiolis and
Labropoulos \cite{Cot-Lab3}.\\
We remind that, given $(\widetilde M,g)$ a Riemannian manifold
(complete or not, but connected), we denote by $I(\widetilde M,
g)$ its group of isometries. Let $(M,g)$ be a compact
$n-$dimensional, $n\geq 3$, Riemannian manifold with boundary
$G-$invariant under the action of a subgroup $G$ of the isometry
group $I(M,g)$. We assume that $(M,g)$ is a smooth bounded open
subset of a slightly larger Riemannian manifold $\left(
{\widetilde M,g} \right)$ (see \cite{Li-Zhu}), invariant under the
action of a subgroup $G$ of the isometry group of $\left(
{\widetilde M,g} \right) $.

The first results we need are the following two properties.
\begin{lemma}\label{L6.2}
\emph{(\cite{Heb-Vau})} Let $ \left( {\widetilde M,g} \right)$ be
a Riemannian $n$-manifold (complete or not), and let $G$ be a
compact subgroup of $I\left( {\widetilde M,g} \right)$. Let  $ P
\in \widetilde M$ and set $k=\mbox{dim}\, O_{P}$. Assume $k\geq1$.
There exists a coordinate chart $\left(\Omega,\xi\right)$ of
$\widetilde M$ at $P$ such that the following properties hold:
\begin{enumerate}
    \item $\xi \left( \Omega  \right) = U \times W$, where $U$ is
some open subset of $\;\mathbb{R}^k$ and $W $ is some open subset
of $ \;\mathbb{R}^{n - k}. $
    \item For any $Q\in \Omega,\;U \times
\Pi_2\left(\xi(Q)\right)\subset \xi \left(O_{Q}\cap\Omega\right)$,
where $\Pi_2:\mathbb{R}^k\times\mathbb{R}^{n-k}\rightarrow
\mathbb{R}^{n-k}$ is the second projection.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{L6.3}
\emph{(\cite{Heb-Vau})} Let $M$ be a compact subset of
$\widetilde{M}$ covered by a finite number of charts $(\Omega_m,
\xi_m)$, $m=1,...,M$ and  $k = \min _{P \in \widetilde{M}} \dim
O_P \geq 1$. The following properties are valid:
\begin{enumerate}
    \item $\xi_m \left( \Omega_m  \right) = U_m \times W_m$, where
          $U_m$ is some open subset of $\mathbb{R}^{k_m }$ and $W_m$
          is some open subset of $\mathbb{R}^{n - k_m} $ and
          $k_m\in \mathbb{N}$ satisfies $k\leq k_m < n$.
    \item $U_m$ and $W_m$ are bounded, and $W_m$ has smooth boundary.
    \item For any $Q\in \Omega_m,\;U_m \times \Pi_2\left(\xi_m(Q)\right)\subset \xi_m
          \left(O_{Q}\cap\Omega_m\right)$, where
          $\Pi_2:\mathbb{R}^{k_m}\times\mathbb{R}^{n-k_m}\rightarrow
          \mathbb{R}^{n-k}$ is the second projection.
    \item There exists $\varepsilon_m>0$ with
          $(1-\varepsilon_m)\delta_{ij}\leq g_{ij}^m\leq
          (1+\varepsilon_m)\delta_{ij}$
          as bilinear forms, where the $g_{ij}^m$'s are the components
          of $g$ in $(\Omega_m, \xi_m)$.
\end{enumerate}
\end{lemma}

Let $ P \in M$ and $O_{ P}=\{\tau ( P), \tau \in G \} $ be its
orbit of dimension $k$, $0 \leqslant k < n$. According to
(\cite{Heb} \S \,9, \cite{Fag1}) the map $\Phi :G \to O _{ P} $
defined by $\Phi \left( \tau \right) = \tau \left( { P} \right)$,
is of rank $k$  and there exists a submanifold $H$ of $G$ of
dimension $k$ with $Id \in H$, such that $\Phi $ restricted to $H$
is a diffeomorphism from $H$ onto its image denoted
$\mathcal{V}_{P } $.

Let $ N$ be a submanifold of $ M$ of dimension $(n-k)$, such that
$T_P \Phi \left( H \right) \oplus T_P  N = T_P  M$. Using the
exponential map at $ P$, we build a $\left( {n - k} \right)$-dimensional submanifold $\mathcal{W}_{P} $ of $ N$, orthogonal to
$O_{ P} $ at ${P}$ and such that for any $ Q \in \mathcal{W}_{P}
$, the minimizing geodesics of $ \left( { M,g} \right)$ joining $
P$ and $ Q$ are all contained in $\mathcal{W}_{ P} $.

Let $\Psi:H \times \mathcal{W}_{ P} \to M$ be the map defined by
$\Psi \left( {\tau , Q} \right) = \tau \left( { Q} \right)$. Using
the local inverse theorem, there exist a neighborhood
$\mathcal{V}_{\left( {Id, P} \right)} \subset H \times
\mathcal{W}_{ P} $ of $\left( {Id, P} \right)$ and a neighborhood
$\mathcal{M}_{ P}  \subset M$ such that $\Psi ^{ - 1} =
\left({\Psi _1 \times \Psi _2 } \right)$ from $\mathcal{M}_{ P}$
onto $\mathcal{V}_{\left( {Id, P} \right)} $ is a diffeomorphism.

Up to restricting $\mathcal{V}_{ P}$, we choose a normal chart
$\left( {\mathcal{V}_{ P} ,\varphi _1 } \right)$ around $ P$ for
the metric $\widetilde{g}$ induced on $O_{ P} $, with
$\varphi_1\left({\mathcal{V}_{ P} } \right) = U \subset
\mathbb{R}^k $. In the same way, we choose a geodesic normal chart
$\left( {\mathcal{W}_{ P} ,\varphi _2 } \right)$ around $ P$ for
the metric $\tilde{ \tilde {g}}$ induced on $\mathcal{W}_{ P} $,
with $\varphi _2 \left( {\mathcal{W}_{ P} } \right) = W \subset
\mathbb{R}^{n - k} $.

We denote  $\xi _1  = \varphi _1 \circ \Phi \circ \Psi _1 $,
$\xi _2  = \varphi _2 \circ  \Psi _2 $, $\xi = \left( {\xi _1 ,\xi
_2 } \right)$ and $\Omega = \mathcal{M}_{ P} $.

>From the above and due to Lemma \ref{L6.3} the following property
holds, see Faget \cite{Fag1}.

\begin{lemma}\label{L6.4}
Let $ \left( {M,g} \right)$ be a compact Riemannian $n$-manifold
with boundary, $G$ be a compact subgroup of $I\left( { M,g}
\right)$, $ P \in  M$ with orbit of dimension $k$, $0 \leqslant k
< n$. Then, there exists a chart $\left( {\Omega ,\xi } \right)$
around $P$ such that the following properties are satisfied:
\begin{enumerate}
    \item $\xi \left( \Omega  \right) = U \times W$, where $U
           \subset \mathbb{R}^k $ and $W \subset \mathbb{R}^{n - k} $.
    \item $U$, $W$ are bounded, and $W$ has smooth boundary.
    \item  $\left( {\Omega ,\xi } \right)$ is a normal chart of
          $ M$ around of $P$, $\left({\mathcal{V}_{ P}, \varphi _1 } \right)$ is a normal
          chart around of $ P$ of submanifold $O_{P} $ and $\left( {\mathcal{W}_{ P} ,\varphi _2 } \right)$ is
          a normal geodesic  chart around of $ P$ of submanifold $\mathcal{W}_{ P}$.
    \item For any $\varepsilon>0$, $\left({\Omega,\xi }\right)$ can be chosen such that:
          \[1 - \varepsilon \leq \sqrt {\det \left( {g_{ij} } \right)}  \leq 1 + \varepsilon
          \,\, \mathrm{on}\,\, \Omega , \,\,\mathrm{for}\; 1 \leq i,j \leq n\]
          \[1 - \varepsilon  \leq \sqrt {\det \left( {\tilde g_{ij} } \right)}
            \leq 1 + \varepsilon\,\,\mathrm{on} \,\,\mathcal{V}_{ P} , \,\,\mathrm{for
          }\; 1 \leq i,j \leq k.\]
    \item For any $u \in C_G^\infty(M)$, $u\circ\xi^{- 1}$ depends only on $W$ variables.
\end{enumerate}
\end{lemma}

We say that we choose a neighborhood of $O_P$ when we choose
$\delta>0$ and we consider $O_{\!P,\; \delta }  = \left\{ {Q \in
\widetilde M: d(Q,O_P ) < \delta } \right\}$. Such a neighborhood
of $O_P$ is called \emph{tubular neighborhood}.

Let $P \in M$ and $O_P $ be its orbit of dimension $k$. Since the
manifold $M$ is included in $\widetilde M$, we can choose a normal
chart $\left( {\Omega_P ,\xi_P } \right)$ around $P$ such that
Lemma \ref{L6.4} holds for some $\varepsilon_0>0$. For any $Q =
\tau \left( P \right) \in O_P $, where $\tau  \in G$, we build a
chart around $Q$, denoted by $\left( {\tau \left( \Omega_P
\right)\!, \xi_P  \circ \tau ^{ - 1} } \right)$ and ``isometric''
to $\left( {\Omega_P ,\xi_P } \right)$. $ O_P $ is then covered by
such charts. We denote by $\left( {\Omega _{P,m} } \right)_{m =
1,...,M} $ a finite extract covering. Then we can choose
$\delta>0$ small enough, depending on $P$ and $\varepsilon_0$ such
that the tubular neighborhood $ O_{\!P,\; \delta }  = \left\{ {Q
\in \widetilde M: d(Q,O_P ) < \delta } \right\}$, (where $d\left(
{ \cdot ,O_P } \right)$ is the
distance to the orbit) has the following properties:\\
$(i)\;$ $\overline { O_{\!P,\; \delta }} $ is a
submanifold of $\widetilde M$ with boundary;\\
$(ii)\;$ $d^2 \left( { \cdot ,O_P } \right)$,  is a $C^\infty $
function on $ O_{\!P,\; \delta } $; \\
$(iii)\;$ $ O_{\!P,\; \delta } $ is covered by $\left(
{\Omega _m } \right)_{m = 1,...,M} $.

Clearly, $M$ is covered by $\bigcup\nolimits_{P \in M} O_{\!P,\;
\delta } $. We denote $\left( {O_{j, \,\delta } } \right)_{j =
1,...,J} $ a finite extract covering of $M$, where all $O_{j,
\,\delta } $'s are covered by $\left( {\Omega _{jm} } \right)_{m =
1,...,M_j } $. Therefore:
$$
M\subset \bigcup\limits_{j = 1}^J
{\bigcup\limits_{m = 1}^{M_j } {\Omega _{jm} } }=\bigcup\limits_{m
= 1}^{\sum\nolimits_{j = 1}^J {M_j } }{\Omega _i } .
$$ So we
obtain a finite covering of $M$ consisting of $\Omega_i$'s, $i =
1,...,\sum\nolimits_{j = 1}^J {M_j } $. We
choose  such a covering in the following way:\\
$(i)\;$ if $P$ lies in the interior of $M$, then there exist $j,\;
1\leq j\leq J$ and $m,\; 1\leq m\leq M_j$ such that the tubular
neighborhood $O _{j,\, \delta} $ and $\Omega _{jm} $, with
$P\in\Omega _{jm} $,  lie entirely  in the interior of $M$, that
is, if $P \in M\backslash \partial M$, then $O_{j, \delta}\subset
M\backslash\partial M $ and $\Omega_{jm} \subset M\backslash
\partial M$;\\
$(ii)\;$ if $P$ lies on the boundary  $\partial M$ of $M$, then
there is some $j,\; 1\leq j\leq J$  such that the tubular
neighborhood $O _{j,\, \delta} $ intersects the boundary
$\partial M$ and an $m,\; 1\leq m\leq M_j$ exists, such that
$\Omega _{jm} $, with $P\in\Omega _{jm} $, intersects a part of
the boundary $\partial M$. Then the $\Omega _{jm} $ cover a patch
of the boundary of $M$ and the whole of the  boundary is covered
by charts around $P \in\partial M$.


Let $N$ denote the projection of the image of $M$ through the
charts $\left( {\Omega _{jm} ,\xi _{jm} } \right)$, ${j = 1,...,J
} $, ${m = 1,...,M_j } $,  on $\mathbb{R}^{n - k} $. Then $\left(
{N,\bar g} \right)$  is a $(n-k)$-dimensional compact submanifold with boundary
of $\mathbb{R}^{n - k} $  and $N$ is covered by
$\left( {W_i } \right)$, $i = 1,...,\sum\nolimits_{j = 1}^J {M_j }
$, where $W_i $ is the component of $\xi _i \left( {\Omega _i }
\right)$ on $\mathbb{R}^{n - k} $ for all $i =
1,...,\sum\nolimits_{j = 1}^J {M_j }$. Let $p $  be the projection
of $\xi _i \left( {P } \right),P \in M$ on $\mathbb{R}^{n - k} $.
Thus, one of the following properties holds:\\
$(i)\;$ if $p  \in N\backslash \partial N$ then $W_i \subset
N\backslash
\partial N$ and $W_i$ is a normal geodesic neighborhood with
normal geodesic coordinates $\left( {y_1 ,...,y_{n - k} }
\right)$;\\
$(ii)\;$ if $p  \in \partial N$ then $W_i$ is a Fermi
neighborhood with Fermi coordinates $\left( {y_1 ,...,y_{n - k - 1} ,t} \right)$.\\
In these neighborhoods of $N$ we have:
$$
1 - \varepsilon _0 \leqslant \sqrt {\det \left( {\bar g_{ij} }
\right)}  \leqslant 1 + \varepsilon _0\,\,\mathrm{for}\,\,1\leq
i,j\leq n-k,$$ where $\varepsilon_0 $ can be as small as we want,
depending on the chosen covering.\\
Set:
$$
O_j  =  O_{j,\, \delta} = \left\{ {Q \in \widetilde M: d(Q,O_{P
_j } ) < \delta } \right\} \quad \mathrm{and} \quad \left( {\Omega
_{jm} ,\xi _{jm} } \right) = \left( {\Omega _m ,\xi _m } \right).
$$

\begin{lemma}\label{L6.5}
$\mathrm{(see\; \cite{Fag2})}$ Let $(M,g)$ be a  compact
Riemannian $n$-dimensional manifold, $G$ be a compact subgroup of the isometry
group of $M$. Then there exists an orbit of minimum dimension $k$ and
of minimum volume.
\end{lemma}

\begin{lemma}\label{L6.6}
\noindent  Let $O_j  = \{{Q\in \widetilde M :
d(Q,O_{P_j})<\delta}\}$ be an arbitrary tubular neighborhood of
$M$, $V_j  = Vol\left( {O_j } \right)$,
$\phi=\upsilon\circ\xi^{-1}$ and $c$ be a positive constant. Then
for any $\upsilon \in H_{1,G}^1 \left( {O_j \cap \partial M}
\right)$, $\upsilon \geqslant 0$ the following inequalities are
valid:
\begin{align}\label{E79}
&(a)\quad\left( {1 - c\varepsilon _0 } \right)V_j \int_{\partial
N} {\phi\, ds_{\bar g} }  \leqslant \int_{\partial M} {\upsilon
dS_g }  \leqslant \left( {1 + c\varepsilon _0 } \right)V_j
\int_{\partial N} {\phi\, ds_{\bar g} };\\\label{E80}
&(b)\quad\left( {1 - c\varepsilon _0 } \right)V_j \int_N { \phi}\,
d\upsilon _{\bar g}   \leqslant \int_M { \upsilon }   dV_g
\leqslant \left( {1 + c\varepsilon _0 } \right)V_j \int_N {
\phi}\, d\upsilon _{\bar g};\\\label{E81}
&(c)\quad\left({1-c\varepsilon_0} \right)V_j \!\int_N \!
{\left|{\nabla _{\bar g} \phi } \right| d\upsilon _{\bar g} }\!
\leqslant\! \int_M \!{\left| {\nabla _g \upsilon } \right| dV_g
}\!\leqslant\!\left({1+c\varepsilon_0} \right)\!V_j\! \int_N
\!\!{\left| {\nabla _{\bar g} \phi} \right| d\upsilon _{\bar g}. }
\end{align}
\end{lemma}
The proof of Lemma \ref{L6.6} is provided in Appendix.

The following theorem  concerns  the exact value  of the first
best constant  of the classical Sobolev inequality for $p=1$, in
the case that the manifold is invariant under the action of a
compact group $G$ of the isometries without finite subgroup.

\begin{theorem}\label{T6.3}
Let $(M,g)$ be a smooth, compact  Riemannian $n$-dimensional manifold with
boundary, $n\geq 3$,  invariant under the action of a subgroup $G$
of the isometry group $I(M,g)$. Let $k$ denote  the minimum orbit
dimension of $G$ and let $V$ denote  the minimum of the volume of
the $k$-dimensional orbits. Then, for any $\varepsilon>0$ there
exists a real constant $B$ such that for all $u \in {H}^1_{1,G}
\left( M \right)$ the following inequality holds:
\begin{equation}\label{E82}
\left( {\int_M {\left| u \right|^{p^*} d\upsilon _g } }
\right)^{1/p^*} \leqslant \left(2^{1/{(n-k)}} {K_G + \varepsilon }
\right) \int_M {\left| {\nabla u} \right| d\upsilon _g }  + B
\int_M {\left| u \right| d\upsilon _g },
\end{equation}
where $p^*=\frac{\textstyle n-k}{\textstyle n-k-1}$ and  $K_G =
\frac{\textstyle K(n - k,1)}{\textstyle V^{1/(n - k) }}$.
\smallbreak\noindent Moreover, $2^{1/{(n-k)}} K_G $ is the best
constant for this inequality.
\end{theorem}

Regarding the existence of a second best constant of the classical
Sobolev inequality, with  $p=1$, for reasons similar to those of
the general case it cannot be formulated a global theorem devoted
to the calculation of it. However, in some cases this constant can
be computed. For example, as in the case of the solid torus (see
Theorem \ref{T4.1}).

We present now our last two theorems in which  the exact values of
the best constants for trace Sobolev inequalities are calculated
for $p=1$,  in the case that the manifold is invariant under the
action of a compact group $G$ of the isometries without finite
subgroup, when they exist.

\begin{theorem}\label{T6.4}
Let $(M,g)$ be a smooth, compact Riemannian $n$-manifold with
boundary, $n\geq 3$,  invariant under the action of a subgroup $G$
of the isometry group $I(M,g)$. Let $k$ denote  the minimum orbit
dimension of $G$ and let $N$ be the compact manifold with boundary
which is the projection of  $M$ on $\mathbb{R}^{n-k}$. Then, for
any $\varepsilon>0$  there exists a real constant $\tilde B$ such
that for all $u \in {H}^1_{1,G} \left( M \right)$ the following
inequality holds:
\begin{equation}\label{E83}
\int_{\partial M} {\left| u \right| ds_g } \leqslant
(1+\varepsilon) \int_M {\left| {\nabla u} \right| d\upsilon _g } +
\tilde B \int_M {\left| u \right| d\upsilon _g}.
\end{equation}
In addition, $1=\tilde K\left(n-k,1\right)$ is the best first
constant for this inequality.
\end{theorem}

\begin{theorem}\label{T6.5}
Let $(M,g)$ be a smooth, compact Riemannian $n$-manifold with
boundary, $n\geq 3$,  invariant under the action of a subgroup $G$
of the isometry group $I(M,g)$. Let $k$ denote  the minimum orbit
dimension of $G$ and let $N$ be the compact manifold with boundary
which is the projection of  $M$ on $\mathbb{R}^{n-k}$. If $N$
is connected, then there exists a real constant $\tilde A$ such
that for all $u \in {H}^1_{1,G} \left( M \right)$ the following
inequality holds:
\begin{equation}\label{E84}
\int_{\partial M} {\left| u \right| ds_g } \leqslant \tilde A
\int_M {\left| {\nabla u} \right| d\upsilon _g } + \frac{|\partial
N|}{|N|} \int_M {\left| u \right| d\upsilon _g}.
\end{equation}
In addition,,  $|\partial N|/|N|$ is the best second constant for
this inequality.
\end{theorem}

\begin{remark}\label{R6.2}
\emph{If the manifold $N$ is not connected the result of the
Theorem \ref{T6.5} concerning the second best constant fails (see Motron
\cite{Mot}).}
\end{remark}

\noindent{{\bf Proof of Theorem \ref{T6.5}.} Let $( {O_{j,
\,\delta } } )_{j = 1,...,J} $ be a finite covering of $M$ and
$(\eta_j)_{j=1,...,J}$ be a partition of unity associated to this
covering. Then by Proposition 3.10 in \cite{Mot} and Lemma
\ref{L6.6}, for any $u\in H^1_{1,G}(M)$, if we set
$\eta_j|u|\circ\xi=|\phi_j|$, we obtain:
\begin{eqnarray}\label{E85}
\nonumber\int_{\partial M} {\left| u \right|ds_g } & =&
\int_{\partial M} {\left( {\sum\limits_{j = 1}^J {\eta_j } }
\right)\left| u \right|ds_g }  = \sum\limits_{j = 1}^J
{\int_{\partial M} {\left( {\eta_j \left| u \right|} \right)ds_g } }  \hfill \\
\nonumber& \leqslant & \sum\limits_{j = 1}^J {\left( {1 +
\varepsilon } \right)V_j
\int_{\partial N} {\left| {\phi _j } \right|ds_{\tilde g} } }  \hfill \\
\nonumber& \leqslant &  \left( {1 + \varepsilon }
\right)\sum\limits_{j = 1}^J {V_j \left( {\tilde A\int_N {\left|
{\nabla \phi _j } \right|dv_{\tilde g} }  + \frac{{\left|
{\partial N} \right|}}
{{\left| N \right|}}\int_N {\left| {\phi _j } \right|dv_{\tilde g} } } \right)}  \hfill \\
\nonumber& \leqslant & \left( {1 + \varepsilon }
\right)\sum\limits_{j = 1}^J {V_j \left( {\tilde A\frac{1}
{{\left( {1 - \varepsilon } \right)V_j }}\int_M {\left| {\nabla
\left( {\eta_j \left| u
\right|} \right)} \right|dv_g} } \right)}  \hfill \\
\nonumber&&+ \left( {1 + \varepsilon } \right)\sum\limits_{j =
1}^J {V_j \left( { \frac{1} {{\left( {1 - \varepsilon } \right)V_j
}}\frac{{\left| {\partial N} \right|}}
{{\left| N \right|}}\int_M {\left( {\eta_j \left| u \right|} \right)dv_g } } \right)}  \hfill \\
& \leqslant & \frac{{1 + \varepsilon }} {{1 - \varepsilon
}}\left(\tilde A\int_M {\left| {\nabla u} \right|dv_g }  +
\frac{{\left| {\partial N} \right|}} {{\left| N \right|}}\int_M
{\left| u \right|dv_g } \right)
\end{eqnarray}
Relation (\ref{E85}) implies  that $\tilde B_{1,G}(M)\geq  |\partial
N|/|N| $. In particular $ \mathcal{\tilde B}_{1,G}(M)\geq
|\partial N|/|N|$.

Suppose by contradiction that for any $\alpha\in\mathbb{N}$ there
exists $u_\alpha\in H_{1,G}^1(M)$ such that:
\begin{eqnarray}\label{E86}
\int_{\partial M} {\left| {u_\alpha }\right|ds_g \geqslant
\alpha\int_M {\left| {\nabla u_\alpha } \right|}dv_g +   \frac{
|\partial N |} {{\left| N \right|}}\int_M  \left| {u_\alpha
}\right|}dv_g.
\end{eqnarray}
Without loss of generality we can assume that all the functions
$u_\alpha$ are defined in the orbit $O_j$. Thus by (\ref{E86}), if
we set $u_\alpha \circ \xi_j=(\phi_j)_\alpha$, we deduce that:
\begin{eqnarray*}
\left( {1+\varepsilon } \right)V_j \int_{\partial N} {\left|
{(\phi_j)_\alpha} \right|ds_{\tilde g} }\!\!\!
&\geqslant&\!\!\!\left( {1 - \varepsilon }\! \right)V_j \left(\!
{\alpha\!\!\int_N {\!\left| {\nabla (\phi_j)_\alpha }
\right|dv_{\tilde g} }\!   +\! \frac{{\left| {\partial N}
\right|}}{{\left| N \right|}}\int_N {\left| {(\phi_j)_\alpha }
\right|dv_{\tilde g} } } \right),
\end{eqnarray*}
\begin{eqnarray*}
\int_{\partial N} {\left| {(\phi_j)_\alpha } \right|ds_{\tilde g}
} & \geqslant &\frac{{1 - \varepsilon }} {{1 + \varepsilon
}}\,\left( {\alpha\int_N {\left| {\nabla (\phi_j)_\alpha }
\right|dv_{\tilde g} } + \frac{{\left| {\partial N} \right|}}
{{\left| N \right|}}\int_N {\left| {(\phi_j)_\alpha}
\right|dv_{\tilde g} } } \right),
\end{eqnarray*}
\begin{eqnarray}\label{E87}
\int_{\partial N} \!\!{\left| {(\phi_\alpha)_j } \right|ds_{\tilde
g} }  & \geqslant & (\!\alpha-1)\!\int_N\!{\left| {\nabla (
\phi_j)_\alpha } \right|dv_{\tilde g} } + \frac{{1-\varepsilon }}
{{1+\varepsilon }}\frac{{\left| {\partial N} \right|}} {{\left| N
\right|}}\!\int_N \!{\left| {(\phi_j)_\alpha } \right|dv_{\tilde
g}}.
\end{eqnarray}
The  inequality  (\ref{E87}) is false since $\varepsilon$ can be
chosen arbitrarily small and since the constant $|\partial N|/|N|$
is optimal (see  \cite{Mot}) for the inequality:
\[
\int_{\partial N}|\phi|ds_{\tilde g}  \leq \tilde A \int_N |\nabla
\phi|dv_{\tilde g} + \frac{|\partial N|}{|N|}\int_N
|\phi|dv_{\tilde g}.
\]
The proof is now complete.  \mbox{}$\Box$

\smallskip
We omit the proofs of Theorems \ref{T6.3} and  \ref{T6.4}
since they rely on similar arguments  in the case of
the torus, in combination with Lemmas \ref{L6.3}-\ref{L6.6}.

\begin{remark}\label{R6.3}
\emph{We cannot formulate a global theorem that
concerns  the trace Sobolev inequality on manifolds with
boundary in the presence  of symmetries, namely to establish  an
inequality where to the positions of $\tilde A$ and $\tilde B$ to
put together the best constant $1=\tilde K\left(n-k,1\right)$ and
$|\partial N|/|N|$. In some cases, such as on the solid torus or
on the disk of $\mathbb{R}^2$, there are extremals for this
inequality (see Theorem \ref{T4.1}).}
\end{remark}

\begin{remark}\label{R6.4}
\emph{The parameter $\varepsilon$ that appears in Theorems
\ref{T4.1}, \ref{T4.2}, \ref{T6.3} and \ref{T6.4} controls in some
sense the thinness of the cover that we use in each case through
the related partition of unity. Thus, its existence is absolutely
necessary because we do not know if the inequalities are valid
without this parameter. Although in some cases,  Sobolev
inequalities exist without $\varepsilon$ (see, e.g.,
\cite{Cot-Lab1, Dru, Fag2, Heb-Vau2}), but in general we cannot
make it disappear. }
\end{remark}

\appendix{\noindent\textbf{\Large{Appendix}}\\
\addcontentsline{toc}{section}{\textbf{Appendix}}

\noindent{\bf{Proof of Lemma \ref{L4.1}.}} Suppose by
contradiction, that  there exist $\tilde A' <  \tilde K (2,1)$ and
$\tilde B'$ such that for all $ u \in H_{1,G}^1 (T)$ the following
inequality holds:
 \begin{equation}\label{E88}
\int_{\partial T} {|u|} dS \leq \tilde A'\int_T {|\nabla u|}
dV+\tilde B'\int_T | u| dV.
\end{equation}

Consider a transformation of the disc  $F:D \to \mathbb{R}_ + ^2
$. Such a transformation is, for example, $$F\left( {t,s} \right)
= \left( {\frac{{\textstyle 4t}} {\textstyle{t^2 + \left( {1 + s}
\right)^2 }},\frac{\textstyle{2\left( {1 - t^2  - s^2 } \right)}}
{\textstyle{t^2 + \left( {1 + s} \right)^2 }}} \right),$$ see
Escobar \cite {Esc}. Choose a finite covering of $\bar D$
consisting of discs $D_k$, centered on $p_k$, such that:\\$(a)$ If
$p_k \in D $, then entire $D_k$ lies in $D$;\\$(b)$ if $p_k\in
\partial D$, then $D_k$ is a Fermi neighborhood.

\noindent In these neighborhoods we have:
\begin{equation}\label{E89}
1 - \varepsilon _0  \leqslant \sqrt {\det \left( {\tilde
g_{\alpha\beta} } \right)}  \leqslant 1 + \varepsilon _0.
\end{equation}
Fix now a point $P_0 \in \partial T$  that belongs to the orbit of
minimum range $l-r$. For any $\varepsilon_0>0$  we can choose $
\delta  = \varepsilon_0 (l - r) < 1$ and $T_\delta   = \left\{ {Q
\in \mathbb{R}^3 :d(Q,O_{P_0 } ) < \delta } \right\}$ such that if
$I \times U \subset I \times D$ is the image of a neighborhood of
$P_0  \in \partial T$ through the chart $\xi$ of $T$ and $V
\subset \mathbb{R}_ + ^2 $ is  the image of $U$ through $F$, then
(\ref{E89}) holds. It follows by (\ref{E88}) that  for any $u\in
C_0^\infty(T_\delta)$ we have:
$$
 \int_{\partial T_\delta  }
\left| u \right| dS \leqslant \tilde A'\int_{T_\delta  } \left|
{\nabla u} \right|dV+\tilde B'\int_{T_\delta  } | u| dV.
$$
Relations  (\ref{E16}-\ref{E18}) yield successively:
\begin{eqnarray*}
\int_{\partial D} {\left| \phi \right|}  (l - r + \delta t)dt
\leqslant \tilde A'\int_{D} {\left| {\nabla \phi } \right|} (l - r
+ \delta t)dtds+  \tilde B'\int_{D} {\left|\phi  \right|} (l - r +
\delta t)dtds;
\end{eqnarray*}
\begin{eqnarray*} {\left( {1 - \varepsilon _0 } \right)\int _{F(\partial D)}
{\left( {\left| \phi \right| \sqrt {\tilde g} } \right)} \circ F^{
- 1} dx' } \!\!\!& \le& \!\!\!( 1 + \varepsilon _0 )\tilde
A'\int_{F(D)} {\left( {\left| {\nabla \phi } \right|  \sqrt
{\tilde g} } \right) \circ F^{ - 1} }dx\\
&+&\!\!\!(1 + \varepsilon _0 )\delta \tilde B'\int_{F(D)} {\left(
{\left| \phi \right|  \sqrt {\tilde g} } \right) \circ F^{ - 1}
}dx,
\end{eqnarray*}
$$
 \left( 1 - \varepsilon_0   \right)^2
 \int _{\partial \mathbb{R}_ + ^2 }  \left| \Phi
\right| dx'    \leqslant \left(1 + \varepsilon_0 \right)^2\left(
\tilde A'  \int_{\mathbb{R}_ + ^2 }  \left|  \nabla \Phi \right|
dx+\delta \tilde B'\int_{\mathbb{R}_ + ^2 } | \phi | dx\right);
$$
\begin{equation}\label{E90}
 \int _{\partial \mathbb{R}_ + ^2 }  \left| \Phi
\right| dx'    \leqslant \left( {\frac{{1 + \varepsilon_0 }}{{1 -
\varepsilon_0 }}} \right)^2 \left( \tilde A'  \int_{\mathbb{R}_ +
^2 }  \left|  \nabla \Phi \right| dx+\delta \tilde
B'\int_{\mathbb{R}_ + ^2 } | \phi | dx\right).
\end{equation}
By (\ref{E90}) we deduce that for $\varepsilon_0$ small enough,
the following inequality holds:
\begin{equation*}
\tilde A''=\left( {\frac{{1 + \varepsilon_0 }}{{1 - \varepsilon_0
}}} \right)^2 \tilde A' < \tilde K(n,1)=1
\end{equation*}
So, for  $\varepsilon_0$ small enough  and for all $\Phi  \in
C_0^\infty \left( D \right)$ we have:
\begin{equation}\label{E91}
{\int_{\partial \mathbb{R}_ + ^2 } {\left| \Phi \right|}  dx'}
\leqslant \tilde A''\int_{ \mathbb{R}_ + ^2} {\left| {\nabla \Phi
} \right|} dx+ \tilde B''\int_{ \mathbb{R}_ + ^2} {\left| { \Phi
}\right|} dx
\end{equation}
Let $\Psi  \in C_0^\infty  \left( {\mathbb{R}_ + ^2 } \right)$ and
$\Psi  _\lambda  (x) = \lambda  \Psi  (\lambda x),\,\lambda
> 0$. Then:
\begin{align*}
&\int_{\partial \mathbb{R}^2_+} {\left| \Psi_\lambda \right| dx' }
=\int_{\partial \mathbb{R}^2_+} {\left| \Psi\right| dx' },\quad
{\int_{\mathbb{R}^2_+}\left| {\nabla \Psi_\lambda} \right| dx
}={\int_{\mathbb{R}^2_+}\left| {\nabla \Psi} \right| dx },\quad
\mathrm{and} \\&\int_{ \mathbb{R}^2_+} {\left|\Psi_\lambda \right|
dx} = \frac{1}{\lambda}\int_{ \mathbb{R}^2_+}{\left| \Psi\right|
dx }.\end{align*}
Thus since $\Psi_\lambda \in C_0^\infty  \left(
D \right)$ for $\lambda>0$ sufficiently large, relation
(\ref{E91}) yields the following inequality:
\[
{\int_{\partial \mathbb{R}_ + ^2 } {\left| \Psi \right|} dx'}
  \leqslant A''\int_{ \mathbb{R}_ + ^2 } {\left|\nabla\Psi
\right|}  dx + B''\frac{1}{\lambda}\int_{ \mathbb{R}_ + ^2 }
{\left|\Psi \right|} dx.
\]
Taking $\lambda\to\infty$ we obtain that the inequality:
\[
{\int_{\partial \mathbb{R}_ + ^2 } {\left| \Psi \right|} dx'}
  \leqslant A''\int_{ \mathbb{R}_ + ^2 } {\left|\nabla\Psi
\right|}  dx
\]
holds for all $\Psi   \in C_0^\infty  \left( {\mathbb{R}_ + ^2 }
\right)$, with $\tilde A''<1$. This contradicts the inequality
(\ref{E33}), which asserts that $\tilde K(n,1)=1$ is the best
constant for the Sobolev trace inequality in $\mathbb{R}_ + ^2 $.
Then, the lemma is proved. \mbox{} $\Box$

\smallbreak \noindent{\bf{Proof of Lemma \ref{L6.1}.}}$\,$
Suppose by contradiction that there exist a Riemannian
$n$-manifold $(M,g)$ and real numbers $ A<1= \tilde
K\left(n,1\right)$ and $\tilde B$ such that the inequality
(\ref{E19}) is true  for all $u\in H_1^1(M)$. Let $P_0\in\partial
M$. Given $\varepsilon >0$,  let $B_\delta(0)\subset
\mathbb{R}_+^n$ be the imagine of a convex neighborhood centered
at $P_0$ through a chart $(\Omega, \xi)$ of $M$, which can be
chosen such that
$$1-\varepsilon  \leq \sqrt{\mathrm{det}(g_{ij})} \leq
1+\varepsilon.$$
Thus by (\ref{E22}), if we choose $\varepsilon$
small enough, it follows that there are real numbers $A'<1$
and $B'$ such that for all $u\in C_0^\infty(B_0(\delta))$,
\begin{equation}\label{E92}
\int_{\partial \mathbb{R}^n_+} {\left| u \right| dx' }  \le
A'{\int_{\mathbb{R}^n_+}\left| {\nabla u} \right| dx }+
B'{\int_{\mathbb{R}^n_+}\left|  u \right| dx }.
\end{equation}

Fix $u\in C_0^\infty(\mathbb{R}^n_+)$ and  set
$u_\lambda(x)=\lambda^{n-1}u(\lambda x)$, where $\lambda$ is a
positive real number. If we choose  $\lambda$  sufficiently large
then $u_\lambda\in C_0^\infty(B_0(\delta))$, and thus by
(\ref{E92}) we obtain:
\begin{equation}\label{E93}
\int_{\partial \mathbb{R}^n_+} {\left| u_\lambda \right| dx' } \le
A'{\int_{\mathbb{R}^n_+}\left| {\nabla u_\lambda} \right| dx }+
B'{\int_{\mathbb{R}^n_+}|  u_\lambda| dx }.
\end{equation}
By rescaling we obtain:
\begin{align*}
&\int_{\partial \mathbb{R}^n_+} {\left| u_\lambda \right| dx'
}=\int_{\partial \mathbb{R}^n_+} {\left| u\right| dx' },\quad
{\int_{\mathbb{R}^n_+}\left| {\nabla u_\lambda} \right|
dx}={\int_{\mathbb{R}^n_+}\left| {\nabla u} \right| dx }\quad
\mathrm{and} \\&\int_{ \mathbb{R}^n_+} {\left| u_\lambda \right|
dx} = \frac{1}{\lambda}\int_{ \mathbb{R}^n_+}{\left| u\right| dx }\,.
\end{align*}
Thus by (\ref{E93}), we deduce that:
\begin{equation}\label{E94}
\int_{\partial \mathbb{R}^n_+} {\left| u \right| dx' } \le
A'{\int_{\mathbb{R}^n_+}\left| {\nabla u} \right| dx
}+B'\frac{1}{\lambda}\int_{ \mathbb{R}^n_+}{\left| u\right| dx }.
\end{equation}
Taking $\lambda\to\infty$ in (\ref{E94})  we obtain that for all
$u\in C^\infty_0(\mathbb{R}_+^n)$ the following inequality holds
\begin{equation}\label{E95}
\int_{\partial \mathbb{R}^n_+} {\left| u \right| dx' } \le
A'{\int_{\mathbb{R}^n_+}\left| {\nabla u} \right| dx },
\end{equation}
with $A'<1$. This contradicts the inequality (\ref{E22}), which establishes
 that  $ 1=\tilde K\left(n,1\right) $ is the
best constant for the Sobolev trace inequality in $\mathbb{R}_+^n$
(see Motron \cite{Mot} and Park \cite{Par}), and the lemma is
proved. \mbox{} $\Box$

\smallbreak \noindent{{\bf Proof of Lemma \ref{L6.6}.} $(a)$  Let
$P \in \partial M$, $O_{P } $ be its orbit and $\upsilon  \in
H_{1,G}^1 \left( {O_j  \cap M} \right)$, $\upsilon \geqslant 0$.
Then, by Lemma 3.3 (4) in Cotsiolis and Labropoulos
\cite{Cot-Lab3} it follows that:
\begin{eqnarray}
\nonumber \int _{\partial M} {\upsilon ds_g }  &=& \int _{\partial
M} {\left( {\sum\limits_{m = 1}^{M_j} {\beta _m } } \right)}
\upsilon dS_g = \sum\limits_{m = 1}^{M_j} {\int _{\partial M}
{\beta _m \upsilon dS_g }   } = \sum\limits_{m = 1}^{M_j} {\int
_{\partial M \cap \Omega _m } {\beta _m
\upsilon dS_g } }   \\
\nonumber & = & \sum\limits_{m = 1}^{M_j} {\int _ {\xi \left(
{\partial M \cap \Omega _m } \right)} {\sqrt {\det \left(
{g_{kl}^m } \right)}
\beta _m \upsilon  \circ \xi _m^{ - 1} dxds_{\bar g} } } \\
\nonumber & = & \sum\limits_{m = 1}^{M_j} {\int _{U_m \times
\left( {\partial N \cap W_m } \right)} {\sqrt {\det \left(
{g_{kl}^m }
\right)} \beta _m \upsilon  \circ \xi _m^{ - 1} dxds_{\bar g} } }  \\
\label{E96} & \leqslant & \left( {1 + \varepsilon _0 }
\right)\sum\limits_{m = 1}^{M_j} {\int _{U_m  \times \left(
{\partial  N \cap W_m } \right)} {\beta _m \upsilon  \circ \xi
_m^{ - 1} dxds_{\bar g} }}.
\end{eqnarray}
Since for each $m$, $\beta _m  \circ \xi _m^{ - 1} $ is
independent of the $W_m $'s variables, we denote by $\beta _{1m} $
the function $\beta _m  \circ \xi _m^{ - 1} $ and regard this
function as defined on $U_m $. In the same way,  we denote by
$\upsilon _{2m} $ the function $\upsilon  \circ \xi _m^{ - 1} $
which is considered  as defined on $W_m $, since according to
 \cite[Lemma 3.3]{Cot-Lab3} it depends only on the $W_m $'s
variables. Thus by relation (\ref{E96}), we obtain:
\begin{equation}\label{E97}
\int _{\partial M} {\upsilon dS_g }  \leqslant \left( {1 +
\varepsilon _0 } \right)\sum\limits_{m = 1}^{M_j} {\int _{U_m }
{\beta _{1m} } dx\int _{\partial N \cap W_m } {\upsilon _{2m}
ds_{\bar g} } }.
\end{equation}
As the charts $\left( {\Omega _m ,\xi _m } \right)$ are isometric
to each other and since $\upsilon$ is $G$-invariant, $
\int_{\partial N \cap W_m } {\upsilon _{2m} ds_{\bar g} }$ does
not depend on $m$. Thus, relation (\ref{E97}) leads to:
\begin{equation}\label{E98}
\int _{\partial M} {\upsilon dS_g }  \leqslant \left( {1 +
\varepsilon _0 } \right)\int _{\partial N \cap W} {\upsilon _2
ds_{\bar g} } \sum\limits_{m = 1}^{M_j} {\int _{U_m } {\beta _{1m}
} dx}.
\end{equation}
Moreover, according to Lemma 3.3 (3) Cotsiolis and Labropoulos
\cite{Cot-Lab3} we have:
\begin{equation}\label{E99}
\left( {1 - \varepsilon _0 } \right)\int _{U_m } {\beta _{1m} } dx
\leqslant \int _{U_m } {\beta _{1m} \sqrt {\det \left( {\tilde
g_{kl}^m } \right)} }  \circ \varphi _{1m}^{ - 1} dx = \int
_{\mathcal{V}_j } {\beta _{1m} }  \circ \varphi _{1m}^{ - 1}
d\upsilon_{\widetilde{g}}.
\end{equation}
Finally, by (\ref{E98}) and (\ref{E99}) we deduce that:
\begin{eqnarray}\label{E100}
\nonumber\int _{\partial M} {\upsilon dS_g } & \leqslant &
\frac{{1 + \varepsilon _0 }} {{1 - \varepsilon _0 }}\int
_{\partial N \cap W} {\upsilon _2 ds_{\bar g} } \sum\limits_{m =
1}^{M_j} {\int _{\mathcal{V}_{jm} } {\beta _{1m} }
\circ \varphi _{1m}^{ - 1} d\upsilon_{\widetilde{g}} }   \\
\nonumber&= & \frac{{1 + \varepsilon _0 }} {{1 - \varepsilon _0
}}\int _{\partial N \cap W} {\upsilon _2 ds_ {\bar g} } \int _{O_j
} {\sum\limits_{m = 1}^{M_j} {\beta _{1m} } }
\circ \varphi _{1m}^{ - 1} d\upsilon_{\widetilde{g}} \\
\nonumber & = & \frac{{1 + \varepsilon _0 }} {{1 - \varepsilon _0
}}V_j \int _{\partial N \cap W_j } {\upsilon
_2 ds_{\bar g} }  \\
\nonumber& = & \frac{{1 + \varepsilon _0 }} {{1 - \varepsilon _0
}}V_j \int _{\partial N} {\upsilon _2 ds_{\bar g} }\\
& \leqslant &\left( {1 + c_1 \varepsilon _0 } \right)V_j \int
_{\partial N} {\upsilon _2 ds_{\bar g} },
\end{eqnarray}
where $c_1  \geqslant 2/\left( {1 - \varepsilon _0 } \right)$.

Following the same arguments as in the proof of (\ref {E100}) we
can show that:
\begin{equation}\label{E101}
\int _{\partial M} {\upsilon dS_g }  \geqslant \left( {1 - c_2
\varepsilon _0 } \right)V_j \int _{\partial N} {\upsilon _2
ds_{\bar g} },
\end{equation}
where $c_2  \geqslant 2/\left( {1 + \varepsilon _0 } \right)$.\\
Set now $c\geq \max\left(c_1,c_2\right)$ and the first part  of
the proposition is proved. \smallbreak\noindent$(b)$ The proof of
this part is analogous  to the proof of $(a)$.
\smallbreak\noindent$(c)$ Under the same considerations as in the
part $(a)$  and always in the same spirit we get successively:
\begin{eqnarray}\label{E102}
\nonumber\int _M {\left| {\nabla _g \upsilon } \right|  dV_g } &=&
\int _M {\left( {\sum\limits_{m = 1}^{M_j} {\beta _m } } \right)}
\left| {\nabla _g \upsilon } \right|  dV_g  = \sum\limits_{m =
1}^{M_j} {\int _M {\beta _m \left| {\nabla _g \upsilon } \right|  dV_g } } \\
\nonumber&= & \sum\limits_{m = 1}^{M_j} {\int _{M \cap \Omega _m }
{\beta
_m  \left| {\nabla _g \upsilon } \right|^p dV_g } }\\
\nonumber&= & \sum\limits_{m = 1}^{M_j} {\int _{\xi \left( {M \cap
\Omega _m }   \right)} {\sqrt {\det \left( {g_{kl}^m } \right)}
\beta _m \left|   {\nabla _g \upsilon } \right|   \circ \xi _m^{ -
1}
dxd\upsilon   _{\bar g} } }   \\
\nonumber& = &  \sum\limits_{m = 1}^{M_j} {\int _{U_m  \times
\left( {N \cap W_m }    \right)} {\sqrt {\det \left( {g_{kl}^m }
\right)} \beta _m \left|    {g_m^{kl} \partial _k \partial _l
\upsilon }
\right|  \circ \xi    _m^{ - 1} dxd\upsilon _{\bar g} } }   \\
\nonumber&  \geqslant & \frac{{1 - \varepsilon _0 }} {{1 +
\varepsilon _0 }}\sum\limits_{m = 1}^{M_j} {\int _{U_m  \times
\left( {N \cap W_m } \right)} {\left( {\beta _m  \circ \xi _m^{ -
1} } \right)\left| {\nabla _e \left( {\upsilon  \circ \xi _m^{ -
1} } \right)}
\right|  dxd\upsilon _{\bar g} } }\\
\nonumber& \geqslant & \frac{{\left( {1 - \varepsilon _0 }
\right)^2 }} {{1 + \varepsilon _0 }}\sum\limits_{m = 1}^{M_j}
{\int _{U_m \times \left( {N \cap W_m } \right)} {\left( {\beta _m
\circ \xi _m^{ - 1} }  \right)\left| {\bar g_m^{kl} \partial _k
\partial _l \left( {\upsilon \circ \xi _m^{ - 1} } \right)}
\right|  dxd\upsilon _{\bar g} } } \\
\nonumber& \geqslant & \frac{{\left( {1 - \varepsilon } \right)^2
}} {{1 + \varepsilon }}\sum\limits_{m = 1}^{M_j} {\int _{U_m
\times \left( {N \cap W_m } \right)} {\left( {\beta _m \circ \xi
_m^{ - 1} } \right)\left| {\nabla _{\bar g} \left( {\upsilon \circ
\xi _m^{ -
1}} \right)} \right|  dxd\upsilon _{\bar g} } }.\\
\end{eqnarray}
Since $\upsilon  \circ \xi _m^{ - 1}$ depends only on the  $W_m
$'s variables, we have $ \left| {\nabla _{\bar g} \left( {\upsilon
\circ \xi _m^{ - 1} } \right)} \right| = \left| {\nabla _{\bar g}
\upsilon _2 } \right|$ and by (\ref{E102}) we obtain
\begin{eqnarray}\label{E103}
\nonumber\int _M {\left| {\nabla _g \upsilon } \right|  dV_g } \,
\nonumber& \geqslant & \frac{{\left( {1 - \varepsilon _0 }
\right)^2 }} {{1 + \varepsilon _0 }}\sum\limits_{m = 1}^{M_j}
{\left( {\int _{U_m } {\beta _{1m} dx} } \right)} \left( {\int _{N
\cap W_m }
 {\left| {\nabla _{\bar g} \upsilon _{2m} } \right|  d\upsilon _{\bar g}
 } } \right)  \\
\nonumber& = &\frac{{\left( {1 - \varepsilon _0 } \right)^2 }} {{1
+ \varepsilon _0 }}\sum\limits_{m = 1}^{M_j} {\left( {\int _{U_m }
{\beta _{1m} dx} } \right)} \left( {\int _N {\left| {\nabla _{\bar
g}
\upsilon _2 } \right|  d\upsilon _{\bar g} } } \right)  \\
\nonumber& = &\frac{{\left( {1 - \varepsilon _0 } \right)^2 }} {{1
+ \varepsilon _0 }}\int _N {\left| {\nabla _{\bar g} \upsilon _2 }
\right|  d\upsilon _{\bar g} }  \\& \geqslant& \left( {1 - c_3
\varepsilon _0 } \right)\int _N {\left| {\nabla _{\bar g} \upsilon
_2 } \right|  d\upsilon _{\bar g} },
\end{eqnarray}
where $
 c_3  \geqslant \frac{\textstyle 1} {\textstyle{\varepsilon _0 }}\left( {1 -
\frac{\textstyle{\left( {1 - \varepsilon _0 } \right)^2 }}
{\textstyle{1 + \varepsilon _0 }}} \right) $.

Inequality (\ref{E97}) is the first part of inequality
(\ref{E89}). Following the same arguments we can establish
second part of this inequality. This ends the proof of Lemma.
 \mbox{ } $\Box$



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\end{document}