\documentclass[11pt]{article} \usepackage{amssymb,amsmath} \usepackage[latin1]{inputenc} \usepackage{amsfonts} \usepackage{mathrsfs} \usepackage[british,english]{babel} \usepackage{amsthm} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \textwidth=6in \textheight=9.5in \topmargin=-0.5cm \oddsidemargin=0.5cm \evensidemargin=0.5cm \usepackage{etex} \usepackage{lmodern} \usepackage[T1]{fontenc} \usepackage{textcomp} \usepackage{amstext} %\usepackage{booktabs} %\usepackage{microtype} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\usepackage{kerkis} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}{Definition}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{corollary}{Corollary}[section] \newtheorem{remark}{Remark}[section] \newtheorem{example}{Example}[section] \newtheorem{thebibliography{99}}{Thebibliography}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallbreak {\noindent}{\textbf{Keywords:} Sobolev inequalities; best constants; limit case; manifolds with boundary; symmetries; solid torus.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \smallbreak\noindent{\textbf{Mathematics Subject Classification (2010):} 46E35; 41A44; 35B33. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\noindent\rule{13.8cm}{0.4pt} %\newpage \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
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 $1
1$, 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 $1
0$ 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: \[ 00$ 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 $1
0$ 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 $1
2$. \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) $. 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