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\title{\bf {\Large Sharp constants in Poincar\'e, Steklov \\ and related
inequalities (a survey)}}

\author{{\bf Nikolay Kuznetsov$^\dag$ and Alexander Nazarov$^\ddag$}}

\date{On the occasion of the 150th anniversary of V.\,A. Steklov's birth}

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\footnotetext {$^\dag$ Laboratory for Mathematical Modelling of Wave Phenomena,
Institute for Problems in Mechanical Engineering, Russian Academy of Sciences,
St.\,Petersburg. 

\noindent E-mail: nikolay.g.kuznetsov@gmail.com

\indent\indent$^\ddag$ Laboratory of Mathematical Physics, St.\,Petersburg
Department of the Steklov Mathematical Institute and Department of Mathematical
Physics, Faculty of Mathematics and Mechanics, St.\,Petersburg State University.
Supported by the RFBR grant 14-01-00534 and by the St.\,Petersburg University grant
6.38.670.2013. E-mail: al.il.nazarov@gmail.com}

\renewcommand{\thefootnote}{\arabic{footnote}} 

\vspace{-2mm}

The 150th anniversary of the birth of the outstanding Russian mathematician Vladimir
Andreevich Steklov falls on 9 January 2014. All over the world, researchers in all
areas of mathematics know this name. Indeed, widely known mathematical institutes in
Moscow and St.\,Petersburg are named after Steklov (before the unfortunate recent
reform of the Russian Academy of Sciences, they were among its leading
institutions). This commemorates the fact that he was the founding father of their
predecessor\,---\,the Physical-Mathematical Institute established in 1921 in
Petrograd (now St.\,Petersburg). Steklov was the first director of the institute
until his unexpected and untimely death on 30 May 1926. Meanwhile, Steklov's
scientific contributions (in particular, to analysis, mathematical physics and
mechanics) are less known even in present-day Russia. (The reason might be that his
papers were published mainly in French.) In this paper, we describe the work of
Steklov and his contemporaries on inequalities of mathematical physics and some
further advances concerning sharp constants in these inequalities. Steklov's results
in other areas and their development are presented in the recent papers \cite{EMS}
and \cite{Not}.

\subsection*{The work of Poincar\'e, Steklov and his disciples}

In this section, we outline the early work on inequalities with sharp constants.


\subsubsection*{One-dimensional inequalities of V.\,A. Steklov, \\ J.\,D.
Tamarkin and N.\,M. Krylov}

In 1896, Steklov \cite{S1} proved that the following inequality
\begin{equation} 
\int\limits_0^\ell u^2 (x) \, \D x \le \left( \frac{\ell}{\pi} \right)^2
\int\limits_0^\ell [u'(x)]^2 \, \D x  \label{Pob_1}
\end{equation}
holds for all functions which are continuously differentiable on $[0, \ell]$ and
have zero mean there. For this purpose he used the closedness equation for the
Fourier coefficients of $u$; the corresponding system is $\{ \cos \, (k \pi x /
\ell) \}_{k=0}^\infty$ normalised on $[0, \ell]$. (One finds similar considerations
in \cite[Ch.~5, Sect.~11]{BeBe}, where the so-called Wirtinger inequality is
proved.) Steklov's extensive work on the closedness equation lasted for 30 years
until his death. For this reason A.~Kneser \cite{Kneser} referred to this equation
as ``Steklov's favorite formula''. It should be mentioned that Steklov introduced
the term {\it closedness equation} for general orthonormal systems much later (see
the brief announcement \cite{S3} and the full-length paper \cite{S4} published in
1910 and 1911, respectively).

Inequality (\ref{Pob_1}) was among the earliest inequalities with sharp constant
that appeared in mathematical physics. Steklov applied it to justifying the Fourier
method for initial-boundary value problems for the heat equation in two dimensions
with variable coefficients independent of time. Later, he also justified the Fourier
method for the wave equation under similar assumptions. The fact that the constant
in (\ref{Pob_1}) is sharp was emphasized by Steklov in \cite[pp.~294--296]{S2},
where he gave an alternative proof of this inequality. Another result proved in
\cite[pp.~292--294]{S2} says that (\ref{Pob_1}) is true for continuously
differentiable functions vanishing at the interval's end-points, and again the
constant is sharp. (It is worth mentioning that the latter result appeared in the
widely known book \cite[Sect. 7.7]{HLP} without any reference concerning its
authorship.) A further generalization of inequality (\ref{Pob_1}) was given by
Steklov in \cite{S6}. In the monograph \cite{MonPob}, the generalized form of
(\ref{Pob_1}) is given along with proofs for both types of assumptions about $u$.

Mitrinovi\'c {\it et al.} \cite[Ch. II]{MPF} investigated the history of
(\ref{Pob_1}) and related inequalities. This 48 pages long chapter entitled ``An
Inequality Ascribed to Wirtinger and Related Results'' includes more than 200
references. In particular, the authors cite \cite{S2} along with Steklov's note
published under the same title in {\it Comptes Rendus} in 1898. Moreover, it is said
that Steklov proved (\ref{Pob_1}) under both conditions guaranteeing its validity;
the generalization obtained in \cite{S6} is also mentioned. However, the first proof
of (\ref{Pob_1}) for functions vanishing at the interval's end-points is ascribed to
L.~Scheeffer (see \cite[p.~67]{MPF}). Indeed, his paper \cite{Schee} was published
posthumously as early as 1885 (the author died that year aged 26), but it is
inaccurate to think that (\ref{Pob_1}) is a result of this note concerned with the
simplest problem of variational calculus. Applying the so-called Jacobi
transformation to the second variation, Scheeffer obtained as an intermediate
formula an identity from which inequality (\ref{Pob_1}) immediately follows.
Unfortunately, he, unlike Steklov, did not notice the importance of this inequality
and it is not even written explicitly in \cite{Schee}.

\vspace{2mm}

Let us turn to results obtained by Steklov's disciples. In his article \cite{Tam}
published in 1910, J.\,D. Tamarkin (at that time he was a student whom Steklov made
interested in boundary value problems of mathematical physics; see \cite{Hil} and
\cite{EMS}) generalized (\ref{Pob_1}) in the following way. Multiplying two
inequalities of this form and combining both conditions imposed on $u$, he proved
that for every function $u \in {\cal C}^2 ([0, \ell])$, satisfying the conditions
\begin{equation} 
u (0) = u (\ell) \quad \mbox{and} \quad \int\limits_0^\ell u (x) \, \D x = 0 ,
\label{tam_1}
\end{equation}
the following inequality holds:
\begin{equation} 
\int\limits_0^\ell u^2 (x) \, \D x \leq \left( \frac{\ell}{\pi} \right)^4
\int\limits_0^\ell [u'' (x)]^2 \, \D x . \label{tam_2}
\end{equation}
It allowed Tamarkin to apply Steklov's method for studying the transversal
vibrations of a homogeneous elastic rod.

Note that the constant in (\ref{tam_2}) is not sharp and this drawback was
exterminated by N.\,M. Krylov\,---\,another disciple of Steklov. (He graduated from
the St.\,Petersburg Institute of Mines in 1902 and after studies in Paris and Pisa
in 1908--1910 completed his mathematics education through personal contacts with
Steklov and by reading his articles.) In his paper \cite{Kry} published in 1915,
Krylov proved that the inequality
\begin{equation} 
\int\limits_0^\ell u^2 (x) \, \D x \le \left( \frac{\ell}{2 \pi} \right)^4
\int\limits_0^\ell [u'' (x)]^2 \, \D x  \label{kr}
\end{equation}
holds for any function with the following properties:

\vspace{2mm}

$\bullet$ conditions (\ref{tam_1}) are fulfilled;

$\bullet$ $u'$ is absolutely continuous and its Fourier expansion converges uniformly
on $[0, \ell]$;

$\bullet$ $u''$ existing almost everywhere is square integrable.

\vspace{2mm}

\noindent It is clear that the constant in (\ref{kr}) is sharp which fact was
emphasized by Krylov. His proof of this inequality is based on the completeness
equation for trigonometric functions, but it is applied in more sophisticated way
than in the cases considered by Steklov. Inequality (\ref{kr}) also holds for
functions vanishing at the interval's end-points (see \cite{KTT}), but this result
was proved only in 1955.


\subsubsection*{An inequality ascribed to Wirtinger}

What is presented here confirms the Arnold Principle \cite{Arnold}: ``If a notion
bears a personal name, then this name is not the name of the discoverer''.

\vspace{2mm}

In the same paper \cite{Kry}, Krylov notes that if $u$ satisfies only the first two
of the above listed conditions, then his method gives the following inequality:
\begin{equation} 
\int\limits_0^\ell u^2 (x) \, \D x \le \left( \frac{\ell}{2 \pi} \right)^2
\int\limits_0^\ell [u' (x)]^2 \, \D x .  \label{alm}
\end{equation}
Again, the constant is sharp and less than that in (\ref{Pob_1}) which is a
consequence of the first condition (\ref{tam_1}) added to the second one. Krylov
also mentions that for $u \in {\cal C}^2 ([0, \ell])$ (this is more restrictive than
the second condition imposed by Krylov) satisfying conditions (\ref{tam_1})
inequality (\ref{alm}) was obtained by E. Almansi \cite{A} in 1905 in connection
with his investigation of stability of the equilibrium of the Plateau figures in
capillary theory.

However, in accordance with the Arnold Principle, inequality (\ref{alm}) for
functions satisfying (\ref{tam_1}) is usually referred to as {\it Wirtinger's
inequality}. No wonder that in Blaschke's book \cite[p.~105]{Bla} (\ref{alm}) is
ascribed to Wirtinger because the latter was Blaschke's teacher. One finds the same
attribution and reference to Blaschke's book in \cite[Sect. 7.7]{HLP} and in
\cite{BeBe}, where Sections 10--13 of Chapter 5 are devoted to this and related
inequalities. The question of priority of Wirtinger was discussed by Mitrinovi\'c
and Vasi\'c in 1969 in their interesting article \cite{MV} and again in \cite[Ch.
II]{MPF}. One finds only the reference \cite{A} in \cite{MV}, but in \cite{MPF} the
results obtained by Steklov, Tamarkin and Krylov are also presented.


\subsubsection*{The Poincar\'e and Steklov inequalities}

In the same volume of the {\it Communications of the Kharkov Mathematical Society}\/
in which inequality (\ref{Pob_1}) was published, the Steklov's paper \cite{S0} had
appeared even a little bit earlier. (One has to keep in mind that another article
having the same title as \cite{S0}, namely \cite{S5}, was published in 1897.) In
\cite{S0}, he considered the following analogue of (\ref{Pob_1}):
\begin{equation}
\int\limits_\Omega u^2 \, \D x \le C \int\limits_\Omega |\nabla u|^2 \, \D x .
\label{Pob_1'}
\end{equation}
Here $\nabla$ stands for the gradient operator and the integral on the right-hand
side is called the Dirichlet integral. Assuming that $\Omega$ is a bounded
three-dimensional domain whose boundary is piecewise smooth and $u$ is a real ${\cal
C}^1$-function on $\bar \Omega$ with zero mean, Steklov found that the sharp
constant in (\ref{Pob_1'}) is $\lambda_1^{-1}$, where $\lambda_1$ is the smallest
positive eigenvalue of the Neumann Laplacian in $\Omega$:
\[ -\Delta u = \lambda u \quad \mbox{in} \quad \Omega; \qquad  
\frac{\partial u}{\partial {\bf n}} = 0 \quad \mbox{on} \quad \partial\Omega .
\]
Here $\partial / \partial{\bf n}$ stands for differentiation with respect to the
exterior unit normal.

Under the same assumptions about $u$, inequality (\ref{Pob_1'}) was first proved by
H.~Poincar\'e \cite{Poin1} in 1890 provided $\Omega$ is a smooth, convex domain. He
also estimated $C$ from above for this class of domains. Moreover, he demonstrated
that if a homogeneous, isotropic body occupies a domain for which (\ref{Pob_1'}) is
valid, then solutions of the heat equation in this domain tend to the equilibrium
temperature distribution at the exponential rate. In the second article
\cite{Poin2} published by Poincar\'e on this topic in 1894, he obtained that
(\ref{Pob_1'}) is true provided $\Omega$ is the union of a finite number of smooth,
convex domains. Moreover, he improved and extended his estimate of 1890 for smooth,
convex domains; namely, he obtained that
\[ C \le \frac{9 ({\rm diam}\,\Omega)^2}{16} \quad \mbox{in three and} \quad C \le 
\frac{7 ({\rm diam}\,\Omega)^2}{24} \quad  \mbox{in two dimensions.}
\]
Here ${\rm diam}\,\Omega$ is the diameter of $\Omega$, that is, its maximal chord.

In his article \cite{S5} published in 1897, Steklov proved the following new results
about (\ref{Pob_1'}). First, this inequality is valid provided $u$ is a real ${\cal
C}^1$-function on $\bar \Omega$ vanishing on $\partial \Omega$; again $\Omega$ is
supposed to be a bounded three-dimensional domain whose boundary is piecewise
smooth. Second, under these assumptions the sharp constant in (\ref{Pob_1'}) is
equal to $\lambda_1^{-1}$, but in this case $\lambda_1$ is the smallest eigenvalue
of
\begin{equation}
-\Delta u = \lambda u \quad \mbox{in} \quad \Omega , \qquad  u=0 \quad \mbox{on}
\quad \partial\Omega ; \label{Dirichlet}
\end{equation}
that is, of the Dirichlet Laplacian in $\Omega$.


\subsection*{Further development}

The problem of finding and estimating sharp constants in functional inequalities attracted much
attention from those who work in theory of functions and mathematical physics (see,
for example, the classical monographs \cite{HLP} and \cite{PS}). More than thirty
years ago, the role of sharp constants was emphasized in the book \cite{Mih1} by
S.\,G.~Mikhlin. Let us quote the review \cite{Peet}: %of its original German version.
\begin{quote}
[This book] is devoted to appraising the (best) constants\,---\,exact results or
explicit (numerical) estimates\,---\,in various inequalities arising in
``analysis'' (=PDE). [\dots] This is a most original work, a bold attack in a
direction where still very little is known.
\end{quote} 
Our aim is to outline main achievements in this area. Since integral inequalities
(as well as integration by parts) are at the heart of theory of differential
equations arising in mathematical physics, one might expect that the interest to
sharp constants in these inequalities will only intensify in the future.


\subsubsection*{Scope of this section and preliminary material}

We restrict ourselves to the direct generalizations of (\ref{Pob_1}) and
(\ref{Pob_1'}), that is, to inequalities of the following form:
\begin{equation}
\|u\|_{L^q (\Omega)} \le C \, \|\nabla u\|_{L^p(\Omega)} . \label{q-p}
\end{equation}
Here $\Omega$ is a domain in $\RR^n$, $n \geq 1$, whereas $p, q \geq 1$ satisfy the
following restrictions: 
\begin{eqnarray*}
&& q \le p^* = \frac{np}{n-p} , \quad\, \mbox{if} \quad 1 \le p<n ; \\
&& q < \infty , \qquad\qquad\quad \mbox{if} \quad p=n>1 ; \\
&& q \le \infty , \qquad\qquad\quad \mbox{if} \quad p>n \quad \mbox{or} \quad n=1 .
\end{eqnarray*}
It is assumed that $u$ belongs to $L^{1,p} (\Omega)$, that is, $u \in L^p_{loc}
(\Omega)$, its Sobolev derivatives of the first order belong to $L^p (\Omega)$, and
$\|\nabla u\|_{L^p(\Omega)}$ is the norm of $|\nabla u|$ in  $L^p (\Omega)$.

Weighted inequalities\,---\,the Hardy inequality and its generalizations such as the
Hardy--Sobolev inequality, the Maz'ya inequality, the Caffarelly--Kohn--Nirenberg
inequality\,---\,are beyond our scope. We also do not consider inequalities
involving derivatives of higher order which received much attention during the past
few years.

If $u$ vanishes on $\partial \Omega$ (this is understood as follows: $u$ can be
approximated in the norm $\|\nabla u\|_{L^p (\Omega)}$ by smooth functions having
compact support in $\Omega$), then (\ref{q-p}) is true with some positive constant
$C$ for any domain of finite volume\footnote{This condition is not sharp; in the
recent papers \cite{JMV} and \cite{JMV1}, the necessary and sufficient condition is
given for the validity of (\ref{q-p}) with $p=q$.} and for an arbitrary domain in the
critical case $p<n$, $q=p^*$. For these functions, inequality (\ref{q-p}) often
appears under various names for different values of $p$ and $q$. In particular, it
is referred to as:

\vspace{2mm}

$\bullet$ the {\it Steklov inequality} when $p=q=2$;

$\bullet$ the {\it Friedrichs inequality} when $p=q$;

$\bullet$ the {\it Sobolev inequality} when $p<n$, $q=p^*$.

\vspace{2mm}

\noindent Note that a slightly different inequality was obtained by K.-O.
Friedrichs \cite{Frie} under the assumption that $\Omega \subset \RR^2$. Namely, his
inequality is as follows:
\begin{equation}
\int\limits_\Omega u^2 \, \D x \le C \left[\, \int\limits_\Omega |\nabla u|^2 \, \D
x + \int\limits_{\partial \Omega} u^2 \, \D S \right] , \label{Fried}
\end{equation}
where $\D S$ denotes the element of length of $\partial \Omega$. Generally speaking,
(\ref{Fried}) holds for all bounded domains in $\RR^n$ ($\D S$ denotes the element
of area when $n > 2$), for which the divergence theorem is true (see
\cite[p.~24]{Maz}). Furthermore, the Sobolev inequality was proved by S.\,L. Sobolev
himself only for $p>1$ and E. Gagliardo proved it for $p=1$ (see \cite{Sob} and
\cite{Ga1}, respectively).

Inequality (\ref{q-p}) for $u$ with zero mean value over $\Omega$ is equivalent to
the following
\begin{equation}
\| u - \langle u \rangle \|_{L^q (\Omega)} \le C \|\nabla u \|_{L^p (\Omega)} ,
\quad \langle u \rangle = \frac{\int_\Omega u (x) \, \D x}{{\rm meas}_n \, \Omega}
\quad \mbox{for all} \ \ u \in L^{1,p} (\Omega) . \label{Poi}
\end{equation}
Here the $n$-dimensional measure of $\Omega$ stands in the denominator. Moreover,
some requirements must be imposed on $\Omega$ for the validity of (\ref{Poi}).
Indeed, as early as 1933 O.~Nikod\'ym \cite{Nik} (see also \cite[p.~7]{Maz})
constructed a bounded two-dimensional domain $\Omega$ and a function with the finite
Dirichlet integral over $\Omega$ such that inequality (\ref{Poi}) is not true for
$p=q=2$. Another example of a domain with this property is given in \cite[Ch.~7,
Sect.~8.2]{CoHi} (see also \cite[Sect.~6.10.3]{Maz}). On the other hand, if $p=q$,
then (\ref{Poi}) (it is called the {\it Poincar\'e inequality}\/ in this case) is
valid for all domains such that their boundaries are locally graphs of continuous
functions in Cartesian coordinates (see, for example, the classical book \cite{CoHi}
by R.~Courant and D.~Hilbert for the proof which can be easily extended from $p=2$
to any $p$).

Furthermore, if $p<n$ and $q=p^*$, then (\ref{Poi}) (it is called the {\it
Poincar\'e--Sobolev inequality}\/ in this case) holds for any bounded
$n$-dimensional Lipschitz domain. Moreover, the inequality is true provided $\Omega$
belongs to the class of so-called {\it John's domains}\/ as was proved by B.
Bojarski \cite{Boj}. We recall that this class was introduced by F. John \cite{J}
and domains belonging to it are more general than the Lipschitz ones. Finally, if $q
\neq p^*$, then (\ref{Poi}) holds if and only if $L^{1,p} (\Omega)$ is continuously
embedded into $L^{q}(\Omega)$. This was established by J. Deny and J.-L. Lions
\cite{DL} for $p=q$, whereas the general case was considered in \cite{NP1}.

Thus, the major point to be clarified about inequality (\ref{Poi}) is smoothness of
$\partial \Omega$. To a great extent, this was made by V.\,G. Maz'ya in his
comprehensive monograph {\it Sobolev Spaces} in which he presented his own results
and surveyed those of other authors. (Originally this book was published in Russian
in 1985 by the Leningrad State University. Recently, the 2nd revised and augmented
English edition \cite{Maz} appeared; its bibliography exceeds 800 entries.
Moreover, several sections deal with the question of exact constants in some
inequalities.) Proofs of basic facts concerning inequality (\ref{Poi}) can be also
found in the recent textbook \cite{NP}; its English translation is currently in
preparation.

Almost everything known about sharp constants in various versions of inequality
(\ref{q-p}) mainly comes under one of the following four conditions:

\vspace{2mm}

$\bullet$ $p=q=2$ (the quadratic case);

$\bullet$ $\Omega=(0,\ell)$ (the one-dimensional case);

$\bullet$ $p<n$, $q=p^*$ (the critical case);

$\bullet$ $p=q=1$ (the ``geometric'' case).


\subsubsection*{The quadratic case}

It was mentioned above that the sharp constant in (\ref{q-p}) is $\lambda_1^{-1/2}$
in the quadratic case. Here $\lambda_1 = \lambda^{D}_1 \big( \lambda^{N}_1 \big)$ is
the smallest positive eigenvalue of the Dirichlet (Neumann, respectively) Laplacian
for the Steklov (Poincar\'e, respectively) inequality. Explicit values of these
eigenvalues are known only for several particular domains. Among them, one finds the
following (see \cite{PS}):

\vspace{2mm}

$\bullet$ Rectangle $a \times b$: $\lambda_1^{D} = \left( \frac{\pi}{a} \right)^2 +
\left( \frac{\pi}{b} \right)^2$, $\lambda_1^{N} = \left[ \frac{\pi}{\max\{a,b\}}
\right]^2$;

$\bullet$ $45^\circ$ right triangle: $\lambda_1^{D} = 5\left( \frac{\pi}{a}
\right)^2$, $\lambda_1^{N} = \big( \frac{\pi}{a} \big)^2$, where $a$ is the leg
length;

$\bullet$ $30^\circ$ right triangle: $\lambda_1^{D} = \frac{112}{9} \big(
\frac{\pi}{a} \big)^2$, $\lambda_1^{N} = \frac{16}{3} \left( \frac{\pi}{a}
\right)^2$, where $a$ is the hypotenuse length;

$\bullet$ Equilateral triangle: $\lambda_1^{D} = \frac{16}{3} \left( \frac{\pi}{a}
\right)^2$, $\lambda_1^{N} = \frac{16}{9} \big( \frac{\pi}{a} \big)^2$, where $a$ is
the side length;

$\bullet$ Disk of the radius $a$: $\lambda_1^{D} = \left( \frac{j_{0,1}}{a}
\right)^2$, $\lambda_1^{N} = \left( \frac{j_{1,1}}{a} \right)^2$.

\vspace{2mm}

\noindent Here $j_{0,1}$ $(j_{1,1})$ is the first positive zero of the Bessel
function $J_0$ ($J_1$, respectively). The Dirichlet and Neumann eigenvalues for
sectors and annuli can also be expressed in terms of Bessel functions. 

Furthermore, there are simple formulae for the fundamental eigenvalues in domains
that are Cartesian products of two domains of different dimensions. Let 
$\Omega_1 \subset \RR^m$ and $\Omega_2 \subset \RR^n$ be bounded domains.
If the fundamental Dirichlet and Neumann eigenvalues in $\Omega_j$ $(j=1,2)$ are
$\lambda_1^{(j),D}$ and $\lambda_1^{(j),N}$, respectively, then
\[ \lambda_1^D = \lambda_1^{(1),D} + \lambda_1^{(2),D} \quad \mbox{and} \quad
\lambda_1^N = \min\{\lambda_1^{(1),N}, \lambda_1^{(2),N}\}
\]
are the corresponding eigenvalues in $\Omega = \Omega_1 \times \Omega_2$.

It is worth mentioning that the Dirichlet eigenvalues have the following integral
representation (see \cite{Rel}):
\begin{equation}
\lambda_k^D = \frac{1}{4} \int\limits_{\partial \Omega} \left( \frac{\partial
u_k}{\partial {\bf n}} \right)^2 \frac{\partial |x|^2}{\partial {\bf n}} \, \D S ,
\quad k = 1,2,\dots \, , \label{R}
\end{equation}
where $u_k$ is the $k$th eigenfunction normalized in $L^2 (\Omega)$. Formula
(\ref{R}) with $k=1$ allows us to express the sharp constant in the Steklov
inequality in terms of the normalized fundamental eigenfunction of the Dirichlet
Laplacian.

In order to estimate $\lambda_1^{D}$ one can use its monotonicity with respect to
domain variation and the Steiner symmetrization (see \cite{PS}). In particular,
among all quadrilaterals of the same area the least value of $\lambda_1^{D}$ is
delivered by the square, whereas the equilateral triangle has the least value of
$\lambda_1^{D}$ among all triangles of the same area (see \cite{Fre}). Finally, a
ball in $\RR^n$ has the least value of $\lambda_1^{D}$ among all figures of the same
area/volume. In 1877, the two-dimensional version of the last assertion was
conjectured by Lord Rayleigh (see \cite[pp.~339--340]{Ray}). It was proved
independently by G. Faber \cite{Fa} and E. Krahn \cite{Kr1}, \cite{Kr2}. It must be
emphasized that all estimates involving symmetrization for their derivation are true
for arbitrary $p$ and $q$. Thus, under the condition that $u$ vanishes on $\partial
\Omega$ the sharp constant in (\ref{q-p}) has the largest value for a ball in
$\RR^n$ (comparing other domains of the same area/volume). Unfortunately, bounds for
sharp constants are implicit unless $p=q=2$.

Less is known about estimates of the first positive Neumann eigenvalue. The
classical result of G. Szeg\H o \cite{Sze} ($n=2$) and H.\,F. Weinberger \cite{Wb}
(higher dimensions) says that a ball in $\RR^n$ has the largest value of
$\lambda_1^{N}$ among all domains of the same area/volume (see also \cite{AB}).
Analogous result for triangles was obtained recently in \cite{LS}. 

A global lower bound for $\lambda_1^{N}$ was obtained for {\it convex} domains by
L.\,E. Payne and H.\,F. Weinberger \cite{PW} ($n=2$) and by M. Bebendorf \cite{Be}
($n\ge3$); namely, $\lambda_1^{N} > \big( \frac{\pi}{{\rm diam}\,\Omega} \big)^2$
unless $n=1$ when $\Omega$ is an interval. A generalization of this result for
arbitrary $p=q>1$ was established recently in \cite{FNT} (see also \cite{ENT} and
\cite{V}).

There are also inequalities between the Dirichlet and Neumann eigenvalues (see, for
example, the recent paper \cite{Fi}, where background is also briefly described).
Furthermore, it is shown in \cite{Sta} that if (\ref{Poi}) holds in $\Omega_1
\subset \RR^m$ and $\Omega_2 \subset \RR^n$ with an arbitrary $p=q$ and the sharp
constants $C_1$ and $C_2$, respectively, then the sharp constant in the same
inequality in $\Omega_1 \times \Omega_2$ is less than or equal to $\sqrt{2} \,
(C_1+C_2)$. We also mention the recent survey \cite{Pen} where, in particular, the
results on the upper estimates for sharp constants in the Poincar\'e inequality in
quadratic case on surfaces without boundary are collected.


\subsubsection*{The one-dimensional case}

Without loss of generality we assume that $\Omega = (0,1)$ and begin with the case
when $u$ vanishes at the end-points for which the sharp constant in (\ref{q-p}) is as
follows:
\begin{equation}
C = C_1 (p,q) = \frac{\mathfrak F \left( q^{-1} + p'^{-1} \right)}{2 \, \mathfrak F
\left( q^{-1} \right) \mathfrak F \left( p'^{-1} \right)} \, , \label{Schmidt}
\end{equation}
where $\mathfrak F (s) = \frac{\Gamma (s+1)}{s^s}$ and $p' = \frac{p}{p-1}$ is the
H\"older conjugate exponent to $p$. This constant was obtained by E.~Schmidt
\cite{Sch} as early as 1940 (the case $p=q$ was considered even earlier by
V.\,I.~Levin \cite{Lev}; see also \cite[Sect. 7.6]{HLP}). This classical result
still remains unnoticed by some researchers. It was rediscovered in 2002 (see
\cite{BY}), whereas its particular case considered in \cite{Bo} was recently
referred to as ``the best one in the literature'' (see \cite{Aga}).

The function $U$ delivering the extremal value (\ref{Schmidt}) is symmetric with
respect to $x - \frac{1}{2}$, can be expressed in quadratures and is usually
referred to as the Lindqvist $\cos_{p,q}$ function (see \cite{Lind}). Besides, it is
well known in the stability theory as the Lyapunov cosine being introduced (for
$p=2$, $q=2m$, $m\in\NN$) by A.\,M.~Lyapunov in 1893 (see \cite{Lyap}). \medskip

The one-dimensional Poincar\'e-type inequality has even more complicated story. It
took several years after the pioneering paper \cite{DGS}\footnote{In \cite[Sect.
1.1.19]{Maz}, the first result for $p=q$ is attributed to A.~Stanoyevitch. However,
the proof in his PhD thesis (1990) turned out to be incorrect.} to establish the
following result (see \cite{BKN}, \cite{Naz} and also the recent paper \cite{GN} for
a more general problem and a historical survey).

\vspace{2mm}

\noindent {\it Let $n=1$ and $\Omega=(0, 1)$. If $q \le 3 p$, then the sharp con\-stant
in $(\ref{Poi})$ is equal to $C_1(p,q)$ defined by $(\ref{Schmidt})$, whereas the
corresponding extremal function $V$ is as follows:
\[ V(x) = \left\{ \begin{array}{rr} U \Big( x+\frac{1}{2} \Big) \quad {\rm when}\ x
\le \frac{1}{2} , \\ - U \Big( x-\frac{1}{2} \Big) \quad {\rm when}\ x \geq
\frac{1}{2} , \end{array} \right.
\]
where $U$ is Schmidt's function. In particular, $V$ is antisymmetric with respect to
$x - \frac{1}{2}$. The constant in $(\ref{Poi})$ is greater than $C_1(p,q)$ and $V$
has no symmetry provided $q > 3 p$.}

\vspace{2mm}

Note that a particular case $q\le 2p$ considered in \cite{DGS} was also rediscovered
in 2004 (see \cite{BY1}).


\subsubsection*{The critical case}

First we note that the sharp constant in the Sobolev inequality is invariant with
respect to dilations of the domain $\Omega$. Since it is obviously monotone with
respect to inclusion of domains, in fact, it is independent of $\Omega$.

In 1960, V.\,G.~Maz'ya \cite{Maz60} and H.~Federer and W.\,H.~Fleming \cite{FF}
found the sharp constant in the Sobolev inequality with $p=1$. Its value is as
follows:
\[ \omega_{n-1}^{-\frac 1n}\cdot n^{\frac {1-n}n} , \quad \mbox{where} \ \ 
\omega_{n-1} = \frac{2\pi^{\frac n2}}{\Gamma \big( \frac n2 \big)} ,
\]
the latter is equal to the $(n-1)$-dimensional measure of the unit sphere in
$\RR^n$.

It was G.~Rosen \cite{Ros}, who made the next step in 1971. Namely, he proved that
the exact constant in the Sobolev inequality for $n=3$, $p=2$ (and $q=6$) is
$2^{\frac 23}3^{-\frac 12} \pi^{\frac 23} \approx 0.4273$.

Four years later, T.~Aubin \cite{Aub} and G.~Talenti \cite{Tal} independently
considered the case of arbitrary $n \geq 2$ and $1<p<n$. It is worth emphasizing
that the Bliss inequality \cite{Bl} and symmetrization\,---\,the key ingredients of
the proof\,---\,were known for a long time before that. The corresponding sharp
constant is equal to
\begin{equation} 
C_2 (n,p) = \omega_{n-1}^{-\frac{1}{n}} \, n^{-\frac{1}{p}} \Big( \frac{p-1}{n-p}
\Big)^{\frac{1}{p'}} \left[ \mathfrak B \Big( \frac{n}{p} , \frac{n}{p'} + 1 \Big)
\right]^{-\frac{1}{n}} , \label{Sob}
\end{equation}
where $\mathfrak B$ stands for the Euler beta function. This constant is {\it not
attained} unless $\Omega = \RR^n$. In the paper \cite{CNV} published ten years ago,
the constant $C_2(n,p)$ was obtained by virtue of the mass transportation approach
(the generalized Monge--Kantorovich problem).

The situation is again more complicated for the Sobolev--Poincar\'e inequality. It
is known that for any John domain the sharp constant is greater than or equal to
$2^{\frac 1n}\cdot C_2(n,p)$, where $C_2(n,p)$ is defined by (\ref{Sob}). Moreover,
if $\Omega$ is a ${\cal C}^2$-domain and $C$ in (\ref{Poi}) is strictly greater than
$2^{\frac 1n}\cdot C_2(n,p)$, then the sharp constant {\it is attained} for this
$\Omega$. In particular, for any bounded ${\cal C}^2$-domain there exists $\beta>0$
such that the sharp constant in the Sobolev--Poincar\'e inequality is attained when
$1 < p < \frac{n+1}{2} + \beta$ (see \cite{DN} for the proof; the case $p=2$ was
considered earlier in \cite{AM} and \cite{Wa}). In the survey article
\cite{NazSurv}, the question when the sharp constant is attainable is discussed for
various critical inequalities.


\subsubsection*{The ``geometric'' case}

In some sense, the case $p = 1$ is simpler than those considered above. Indeed, by
linearity one rearranges the gradient along level lines and then uses the coarea
formula. On the other hand, since $L^1 (\Omega)$ is not reflexive, the sharp
constant usually is not attained in $L^{1,1} (\Omega)$, and one has to solve the
corresponding extremal problem in the space $BV (\Omega)$ of functions with bounded
variation.

Results outlined in this section originate from J. Cheeger's insight dating back to
his pioneering paper \cite{Ch}. Twenty years later, A. Cianchi \cite{Ci} obtained
the following expression for the sharp constant in the Poincar\'e inequality for
$p=q=1$:
\[ C = \sup_{F \subset \Omega}\, \frac{2}{{\rm meas}_{n-1}\, F}\, 
\frac{{\rm meas}_n\, E \cdot {\rm meas}_n (\Omega \setminus \bar E)}{{\rm meas}_n\,
\Omega} \, .
\]
Here $F$ ranges over all surfaces dividing $\Omega$ into two connected subsets $E$
and $\Omega \setminus \bar E$; ${\rm meas}_{n-1} F$ is the $(n-1)$-dimensional
measure of $F$ (generally speaking, its Hausdorff measure).

For functions vanishing on $\partial \Omega$ the sharp constant in (\ref{q-p}) with
$p=q=1$ was found by L. Lefton and D. Wei \cite{LW} (see also \cite{KF}):
\[ C = \sup_{E \subset \Omega} \frac{{\rm meas}_n\, E}{{\rm meas}_{n-1}\, \partial E}. 
\]
Here $E$ ranges over all subsets of $\Omega$ such that $\partial E \cap \partial
\Omega = \emptyset$. 

Let $C (p)$ denote the sharp constant in the Friedrichs inequality, that is, in
inequality (\ref{q-p}) with $p=q$ valid for $u$ vanishing on $\partial \Omega$. The
following estimate $C (p) \leq p \cdot C (1)$ was also proved in \cite{LW}; the case
$p=2$ was considered earlier by J. Cheeger \cite{Ch}.

Concerning estimates of the constant in the Poincar\'e inequality, we mention the
sharp inequality $C (1) \leq \frac{{\rm diam}\,\Omega}{2}$ obtained by G. Acosta
and R. Dur\'an \cite{AcDu} for {\it convex domains}. This $L^1$-analogue of the
Payne--Weinberger--Bebendorf estimate is widely used in studies of finite element
approximations. A survey of related results for the ``geometric'' case can be found,
for example, in \cite{Pa} and in \cite{Ste}.


\subsubsection*{The ``twisted'' Steklov--Poincar\'e inequality}

Let us consider inequality (\ref{q-p}) for functions $u$ satisfying {\it both}\/
Steklov and Poincar\'e conditions, that is, vanishing on $\partial \Omega$ and
having zero mean value, respectively. We will refer to the corresponding inequality
as the ``twisted'' Steklov--Poincar\'e. 

\vspace{2mm}

In the one-dimensional case, this estimate arises (mainly for $p=2$) in various
applications. We mention just two of them: estimating the fundamental eigenvalue in
the Lagrange problem about the shape of strongest column (see, for example,
\cite{EK}); the characterization problem in nonparametric statistics (see, for
example, \cite[Sect.~6.2]{Ni}). It is noted in \cite{Naz} that the sharp constant in
the twisted inequality is the one-half of the sharp constant in the usual Poincar\'e
inequality (\ref{Poi}) (cf. (\ref{Pob_1}) and (\ref{alm}) as well). \medskip

In the quadratic case considered by L. Barbosa and P. B\'erard (\cite{BaBe}), it was
shown that the sharp constant is equal to $\lambda_1^{-1/2}$, where $\lambda_1 =
\lambda_1^T$ is the smallest positive eigenvalue of the following ``twisted''
Dirichlet problem:
\[ -\Delta u = \lambda u - \langle \Delta u \rangle \ \ \mbox{in} \ \Omega; \quad 
u=0 \ \ \mbox{on} \ \partial \Omega . 
\]
In \cite{BaBe}, it was proved that this problem has the following property along
with some others. Its spectrum interlaces with that of problem (\ref{Dirichlet}). In
particular, this implies that
\begin{equation}
\lambda_1^D \leq \lambda_1^T \leq \lambda_2^D . \label{Dir-twist}
\end{equation}
In their paper \cite{FrH}, P. Freitas and A. Henrot showed that if $\Omega$ is a
pair of equal disjoint balls, then $\lambda_1^T$ has the least value comparing with
those for open sets of the same area/volume. Note that both inequalities
(\ref{Dir-twist}) are equalities for this $\Omega$. In the recent paper \cite{CHP},
Henrot and his coauthors tried to obtain a similar result for a more general range
of values of $p$ and $q$, but, unfortunately, there is a gap in their proof as is
shown in \cite{Naz2}. \medskip

The case $p=1$ was considered recently in \cite{BDNT}. As in the quadratic case, a
pair of disjoint balls yields the largest sharp constant among all open sets of
given area/volume. However, their radii depend on $q$; namely, if $q$ is close to
$1$, then the optimal set consists of two equal balls, whereas two different balls
give the optimal set for $q$ close to $1^* = \frac{n}{n-1}$.


\subsubsection*{The ``boundary'' Poincar\'e inequality}

Let $\Omega$ be a bounded Lipschitz domain in $\RR^n$, $n \geq 2$, and let $G$ be an
open part of $\partial \Omega$ possibly coinciding with $\partial \Omega$. Then the
``boundary analogue'' of inequality (\ref{Poi}) is as follows:
\begin{equation}
\| u - \langle u \rangle_G \|_{L^q(G)} \le C \|\nabla u \|_{L^p(\Omega)} , \quad
\langle u \rangle_G = \frac{\int_G u (x) \, \D S}{{\rm meas}_{n-1} \, G} .
\label{PoinBd}
\end{equation}
This inequality holds for $u \in L^{1,p} (\Omega)$ provided
\begin{eqnarray*}
&& q \le p^{**} = \frac{(n-1)p}{n-p} , \quad\ \mbox{if} \quad 1 \le p < n ; \\ &&
q <\infty , \qquad\qquad\qquad\quad \mbox{if} \quad p=n ; \\ && q \le \infty ,
\qquad\qquad\qquad\quad \mbox{if} \quad p > n .
\end{eqnarray*}

In the {\it quadratic} case (that is, $p=q=2$), the sharp constant in (\ref{PoinBd})
is again equal to $\lambda_1^{-1/2}$, but now $\lambda_1 = \lambda_1^S$ is the
smallest positive eigenvalue of the following mixed (unless $G = \partial \Omega$)
Steklov problem: 
\[ \Delta u = 0 \ \ \mbox{in} \ \Omega , \quad \frac{\partial u}{\partial {\bf n}} = \lambda u 
\ \ \mbox{on} \ G , \quad \frac{\partial u}{\partial {\bf n}} = 0 \ \ \mbox{on} \
\partial \Omega \setminus G .
\]
We recall that ${\bf n}$ is the exterior unit normal existing almost everywhere on
$\partial \Omega$. For $n = 2$ $(n = 3)$ and particular choices of $\Omega$ and $G$
the eigenvalues of the above problem give {\it sloshing frequencies}\/ of the free
oscillations of a liquid in a channel (container, respectively); see, for example,
\cite[Ch. IX]{Lamb}.

In \cite{NRep}, $\lambda_1^S$ is found for several simple domains with different
sets chosen as $G$. For example, let $\Omega$ be a $45^\circ$ right triangle with leg
equal to $a$, then:

\vspace{2mm}

$\bullet$ if $G$ is the hypotenuse, then $\lambda_1^S = \frac{\sqrt 2}{a}$;

$\bullet$ if $G$ is a leg, then $\lambda_1^S = \frac{z_1^{(1)} \tanh z_1^{(1)}}{a}
\approx \frac{2.3236}{a}$, where $z_1^{(1)}$ is the smallest positive zero of $\tan
z + \tanh z = 0$;

$\bullet$ if two legs form $G$, then $\lambda_1^S = \frac{2 z_1^{(2)} \tanh
z_1^{(2)}}{a} \approx \frac{1.3765}{a}$, where $z_1^{(2)}$ is the smallest positive
zero of $\tan z \cdot \tanh z = 1$.

\vspace{2mm}

\noindent In \cite{NRep} (see also \cite{Rep}), some applications of sharp constants
in (\ref{Poi}) and (\ref{PoinBd}) are considered. These applications concern
quantitative analysis of solutions and a posteriori error estimation for partial
differential equations.

When $G = \partial \Omega$, the estimate analogous to that obtained by
Szeg\H{o}--Weinber\-ger was found by R. Weinstock \cite{Ws} ($n=2$) and by F. Brock
\cite{Br} (higher dimensions). Namely, a ball in $\RR^n$ has the largest value of
$\lambda_1^{S}$ among all domains of the same area/volume.\medskip

It should be emphasized that in the {\it critical}\/ case (that is, $p<n$,
$q=p^{**}$) the sharp constant in (\ref{PoinBd}) is related to that in the following
{\it trace Sobolev inequality} for the half-space $\RR^n_+ = \{ x \in \RR^n\,: \ x_n
> 0 \}$:
\begin{equation}
\| u (\cdot,0) \|_{L^{p^{**}} (\RR^{n-1})} \le C_3(n,p) \, \| \nabla u
\|_{L^p(\RR^n_+)} , \quad u \in L^{1,p} (\RR^n_+) . \label{trSob}
\end{equation}
In particular, $C_3(n,p) = 1$ for $p=1$ which follows from \cite[Sect.~1.3]{Maz88}.

J.\,F. Escobar \cite{E} conjectured that if $p>1$ in (\ref{trSob}), then the
extremal function is equal to $|x-x^*|^{-(n-p) / (p-1)}$, where $x^* \notin \RR^n_+$
is arbitrary, but he proved this assertion only for $p=2$. Afterwards, the general
case was established in the remarkable paper \cite{Nt} based on the mass
transportation approach (see also \cite{Naz1}). This result implies that
\[ C_3 (n,p) = \left( \frac{p-1}{n-p} \right)^{\frac{1}{p'}} \Big[ 
\frac{\omega_{n-2}}{2} \, \mathfrak B \Big( \frac{n-1}{2}, \frac{n-1}{2(p-1)} \Big)
\Big]^{- \frac{1}{(n-1) p'}} .
\]

As for the Sobolev--Poincar\'e inequality, the following is true in the critical
case. The sharp constant in (\ref{PoinBd}) is greater than or equal to $C_3(n,p)$
for Lipschitz domains. Moreover, if $\Omega$ is a ${\cal C}^2$-domain and
$C>C_3(n,p)$, then the sharp constant {\it is attained} for this $\Omega$. In
particular, for any bounded ${\cal C}^2$-domain in $\RR^n$, $n \geq 3$, there exists
$\delta > 0$ such that the sharp constant is attained for $1 < p <
\frac{n+1}{2}+\delta$ (see \cite{NRez} for the proof).\medskip

Finally, we mention two recent papers dealing with the {\it ``geometric''}\/ case
$p=q=1$. In the first of them \cite{Ci1}, A. Cianchi obtained the following formula
for the sharp constant in (\ref{PoinBd}):
\[ C = \frac{2}{{\rm meas}_{n-1}\, \partial\Omega}\, \sup_{E \subset \Omega}\,
\frac{{\rm meas}_{n-1}\, (\partial E \cap \partial \Omega) \cdot {\rm meas}_{n-1}
(\partial \Omega \setminus \partial E)} {{\rm meas}_{n-1}\, \partial E \cap \Omega}
\, ,
\]
where $E$ ranges over all subdomains of $\Omega$ with Lipschitz boundary. He also
found the sharp constant for balls. It turns out that for $n \ge 3$ the optimal
choice of $E$ is a half-ball and $C = \frac{(n-1) \omega_{n-1}}{2 \omega_{n-2}}$,
whereas $C=2$ and the supremum is {\it not attained} for $n=2$.

In the second paper \cite{CFNT}, it is proved that the least sharp constant for
Lipschitz domains is attained for balls. (Note that the ``geometric'' case here is,
at the same time, the critical one, and so the sharp constant depends on the shape
of $\Omega$, but not on its size.) Moreover, for $n \ge 3$ balls are the only
optimal domains, whereas if $n=2$, then some nearly circular stadium-shaped domains
yield the same value of the sharp constant.


\vspace{2mm}

\noindent {\bf Acknowledgement.} The authors are grateful to Professor Sergey
Poborchi for many useful discussions on the subject of the paper.


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}
\end{thebibliography}


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