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\def\moyenne{\not\hskip -4pt\int}
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\centerline{\bf HOW TO RECOGNIZE CONSTANT FUNCTIONS.}

\centerline{\bf CONNECTIONS WITH SOBOLEV SPACES}

\medskip

\centerline{\bf Ha\"\i m Brezis}

\vskip 1cm

\centerline{Dedicated to Mark Visik with esteem and
friendship}

\bigskip

\noindent {\bf 1. Introduction}
\medskip

Most of the ideas in this paper are coming from a
series of recent collaborations with J. Bourgain, Y.
Li, P. Mironescu and L. Nirenberg (see J. Bourgain, H.
Brezis and P.~Mironescu [1], [2], [3], [4], H. Brezis
and L. Nirenberg [1], H. Brezis, Y. Li, P. Mironescu
and L. Nirenberg [1]). However we will adopt here on
slightly different presentation and provide some
simplified proofs.

\medskip

The starting point is the following

\medskip

\noindent {\bf Proposition 1.} {\sl Let $\Omega$ be a
connected open set in $\R^N$ and let $f : \Omega
\rightarrow \R$ be a measurable function such that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)| \over |x -
y|^{N+1}} dx \, dy < \infty,\leqno(1)
$$
then $f$ is a constant.}

\medskip

The original motivation for such a proposition was
twofold:

\medskip

(i) {\sl Uniqueness of lifting.} Given a (measurable)
function $u : \Omega \rightarrow \C$ such that $|u| =
1$ a.e., there are many liftings $\varphi$, i.e., $u =
e^{i\varphi}$. If $\varphi_1$, $\varphi_2$ are 2
liftings then
$$
k(x) = {1 \over 2\pi} \left(\varphi_1(x) -
\varphi_2(x)\right) : \Omega \rightarrow \Z.
$$
Under further assumptions one may hope to prove that
$k$ is a {\sl constant} function. For example, if
$\varphi_1$, $\varphi_2$ are continuous and $\Omega$
is connected, then $k$ is constant. The message I wish
to convey is that the continuity assumption can be
replaced by a different type of condition, such as
(1), which is much more natural in the framework of
Sobolev spaces (see Remark 3).

\medskip

(ii) {\sl A degree theory for classes of discontinuous
maps.} The possibility of defining a degree for maps
in Sobolev spaces (see H. Brezis and J.M. Coron [1],
H. Brezis, Y. Li, P. Mironescu and L. Nirenberg [1]),
is based on the fact deg $h_t(\cdot)$ remains constant
along a homotopy $h_t(\cdot)$, as $t$ varies in $[0,1]$
(or more generally in a connected parameter space
$\Lambda$). Such a conclusion holds possibly in
situations where the dependence in $t$ need not be
continuous.

\medskip

\noindent {\sl Remark 1.} The conclusion of Proposition 
1 is easy to state, but I do not know a direct,
elementary, proof. Our proof is not very complicated
but requires an ``excursion'' via the Sobolev spaces.

\medskip

\noindent {\sl Remark 2.} The connectedness assumption
is of course needed. The conclusion of Proposition
1 still holds if in (1) $N+1$ is replaced by $q \geq
N+1$. Indeed, it suffices to prove Proposition 1 when
$\Omega$ is a ball $B$ (and complete the general case
via connectedness); then
$$
{1 \over |x - y|^{N+1}} \leq {C \over |x - y|^q}
\qquad \forall x,y \in B.
$$
(However the conclusion still holds in some
non connected domains, for example $\Omega = G
\backslash \Sigma$ where $G$ is connected and $\Sigma$
is closed with meas $\Sigma = 0$. It would be
interesting to study non connected domains where the
conclusion of Proposition 1 holds).

\medskip
  On the other hand, if in (1) $N+1$ is
replaced by $q < N+1$, then the conclusion fails.
Indeed, for {\sl any} Lipschitz function on $B$ one has
$$
\int_B \int_B {|f(x) - f(y)| \over |x - y|^q} dx \,
dy \leq C \int_B \int_B {dx \, dy \over |x -
y|^{q-1}} < \infty
$$
since $q < N+1$.

\medskip

There are many consequences and variants of
Proposition 1. Here are a few.

\medskip

\noindent {\bf Corollary 1.} {\sl Assume $\Omega$ is a
connected open set in $\R^N$, and let $f : \Omega
\rightarrow \Z$ be a measurable function such that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+1}} dx \, dy < \infty,\leqno(2)
$$
for some $1 \leq p < \infty$, then $f$ is a constant.}

\medskip

\noindent {\sl Proof.} Observe that
$$
|f(x) - f(y)|^p \geq |f(x) - f(y)|
$$
since $f(x) - f(y) \in \Z$.

\medskip

\noindent {\sl Remark 3.} When $p > 1$, condition (2)
says that $f$ belongs to the fractional Sobolev space
$W^{s,p}$ (see e.g. Adams [1]) with $s = 1/p$.
Therefore, we may assert that any function in
$W^{s,p}(\Omega;\Z)$ with $sp \geq 1$ is a constant.
Note that the condition $sp \geq 1$ is {\sl
considerably weaker} than the condition $sp > N$ which
implies (via the Sobolev embedding theorem) that $f$
is continuous. Corollary 1 is originally due to R.
Hardt, D. Kinderlehrer and F.H. Lin [1] (Lemma 1.1)
when $p = 2$ and $s = 1/2$ (they attribute it to
Wiener when $N = 2$). Bethuel and Demengel [1] had
obtained a similar conclusion under the stronger assumption $sp
>  1$.

\medskip

\noindent {\bf Corollary 2.} {\sl Assume $\Omega$ is
a connected open set in $\R^N$ and $A$ is any
measurable subset such that
$$
\int_A \int_{^c\!A} {dx \, dy \over |x - y|^{N+1}} <
\infty\leqno(3)
$$
then either meas$(A) = 0$ or meas$(\Omega\backslash A)
= 0$.}

\medskip

It suffices to apply Proposition 1 to $f = \chi_A$,
the characteristic function of $A$. Note that in (3),
$(N+1)$ is again optimal. If $A$ is any subset of
$\Omega$ with smooth boundary, then (3) holds if
$(N+1)$ is replaced by any $q < N+1$ (it suffices to
consider the case where $\partial A$ is flat and to
make an explicit computation).

\medskip

Now some variants of Proposition 1.

\medskip

\noindent {\bf Proposition 2.} {\sl Assume $\Omega$ is
a connected open set in $\R^N$ and $f : \Omega
\rightarrow \R$ is a measurable function such that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+p}} dx \, dy < \infty,\leqno(4)
$$
for some $1 \leq p < \infty$, then $f$ is constant.}

\medskip

\noindent [Proposition 1 corresponds to the case $p =
1$].

\medskip

Still a further generalization

\medskip

\noindent {\bf Proposition 3.} {\sl Assume $\Omega$ is
a connected open set in $\R^N$ and $f : \Omega
\rightarrow \R$ is a measurable function such that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^p} \psi(|x-y|) dx \, dy < \infty,\leqno(5)
$$
where $p \geq 1$ and $\psi \in L^1_{loc}(0,\infty)$,
$\psi \geq 0$ satisfies
$$
\int^1_0 \psi(r) r^{N-1}dr = \infty,\leqno(6)
$$
then $f$ is a constant.}

\medskip

\noindent [Proposition 2 corresponds to the case
$\psi(r) = r^{-N}$].

\medskip

Here is one important generalization of Proposition 2.

\medskip

\noindent {\bf Proposition 4.} {\sl Assume $\Omega$ is
a connected open set in $\R^N$ and $f : \Omega
\rightarrow \R$ is a measurable function such that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+p-\varepsilon}} dx \, dy =o \left({1 \over
\varepsilon}\right) \; {as} \; \varepsilon
\rightarrow 0,\leqno(7)
$$
i.e.,
$$
\lim_{\varepsilon \rightarrow 0} \varepsilon
\int_\Omega \int_\Omega{|f(x) - f(y)|^p \over |x -
y|^{N+p-\varepsilon}} dx \, dy = 0\leqno(7')
$$
for some $p \geq 1$, then $f$ is a constant.}


\medskip

\noindent {\sl Remark 4.} Assumption (7) is clearly
much weaker than (4) (when $\Omega$ is bounded) which
says that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+p-\varepsilon}} dx \, dy = 0(1)  \; \hbox{as} \;
\varepsilon \rightarrow 0,
$$
On the other hand (7) is optimal since for any
Lipschitz function $f$ on $\Omega$
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+p-\varepsilon}} dx \, dy = 0\left({1 \over
\varepsilon}\right)\leqno(8)
$$
because
$$
\int^1_0 {1 \over r^{N-\varepsilon}} r^{N-1} dr = {1
\over \varepsilon}.
$$

\medskip

Here is a final generalization, which brings us closer
to the connection with Sobolev spaces.

\medskip

\noindent {\bf Theorem 1.} {\sl Assume $\Omega$ is a
connected open set in $\R^N$ and $f : \Omega
\rightarrow \R$ is a measurable function. Let
$(\rho_\varepsilon)_{\varepsilon> 0}$ be a
sequence of radial mollifiers, i.e.
$$
\rho_\varepsilon \in L^1_{loc}(0,\infty),
\quad \rho_\varepsilon \geq 0,\leqno(9)
$$
$$
\int^\infty_0 \rho_\varepsilon(r) r^{N-1} dr
= 1 \qquad \forall \varepsilon > 0,\leqno(10)
$$
$$
\hbox{for every} \; \delta > 0, \qquad
\lim_{\varepsilon \rightarrow
0} \int^\infty_\delta \rho_\varepsilon(r)
r^{N-1} dr = 0.\leqno(11)
$$
Assume that, for some $p \geq 1$,
$$\lim_{\varepsilon \rightarrow 0}
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^p}\rho_\varepsilon(|x - y|) dx \, dy =
0.\leqno(12)
$$
Then $f$ is a constant.}

\medskip

Note that Proposition 4 is a consequence of
Theorem 1 when choosing
$$
\rho_\varepsilon(r) =
\left\{\eqalign{
&\varepsilon r^{-N+\varepsilon}, \qquad r < 1\cr
&0\qquad\;\;\;,\qquad r > 1.\cr
}\right.
$$
And Proposition 3 is also a consequence of
Theorem 1 when choosing
$$
\rho_\varepsilon(r) = \left\{\eqalign{
&0 \qquad \;\,\qquad \hbox{if} \; r < \varepsilon\cr
&a_\varepsilon \psi(r) \qquad \hbox{if} \;
\varepsilon < r < 1\cr
&0 \qquad \;\,\qquad \hbox{if} \; r > 1,\cr
}\right.$$
where
$$
a_\varepsilon = \left(\int^1_\varepsilon \psi(r)
r^{N-1} dr\right)^{-1} \rightarrow 0 \quad
\hbox{as} \; \varepsilon \rightarrow 0.\leqno(13)
$$
Note that, in view of (5),
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^p}\rho_\varepsilon(|x - y|) dx \, dy \leq
Ca_\varepsilon \rightarrow 0 \; \hbox{as} \;
\varepsilon \rightarrow 0, \; \hbox{by (13)}.
$$

The proof of Theorem 1 involves an excursion into
Sobolev spaces which we will now describe.

\vskip 1cm

\noindent {\bf 2. A new characterization of Sobolev
spaces}

For simplicity, we start with the case of all of
$\R^N$. Let $f \in L^p(\R^N)$, $1 < p < \infty$. It is
well-know (see e.g. H. Brezis [1], Proposition IX.3)
that if
$f \in W^{1,p}(\R^N)$ %%if and only if
then
$$
\int_{\R^N} |f(x+h) - f(x)|^p dx \leq |h|^p  \int_{\R^N}
|\nabla f|^p dx \;
\quad  \hbox{ for every} \; h\in
\R^N.\leqno(14)
$$
And conversely, if $f \in L^p(\R^N)$ and if there
exists a constant $C$ such that
$$
\int_{\R^N} |f(x+h) - f(x)|^p dx \leq C |h|^p \;
\hbox{ as } \; h \rightarrow 0,
\leqno(15)$$
then $f \in W^{1,p}(\R^N)$.

When $p = 1$, $W^{1,1}$ should be replaced by $BV$, the
space of functions in $L^1$ who's derivatives (in the
sense of distributions) are bounded Radon measures;
thus $f \in BV$ if and only if
$$
\int_{\R^N} |f(x+h) - f(x)| dx \leq C|h| \;
\hbox{as} \; |h| \rightarrow 0,
\leqno(16)
$$
%%and then one can choose $C=\int|\nabla f|dx$ in (16).
and then (16) holds for all $h\in \R^N$ with $C=\int|\nabla f|dx$.
In particular, if
$\rho_\varepsilon$ satisfies (9), (10) and $f \in 
W^{1,p}$, we have
$$
\int_{\R^N} \rho_\varepsilon(|h|)dh \int_{\R^N}
{|f(x+h) - f(x)|^p \over |h|^p} dx \leq C \; \hbox{as}
\; \varepsilon \rightarrow 0,\leqno(17)
$$
since
$$
\int_{\R^N} \rho_\varepsilon(|h|)dh = \sigma_N
\int^\infty_0 \rho_\varepsilon(r) r^{N-1} dr = \sigma_N
$$
where $\sigma_N = |S^{N-1}|$.

Changing variables in (17) yields
$$
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|)dx \, dy \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0.\leqno(18)
$$
Similarly, if $f \in BV$, we have
$$
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)| \over |x -
y|} \rho_\varepsilon(|x - y|)dx \, dy \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0.\leqno(19)
$$

The heart of the matter is that (18) (resp. (19)) gives
a characterization of $W^{1,p}$ when $p > 1$ (resp.
$BV$).

\medskip

\noindent {\bf Theorem 2.}  {\sl Assume $f \in
L^p(\R^N)$ satisfies (18) with $p > 1$. Let
$(\rho_\varepsilon)$ be as in (9)-(10)-(11). Then $f
\in W^{1,p}$ and
$$
\lim_{\varepsilon \rightarrow 0}
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|)dx \, dy = K_{p,N}
\int_{\R^N} |\nabla f|^p dx \leqno(20)
$$
where $K_{p,N}$ depends only on $p$ and $N$.}

\medskip

Similarly for $p = 1$ we have

\medskip

\noindent {\bf Theorem 3.} {\sl Assume $f \in
L^1(\R^N)$ satisfies (19). Let $(\rho_\varepsilon)$ be
as in (9)-(10)-(11). Then $f \in BV$ and
$$
\lim_{\varepsilon \rightarrow 0}
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)| \over |x -
y|} \rho_\varepsilon(|x - y|)dx \, dy = K_{1,N}
\int_{\R^N} |\nabla f| dx \leqno(21)
$$
where the right-hand side denote the total mass of the
measure $\nabla f$.}


\medskip

An interesting consequence of Theorem 3 is the 
following

\medskip

\noindent {\bf Corollary 3.}
{\sl Let $A$ be a bounded 
measurable set in $\R^N$. Then $A$ has finite 
perimeter (in the sense of De Giorgi) if and only if
$$
\int_A \int_{^c\!A} {1 \over |x - y|} \rho_\varepsilon
(|x - y|)dx \, dy 
 \leq C \; \hbox{ as } \; \varepsilon 
\rightarrow 0
$$
and then
$$
\lim_{\varepsilon \rightarrow 0} \int_A \int_{^c\!A} 
{1 \over |x - y|} \rho_\varepsilon
(|x - y|)dx \, dy = K_{1,N} \hbox{Per}(A).
\leqno(22)
$$}

\medskip

\noindent {\sl Proof of Theorem 2.} The original proof
of Theorem 2 is to be found in Bourgain, Brezis and
Mironescu [3]. We present here a simpler argument
suggested by E. Stein [1]. Assume $f \in L^p$
satisfies (18) an let $(\gamma_\delta)$ be any
sequence of smooth mollifiers. Set
$$
f_\delta = \gamma_\delta \star f.
$$
Note that (18) still holds when $f$ is replaced by its
translates $(\tau_h f)(x) = f(x+h)$. Also, (18) is
stable under convex combinations and thus $f_\delta$
satisfies (18) with the {\sl same} constant $C$, i.e.,
we have
$$
\int_{\R^N} \int_{\R^N} {|f_\delta(x) - f_\delta(y)|^p
\over |x-y|^p} \rho_\varepsilon(|x - y|)dx \, dy \leq
C\leqno(23)
$$
where $C$ is {\sl independent} of $\varepsilon$ {\sl
and} $\delta$.

Next, let $g \in C^2(\R^N)$ be such that
$$
\int_{\R^N} \int_{\R^N} {|g(x) - g(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|) dx \, dy \leq C \;
\hbox{ as } \; \varepsilon \rightarrow 0,\leqno(24)
$$
where $\rho_\varepsilon$ satisfies (9), (10), (11). We
claim that
$$
\int_{\R^N} |\nabla g(x)|^p dx \leq
C/K_{p,N},\leqno(25)
$$
with $C$ taken from (24) and
$$
K_{p,N} = \int_{S^{N-1}} |(\sigma \cdot e)|^p d\sigma,
\quad e \in S^{N-1}.\leqno(26)
$$

\medskip

\noindent {\sl Proof of (25).} Let $K$ be any compact
subset of $\R^N$. For $x \in K$ and $|h| \leq 1$ we
have
$$
|g(x+h) - g(x) - h\cdot \nabla g(x)| \leq C_K
|h|^2.\leqno(27)
$$
From (24) we have
$$
\int_K dx \int_{|h| \leq 1} {|g(x+h) - g(x)|^p \over
|h|^p} \rho_\varepsilon(|h|) dh \leq C.\leqno(28)
$$
By (27) we have
$$
|h \cdot \nabla g(x)| \leq |g(x+h) - g(x)| + C_K |h|^2
$$
and therefore, for every $\theta > 0$
$$
|h \cdot \nabla g(x)|^p \leq (1 + \theta) |g(x+h) -
g(x)|^p + C_{\theta,K} |h|^{2p}.
$$
Combining this with (28) yields
$$
\int_K dx \int_{|h| \leq 1} {|(h \cdot \nabla g(x))|^p
\over |h|^p} \rho_\varepsilon(|h|)dh \leq (1+\theta)C
+ C_{\theta,K} |K| \int_{|h| \leq 1} |h|^p
  \rho_\varepsilon(|h|)dh.\leqno(29)
$$
But, for any vector $V \in \R^N$,
$$
\int_{|h| \leq 1} {|(h \cdot V)|^p \over
|h|^p} \rho_\varepsilon(|h|)dh = K_{p,N} |V|^p
\int^1_0 \rho_\varepsilon(r) r^{N-1} dr.
$$
On the other hand, it is clear from (10) and (11) that
$$
\lim_{\varepsilon \rightarrow 0} \int_{|h| \leq
1} |h|^p \rho_\varepsilon(|h|)dh = 0.
$$
Passing to the limit as $\varepsilon \rightarrow 0$ in
(29) we find
$$
K_{p,N} \int_K |\nabla g(x)|^p dx \leq
(1+\theta)C.\leqno(30)
$$
Since (30) holds for every $\theta > 0$ and every
compact set $K$ (with $C$ independent of $\theta$ and
$K$) we obtain (25), that is,
$$
K_{p,N} \int_{\R^N}  |\nabla g(x)|^p dx \leq
\liminf_{\varepsilon \rightarrow 0} \int_{\R^N} 
\int_{\R^N} {|g(x) - g(y)|^p \over |x - y|^p} 
\rho_\varepsilon (|x - y|)dx \, dy.\leqno(31)
$$
On the other hand, if $g \in C^2_0(\R^N)$ we have, as 
above,
$$
|g(x +h) - g(x)| \leq |h \cdot \nabla g(x)| + C'|h|^2 
\qquad \forall \, x \in \R^N, \; \forall\,h\in \R^N.
$$
Hence
$$
|g(x+h)-g(x)|^p\leq (1+\theta)|h \cdot \nabla g(x)|^p+
C'_{\theta}|h|^{2p}  . 
%\quad \forall\,x\in \R^N \;\forall\,h\in \R^N.
$$
We multiply this by $\rho_\varepsilon(|h|)/|h|^p$
and integrate over the set $\{(x,h)\in\R^{2N}:x \hbox{ or } x+h \in {\supp}g\}$
to obtain
$$
\int_{\R^N} dx \int_{\R^N} {|g(x+h) - g(x)|^p \over
|h|^p} \rho_\varepsilon(|h|) dh \leq 
\hbox{\hskip 4cm}
$$
\vskip -5mm
$$
\hbox{\hskip 2cm}
(1+\theta) \int_{\R^N} K_{p,N}|\nabla g(x)|^p dx+
2C'_{\theta}|{\supp}g|\int_{\R^N}|h|^p \rho_\varepsilon(|h|)dh.
$$
We first let $\varepsilon \rightarrow 0$ and then
$\theta \rightarrow 0$. This yields
$$
\limsup_{\varepsilon \rightarrow 0}
\int_{\R^N} dx \int_{\R^N} {|g(x + h) - g(x)|^p \over
|h|^p} \rho_\varepsilon(|h|) dh \leq K_{p,N}
\int_{\R^N} |\nabla g(x)|^p dx.\leqno(32)
$$  
Combining (31) and (32) yields, for every $g \in    
C^2_0(\R^N)$,                                       
$$                                                  
\lim_{\varepsilon \rightarrow 0}
\int_{\R^N} \int_{\R^N} {|g(x) - g(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|)dx \, dy = K_{p,N}
\int_{\R^N} |\nabla g(x)|^p dx.
$$
Since $C^2_0(\R^N)$ is dense in $W^{1,p}(\R^N)$, it is
easy to conclude (using (14)) that (20) holds for
every $f \in W^{1,p}(\R^N)$.
 

\medskip

We may now complete the proof of Theorem 2. 
Assuming $f \in L^p(\R^N)$ satisfies (18) and
applying  Claim (25) to 
$g = f_\delta$ we see that
$$
\int_{\R^N} |\nabla f_\delta|^p dx\leq
C/K_{p,N},\leqno(33)
$$
where $C$ comes from (18).
Finally, we pass to the limit in (33) as $\delta \rightarrow 0$ and obtain $f
\in W^{1,p}$.
% with
%$$
%\int_{\R^N} |\nabla f|^p dx\leq C/K_{p,N}.
%$$
%%%%%%%%%%%%%%%%%%%%%%

\medskip


\noindent {\sl Proof of Theorem 3.} If $f \in 
L^1(\R^N)$ and satisfies (19) and we proceed as above 
we are led to
$$
\int_{\R^N} |\nabla f_\delta| dx\leq C/K_{1,N}.
$$
Therefore $f \in BV$ and
$$
\int_{\R^N} |\nabla f| dx\leq C/K_{1,N}.
$$
In other words we have proved that
$$
K_{1,N} \int_{\R^N} |\nabla f| dx\leq
\liminf_{\varepsilon \rightarrow 0}
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)| \over |x -
y|} \rho_\varepsilon(|x - y|)dx \, dy.\leqno(34)
$$
On the other hand it is easy to see, using (16), that 
for 
$f 
\in BV$
$$
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)| \over |x - y|}
\rho_\varepsilon(|x - y|)dx \, dy \leq \tilde K_N
\int_{\R^N} |\nabla f| dx.\leqno(35)
$$

Unfortunately the constant $\tilde K_N$ in (35) is not
the same as $K_{1,N}$. It is also clear that (21)
holds when $f \in C^2_0(\R^N)$.  However we cannot
conclude easily that (21) holds for every $f \in BV$
since $C^2_0(\R^N)$ is {\sl not} dense in $BV$.

%%This requires an additional argument due to J. Davila [1].
It remains to be shown that, for every $f \in BV(\R^N)$
$$
\limsup_{\varepsilon \rightarrow 0}
\int_{\R^N} \int_{\R^N} {|f(x) - f(y)| \over |x - y|}
\rho_\varepsilon(|x-y|)dx \, dy \leq K_{1,N}
\int_{\R^N} |\nabla f|dx.
$$
This has been established by J. Davila
[1] using new ideas which are not presented here.
\medskip

\noindent {\sl Remark 5.}  There are statements similar
to Theorem 2 and Theorem 3 when $\R^N$ is replaced by
a {\sl smooth} bounded domain $\Omega$ in $\R^N$.
However the same conclusion {\sl fails} for a general
bounded domain $\Omega$ if $\partial\Omega$ is {\sl
not smooth}. It is still {\sl true} (for a general
$\Omega$) that
$$
K_{p,N} \int_\Omega |\nabla f|^p \leq
\liminf_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega {|f(x) - f(y)|^p \over |x - y|^p}
\rho_\varepsilon(|x - y|)dx \, dy.\leqno(36)
$$
However, it may happen for $p > 1$ that $f \in
W^{1,p}(\Omega)$ (so that the left hand side in (36)
is {\sl finite}) while the right-hand side in (36) is
{\sl infinite}. Here is such an example. Let $\Omega =
D \backslash\Sigma$ where $D$ is a disc (in $\R^2$)
and $\Sigma$ is a slit. Let $f$ be a smooth function
in $\Omega$ which is discontinuous across the slit
(for example two different constants on each side of
the slit). Clearly $f \in W^{1,p}(\Omega)$, but the
RHS in (36) is infinite. This is so because
$$
\int_\Omega \int_\Omega ... = \int_D \int_D ...
$$
and if the RHS in (36) were finite we would conclude
that $f \in W^{1,p}(D)$ (by Theorem 2), which is
obviously wrong. This example suggests the following

\medskip

\noindent {\sl Open problem 1.} Let $\Omega \subset
\R^N$ be a bounded connected set (not necessarily
smooth). Let $\delta(x,y)$ denote the {\sl geodesic
distance} in $\Omega$. Let $f \in L^p(\Omega)$ be such
that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over
\delta(x,y)^p} \rho_\varepsilon(\delta(x,y))dx\, dy
\leq C \; \hbox{as} \; \varepsilon \rightarrow 0.
$$
Does it follow that $f \in W^{1,p}$ and if so, does
one have
$$
\lim_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega {|f(x) - f(y)|^p \over \delta(x,y)^p}
\rho_\varepsilon(\delta(x,y))dx \, dy = K_{p,N}
\int_\Omega |\nabla f|^p dx ?
$$

\noindent {\sl Remark 6.} The characterization of
$W^{1,p}$ (resp. $BV$) given by Theorem 2 (resp. 3)
suggests a definition of Sobolev spaces for maps $f :
M \rightarrow \tilde M$ between metric spaces, where
$M$ is equipped with a measure $\mu$, namely
$$
\int\int  {\tilde d(f(x),f(y))^p \over d(x,y)^p}
\rho_\varepsilon(d(x,y))d\mu(x)d\mu(y) \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0.
$$
Note that assumptions (10) and (11) involve the notion
of a dimension $N$ but this can be done easily by
considering $\displaystyle{\lim_{r \rightarrow 0}}
|log \, \mu(B_r(x))|/|log \, r|$. It would be
interesting to study the properties of such maps
(Sobolev imbeddings, etc...) and to compare this
notion with other definitions (see Korevaar and Schoen
[1], P. Hajlasz and P.Koskela [1], L. Ambrosio and 
P.Tilli [1] and the numerous references in these works).

\medskip

\noindent {\sl Remark 7.} There are variants of
Theorems 2 and 3 when $\Omega$ is a {\sl smooth}
bounded domain in $\R^N$. For example, we have

\medskip

\noindent {\bf Theorem 2'.} {\sl Assume $f \in
L^p(\Omega)$ satisfies
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|)dx \, dy \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0,\leqno(37)
$$
with $\rho_\varepsilon$ as in (9), (10), (11). Then $f
\in W^{1,p}(\Omega)$ and
$$
\lim_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega {|f(x) - f(y)|^p
\over |x - y|^p} \rho_\varepsilon(|x - y|)dx \, dy =
K_{p,N} \int_\Omega |\nabla f|^p.\leqno(38)
$$}

\medskip

\noindent {\sl Sketch of proof.} First assume that
(37) holds. By a standard technique of reflection
across the boundary and multiplication by a cut-off
one constructs a function $\tilde f$ on $\R^N$, with
compact support, such that $\tilde f = f$ on $\Omega$
and satisfying
$$
\int_{\R^N} \int_{\R^N} {|\tilde f(x) - \tilde f(y)|^p
\over |x - y|^p} \rho_\varepsilon(|x - y|)dx \, dy
\leq C' \;
\hbox{as} \; \varepsilon \rightarrow 0,\leqno(39)
$$
By Theorem 2 we conclude that $\tilde f \in
W^{1,p}(\R^N)$ and thus $f \in W^{1,p}(\Omega)$.

Next one shows that if $f \in C^2(\overline\Omega)$,
then
$$
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^p} \rho_\varepsilon(|x - y|)dx \, dy \leq C(\Omega)
\int_\Omega |\nabla f|^p dx.\leqno(40)
$$
Finally one proves that if $f \in C^2(\overline\Omega)$
$$
\lim_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega {|f(x) - f(y)|^p
\over |x - y|^p} \rho_\varepsilon(|x - y|)dx \, dy =
K_{p,N}
\int_\Omega |\nabla f|^p dx.\leqno(41)
$$

The conclusion of Theorem 2' follows from an easy
density argument.

\medskip

\noindent {\sl Remark 8.} There are several choices
for $\rho_\varepsilon$ which are of interest. Here are
a few

\medskip

A) {\sl Choice 1}
$$
\rho_\varepsilon(r) =
\left\{\eqalign{
{\varepsilon \over r^{N-\varepsilon}} \qquad &0 < r <
1\cr
0 \qquad &r > 1.\cr
}\right.
$$

This choice yields

\medskip

\noindent {\bf Corollary 4.} {\sl Assume $\Omega$ is a
smooth bounded domain in $\R^N$. Let $f \in
L^p(\Omega)$ be such that
$$
\varepsilon \int_\Omega \int_\Omega {|f(x) - f(y)|^p
\over |x - y|^{N+p-\varepsilon}} dx \, dy \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0,
$$
then $f \in W^{1,p}(\Omega)$
and
$$
\lim_{\varepsilon \rightarrow 0} \varepsilon
\int_\Omega \int_\Omega {|f(x) - f(y)|^p \over |x -
y|^{N+p-\varepsilon}} dx \, dy = K_{p,N} \int_\Omega
|\nabla f|^p.\leqno(42)
$$}

Recall that the standard fractional Sobolev space
$W^{s,p}$, $0 < s < 1$, $1 < p < \infty$, is equipped
with Gagliardo (semi) norm
$$
\|f\|^p_{W^{s,p}} = \int_\Omega \int_\Omega {|f(x) -
f(y)|^p \over |x - y|^{N+sp}} dx \, dy.\leqno(43)
$$

It is well-known that $\|f\|_{W^{s,p}}$ does {\sl
not}  converge to $\|f\|_{W^{1,p}}$ as $s \uparrow 1$;
in fact it converges to $\infty$ (unless $f$ is
constant) by Proposition 2. However in view of
Corollary 4 we may now assert that
$$
\lim_{s \uparrow 1} (1-s) \|f\|^p_{W^{s,p}} =
{K_{p,N} \over p} \int_\Omega |\nabla f|^p.\leqno(44)
$$
This ``reinstates''  $W^{1,p}$ as a continuous limit
of $W^{s,p}$ as $s \uparrow 1$ provided one uses the
norm $(1-s)^{1/p} \|f\|_{W^{s,p}}$ on $W^{s,p}$.

\medskip

B) {\sl Choice 2}
$$
\rho_\varepsilon(r) =
\left\{\eqalign{
{N \over \varepsilon^N} \qquad &\hbox{if} \; r <
\varepsilon\cr
0  \qquad &\hbox{if} \; r >
\varepsilon\cr
}\right.
$$
This choice yields
$$
\lim_{\varepsilon \rightarrow 0} {1 \over
\varepsilon^N} 
%{\int_\Omega \int_\Omega \atop\scriptstyle |x - y| < \varepsilon} 
\mathop{\int_\Omega \int_\Omega}_{ |x - y| < \varepsilon} 
{|f(x) - f(y)|^p \over |x - y|^p} dx \,
dy = {K_{p,N} \over N} \int_\Omega |\nabla
f|^p.\leqno(45)
$$
A variant is
$$\rho_\varepsilon(r) =
\left\{\eqalign{
{(N+p)r^p \over \varepsilon^{N+p}} \qquad & r <
\varepsilon\cr
0 \qquad & r > \varepsilon\cr
}\right.
$$
and then we have
$$
\lim_{\varepsilon \rightarrow 0} \; {1 \over
\varepsilon^{N+p}} 
%{\int_\Omega \int_\Omega\atop\scriptstyle |x - y| < \varepsilon} 
\mathop{\int_\Omega \int_\Omega}_{|x - y| < \varepsilon} 
|f(x) - f(y)|^p dx \, dy = {K_{p,N} \over
(N+p)} \int_\Omega |\nabla f|^p.
\leqno(46)
$$
Still another choice yields
$$
\lim_{\varepsilon \rightarrow 0} \, {1 \over
\varepsilon^{N+p}} 
%{\int_\Omega \int_\Omega
%\atop\scriptstyle \varepsilon < |x - y| < 2\varepsilon}
\mathop{\int_\Omega \int_\Omega}_{\varepsilon < |x - y| < 2\varepsilon}
 |f(x) - f(y)|^p dx \, dy = \tilde
K_{p,N} \int_\Omega |\nabla f|^p.\leqno(47)
$$

C) {\sl Choice 3}
$$
\rho_\varepsilon(r) =
\left\{\eqalign{
0 \qquad &r < \varepsilon\cr
{1 \over |log \, \varepsilon| r^N} \qquad &\varepsilon
< r < 1\cr
0 \qquad &r > 1.\cr
}\right.
$$
This choice yields
$$
\lim_{\varepsilon \rightarrow 0} \, {1 \over |log \,
\varepsilon|} 
%{\int_\Omega \int_\Omega\atop\scriptstyle |x - y| > \varepsilon} 
\mathop{\int_\Omega \int_\Omega}_{|x - y| > \varepsilon} 
{|f(x) - f(y)|^p \over |x - y|^{N+p}} dx
\, dy = K_{p,N} \int_\Omega |\nabla \, f|^p.\leqno(48)
$$

D) {\sl Choice 4}
\smallskip
Let $\gamma \in L^1_{loc}(0,+\infty)$, $\gamma \geq
0$, be such that
$$
\int^\infty_0 \gamma(r) r^{N+p-1} dr = 1.
$$
Choosing
$$
\rho_\varepsilon(r) = {1 \over \varepsilon^{N+p}} \;
\gamma \left({r \over \varepsilon}\right) r^p
$$
yields
$$
\lim_{\varepsilon \rightarrow 0} {1 \over
\varepsilon^{N+p}} \int_\Omega \int_\Omega |f(x) -
f(y)|^p \; \gamma\left({|x-y| \over
\varepsilon}\right) dx \, dy = K_{p,N} \int_\Omega
|\nabla f|^p,
$$
for every $f \in W^{1,p}$ (with $p > 1$) and for every
$f \in BV$ (with $p = 1$). Applying this in the BV
case with $f = \chi_A$ we obtain a new {\sl
characterization} of sets of {\sl finite perimeter}.
Namely a measurable set $A \subset \Omega$ has finite
perimeter if and only if
$${1 \over \varepsilon^{N+1}} \int_A  \int_{^c \! A}
\gamma \left({|x - y| \over \varepsilon}\right) dx \,
dy \leq C \; \hbox{as} \; \varepsilon \rightarrow 0,
$$
and then
$$
\lim_{\varepsilon \rightarrow 0} {1 \over
\varepsilon^{N+1}} \int_A  \int_{^c \! A}
\gamma \left({|x - y| \over \varepsilon}\right) dx \,
dy = K_{1,N} \hbox{Per}(A).
$$

\vskip 1cm

\noindent {\bf 3. Back to constant functions}

All the results of Section 1 are immediate
consequences of the statements of Section 2 applied
in a ball $B \subset \Omega$. One concludes that $f$
is constant on $B$ and then that $f$ is constant on
$\Omega$ since $\Omega$ is connected.

Note that the assumption
$$
\lim_{\varepsilon \rightarrow 0} \int_B \int_B {|f(x)
- f(y)| \over |x - y|} \rho_\varepsilon (|x - y|)dx \,
dy = 0\leqno(49)
$$
implies first that $f \in BV$ and then that $\nabla f
= 0$, so that $f$ is a constant.

By {\sl contrast}, when $p > 1$, and $f$ {\sl takes
its values into} $\Z$ it suffices to assumes that
$$
\int_B \int_B {|f(x) - f(y)|^p \over |x - y|^p}
\rho_\varepsilon(|x - y|) dx \, dy \leq C \;
\hbox{as} \; \varepsilon \rightarrow 0.\leqno(50)
$$
Indeed, (50) implies that $f \in W^{1,p}$ (attention
when $p = 1$, (50) only implies that $f \in BV$).
Then, one may use the fact that $f$ takes its values
into $\Z$ to conclude that $f$ is constant. The
argument is the following: write
$$
\Omega = \bigcup_{k \in \Z}  A_k
$$
where $A_k = \{x \in \Omega ; f(x) = k\}$ and use a
well-known result of Stampacchia (see e.g. Lemma 7.7
in Gilbarg--Trudinger [1]) asserting that 
$\nabla f = 0$ a.e. on $A_k$. Hence $\nabla f = 0$ 
a.e. on 
$\Omega$.

Alternatively, one may deduce from (50) and assumption
$f : \Omega \rightarrow \Z$, that
$$
\int_\Omega \int_\Omega {|f(x) - f(y)| \over |x - y|}
\; {\rho_\varepsilon(|x - y|) \over |x - y|^{p-1}} dx
\, dy \leq C.
$$
This yields easily
$$
\lim_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega {|f(x) - f(y)| \over |x - y|}
\rho_\varepsilon(|x - y|) dx\, dy= 0
$$
and thus $f$ is a constant.


There are interesting extensions of some of the above 
results where the ratio
$$
|f(x) - f(y)|^p\over|x-y|^p
$$ 
is replaced by a more general expression 
$$
\omega\displaystyle{\left({|f(x) - f(y)| \over
|x-y|}\right).}
$$
  Here are two results due to R. Ignat, 
V. Lie  and A. Ponce [1].

\medskip

\noindent  {\bf Theorem 4.} {\sl Assume $\omega :
[0,\infty) \rightarrow [0,\infty)$ is a continuous 
function such that $\omega(0) = 0$, $\omega(t) > 0 \;
\forall t > 0$ and
$$
\int^\infty_1 {\omega(t) \over t^2} dt = \infty.\leqno(51)
$$
Assume $f \in L^1(\Omega)$ satisfies
$$
\int_\Omega \int_\Omega \omega \left({|f(x) - f(y)| 
\over |x - y|}\right) {dx \, dy \over |x - y|^N} <  
\infty,                                             
$$
then $f$ is a constant.}

\medskip

\noindent  {\bf Theorem 5.} {\sl Assume $\omega :
[0,\infty) \rightarrow [0,\infty)$ is a continuous
function such that $\omega(0) = 0$ and
$$
\lim_{t \rightarrow \infty} {\omega(t) \over t} = 
\alpha > 0.                                       
$$
Assume $f \in L^1(\Omega)$ satisfies
$$
\int_\Omega \int_\Omega \omega \left({|f(x) - f(y)|    
\over |x-y|}\right) \rho_\varepsilon (|x - y|) dx \, dy
\leq C \; \hbox{as} \; \varepsilon \rightarrow 0.      
$$
Then $f \in BV$ and
$$
\lim_{\varepsilon \rightarrow 0} \int_\Omega
\int_\Omega \omega \left({|f(x) - f(y)| 
\over |x-y|}\right) \rho_\varepsilon (|x - y|) dx \, dy
= \int_\Omega \overline\omega(|\nabla f_{ac}|)dx +
\alpha K_{1,N} \int_\Omega|\nabla f_s|dx,
$$
where $\overline\omega(t) =
\displaystyle{\int_{S^{N-1}}} \omega(t|\sigma \cdot 
e|)d\sigma$ and $\nabla f = \nabla f_{ac} + \nabla 
f_s$ is the Radon--Nikodym decomposition of $\nabla f$.}


\medskip

Here is still another  open problem:

\medskip

%\noindent {\sl Open problem 2.} Let $\Omega$ be a
%(smooth) connected, bounded domain in $\R^N$. Let $f :
%\Omega \rightarrow \R$ be a measurable (or even
%continuous) function. Let $\omega : [0,\infty)
%\rightarrow [0,\infty)$ be a continuous function such
%that $\omega(0) = 0$ and $\omega(t) > 0$ for $t > 0$.
%Assume that
%$$
%%\lim_{\varepsilon \rightarrow 0} 
%\int_\Omega
%\int_\Omega \omega\left({|f(x) - f(y)| \over |x - y|}\right) 
%%\rho_\varepsilon(|x - y|)dx \, dy = 0.
%{dx \, dy \over |x - y|^N} < \infty.
%$$
%Can one conclude that $f$ is a constant?


\noindent {\sl Open problem 2.} Let $\Omega$ be a
(smooth) connected, bounded domain in $\R^N$. Let $f :
\Omega \rightarrow \R$ be a continuous
(or even  H\"older continuous) function. Let $\omega : [0,\infty)
\rightarrow [0,\infty)$ be a continuous function such
that $\omega(0) = 0$ and $\omega(t) > 0$ for $t > 0$.(Here (51) might fail).
Assume that
$$
\int_\Omega
\int_\Omega \omega\left({|f(x) - f(y)| \over |x -
y|}\right) {1 \over |x-y|^N} \; dx \, dy < \infty.
$$
Can one conclude that $f$ is a constant?




%%Note that if is {\sl convex} (e.g. $\omega(t) = t^p$,
%%$p \geq 1$), the answer is positive because one may
%%argue as in Section 2. The problem seems to be open
%%even when $\omega(t) = t^\alpha$ with $0 < \alpha < 1$.

\vskip 1cm

\noindent {\bf 4. Another approach. Connection with
VMO}


We first recall the definition of VMO$(\Omega;\R)$ (= 
vanishing mean oscillation). We say that a function $f
\in VMO(\Omega ; \R)$ if $f \in L^1_{loc}(\Omega;\R)$ 
satisfies
$$
\lim_{\varepsilon \rightarrow 0} {1 \over 
|B_\varepsilon(x)|^2} \int_{B_\varepsilon(x)}
\int_{B_\varepsilon(x)} |f(y) - f(z)|dy \, dz = 0
\quad \hbox{ uniformly for } x \in \Omega.
$$


   Let $\Omega$ be a connected (smooth) open 
set in
$\R^N$ and let $f \in VMO(\Omega;\Z)$. Then $f$ is a
constant. This was already observed in
Brezis--Nirenberg [1] (Section I.5, part 2). Indeed if
we set
$$
\overline f_\varepsilon(x) = {1 \over
|B_\varepsilon(x)|} \int_{B_\varepsilon(x)} f(y)dy
$$
then dist$(\overline f_\varepsilon(x),\Z) \rightarrow
0$ {\sl uniformly} in $\Omega$ (see Brezis--Nirenberg
[1], Section I.1) and thus there is some constant 
$k_\varepsilon \in \Z$ such that $|\overline
f_\varepsilon(x) - k_\varepsilon| \rightarrow 0$
uniformly in $\Omega$. Hence $f$ is a constant.

%$\overline f_\varepsilon$
%is constant (because $\Omega$ is connected). Hence
%$\overline f_\varepsilon$ is constant and so is $f$.

Functions in $W^{s,p}(\Omega)$ belong to $VMO(\Omega)$
provided $sp \geq N$ (see Brezis--Nirenberg [1],
Section I.2). Therefore one cannot apply directly this
argument in our setting which corresponds roughly
speaking to $sp \geq 1$. However one may use an
argument of {\sl reduction to dimension one} already
used in Bourgain--Brezis--Mironescu [2].

Assume for simplicity that $\Omega$ is a square in
$\R^2$. Let $f \in W^{s,p}(\Omega)$. Then, the
restrictions $f(x_1,\cdot)$ and $f(\cdot,x_2)$ still
belong to $W^{s,p}(I)$ for a.e. $x_1$ and a.e. $x_2$
(where $I$ is an interval) (see e.g. Brezis, Li, Mironescu
and Nirenberg [1], Section 2).

This observation is very useful when combined with the
following measure theoretical tool:

\medskip

\noindent {\bf Lemma} (see e.g. Brezis, Li,
Mironescu and Nirenberg [1], Lemma 2). {\sl Assume
that $f : \Omega \rightarrow \R$ is measurable.
Suppose that for a.e. $x_1$, $f(x_1,\cdot)$ and for
a.e. $x_2$, $f(\cdot,x_2)$ are constant functions.
Then $f$ is a constant.}

\medskip

The considerations above yield an alternative proof of
Corollary 1 when $p > 1$. Indeed, if $p > 1$, (2) says
that $f\in W^{s,p}(\Omega)$ where $s = 1/p$. The
restrictions of $f$ to almost every line still belong
to $W^{s,p}$ with $s = 1/p$. Hence these restrictions
are VMO.

Therefore, if $f : \Omega \rightarrow \Z$ one may
conclude that the restrictions of $f$ to almost every
line are constant. The above lemma allows to conclude
that $f$ is constant.

The preceding argument also gives

\medskip

\noindent  {\bf Theorem 6.} {\sl Assume $\Omega
\subset \R^N$ is connected and let $f : \Omega
\rightarrow \Z$ be a measurable function such that $f
= f_0 + f_1 + f_2 + ... + f_k$ where $f_0 \in
W^{1,1}(\Omega ; \R)$ and $f_i \in
W^{s_i,p_i}(\Omega;\R)$ with $s_i p_i \geq 1$ for $i =
1,2,...,k$. Then $f$ is a constant.}

\medskip

\noindent {\sl Open problem 3.} Is there a simple
intrinsic assumption on $f$ which can replace the
decomposition assumption $f = f_0  + f_1 + f_2 + ... +
f_k$? Is there an elegant way to unify Theorem 6 with
the results of Section 1?

\medskip

Another interesting direction of research is

\medskip

\noindent {\sl Open problem 4.} Find estimates for
$$
\|f - \moyenne f\|
$$
in terms of the quantities appearing throughout
the paper and which would imply that $f$ is constant in
various situations. The reader may find some results
in that direction in Bourgain, Brezis and Mironescu
[4] (see also Maz'ya and Shaposhnikova [1]).

\vskip 1cm

\noindent {\bf References}

\noindent R.A. Adams [1], {\sl Sobolev spaces}, Acad.
Press (1975).

\smallskip

\noindent L. Ambrosio and P. Tilli [1], {\sl Selected
topics on ``Analysis in metric spaces''}, Lecture
Notes, Scuola Normale Superiore Pisa (2000).

\smallskip

\noindent F.Bethuel and F. Demengel [1], Extensions
for Sobolev mappings between manifolds, Cal. Var. PDE
{\bf 3} (1995), 475--491.

\smallskip

\noindent J. Bourgain, H. Brezis and P. Mironescu
[1], Lifting in
Sobolev spaces, J. Analyse Math. {\bf 80} (2000), p.
37-86.


\noindent [2], On the structure of
the Sobolev space $H^{1/2}$ with values into
the circle, C. R. Acad. Sc. {\bf 331} (2000), p. 119--124.

\noindent [3], Another look at
Sobolev spaces, in {\sl Optimal Control and
Partial Differential Equations} (J.L. Menaldi, E. Rofman et A. Sulem,
eds) a volume in honour of A. Bensoussan's
60th birthday, IOS Press, 2001, p. 439--455.


\noindent [4], Limiting embedding
theorems for $W^{s,p}$ when $s \uparrow 1$ and
applications, J. Analyse Math.  (to appear).

\smallskip

\noindent H. Brezis [1], {\sl Analyse fonctionnelle ;
th\'eorie et applications}, Masson (1983) et Dunod
(1999).


\smallskip

\noindent H. Brezis and J.M. Coron [1], Large solutions for harmonic maps
in two dimensions, Comm.
    Math. 	Phys. {\bf 92} (1983), p. 203--215.



\smallskip

\noindent H. Brezis, Y.Li, P. Mironescu and L.
Nirenberg [1], Degree and
Sobolev spaces, Topological
methods in Nonlinear Analysis {\bf 13} (1999),
p. 181--190.


\smallskip

\noindent H. Brezis and L. Nirenberg [1], Degree theory and BMO, Part I :
compact manifolds without
boundaries, Selecta Math. {\bf 1} (1995), p. 197--263.


\smallskip

\noindent J. Davila [1], On an open question about
functions of bounded variation, Cal. Var. PDE (to
appear).

\smallskip

\noindent D. Gilbarg and N.S. Trudinger [1], {\sl 
Elliptic Partial Differential Equations of Second
Order}, Springer, Second edition, 1983.


\smallskip

\noindent P. Hajlasz and P. Koskela [1], {\sl Sobolev
met Poincar\'e}, Memoirs Amer. Math. Soc. {\bf 145}
(2000).

\smallskip

\noindent R. Hardt, D. Kinderlehrer and F.H. Lin [1],
The variety of configurations of static liquid
crystals, in {\sl Variational Methods} (H. Berestycki,
J.-M. Coron and I. Ekeland eds), Birkhauser 1990.


\smallskip

\noindent R. Ignat, V. Lie  and A. Ponce [1], paper in
preparation.


\smallskip

\noindent N. Korevaar and R. Schoen [1], Sobolev
spaces and harmonic maps for metric space targets,
Comm. in Anal. and Geom., 1 (1993), 561--659.


\smallskip

\noindent V. Maz'ya and T. Shaposhnikova [1], On the
Bourgain, Brezis and Mironescu theorem concerning
limiting embeddings of fractional Sobolev spaces, J. 
Funct. Anal. (to appear).

\smallskip

\noindent E. Stein [1], personal communication.

\vskip 1cm

\noindent Laboratoire Jacques-Louis Lions

\noindent Universit\'e Pierre et Marie Curie

\noindent Bo\^\i te courrier 187

\noindent 4 place Jussieu

\noindent 75252 Paris cedex 05

\smallskip

\noindent email: brezis@ann.jussieu.fr,
brezis@ccr.jussieu.fr





\bye




