\magnification=1200 \newfam\msbmfam \font\tenmsbm=msbm10\textfont\msbmfam=\tenmsbm \font\sevenmsbm=msbm7 \scriptfont\msbmfam=\sevenmsbm \font\fivemsbm=msbm5 \scriptscriptfont\msbmfam=\fivemsbm \mathchardef\R="7852 \mathchardef\N="784E \mathchardef\M="784D \mathchardef\L="784C \mathchardef\C="7843 \mathchardef\Q="7851 \mathchardef\Z="785A \mathchardef\P="7850 \mathchardef\T="7854 \mathchardef\I="7849 \mathchardef\H="7848 \def\moyenne{\not\hskip -4pt\int} \def\supp{\hbox{\rm supp}\,} \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