EXTREMAL FUNCTIONS FOR AN EMBEDDING FROM SOME ANISOTROPIC SPACE, INVOLVING THE “ONE LAPLACIAN”

. In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to 1. We prove also that the extremal functions satisfy a partial diﬀerential equation involving the 1 Laplacian.


1.
Introduction. Anisotropic Sobolev spaces have been studied for a long time, with different purposes. Let us recall that for p = (p 1 , · · · , p N ), and the p i ≥ 1 the space D 1, p (R N ), denotes the closure of D(R N ) for the norm i |∂ i u| pi . The existence of a critical embedding from D 1, p (R N ) into Lp , withp = pi > 1 is due to Troisi, [34]. There is by now a large number of papers and an increasing interest about anisotropic problems. With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces [23], [30] and some more recent regularity results for minimizers of anisotropic functionals, that we will recall below.
Let us note that anisotropic operators bring new problems, essentially when one wants to prove regularity properties. As an example, ( except when N p − N −p − ≥ p ), the property that Ω be Lipschitz does not ensure the embedding W 1, p (Ω) → Lp (Ω). This is linked to the fact that in the absence of further geometric properties of Ω, one cannot provide a continuous extension operator from W 1, p (Ω) in D 1, p (R N ). To illustrate this, see the counterexample in [22], see also [13] for one example when some of the p i are equal to 1, in the context of the present article.
Let us say a few words about the existence and regularity results of solutions to − i ∂ i (|∂ i u| pi−2 ∂ i u) = f , u = 0 on ∂Ω when Ω is a bounded smooth domain in R N . Assuming a convenient assumption on f , the existence of solutions can generally easily be obtained by the use of classical methods in the calculus of variations. But, as a first step in the regularity of such solutions, the local boundedness of the solutions, can fail if the supremum of the p i is too large, let us cite to that purpose [27] , [18] where the authors exhibit a counterexample to the local boundedness when p i = 2 for i ≤ N − 1 and p N > 2 N −1 N −3 . This restriction on p, to ensure the 1136 FRANÇ OISE DEMENGEL AND THOMAS DUMAS local boundedness is confirmed by the results obtained later : let us cite in a non exhaustive way [7], [27], [4]. From all these papers it emanates in a first time that a sufficient condition for a local minimizer to be locally bounded is that the supremum of the p i be strictly less than the critical exponentp . This local boundedness is extended by Fusco Sbordone in [17] to the case where sup p i =p . For further regularity properties of the solutions, as the local higher integrability of the local minimizers for some generalized functionals, see Marcellini in [28], and Esposito Leonetti Mingione [14,15] . Coming back to D 1, p (R N ), and concerning extremal functions, let us recall that in the isotropic case, the first results concerned the case where p i = 2 for all i, in which case the extremal functions are solutions of −∆u = u 2 −1 . The existence and the explicit form of them is completely solved by Aubin [3], and Talenti, [32]. For W 1,p and the isotropic p Laplacian, say −∆ p u = −div(|∇u| p−2 ∇u) the explicit form is also known as the family of radial functions u a,b (r) = (a + br For further results about sharp embedding constant, and a new, elegant approach by using mass transportation the reader can see [6].
Let us now consider the case where the p i can be different from each others, and let us first cite the paper of Fragala Gazzola and Kawohl [16], where the authors prove the existence of extremal functions for some subcritical embeddings in the case of bounded domains.
For the case of R N and the critical case, the existence of extremal functions is proved in [21], when all the p i > 1, and p + := sup p i <p . The authors provide also some properties of the extremal functions, as the L ∞ behaviour, extending in that way the regularity results already obtained for solutions of anisotropic partial differential equation in a bounded domain, with a right hand side sub-critical as in [16], to the critical one. The method uses essentially the concentration compactness theory of P. L. Lions [24,25] adapted to this context, and some other tools developed also in a more general context in [20].
In the case where p + =p and for more general domains than R N the reader can see Vetois, [35]. In this article the author provides also some vanishing properties of the solutions, as well as some further regularity properties of the solutions.
When some of the p i are equal to 1, let us cite the paper of Mercaldo, Rossi, Segura de leon, Trombetti, [29], which proved the existence of solutions in some anisotropic space, with some derivative in the space of bounded measures, for the p-Laplace equation in bounded domains, using the definition of the one Laplacian with respect to the coordinates for which p i = 1. For the existence of extremal functions in the case of R N , and in the best of our knowledge, nothing has been done in the case where some of the p i are equal to 1. Of course in that case these extremal functions have their corresponding derivative in the space M 1 (R N ) of bounded measures on R N . Even if the existence of such extremal can be obtained following the lines in the proof of [21], the partial differential equation satisfied by the extremal cannot be obtained by this existence's result. In order to get it, we are led to consider a sequence of extremal functions for the embedding of D 1, p (R N ) in Lp (R N ) where in p , all the p i > p i and tend to them as goes to zero. Note that one of the difficulties raised by this approximation is that, due to the unboundedness of R N , D 1, p (R N ) is not a subspace of D 1, p (R N ), a problem which does not appear when one works with bounded domains, see [13]. In particular this does not allow to use directly the concentration compactness theory of P.L. Lions, [24]. We will prove both that the best constant for the embedding from D 1, p (R N ) in Lp (R N ) converges to the best constant for the embedding of D 1, p (R N ) into Lp (R N ), and that some extremal u converge sufficiently tightly to some u. Passing to the limit in the partial differential equation satisfied by u one obtains that u is extremal and satisfies the required partial differential equation.
When µ n and µ are in M 1 (Ω) we will say that µ n converges tightly to µ if for any ϕ ∈ C(R N ) and bounded, µ n , ϕ → µ, ϕ . Remark 1. When µ n ≥ 0, the tight convergence of µ n to µ is equivalent to both the two conditions 1) µ n µ vaguely and 2) Ω µ n → Ω µ.
We will frequently use the following density result: there exists u n ∈ D(Ω, R N ) such that (u n ) ± i , ( respectively |u n |), converges tightly to µ ± i , (respectively |µ|). The reader is referred to [12], [11], for further properties on convergence of measures and density of regular functions for the vague and tight topology.
Let N 1 ≤ N ∈ N, and p := (p 1 , · · · , p N ) ∈ R N such that p i = 1 for all 1 ≤ i ≤ N 1 , and p i > 1 for all N 1 + 1 ≤ i ≤ N . Let p + = sup p i , andp, defined by the identity In all the paper we will suppose that p + <p , though some of the results are valid also for p + =p , and we will precise it when it will occur. Let D 1, p (R N ) be the completion of D(R N ) with respect to the norm

Remark 2.
Of course by the equivalence of norms in R N1 this completion coincides with the completion for the norm N i=1 |∂ i u| pi . We now recall the existence of the embedding from D 1, p (R N ) in Lp (R N ), a particular case of the result of Troisi , [34].
and there exists some constant T 0 depending only on p, and N such that for all u ∈ D 1, p (R N ).
We now introduce a weak closure of D(R N ) for the norm (1). Set We also define , for any ϕ ∈ D(R N )} Definition 2.5. We will say that u n ∈ BV p (R N ) converges weakly to u if u n u (weakly) in Lp , ∂ i u n converges vaguely to ∂ i u in M 1 (R N ) when i ≤ N 1 , and ∂ i u n ∂ i u (weakly) in L pi , when i > N 1 . The convergence is said to be tight if furthermore R N |∂ i u n | pi → R N |∂ i u| pi for any i ≥ 1 , and R N |∇ 1 u n | → R N |∇ 1 u|. Remark 3. If u n converges weakly to u, since (u n ) is bounded in Lp , it converges strongly in L q loc for a subsequence, when q <p and then for a subsequence it converges almost everywhere.
Proposition 2. Suppose that p + <p . It is equivalent to say that 1. u ∈ BV p (R N ).
2. There exists u n ∈ D(R N ) which converges tightly to u.
3. There exists u n ∈ D(R N ) which converges weakly to u.
Remark 4. Following the lines in the proof below, but using strong convergence in L 1 of ∂ i u n for i ≤ N 1 , in place of tight convergence, it is clear that Proof. Suppose that 1) holds. We begin by a troncature. For 1 ≤ i ≤ N let α i defined as 1], and for all n ∈ N, In the same manner we have R N |∇ 1 u n − ∇ 1 u| → 0. The second step classically uses a regularisation process. Recall that when µ is a compactly supported measure in From this one derives the tight convergence when goes to zero and n to ∞ of 2) implies 3) is obvious. To prove that 3) implies 1), note that if (u n ) is weakly convergent to u, one has the existence of some constant independent on n so that Then by the embedding in Theorem 2.4, (u n ) is bounded in Lp , and by extracting subsequences from ∇ 1 u n in M 1 (R N , R N1 ) weakly, and from ∂ i u n in L pi weakly for i ≥ N 1 + 1, one gets that the limit u ∈ BV p (R N ).
Remark 5. Using the last proposition, one sees that (2) extends to the functions in BV p (R N ).
Theorem 2.6. Let σ a function with values in R N , such that its projection σ 1 on the first N 1 coordinates, belongs to L ∞ loc (R N , R N1 ), and suppose that for any Then σ · ∇u is a measure, and σ 1 · ∇ 1 u := σ · ∇u − N i=N1+1 σ i ∂ i u is a measure absolutely continuous with respect to |∇ 1 u|, with for ϕ ≥ 0 in C c (R N ) : (3) and u ∈ BV p (R N ), σ · ∇u and σ 1 · ∇ 1 u are bounded measures on R N and one has and . By Proposition 2, there exists u n ∈ D(R N ) such that u n converges tightly to ψu in BV p (R N ). By the classical Green's formula Using the weak convergence of u n towards ψu one gets that (σ · ∇u n )ϕ converges to σ · ∇u, ϕ . By the assumptions on σ i and ∂ i u n , one has The identity ( 4) is easily obtained by letting ϕ go to 1 R N , since all the measures involved are bounded measures.
Note that one has for all i and as soon as is small enough, p + <p . Let us finally define and note for further purposes that λ p =p .
Recall that as a consequence of the embedding of Troisi, [34] one has and there exists some T 0 > 0, such that for all u ∈ D 1, p (R N ), It is clear by Proposition 2 that Adapting the proof in [21] one has the following result Theorem 2.7. There exists u ∈ D 1, p (R N ) non negative which satisfies |u |p = 1 and In the sequel we will use the notation div 1 as the divergence of some N 1 vector with respect to the N 1 first variables.
By multiplying equation ( 7) by u and integrating one has K ≤ l ≤ p + K , and as we will see in Proposition 3 that lim sup K ≤ K, if u is an extremal function for K , |∇ 1 u | 1+ and |∂ i u | p i are bounded independently on , hence one can extract from u a subsequence which converges weakly in BV p . In the sequel we will prove that by choosing conveniently the sequence u , it converges up to subsequence to an extremal function for K.
3. The main results. The main result of this paper is the following : 1) There exists v ∈ D 1, p (R N ), |v |p = 1, an extremal function for K , which converges in the following sense to v ∈ BV p (R N ): v converges to v in the distribution sense, and almost everywhere, pi for all i ≥ 1, and |v|p = 1. Furthermore lim K = K. As a consequence v is an extremal function for K.
2) v satisfies in the distribution sense the partial differential equation : where σ 1 · ∇ 1 v is taken in the sense of Theorem 2.6 and l = lim l l defined in (7).
The proof of Theorem 3.1 is given in the next subsection, and it relies of course on a convenient adaptation of the PL Lions compactness concentration theory. However, due to the fact that the exponents of the derivatives and the critical exponent vary with , we are led to introduce a power v λ of some convenient extremal function, -where λ has been defined in (6)-, and to analyze the behaviour of this new sequence, which belongs to BV p (R N ), and is bounded in that space, independently on , as we will see later.
In a second time we prove that Lemma 3.3. Suppose that u ∈ BV p (R N ), and that |u|p ≤ 1, then

Hint of the proof :
Use u |u|p in the definition of K and the fact that if |u|p ≤ 1, |u| pī p ≥ |u| p + p .
Proposition 3. One has lim sup K ≤ K.
As a consequence any sequence (v ) of extremal functions for K is bounded independently on , in the sense that there exists some positive constant c so that, for Proof. Let δ > 0, δ < 1 2 and let u δ ∈ D 1,p (R N ), ( or BV p (R N )), so that |u δ |p = 1 and By definition of D 1, p (R N ), there exists v δ ∈ D(R N ) such that ||v δ |p − 1| ≤ δ, For small enough one has ||v δ |p − 1| ≤ 2δ. By considering w δ = v δ |v δ |p , one sees that w δ ∈ D(R N ), |w δ |p = 1, and By the Lebesgue's dominated convergence theorem, which concludes the proof since δ is arbitrary.
Proposition 4. Suppose that w ∈ BV p (R N ) satisfies |w |p = 1, and that w → v almost everywhere. Then for small enough |w − v|p ≤ 1.
Proof. If v ≡ 0, there is nothing to prove. If v = 0, using Brezis Lieb Lemma, [5] one has |w − v|p − (|w |p − |v|p ) → 0 which implies that lim sup |w − v|p < 1, hence the result holds. This lemma will be used for w = v λ , where v is some convenient extremal function, given in Lemma 3.4 below, and λ has been defined in ( 6).
Lemma 3.4. Let u be a non negative extremal function for K , so that |u | p = 1. There exists v ≥ 0 which satisfies for all i ≥ 1, and Proof. This proof is as in [21], but we reproduce it here for the reader's convenience. Let α i =p p i − 1, i = 1, · · · , N . For every y = (y 1 , · · · , y N ) ∈ R N , and for any u ∈ D 1, p (R N ), and t > 0, we set Then, we have |u|p = |u t,y |p , Let u be an extremal function for K so that |u |p = 1. As in [21], [25], we recall the definition of the Levy concentration function, for t > 0 : where E(y, t α 1 , · · · , t α N ) is the ellipse defined by with y = (y 1 , · · · , y N ). Since for every > 0, lim t→0 Q (t) = 0, and lim t→∞ Q (t) = 1, there exists t > 0 such that Q (t ) = 1 2 , and there exists y ∈ R N such that Thus, by a change of variable one has for v = u t ,y : Note for further purpose that v is also extremal for K .

Proposition 5.
Let v ≥ 0 be in D 1, p (R N ), bounded in that space, independently on . Then for λ defined in (6), the sequence w = v λ is bounded in D 1, p (R N ).

FRANÇ OISE DEMENGEL AND THOMAS DUMAS
Proof. One has and for all i > N 1 , using the definition in (5)
Theorem 3.5. Let v ∈ D 1, p (R N ), be given by Lemma 3.4, and λ be defined in (6). There exist v ∈ BV p (R N ), some positive bounded measures on R N : τ,τ , µ i ,μ i , for N 1 + 1 ≤ i ≤ N, and ν, a sequence of points x j ∈ R N , and some positif reals ν j , µ i j , τ j ,τ j , j ∈ N, so that for a subsequence 1. v , and v λ converge both to v, almost everywhere and strongly in every L q loc , q <p , and v ∈ BV p (R N ).
4. |v λ |p = |v |p |v|p + ν := |v|p + j ν j δ xj in M 1 (R N ) weakly. 5. One has τ ≥ j τ j δ xj , µ i ≥ j µ i j δ xj , for all i ≥ N 1 + 1, and for any j ∈ N, Proof. 1 The convergence of v λ is clear by using the compactness of the embedding from BV p in L q with q <p , on bounded sets of R N , the analogous for v is also true since q < lim infp . Let us prove the existence ofτ , τ, µ i ,μ i , N 1 + 1 ≤ i ≤ N, and ν. Indeed one has by extracting a subsequence the existence ofτ , since we know that |∇ 1 v| ≤ lim inf |∇ 1 v | 1+ . The existence of τ is obtained from the same arguments. Furthermore, by Hölder's inequality Letting go to zero, since λ goes to 1, one gets thatτ ≥ τ . We argue in the same manner to prove the analogous results for |∂ i (v λ )| pi and |∂ i v | p i . The existence of ν is clear.
We prove in the lines which follow that ν is purely atomic. This is classical, but we reproduce the proof for the convenience of the reader. Let To prove Claim 1, let us define h = (v λ − v). Using (2), We have defined ν and µ i by the following vague convergences : v λ p vp + ν, |∂ i v λ | pi |∂ i v| pi + µ i , and |∇ 1 v λ | |∇ 1 v| + τ . By Bresis Lieb's Lemma, [5], one derives that |h |p * ν, and Using the fact that h tends to 0 in L pi (Supptϕ), for all i, since p i <p , one has |h | pi |∂ i ϕ| pi → 0. Passing to the limit in (10), one gets We then use for i ≥ N 1 + 1 Taking the power 1 N pi and N1 N and multiplying the inequalities, one derives Claim 1.
By (9) one sees that ν is absolutely continuous with respect to µ, with for some constant c and for any borelian set E, Let then h ≥ 0 be µ integrable so that ν = hdµ. Then if x is a density point for µ, ie, so that lim r→0 µ(B(x, r)) = 0, one gets that ν(B(x,r)) µ(B(x,r)) → 0, hence if D is the at most numerable set where µ({x j }) > 0, one has h = 0 in R N \ D. This implies that ν has only atoms that we will denote {x j } j∈N . Note that here the assumption p + <p is essential.
We now prove 5. We still follow the lines in [21]. Let δ > 0 small, In particular for all i ≤ N , when δ goes to zero.
To prove Claim 2, we apply Lemma 3.3 with |v λ φ δ |p ≤ 1 We use hence by (11) when p i = 1 and v λ − v → 0 in L q loc for all q <p , this goes to zero in L 1 when and δ go to zero. For i ≥ N 1 + 1, by the mean value's theorem Using Holder's inequality, ( 11) and v λ − v → 0 in L q loc for all q <p , this goes to zero in L 1 , when and δ go to zero. Claim 2 is proved.
We can now conclude, using the fact that |∇ 1 v| is orthogonal to Dirac masses, as a consequence of the results on the dimension of the support of |∇ 1 v| s , [19], and using the fact that |∂ i v| pi belongs to L 1 , for i ≥ N 1 + 1, that Defining τ j = lim sup δ→0 R N τ φ δ and µ i j = lim sup δ→0 R N µ i φ pi δ , one gets the first part of 5.
To prove the last part of 5, let R > 0 large and ψ R some C ∞ function which is 0 on |x| < R, and equals 1 for |x| > R + 1, 0 ≤ ψ R ≤ 1. It can easily be seen that for any i ≥ N 1 + 1 and for any γ i ≥ 1 and |x|>R+1 |v λ |p ≤ And then by the definition of µ ∞ lim R→+∞ lim sup Let us remark that since v ∈ BV p , one has lim R→+∞ |∇ 1 v|ψ R + N i=N1+1 1 pi |∂ i v| pi ψ pi R + R N |v|p ψp R = 0. We use once more h = v λ − v, which goes to zero in L q loc . Note that since |h |p ≤ 1, one also has |h ψ R |p ≤ 1 and then applying Lemma 3.3 Since ∇ψ R is compactly supported in R < |x| < R + 1, and since p i <p one has
Proof of Theorem 3.1. We take a subsequence v so that with lim K = lim inf K , in the sequel we will still denote it v for simplicity. We are going to prove both that lim sup K = K = lim inf K , ν ∞ = µ ∞ = 0, µ i j = ν j = 0, for all j ∈ N, that for all i, |∂ i v | p i → |∂ i v| pi , tightly on R N , and that lim |∇ 1 (v λ )| = lim |∇ 1 v | 1+ = |∇ 1 v|, tightly on R N . Indeed, using the previous convergences in Theorem 3.5 Using the fact that lim sup K ≤ K, R N τ ≥ j τ j , R N µ i ≥ j µ i j , one gets that we have equalities in place of inequalities everywhere we used them. In particular ∞ , and then always by the assumption p + <p , only one of the positive reals R N |v|p , ν j , ν ∞ , can be different from zero. But this imposes that the only one which is = 0 must be equal to one. By Remark 6, one then gets ν ∞ = 0. On the other hand, let j ∈ N, either x j / ∈ B(0, 1) and then for δ small enough B(xj ,δ) |v |p + B(0,1) |v |p ≤ 1, hence ν j = 0, or x j ∈ B(0, 1) and then ν j ≤ lim B(0,1) |v |p = 1 2 , and once more ν j = 0. One then derives that 1 = |v |p p → |v|p p . By the definition of K one has We have obtained that v is an extremal function, and lim K = K.
We now prove that v satisfies (8). First recall that l ≥ K ≥ 1 p + l , as we can see by multiplying (7) by v the equation, integrating, and using |v |p p = 1. In particular l is bounded. Let us extract from it a subsequence which converges to some l ≥ lim K = K.
Let us define and -with an obvious abuse of notation-σ = (σ 1, , σ N1+1 , · · · , σ N ). Note that σ 1, is bounded in L q loc , for any q < ∞. Indeed, let K be a compact set, one has by Holder's inequality K |σ 1, Using the boundedness of K one gets that σ 1, is bounded in L q loc , hence converges up to subsequence weakly in L q loc to some σ 1 which satisfies for any compact set K, From these convergences, one gets that defining σ = (σ 1 , σ N1+1 , · · · , σ N ), by the definition in Theorem 2.6, σ · ∇v converges to σ · ∇v in the distribution sense. Using loc , one derives that σ 1, ·∇ 1 v converges to σ 1 ·∇ 1 v in D (R N ). Since σ 1, ·∇ 1 v is also bounded in L 1 , this convergence is in fact vague. By lower semi-continuity for the vague topology, for any ϕ ≥ 0 in C c (R N ) This implies that |∇ 1 v| ≤ σ 1 · ∇ 1 v in the sense of measures, and since one always has the reverse inequality, we have obtained that σ 1 · ∇ 1 v = |∇ 1 v|.
We get by passing to the limit in (7) that v satisfies the partial differential equation : Furthermore, multiplying the equation by v and integrating, one gets l ≥ K > 0.
3.2. Proof of Theorem 3.2. We will prove the L ∞ regularity when u is some extremal function which satisfies (8), with l = 1. Indeed one has Lemma 3.6. Let v and v be as in Theorem 3.1. Then Furthermore u converges tightly to u in BV p (R N ).
We do not give the proof of this lemma, which is left to the reader. In the sequel we will consider u and u as in Lemma 3.6.
Lemma 3.7. Suppose that u ∈ BV p is as in Lemma 3.6. Suppose that g is Lipschitz continuous on R, such that g(0) = 0 and g ≥ 0, then g(u) ∈ BV p , with σ 1 · ∇ 1 (g(u)) = |∇ 1 (g(u))|. Furthermore one has the identity Proof. In the following lines, we will use "UTS" to say that the convergence holds up to subsequence. Note that g(u ) ∈ D 1, p (R N ) by the mean value's theorem, since g ∈ L ∞ , and (g(u )) is bounded in that space by the assumptions on u , and then also in BV p loc . Then since u converges to u almost everywhere "UTS" and g is continuous, g(u) ∈ BV p (R N ), and g(u ) converges weakly to g(u) in BV p loc "UTS" . In particular it converges to g(u) in L q loc , "UTS" for all q <p . Let us observe that the sequence of measures σ ·∇(g(u )) converges "UTS" to σ ·∇(g(u)) : Since σ ·∇g(u ) is bounded in L 1 , it is sufficient to prove that it converges in the distribution sense. To check this, let ϕ ∈ D(R N ), take q <p so that for small enough p i < q, then σ → σ in L q loc . Using g(u ) → g(u) in L q loc strongly and "UTS" for all q <p , one has g(u )σ · ∇ϕ → g(u)σ · ∇ϕ. Secondly note that |up −1 g(u )| ≤ |g | ∞ |u |p . By the strong convergence of (u )p in L 1 one can suppose that "UTS" it is dominated by a function h in L 1 , hence so does up −1 g(u ). By the almost everywhere convergence "UTS" of up −1 g(u ) to up −1 g(u) and the Lebesgue's dominated convergence theorem, one gets that for any ϕ ∈ D(R N ), up −1 g(u )ϕ → up −1 g(u)ϕ. We have obtained that σ · ∇(g(u ))ϕ → σ · ∇(g(u))ϕ, for any ϕ in D(R N ), hence also for ϕ in C c (R N ). Furthermore, by lower semicontinuity one has for any ϕ ≥ 0 in C c (R N ), This implies since one also has σ 1 · ∇ 1 (g(u)) ≤ |∇ 1 g(u)|, that σ 1 · ∇ 1 (g(u)) = |∇ 1 (g(u))|.
To get identity (17 ) it is then sufficient to multiply the equation ( 16) by g(u )ϕ, and pass to the limit using the previous convergence. Next one can let ϕ go to 1 R N since all the measures involved are bounded measures. Corollary 1. Let u be as in Lemma 3.6. For any L and a > 0, (u min(u a , L)) ∈ BV p (R N ), σ 1 · ∇ 1 (u min(u a , L)) = |∇ 1 (u min(u a , L))|, and |∇ 1 (u min(u a , L))| + Proof. We use Lemma 3.7 with g(u) = u min(u a , L) and equation (17). Then it is sufficient to observe that We now prove the following Proposition 6. Let u be as in Lemma 3.6, then u ∈ L ∞ .
Proof. This proof follows the lines in [16] and [21]. Once more, we reproduce it here for the sake of completeness. We begin to prove that u ∈ L q for all q < ∞. In the sequel, c denotes some positive constant which does not depend on k nor on a, which can vary from one line to another. Let k to choose later, and write for all p j , ( recall that p j = 1 for j ≤ N 1 ) : up min(u apj , L pj ) = Using Corollary 1, for u min(u apj , L) one gets for all j (1 + a) −pj +1 |∂ j (u min(u a , L))| pj ≤ up min(u apj , L pj ) and then defining I j = ( |∂ j (u min(u a , L))| pj ) Choosing k a so that c(a + 1) Letting L go to ∞ one gets |u a+1 |p ≤ C (|u|p )(1 + a)k a a , taking the power 1 a+1 , one has obtained that for q =p (a + 1), and then u belongs to L q for all q < ∞.
Note that one could consider in place of K = inf u∈D 1, p ,|u|p =1 |∇ 1 u| 1 + N i=N1+1 1 pi |∂ i u| pi pi the infinimum K = inf u∈D 1, p (R N ),|u|p =1 and prove the existence of an extremal function with obvious changes.