Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian"

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Extremal functions for an embedding from some anisotropic space, involving the ”one Laplacian”

Françoise Demengel, Thomas Dumas

To cite this version:

Françoise Demengel, Thomas Dumas. Extremal functions for an embedding from some anisotropic space, involving the ”one Laplacian”. Discrete and Continuous Dynamical Systems - Series A, Amer- ican Institute of Mathematical Sciences, In press. �hal-01887287�

Extremal functions for an embedding from some anisotropic space, involving the ”one Laplacian”

Fran¸coise Demengel, Thomas Dumas ^∗

Abstract

In this paper, we prove the existence of extremal functions for the best con- stant of embedding from anisotropic space, allowing some of the Sobolev ex- ponents 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= (p1,· · · , pN), and thepi ≥1 the space D^1,~p(R^N), denotes the closure of D(R^N) for the norm P

i|∂_iu|_pi. The existence of a critical embedding from D^1,~p(R^N) into L^p¯?, with ¯p¯^? = P ^N

i 1

pi−1 when ^N_p¯ := P

i 1

pi > 1 is due to Troisi, [?].

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 pio- neering works on anisotropic Sobolev spaces [?], [?] 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^{N p}_−p⁻− ≥ p^?), the property that Ω be Lipschitz does not ensure the embedding W^1,~p(Ω),→L^p¯?(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 [?], see also [?] 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 −P

i∂i(|∂_iu|^pi−2∂iu) = f, u = 0 on ∂Ω when Ω is a bounded smooth domain in

∗Laboratoire AGM, UMR 80-88, Universi´e de Cergy Pontoise, 95302 Cergy Pontoise.

R^N. Assuming a convenient assumption onf, 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 thepi is too large, let us cite to that purpose [?] where the author exhibits a counterexample to the local boundedness whenp_i = 2 fori≤N−1 and p_N >2^N−1_N−3, see also [?]. This restriction on ~p, to ensure the local boundedness is confirmed by the results obtained later : let us cite in a non exhaustive way [?], [?], [?]. 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 thep_i be strictly less than the critical exponent ¯p^?. This local boundedness is extended by Fusco Sbordone in [?] to the case where supp_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 [?], and Esposito Leonetti Mingione [?,?] .

Coming back toD^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 [?], and Talenti, [?]. For W^1,p and the isotropicp Laplacian, say−∆_pu=−div(|∇u|^p−2∇u) the explicit form is also known as the family of radial functionsu_a,b(r) = (a+br^p−1p )

p−N

p ,while for the non isotropic p-Laplacian, say for the equation −PN

i=1∂i(|∂_iu|^p−2∂iu) =u^p¯?−1, the explicit solutions are obtained by Alvino Ferrone Trombetti Lions [?] and are given by u_a,b(r) = (a+bPN

i=1|x_i|^p−1p )

p−N

p . For further results about sharp embedding constant, and a new, elegant approach by using mass transportation the reader can see [?].

Let us now consider the case where thep_i can be different from each others, and let us first cite the paper of Fragala Gazzola and Kawohl [?], 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 [?], when all the pi > 1, and p⁺ := suppi < 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 [?], to the critical one. The method uses essentially the concentration compactness theory of P. L. Lions [?, ?] adapted to this context, and some other tools developed also in a more general context in [?].

In the case wherep⁺= ¯p^? and for more general domains thanR^N the reader can see Vetois, [?]. 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 thep_iare equal to 1, let us cite the paper of Mercaldo, Rossi, Segura de leon, Trombetti, [?], 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 thep_i are equal to 1. Of course in that case these extremal functions have their corresponding derivative in the space M¹(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 [?], 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 ofD^1,~p(R^N) inL^p¯?(R^N) where in

~

p, all thep_i > pi and tend to them asgoes 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 [?]. In particular this does not allow to use directly the concentration compactness theory of P.L. Lions, [?]. We will prove both that the best constant for the embedding from D^1,~p(R^N) in L^p¯?(R^N) converges to the best constant for the embedding of D^1,~p(R^N) into L^p¯?(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.

2 Notations, and previous results

2.1 Some measure Theory, definition and properties of the space BV^~p

Definition 2.1. Let Ωbe an open set in R^N, and M(Ω), the space of scalar Radon measures, i.e. the dual of C_c(Ω). Let M¹(Ω) be the space of scalar bounded Radon measures or equivalently the subspace of µ∈M(Ω) which satisfy

R

Ω|µ|= sup_ϕ∈Cc(Ω)hµ, ϕi<∞.

M⁺(Ω)is the space of non negative bounded measures on R^N. Definition 2.2. When µ= (µ₁,· · · , µ_n) we define |µ|= (P

µ²_i)¹² as the measure : For ϕ≥0 in C_c(Ω), h|µ|, ϕi= sup_ψ∈C

c(Ω,R^N),PN 1 ψ²_i≤ϕ²

Phµ_i, ψ_ii.

Let us recall that

Definition 2.3. µn* µvaguely or weakly in M(Ω)if for anyϕ∈ C_c(Ω), hµ_n, ϕi → hµ, ϕi.

Whenµ_n andµare inM¹(Ω)we will say thatµ_n converges tightly toµif for any ϕ∈ C(R^N) and bounded, hµ_n, ϕi → hµ, ϕi.

Remark 2.4. When µn ≥0, the tight convergence of µn to µ is equivalent to both the two conditions 1) µn* µ vaguely and 2)R

Ωµn→R

Ωµ.

We will frequently use the following density result:

Proposition 2.5. If ~µ∈M¹(Ω,R^N) 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 [?], [?], for further properties on convergence of measures and density of regular functions for the vague and tight topology.

LetN₁ ≤N ∈N, and~p:= (p₁,· · · , p_N)∈R^N such thatp_i= 1 for all 1≤i≤N₁, and p_i >1 for all N₁+ 1≤i≤N. Letp⁺= supp_i, and ¯p, defined by the identity

N

¯

p =X 1 pi

=N1+ X

i≥N1+1

1 pi

,

and ¯p^? = _N−¯^Np¯_p, or equivalently here

¯

p^∗ := N

N1+PN i=N1+1

1 pi −1.

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

|u|_~p =|(

N1

X

i=1

(∂_iu)²)¹²|₁+

N

X

i=N1+1

|∂_iu|_pi :=|∇₁u|₁+

N

X

i=N1+1

|∂_iu|_pi, (2.1) where ∇₁u is the N₁ vector (∂₁u,· · · , ∂_N1u), and |u|_pi denotes for i ≥ N₁+ 1 the usual L^pi(R^N) norm.

Remark 2.6. Of course by the equivalence of norms inR^N1 this completion coincides with the completion for the norm PN

i=1|∂_iu|_pi.

We now recall the existence of the embedding from D^1,~p(R^N) in L^p¯?(R^N), a particular case of the result of Troisi , [?].

Theorem 2.7.

D^1,~p(R^N),→L^p¯∗(R^N),

and there exists some constantT₀ depending only on ~p, and N such that

T0|u|_p¯^? ≤

N

Y

i=1

|∂_iu|

1

pNi, and then |u|_p¯^?≤ 1 T_oN





pN1|∇₁u|₁+

N

X

i=N1+1

|∂_iu|_pi



, (2.2) for allu∈ D^1,~p(R^N).

We now introduce a weak closure of D(R^N) for the norm (??). Set

BV^~p(R^N) : = {u∈L^p¯∗(R^N), ∂_iu∈M¹(R^N) for 1≤i≤N₁, and ∂_iu∈L^pi(R^N) forN₁+ 1≤i≤N}.

We also define

BV_loc^~p (R^N) ={u∈ D⁰(R^N), ϕu∈BV^~p(R^N), for any ϕ∈ D(R^N)}

Definition 2.8. We will say that u_n ∈ BV^~p(R^N) converges weakly to u if u_n * u (weakly) in L^p¯?,∂iun converges vaguely to∂iu inM¹(R^N) wheni≤N1, and∂iun*

∂iu (weakly) in L^pi, when i > N1.

The convergence is said to be tight if furthermore R

R^N|∂_iu_n|^pi → R

R^N|∂_iu|^pi for any i≥1 , and R

R^N|∇₁un| →R

R^N|∇₁u|.

Remark 2.9. If u_n converges weakly tou, since(u_n) is bounded inL^p¯?, it converges strongly inL^q_locfor a subsequence, whenq <p¯^?and then for a subsequence it converges almost everywhere.

Proposition 2.10. Suppose that p⁺<p¯^?. It is equivalent to say that 1. u∈BV^~p(R^N).

2. There existsu_n∈ D(R^N) which converges tightly to u.

3. There existsu_n∈ D(R^N) which converges weakly to u.

Remark 2.11. Following the lines in the proof below, but using strong convergence in L¹ of ∂_iu_n for i ≤N₁, in place of tight convergence, it is clear that D^1,~p(R^N) = {u∈L^p¯?(R^N), ∂iu∈L¹(R^N), i≤N1, ∂iu∈L^pi(R^N), i≥N1+ 1}.

Proof. Suppose that 1) holds.

We begin by a troncature. For 1≤i≤N let α_i defined as α_i = p¯^?

pi

−1.

Note thatα_i>0 for alli. Letϕ∈ D(]−2,2[),ϕ= 1 on [−1,1], and for alln∈N, un(x) = Π^N_i=1ϕ( xi

n^αi)u(x).

We denote Cn = Π^N_i=1[−2n^αi,2n^αi]. ( Remark that|C_n|= 4^Nn^PNi=1αi). We need to prove that ∂_iu_n→∂_iuin L^pi(R^N) for alli∈[N₁+ 1, N]. Since

∂iun(x) =u(x)∂i

Π^N_j=1ϕ( xj

n^αj)

+ Π^N_j=1ϕ( xj

n^αj)∂iu(x), it is sufficient to prove thatu∂_i

Π^N_j=1ϕ(_n^x_αj^j )

→0 inL^pi(R^N). By H¨older’s inequality Z

R^N

|u∂_i

Π^N_j=1ϕ( x_j n^αj)

|^pi ≤ c n^αipi(

Z

R^N−1

Z

n^αi≤|xi|≤2n^αi

|u|^p¯?)^p?pi¯ |C_n|^1−p?pi¯

≤ c⁰n^{−αipi+(1−p?pi¯ )}

PN

j=1αjo(1),

which tends to zero, since u∈L^p¯?(R^N) implies thatR

R^N−1

R

n^αi≤|xi|≤2n^αi|u|^p¯? →

n→∞0, and for anyiby the definition ofα_i,α_ip_i≥(1−^p_p¯?ⁱ)P_N

j=1α_j . In the same manner we haveR

R^N|∇₁un−∇₁u| →0. The second step classically uses a regularisation process.

Recall that when~µis a compactly supported measure inR^N, with values inR^N1, when ρ ∈ D(R^N), R

ρ = 1, ρ ≥ 0, and ρ = ¹Nρ(^x), ρ?|~µ| converges tightly to |~µ|, ρ? µ^±_i converges tightly to µ^±_i ,for i≤N₁. From this one derives the tight convergence whengoes to zero andnto∞of|∇₁(ρ? un)|towards|∇₁u|, of|∂_i(ρ? un)|towards

|∂_iu|fori≤N₁, and of∂_i(ρ? u_n) towards ∂_iu inL^pi fori≥N₁+ 1.

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

|∇₁un|₁ +PN

i=N1+1|∂_iun|_pi ≤ C. Then by the embedding in Theorem ??, (un) is bounded inL^p¯?, and by extracting subsequences from∇₁u_ninM¹(R^N,R^N1) weakly, and from∂iuninL^pi weakly fori≥N1+ 1, one gets that the limitu∈BV^p~(R^N).

Remark 2.12. Using the last proposition, one sees that (??)extends to the functions in BV^~p(R^N).

We now enounce a result which extends the definition of the ”Anzelotti pairs”, [?], see also Temam [?], Strang Temam in [?], and [?,?,?].

Theorem 2.13. Letσa function with values inR^N, such that its projectionσ¹ on the first N₁ coordinates, belongs to L^∞_loc(R^N,R^N1), and suppose that for any i≥N₁+ 1, σ ·e_i ∈ L^p

0 i

loc, and that divσ ∈ L

¯ p?

¯ p?−1

loc . Then if u ∈ BV_loc^~p (R^N), one can define a distribution σ· ∇u in the following manner, for ϕ∈ D(R^N),

hσ· ∇u, ϕi=− Z

R^N

divσ(uϕ)− Z

R^N

(σ· ∇ϕ)u.

Then σ · ∇u is a measure, and σ¹ · ∇₁u := σ· ∇u−PN

i=N1+1σ_i∂_iu is a measure absolutely continuous with respect to |∇₁u|, with for ϕ≥0 in C_c(R^N) :

h|σ¹· ∇₁u|, ϕi ≤ |σ¹|_L∞(Supptϕ)h|∇₁u|, ϕi. (2.3) Furthermore when σ¹ ∈L^∞(R^N,R^N1) andσi∈L^p0i(R^N), for any i≥N1+ 1, divσ∈ L

¯ p?

p?−1¯ (R^N) andu∈BV^~p(R^N),σ· ∇u andσ¹· ∇₁u are bounded measures onR^N and one has

Z

R^N

σ· ∇u=− Z

R^N

div(σ)u (2.4)

and

|σ¹· ∇₁u| ≤ |σ¹|_∞|∇₁u|.

Proof. Takeψ∈ D(R^N),ψ= 1 on Supptϕ. Then ifu∈BV_loc^~p (R^N),ψu∈BV^~p(R^N).

By Proposition ??, there existsun∈ D(R^N) such thatun converges tightly toψuin BV^~p(R^N). By the classical Green’s formula

Z

R^N

σ· ∇u_nϕ=− Z

R^N

div(σ)(unϕ)− Z

R^N

(σ· ∇ϕ)u_n. Using the weak convergence of un towardsψu one gets that R

(σ· ∇u_n)ϕ converges to hσ· ∇u, ϕi. By the assumptions on σi and ∂iun, one has R

σi∂iunϕ →R

σi∂iuϕ, fori≥N₁+ 1, hence R

(σ¹· ∇₁u_n)ϕ→ hσ¹· ∇₁u, ϕi. Furthermore, using forϕ≥0,

|R

σ¹· ∇₁unϕ| ≤ |σ¹|_L^∞_(Supptϕ)R

|∇₁un|ϕ→ |σ¹|_L^∞_(Supptϕ)R

|∇₁u|ϕ, one gets (??).

The identity ( ??) is easily obtained by letting ϕ go to 1_R^N, since all the measures involved are bounded measures.

2.2 The approximated space D^1,~p(R^N) Let >0 small, define

a_i = (p_i−1)p_i²

1−(pi−1), _i=p_i+a_i andp_i =p_i(1 +_i). (2.5)

Note that one has for all i ≥ N₁+ 1, ^{pi(1+ i)}

i = ¹⁺ . We define D^1,~p(R^N) as the closure ofD(R^N) for the norm|∇₁v|₁₊+P_N

i=N1+1|∂_iv|_p

i. Then the critical exponent

¯

p^? for this space is defined by _p^N_¯?

= ₁₊^N1 +P

i 1

p_i −1. Note that ¯p^? satisfies N

¯ p^? = N

¯

p^? − N 1 +,

and as soon as is small enough,p⁺ <p¯^?. Let us finally define λ = p¯^?

1 ++ 1, (2.6)

and note for further purposes that λp¯^?= ¯p^?.

Recall that as a consequence of the embedding of Troisi, [?] one has D^{1, ~p}(R^N),→L^p¯?(R^N),

and there exists some T₀>0, such that for allu∈ D^1,~p(R^N), T₀|u|_p¯^?

≤

N

Y

i=1

|∂_iu|

1 N

p_i, and then|u|_p¯^?

≤ 1

N T₀(p

N₁|∇₁u|₁₊+

N

X

i=N1+1

|∂_iu|_p

i) for all u∈ D^{1, ~p}(R^N).

Let us define

K = inf

u∈D^1,~p(R^N),|u|_p?¯

=1



 1

1 +|∇₁u|¹⁺₁₊+

N

X

i=N1+1

1 p_i|∂_iu|^p_pⁱ

i



,

and

K= inf

u∈D^1,~p(R^N),|u|_p?¯ =1



|∇₁u|₁+

N

X

i=N1+1

1 pi

|∂_iu|^p_pⁱ

i



.

It is clear by Proposition??that

K= inf

u∈BV^~p(R^N),|u|_p?¯ =1



 Z

|∇₁u|+

N

X

i=N1+1

1 p_i|∂_iu|^p_pⁱ

i



.

Adapting the proof in [?] one has the following result

Theorem 2.14. There exists u ∈ D^1,~p(R^N) non negative which satisfies |u|_p¯^? = 1 and

K = 1

1 +|∇₁u|¹⁺₁₊+

N

X

i=N1+1

1

p_i|∂_iu|^p_pⁱ i.

Furthermore there exists l >0, so that

−

N1

X

i=1

∂_i(|∇₁u|⁻¹∂_iu)−

N

X

i=N1+1

∂_i(|∂_iu|^pi−2∂_iu) =lu^p¯?−1. (2.7)

In the sequel we will use the notation div1 as the divergence of some N1 vector with respect to the N1 first variables.

By multiplying equation ( ??) by u and integrating one has K ≤ l ≤ p⁺ K , and as we will see in Proposition??that lim supK≤ K,ifuis an extremal function forK,|∇₁u|₁₊ and|∂_iu|_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 sequenceu, it converges up to subsequence to an extremal function for K.

3 The main results

The main result of this paper is the following : Theorem 3.1. Suppose that p⁺= sup_1≤i≤Npi <p¯^?.

1) There exists v ∈ D^1,~p(R^N), |v|_p¯^? = 1, an extremal function for K, which con- verges in the following sense to v ∈BV^~p(R^N): v converges to v in the distribution sense, and almost everywhere, R

R^N|∇₁v|¹⁺ →R

R^N|∇₁v|, R

R^N|∂_iv|^pi →R

R^N|∂_iv|^pi for all i ≥ 1, and |v|_p¯^? = 1. Furthermore limK = K. As a consequence v is an extremal function for K.

2) v satisfies in the distribution sense the partial differential equation :

−div₁(σ¹)−

N

X

i=N1+1

∂i(|∂_iv|^pi−2∂iv) =lv^p¯?−1, σ¹· ∇₁v=|∇₁v| (3.1) where σ¹· ∇₁v is taken in the sense of Theorem ?? and l= liml l defined in (??).

The proof of Theorem??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, - whereλhas been defined in (??)-, and to analyze the behaviour of this new sequence, which belongs toBV^~p(R^N), and is bounded in that space, independently on, as we will see later.

In a second time we prove that

Theorem 3.2. Let v be given by Theorem ??. Then v ∈ L^∞(R^N) and there exists some constant C(|v|_p¯^?) depending on the L^p¯? norm of v and on universal constants, such that |v|_∞≤C(|v|_p¯?).

3.1 Proof of Theorem ??

The proof is the consequence of several lemmata and propositions.

Lemma 3.3. Suppose thatu∈BV^~p(R^N), and that |u|_p¯^? ≤1, then K|u|^p_p¯⁺? ≤ |∇₁u|₁+

N

X

i=N1+1

1 p_i|∂_iu|^p_pⁱ

i

and analogously if u∈ D^1,~p(R^N), |u|_p¯^? ≤1 K|u|^p_p¯⁺?

≤ 1

1 +|∇₁u|¹⁺₁₊+

N

X

i=N1+1

1 p_i|∂_iu|^p_pⁱ

i.

Hint of the proof : Use _|u|^u

¯

p? in the definition ofK and the fact that if |u|_p¯^? ≤1,|u|^p_p¯ⁱ? ≥ |u|^p_p¯⁺?. Proposition 3.4. One has

lim supK ≤ K.

As a consequence any sequence (v) of extremal functions for K is bounded indepen- dently on , in the sense that there exists some positive constant c so that, for all >0, |∇₁v|₁₊,|∂_iv|_p

i ≤c, for all i≥N1+ 1.

Proof. Let δ >0, δ < ¹₂ and let uδ ∈ D^1,p(R^N), ( or BV^~p(R^N)), so that |u_δ|_p¯^? = 1 and

|∇₁uδ|₁+

N

X

i=N1+1

1 pi

|∂_iuδ|^p_pⁱ

i ≤ K+δ.

By definition ofD^1,~p(R^N), there existsvδ∈ D(R^N) such that||v_δ|_p¯^?−1| ≤δ,

|∇₁vδ|₁+

N

X

i=N1+1

1 pi

|∂_ivδ|^p_pⁱ

i ≤ K+ 2δ.

For small enough one has ||v_δ|_p¯^?

−1| ≤ 2δ. By considering w_δ = _|v^vδ

δ|_p?¯

, one sees that w_δ ∈ D(R^N),|w_δ|_p¯^?

= 1, and

|∇₁w_δ|₁+

N

X

i=N1+1

1

p_i|∂_iw_δ|^p_pⁱ

i ≤ |∇₁v_δ|₁ 1−2δ +

N

X

i=N1+1

1 p_i

|∂_iv_δ|^p_pⁱ_i (1−2δ)^pi

≤ 1

(1−2δ)^p+(K+ 2δ).

By the Lebesgue’s dominated convergence theorem,|∇₁w_δ|¹⁺₁₊→ ^|∇_|v^1vδ|1

δ|_p¯∗ , and|∂_iw_δ|^p_pⁱ i →

|∂_ivδ|^pi_pi

|v_δ|^pi_p¯∗ for all i≥N1+ 1 when →0, hence we get lim sup

→0

K≤lim sup

→0



 1

1 +|∇₁w_δ|¹⁺₁₊+

N

X

i=N1+1

1

p_i|∂_iw_δ|^p_pⁱ i



≤ K+ 2δ (1−2δ)^p+, which concludes the proof since δ is arbitrary.

Proposition 3.5. Suppose that w∈BV^~p(R^N) satisfies|w|_p¯^? = 1, and thatw→v almost everywhere. Then for small enough |w−v|_p¯^? ≤1.

Proof. Ifv≡0, there is nothing to prove. Ifv6= 0, using Brezis Lieb Lemma, [?] 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 , wherev is some convenient extremal function, given in Lemma ??below, and λ has been defined in (??).

Lemma 3.6. Let u be a non negative extremal function for K, so that |u|_p^?

= 1.

There exists v≥0 which satisfies

|u|_p¯^∗

= |v|_p¯^∗

= 1, |∇₁u|₁₊=|∇₁v|₁₊, and|∂_iu|_p

i =|∂_iv|_p

i, for all i≥1, and

Z

B(0,1)

v^p¯? = 1 2.

Proof. This proof is as in [?], but we reproduce it here for the reader’s convenience.

Let α_i = ^p_p^¯? i

−1, i = 1,· · · , N. For every y = (y1,· · · , yN) ∈ R^N, and for any u∈ D^{1, ~p}(R^N), and t >0, we set

u^t,y(x) =tu(t^α1(x₁−y₁),· · ·, t^αN(x_N −y_N)).

Then, we have

|u|_p¯^?

=|u^t,y|_p¯^?

,

|∂_iu|_p

i =|∂_iu^t,y|_p

i, for all 1≤i≤N,

|∇₁u|₁₊ =|∇₁u^t,y|₁₊. Let u be an extremal function forK so that |u|_p¯^?

= 1. As in [?], [?], we recall the definition of the Levy concentration function, for t >0 :

Q(t) = sup

y∈R^N

Z

E(y,t^α1,···,t^αN)

|u|^p¯∗, where E(y, t^α1,· · · , t^αN) is the ellipse defined by

{z= (z₁,· · ·, z_N)∈R^N,

N

X

i=1

(z_i−y_i)² t^2αi ≤1}, with y= (y₁,· · ·, y_N), andα_i = ^p_p^¯∗

i

−1 for alli.

Since for every >0, lim

t→0Q(t) = 0, and lim

t→∞Q(t) = 1, there exists t >0 such that Q(t) = ¹₂, and there exists y∈R^N such that

Z

E(y,t^α1,···,t^αN)

|u|^p¯∗(x)dx= 1 2. Thus, by a change of variable one has forv =u^t,y :

Z

B(0,1)

|v|^p¯∗ = 1

2 = sup

y∈R^N

Z

B(y,1)

|v|^p¯?. Note for further purpose thatv is also extremal forK.

Proposition 3.7. Let v ≥0 be in D^1,~p(R^N), bounded in that space, independently on . Then for λ defined in (??), the sequence w=v^λ is bounded in D^1,~p(R^N).

Proof. One has Z

R^N

|∇₁(v^λ)| = λ Z

R^N

v^λ−1|∇₁v|

≤ λ( Z

R^N

|∇₁v|¹⁺)¹⁺¹ ( Z

R^N

v

(λ−1)(1+)

)¹⁺

= λ( Z

R^N

|∇₁v|¹⁺)¹⁺¹ ( Z

R^N

v^p¯?)¹⁺

and for all i > N₁, using the definition in (??) Z

R^N

|∂_i(v^λ)|^pi = λ^pi Z

R^N

v^(λ−1)pi|∂_iv|^pi

≤ λ^pi( Z

R^N

|∂_iv|^pi)

1 1+i(

Z

R^N

v

(λ−1)pi

i )¹⁺ⁱⁱ

= λ^pi( Z

R^N

|∂_iv|^pi)

1 1+i(

Z

R^N

v^p¯?)¹⁺ⁱⁱ. Then R

R^N|∇₁(v^λ)|and R

R^N|∂_i(v^λ)|^pi fori≥N1+ 1 are bounded independently on , by the assumptions.

Letv be given by Lemma??. One has by the definition of λ, Z

R^N

|v|^p¯? = 1 = Z

R^N

|v^λ|^p¯?. Let us define

R→+∞lim lim sup

→0

Z

|x|>R

|v^λ|^p¯?=ν∞, and

R→+∞lim lim sup

→0

Z

|x|>R



|∇₁(v^λ)|+

N

X

i=N1+1

1 pi

|∂_i(v^λ)|^pi



=µ∞, while

R→+∞lim lim sup

→0

Z

|x|>R



 1

1 +|∇₁v|¹⁺+

N

X

i=N1+1

1

p_i|∂_iv|^pi



= ˜µ∞. Remark 3.8. Note that sinceR

B(0,1)|v^λ|^p? = ¹₂, and R

R^N|v^λ|^p? = 1, ν∞≤ ¹₂.

Theorem 3.9. Letv∈ D^1,~p(R^N), be given by Lemma??, andλ be defined in (??).

There exist v ∈BV^p~(R^N), some positive bounded measures on R^N : τ,τ , µ˜ ⁱ,µ˜ⁱ, for N₁ + 1 ≤ i ≤ N, and ν, a sequence of points x_j ∈ R^N, and some positif reals νj, µⁱ_j, τj,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).

2. |∇₁(v^λ)|*|∇₁v|+τ,|∇₁v|¹⁺*|∇₁v|+ ˜τ , with ˜τ ≥τ, in M¹(R^N)weakly.

3. |∂_iv^λ|^pi *|∂_iv|^pi +µⁱ for all i≥N1+ 1, |∂_iv|^pi * |∂_iv|^pi + ˜µⁱ with µ˜ⁱ ≥µⁱ, in M¹(R^N) weakly.

4. |v^λ|^p¯? =|v|^p¯? *|v|^p¯?+ν :=|v|^p¯?+P

jν_jδ_xj in M¹(R^N) weakly.

5. One has τ ≥P

jτ_jδ_xj, µⁱ ≥P

jµⁱ_jδ_xj, for all i≥N₁+ 1, and for any j ∈N, ν

p+

¯ p?

j ≤ _K¹(τj+P

i 1

piµⁱ_j), and ν

p+

¯

∞p? ≤ _K¹µ∞. 6.

|∇₁(v^λ)|₁ +

N

X

i=N1+1

1

p_i|∂_iv^λ|^p_pⁱ

i → Z

R^N

|∇₁v|+

N

X

i=N1+1

1 p_i

Z

R^N

|∂_iv|^pi

+ Z

R^N



τ +

N

X

i=N1+1

1 p_iµⁱ



+µ∞. 7.

1

1 +|∇₁v|¹⁺₁₊ +

N

X

i=N1+1

1

p_i|∂_iv|^p_pⁱ i →

Z

R^N

|∇₁v|+

N

X

i=N1+1

1 p_i

Z

|∂_iv|^pi

+ Z

R^N



˜τ+

N

X

i=N1+1

1 pi

˜ µⁱ



+ ˜µ∞.

8. Z

R^N

|v|^p¯? = 1 = Z

R^N

|v|^λp¯? → Z

R^N

|v|^p¯?+ Z

R^N

ν+ν∞.

Proof. 1 The convergence ofv^λ is clear by using the compactness of the embedding from BV^~p in L^q with q < p¯^? < p¯^?, on bounded sets of R^N, the analogous for v is also true since q <lim inf ¯p^?.