ON THE BEHAVIOUR OF THE SOLUTIONS TO p-LAPLACIAN EQUATIONS AS p GOES TO 1

− div ( |∇up|p−2∇up ) = f in Ω up = 0 on ∂Ω, where p > 1 and Ω is a bounded open set of R (N ≥ 2) with Lipschitz boundary. We analyze the case where Ω is a ball and the datum f is a non-negative radially decreasing function belonging to the Lorentz space LN,∞(Ω) and the case where the datum f belongs to the dual space W−1,∞(Ω). We are interested in finding the pointwise limit of up as p goes to 1 and in proving that such a limit is a solution to the “limit equation” of (1.1), namely:


Introduction
In the present paper we study the behaviour, when p goes to 1, of the solutions u p ∈ W 1,p 0 (Ω) to the problems We analyze the case where Ω is a ball and the datum f is a non-negative radially decreasing function belonging to the Lorentz space L N,∞ (Ω) and the case where the datum f belongs to the dual space W −1,∞ (Ω).We are interested in finding the pointwise limit of u p as p goes to 1 and in proving that such a limit is a solution to the "limit equation" of (1.1), namely: with homogeneous Dirichlet boundary condition.Hence, firstly we study the behaviour of u p when p goes to 1, finding a limit function u, and secondly we prove that such a limit function u is a solution to (1.2).Both aspects of our study have been investigated by several authors.The interest in studying the behaviour of u p comes from an optimal design problem in the theory of torsion and related geometrical problems (see [16] and [17]).The behaviour of u p in the case where the datum f is constant has been studied in [16] by Kawohl, where the author proved that under a suitable smallness assumption on the domain, it results (1.3) lim p→1 u p = 0, while under the assumption that the domain is large enough one has (1.4) lim The behaviour of u p in the case where f is not constant and it belongs to the Lebesgue space L N (Ω) (or to the Lorentz space L N,∞ (Ω)) is studied in [10].In such a paper the authors prove again (1.3) under the assumption that the L N (Ω)-norm (or L N,∞ (Ω)-norm) of the datum f is small enough.
As just pointed out, the second aim of our study consists in proving that the limit function u = lim p→1 u p is a solution to problem (1.2).The limit equation (1.2) have been studied by various authors (see for instance [4], [6], [9], [11], [12] and references there in).Motivations for such an interest are found in the variational approach to image restoration introduced by L. Rudin, S. Osher and E. Fatemi 1 .The definitions of solutions to equation (1.2) typically consider a datum in L N (Ω) or L N loc (Ω); moreover such solutions are functions belonging to the space BV (Ω), which guaranties the existence of a distributional gradient, well defined as a Radon measure.In order to give a meaning to Du |Du| in the limit equation, any definition of solution to (1.2) relies on the existence of a vector field z : Ω → R N , which belongs to L ∞ (Ω; R N ), with z ∞ ≤ 1. Moreover z satisfies the equation − div z = f in the distributional sense and z • Du = |Du|.The boundary condition may be included as z • ν ∈ sign(−u) a.e. on ∂Ω where ν denotes the outward normal unit vector.The expressions z •Du and z •ν have sense thanks to the Anzellotti theory (see [7] or [5]) which defines a Radon measure (z, Du), provides the definition of a weakly trace on ∂Ω to the normal component of z, denoted by [z, ν], and guaranties a Green's formula.Roughly speaking, z plays the role of Du |Du| .In this paper we consider problem (1.2) with data belonging to the Lorentz space L N,∞ (Ω) and to the dual space W −1,∞ (Ω).Let us now explain the reason for which we consider such types of data.The embedding W −1,∞ (Ω) ֒→ W −1,p ′ (Ω) for all p > 1 ensures the existence of an unique weak solution u p ∈ W 1,p 0 (Ω) to problem (1.1) (see [18]).Smallness assumption on the data allows us to prove the existence of a limit function u = lim p→1 u p which belongs to the space BV (Ω).On the other hand, we prove the existence of a vector field z ∈ L ∞ (Ω, R N ) satisfying − div z = f in the sense of distributions.This implies that Observe that we take a datum belonging to W −1,∞ (Ω) and we find a solution in BV (Ω), even if W −1,∞ (Ω) is not the dual space of BV (Ω).This fact yields some difficulties which we completely solve only in the case where the datum f belongs to specific subspaces of W −1,∞ (Ω).
Let us observe that, by the improvement of Sobolev embedding (see for example [1]) and duality arguments, the Lorentz space L N,∞ (Ω) is a subset of the dual space W −1,∞ (Ω).We will consider data belonging to L N,∞ (Ω) which are radially symmetric, without smallness assumptions.Indeed such symmetries allow us to write the expression of the solutions u p and to handle it in order to study the behaviour of u p .
The case where the data do not belong to W −1,∞ (Ω) have been considered in [4], [6], [9].However, in such papers the solutions to the "limit equation" (1.2) are not obtained as limit of u p , except in the case where the data are smooth enough, and the methods employed do not apply in our framework.We explicitly remark that the asymptotic behaviour of u p when the datum f belongs to L 1 (Ω) will be studied by the authors in the forthcoming paper [19], where a notion of solution to the equation (1.2) is introduced.
Finally we summarize the contents of the present paper.After introducing our notation (see Section 2), we begin by studying the case where f is a radially decreasing function defined in a ball Ω and belonging to L N,∞ (Ω), without assuming any smallness condition on its L N,∞ (Ω)-norm (see Section 3).Next we study the case where f belongs to the dual space W −1,∞ (Ω) and we prove that (1.3) holds true under the assumption f W −1,∞ (Ω) < 1.We also prove that if f W −1,∞ (Ω) = 1, then u p tends to a BV -function, and if f W −1,∞ (Ω) > 1, then there is not any BV -function which is the pointwise limit of u p (see Section 4).In general we are not able to prove that u p tends to a solution to problem (1.2) when f W −1,∞ (Ω) = 1.In such a case we have to assume that f belongs to the predual space of BV (Ω) (see Subsection 4.2).
We conclude with few words on the Appendix.First we present some properties of the predual space of BV (Ω) and we prove that its norm as a subspace of the dual of BV (Ω) is exactly the same as the norm of W −1,∞ (Ω).Secondly, we adapt Anzellotti's theory in the framework of the predual of BV (Ω).

Notation
In this section we will introduce some notation which will be used throughout this paper.We will denote by Ω a bounded open subset of R N with Lipschitz boundary.Thus there exists a unit vector field (denoted by ν) normal to ∂Ω and exterior to Ω, defined H N −1 -a.e. on ∂Ω.Here H N −1 denotes the (N − 1)-dimensional Hausdorff measure.Here and in the sequel, |E| denotes the Lebesgue measure of a measurable subset E of R N .
Let u : Ω → R be a measurable function.We denote by µ u the distribution function of u, that is the function The decreasing rearrangement of u is the function u For 1 < q < ∞, the Lorentz space L q,∞ (Ω), also known as Marcinkiewicz or weak-Lebesgue, is the space of Lebesgue measurable functions u such that (2.1) sup t>0 t µ u (t) 1/q < +∞.
It is endowed with the norm is the space of all Lebesgue measurable functions u such that (2.2) endowed with the norm (2.2).It is well-known (cf.[15], [21]) that the following inclusions hold for every ǫ > 0. Finally we recall that the Marcinkiewicz space L N,∞ (Ω) is the dual space of We define M(Ω) as the space of all Radon measures with bounded total variation on Ω and we denote by |µ| the total variation of µ ∈ M(Ω).The space of all functions of finite variation, that is the space of those u ∈ L 1 (Ω) whose distributional derivatives belong to M(Ω), is denoted by BV (Ω).It is endowed with the norm defined by u BV (Ω) = Ω |u| dx+|Du|(Ω), for any u ∈ BV (Ω).Since Ω has Lipschitz boundary, if u belongs to BV (Ω), then the function We explicitly point out that |Du 0 |(R N ) defines an equivalent norm on BV (Ω), which we will use in the sequel.Through the paper, with an abuse of notation, we still denote u 0 by u.
We will denote by S N,p the best constant in the Sobolev inequality (cf.[22]), that is, u p * ≤ S N,p ∇u p , for all u ∈ W 1,p 0 (Ω).We will also write S N instead of S N,1 .It is well-known (cf.[22]), that Sobolev's inequality can be improved in the context of Lorentz spaces (cf.[1]) and, furthermore, by an approximation argument may be extended to BV -functions (see for instance [25]); as a consequence we obtain the continuous embedding We will denote by W −1,q ′ (Ω) the dual space of W 1,q 0 (Ω), 1 ≤ q < ∞.Here we just recall that the norm in W −1,∞ (Ω) is given by It is worth pointing out some remarkable subspaces of W −1,∞ (Ω).One of these is M(Ω)∩W −1,∞ (Ω) whose elements are named Guy David measures in [20].Another subspace is the so-called predual of BV (Ω).Indeed, the space BV (Ω) is the dual of a separable space which will be denoted by Γ(Ω); its elements can be written as f − div F , with (f, F ) ∈ C 0 (Ω; R N +1 ) (see [14], and also [20] and [3]).Since the elements of W −1,∞ (Ω) may be written as f − div F , with (f, F ) ∈ L ∞ (Ω; R N +1 ), we deduce that Γ(Ω) ⊂ W −1,∞ (Ω).Recall that in the 1-dimensional case we have So that it is easy to find examples in any dimension that show all these spaces are different.
Finally we recall that a sequence (µ n ) n in M(Ω) weakly * converges to µ if for every f ∈ C 0 (Ω).We will say that a sequence (u n ) n weakly * converges to u in BV (Ω) if it strongly converges in L 1 (Ω) and (Du n ) n weakly * converges to Du in M(Ω).At least for sufficiently regular domains, this notion corresponds to weak * convergence in the usual sense: that is with respect to σ(BV (Ω), Γ(Ω)).

The radial case
In this section we consider problem (1.1) in the case where the domain Ω is a ball centered at the origin, i.e.Ω ≡ B R = {x ∈ R N : |x| < R} and the datum f is a nonnegative radially decreasing function belonging to the Lorentz space L N,∞ (B R ).Since both the domain and the datum are radially symmetric, it is well-known (see for instance [23]) that the weak solution u p is given by Our aim is to describe the behaviour of u p and the behaviour of |∇u p | p−2 ∇u p as p goes to 1.
We begin by introducing some notation.In what follows we will denote by . Moreover we will denote by N }, and by r 1 and r 2 the radii of the balls centered in the origin having measure s 1 and s 2 respectively: We set N } is empty, and N } is empty.We explicitly remark that it results s 1 ≤ s 2 and hence r 1 ≤ r 2 ≤ R. Therefore the balls B r1 and B r2 centered at the origin and radii r 1 and r 2 respectively are both contained in B R .
In general the limit of solutions u p , as p goes to 1, is finite a.e. in Ω when the datum f is small enough.For instance, if the datum f is constant, that is f * (s) = λ > 0, then (cf. [16]).In this section we analyze the behaviour of u p in a more general case, where f is not constant, without any dependence on the smallness of the datum.Indeed Theorem 3.1 below states that, as p goes to 1, u p diverges in the ball B r1 , that it has a finite non-negative limit (for which we give an upper bound) in the annulus B r2 \B r1 of radii r 1 and r 2 and finally that it converges to zero in the annulus B R \B r2 of radii r 2 and R.
Theorem 3.1.Let u p be the solution to problem (1.1).Then Remark 3.1.We explicitly observe that Theorem 3.1 improves the result proved in [10] where the behaviour of u p is studied just under a smallness assumption on N , then s 1 = 0 and hence the limit function is a.e.finite in B R , as in [10].
Remark 3.2.We point out that the values r 1 and r 2 (or equivalently s 1 and s 2 ) may be different.Indeed consider the function This example allows also to show that the value R − |x| in (3.5) may be attained, i.e. then lim p→1 u p (x) = R − |x| (see Remark 3.2 in [10]).
Proof of Theorem 3.1: From (3.1) we can write, almost everywhere in B r1 , where for almost all x ∈ B r1 .Now we evaluate U p (x), for almost every fixed x ∈ B r1 .Denote by Since |x| < r 1 , it follows that σ x < s 1 .Thus, by definition (3.3) of s 1 , we deduce that N .Note also that (3.3) implies σx < s 1 .Hence, by the continuity of the Therefore, Since the first and the third integral in (3.12) are non-negative, we obtain (3.13)By (3.11) and definition of f * * , we have Thus, the right-hand side of (3.13) tends to +∞ when p goes to 1 and by (3.10) we deduce (3.4) Now we prove (3.5).Using (3.1), we have for almost all x ∈ B r2 \B r1 , where, by (3.3), (3.15) Hence immediately follows that u p is nonincreasing with respect to p; then lim p→1 u p exists and it is non-negative.Furthermore, by (3.14) and (3.15), we get Therefore lim p→1 u p (x) ≤ R − |x|, which gives (3.5).Now we prove (3.6).By the definition of s 2 in (3.3) we deduce that for some ǫ > 0. Thus, for all s ∈ [ŝ, R[.Therefore by (3.1), we have we deduce that lim p→1 u p (x) = 0 uniformly on B ŝ. Thus (3.6) holds true.
Next we turn to describe the behaviour of |∇u p | p−2 ∇u p as p goes to 1. Proposition 3.1.Let u p be the solution to problem (1.1).Denote by z the vector field Then we have and for almost every x ∈ B R .
Moreover it results Hence, the vector field does not depend on p and this implies (3.17).Moreover, since As a straightforward consequence of the definition of z, (3.1) becomes and, changing the integration variable, we finally obtain that the above limit is equal to the measure of the set {|z(x The above results, Theorem 3.1 and Proposition 3.1, allow to prove a stability type result for the "limit equation" (1.2).More precisely we will prove that if the norm of f satisfies suitable smallness assumptions, then u = lim p→1 u p is a solution to the "limit problem" which can be formally written in the sense of the definition given in [4], [6], [5].
and let u p be the solution to problem (1.1), for any 1 < p < ∞.Then u p converges a.e. in B R to a function u ∈ W 1,1 0 (B R ) and there exists a vector field where ν denotes the outer normal to ∂B R ; Remark 3.5.We explicitly observe that the function u given by Theorem 3.2 is a solution to (3.24) in the sense of the definition given in [4], [6], [5]  Moreover, from Proposition 3.1, we deduce that the vector field z defined in (3.16) satisfies the conditions z ∞ ≤ 1 and N , Sobolev's inequality for BV -functions implies that ∇u p ⇀ Du weakly * in the sense of measures.
On the other hand, by (3.16), (3.21) and (3.25), we have u ∈ W 1,1 0 (B R ) and Moreover from (3.17), since u p is a solution to problem (1.1), it follows that Finally, a straightforward calculation shows that, for every Borel set This yields the conclusion.
Remark 3.6.One could think from the results in this section that the set where |z| ≤ 1 is the same as the set where |u| < +∞; this is not the case as the following example shows.
Consider problem (1.1) with datum f (C N |x| N ) defined by It is not difficult to check that the solution is given by Observe that g is an increasing function; thus, if λ ≤ (N −1)C then g(s) < 1 for all s < C N R N and so u p (x) → 0 everywhere in B R .
On the other hand, if λ > Let us compute the vector field z: Since and so |∇u p | p−2 ∇u p does not depend on p, it follows that . Therefore, {|u| < +∞} {|z| ≤ 1} for these values.Finally, we observe that when the limit function blows up at the boundary, it blows up everywhere.

Stability results with W −1,∞ (Ω) data
Consider the nonlinear elliptic problems where p > 1 and the datum µ belongs to W −1,∞ (Ω).Since by duality arguments, W −1,∞ (Ω) is included in W −1,p ′ (Ω) for every p > 1, then the existence and uniqueness of the solution u p ∈ W 1,p 0 (Ω) to problem (4.1) can be proved by classical methods (see, for instance, [18]).In this section we will study the behaviour, as p goes to 1, of these solutions u p to problems (4.1).We prove that, if the norm in W −1,∞ (Ω) of the datum µ is less than 1, then u p converges to the function u ≡ 0; if the norm in W −1,∞ (Ω) of the datum µ is equal to 1, then u p converges to a function u belonging to BV (Ω).Moreover we prove that u p does not converge to any BV -function if the norm in W −1,∞ (Ω) of the datum µ is greater than 1 (see Subsection 4.1 below).As in the previous section, we deduce a stability result for the limit problem (1.2) when the norm in W −1,∞ (Ω) of the datum µ is less than or equal to 1. Actually we does not give a stability result in the case where µ W −1,∞ (Ω) = 1 for all µ ∈ W −1,∞ (Ω).We study such a case when µ belongs to a subspace of W −1,∞ (Ω), namely Γ(Ω), the predual space of BV (Ω) (see Subsection 4.2 below).
From now on, abusing of the terminology, we will say that u p is a sequence and we will consider subsequences of it, as p goes to 1.

General data in
The main theorem of this subsection is the following.If µ W −1,∞ (Ω) < 1, then, as p goes to 1, u p converges to 0 in L q (Ω), with q < N N −1 .If µ W −1,∞ (Ω) = 1, then, up to a subsequence, u p converges to a BVfunction in L q (Ω), with q < and hence there is not any u ∈ BV (Ω) which is the weak * limit of u p .
Proof: Since u p is a weak solution to problem (4.1), the following inequalities hold true Therefore, one always has Assume first that µ W −1,∞ (Ω) < 1.Thus, we can write µ W −1,∞ (Ω) = 1 − ǫ, for a suitable ǫ > 0. By Young inequality, we have (4.4) This estimate implies that there exist u ∈ BV (Ω) and a subsequence, still denoted by u p , satisfying u p → u in L q (Ω), with q < N N −1 , and By (4.4), letting p → 1, we obtain and therefore u = 0. Since u ≡ 0 is the unique limit point, by Sobolev inequality, we actually obtain that lim p→1 u p = 0 in L q (Ω), with q < N N −1 .
On the other hand, if Since by definition we have

The conclusion follows from
Let us prove the following result which describes the behaviour of |∇u p | p−2 ∇u p .Proposition 4.1.Let u p be the solution to problem (4.1), then there exists a vector field z ∈ L ∞ (Ω; R N ) such that, up to subsequences, |∇u p | p−2 ∇u p ⇀ z weakly in L q (Ω) for all 1 ≤ q < +∞, (4.7) Step 1: Proof of (4.7): Arguing as in the proof of Theorem 4.1, we obtain inequality (4.3).Then for every q, 1 ≤ q < p ′ , we have (4.10) It yields that, for any q fixed, the sequence |∇u p | p−2 ∇u p is bounded in L q (Ω; R N ) and then there exists z q ∈ L q (Ω; R N ) such that, up to subsequences, Moreover, by a diagonal argument we can find a limit z that does not depend on q, that is (4.11) |∇u p | p−2 ∇u p ⇀ z in L q (Ω) for all 1 ≤ q < +∞.
Step 2: Proof of (4.8):Since u p is a distributional solution to problem (4.1), it follows that Hence, using (4.11) we obtain that is (4.8).
Step 3: Proof of (4.9):For any fixed h > 0 and p > 1, we denote By (4.10) for q = 1, as p goes to 1 we have (4.14) for some g h ∈ L 1 (Ω) and f h ∈ L 1 (Ω).On the other hand by (4.10) with q = 1 (4.15) Therefore by Hölder's inequality and (4.15) for every fixed Φ∈L ∞ (Ω; R N ), with ||Φ|| ∞ ≤ 1, we have (4.16)By (4.13) and (4.16), for any fixed h > 0 we deduce for every Φ ∈ L ∞ (Ω; R N ) such that ||Φ|| ∞ ≤ 1.By duality we deduce the following estimate for g h for any fixed h > 0. Moreover by definition of the set D p,h , This implies, using the inequality contained in [24, p. 337], the following pointwise estimate for f h |f h | ≤ h, a.e. in Ω, for any fixed h > 0. Therefore f h ∈ L ∞ (Ω; R N ).Applying once again (4.10), we have that is, for some q 0 , This implies Therefore, for every h > 0, we have The above condition on g h gives lim and hence lim From (4.8) and the definition of the norm µ W −1,∞ (Ω) , since we have the reverse inequality follows, and therefore (4.9) is proved.
Remark 4.1.In Theorem 4.1, when µ W −1,∞ (Ω) > 1, we did not state which is the pointwise limit of u p .Nevertheless, an "a posteriori" argument can be done to obtain some kind of limit of the solutions.In fact, we can prove the following claim.
There exists a function v satisfying, up to subsequences, for all 1 ≤ q < +∞.
To prove this claim, we must carefully apply Sobolev's inequality.It is well-known (see, for instance, [25, p. 57 or p. 82]) that a straightforward argument yields a simple connection between S N,p and S N , namely: From (4.10), we deduce 3 , by applying Sobolev's inequality, we get Therefore, taking S N,r(p−1) ≤ 2(N − 1)S N into account, we obtain Now let q satisfy 1 ≤ q ≤ 1 p−1 < r and apply Hölder's inequality, then Since p → 1, it follows that we may consider any q such that 1 ≤ q < +∞; moreover, |u p | p−1 is bounded in L q (Ω) and its bound tends to |Ω| 1 q µ W −1,∞ (Ω) as p → 1.Therefore, (4.17) is a consequence of a diagonal argument.
We also remark that there is some connection between the functions u = lim p→1 u p and v. Indeed, on the set {u = 0} it yields v ≤ lim sup p→1 e (p−1) log |up| ≤ 1 a.e., while on {|u| = +∞} we have v ≥ lim inf p→1 e (p−1) log |up| ≥ 1 a.e.Finally, up a null set, we obtain Observe that Theorem 3.1 implies the existence of a finite limit if and only if As in the previous section, the study of the behaviour, as p goes to 1, of u p and |∇u p | p−2 ∇u p allows to deduce a stability result to the "limit problem" formally written Let us begin by recalling the definition of solution to this problem (see [4], [5], [9] and [12]).To this aim, we need to introduce the following distribution (cf.[7]): Let u be a function belonging to BV (Ω) and let z be a vector field belonging to L ∞ (Ω; R N ) such that div z, in the sense of distributions, belongs to BV (Ω) * , i.e. div z, ϕ = − Ω z • ∇ϕ dx for all ϕ ∈ C ∞ 0 (Ω).Then we define the distribution and div z ∈ BV (Ω) * , all terms in (4.19) make sense.
Moreover as in [5, pp. 126-127] we may define the weak trace of the exterior normal component of z, which will be denoted by [z, ν].As already observed, this means that the function u and the vector field z whose existence has been proved in the previous section yields a solution to problem (3.24).
The announced stability result it is now an easy consequence of Theorem 4.1 and Proposition 4.1.

Data in the predual space of BV (Ω).
In the previous subsection, we have stated a stability result when µ W −1,∞ (Ω) = 1.The case µ W −1,∞ (Ω) = 1 is the most interesting since then lim p→∞ u p defines non-trivial solutions to the "limit problem" (4.18).To check that lim p→∞ u p is indeed a solution to (4.18), apart from passing to the limit, some extension of Anzellotti's theory is required.We refer to the definition of solution to (4.18) given in Definition 4.1.
We are able to extend the Anzellotti theory in some distinguished subspaces of W −1,∞ (Ω).The case of the space of all Guy David measures is cumbersome and will be provided in a forthcoming paper.The case of the predual space Γ(Ω) of BV (Ω) is shown in the Appendix.We point out (see Theorem 5.1 in the Appendix) that if z ∈ L ∞ (Ω; R N ) is a vector field such that its divergence in the sense of distributions belongs to Γ(Ω), then the distribution defined in (4.19) is always a Radon measure.
In this subsection we completely analyze the case where the datum µ belongs to Γ(Ω).The main theorem of this subsection is the following Theorem 4.3.Let µ ∈ Γ(Ω) and let u p be the solution to problem (4.1).Then the following statements are equivalent: 1) Up to subsequences, u p converges a.e., as p goes to 1, to a measurable function u which is a solution to problem (4.18) in the sense of Definition 4.1.
Arguing as in the proof of Theorem 4.1, we obtain inequality (4.5) and then a function u ∈ BV (Ω) such that, up to subsequences, ∇u p ⇀ Du weakly * in the sense of measures, By Green's formula (5.4) in the Appendix below, we get for every w ∈ BV (Ω).
Next, we will only sketch the proof of this fact, which is "inspired" in In order to take limit when p → 1, apply Young's inequality and have in mind (4.26) and (4.27), to get for all w ∈ W 1,2 0 (Ω).By approximating, this inequality holds for all w ∈ W 1,1 0 (Ω).In the following step assume that w ∈ W 1,1 (Ω), take into account (w n ) n (the sequence of Lemma 5.1), pass to the limit as n → ∞ and obtain Finally, given w ∈ BV (Ω), by Proposition 5.1, consider w n ∈ W 1,1 (Ω) satisfying w n | ∂Ω = w| ∂Ω for all n ∈ N, Remark 4.7.The same scheme followed in Theorem 4.3 can be adapted when the datum lives in L N (Ω).Indeed, if µ = f ∈ L N (Ω) and f W −1,∞ (Ω) ≤ 1, then (4.5) holds true and so we may find u ∈ BV (Ω) such that, up to subsequences, Once we have obtained this convergence, we may follow the same proof of the above theorem and get that u p converges to a solution to problem (4.18) in the sense of Definition 4.1 if and only if f W −1,∞ (Ω) ≤ 1.
We point out that the above reasoning cannot be extended to all data belonging to the Marcinkiewicz space L N,∞ (Ω) since, in general, u p bounded in L N N −1 ,1 (Ω) and u p → u a.e. in Ω does not imply u p ⇀ u weakly in L N N −1 ,1 (Ω).Nevertheless, we can handle every datum that belongs to L N,∞ (Ω) by using truncation (see [19]).
Example 4.3.In this example we give an element of the predual Γ(Ω) which does not belong to L N,∞ (Ω).

Appendix
This Appendix contains some simple facts on Γ(Ω) which we prove for the sake of completeness.
A straightforward consequence of conditions ( 2) and ( 3) is Since ǫ is arbitrary, the conclusion follows.
(see Definition 4.1 and Remark 4.3 below).Proof of Theorem 3.2: Since f s ≤ N C 1/N N for all s ∈]0, C N R N [, we have r 1 = 0 and by Theorem 3.1 we have (3.25)0 ≤ u(x) = lim p→1 u p (x) ≤ R − |x|, a.e. in B R .

Theorem 4 . 1 .
Let u p be the solution to problem (4.1).
and the datum µ belongs to L N,∞ (B R ), then Proposition 4.1 and Theorem 4.1 hold true.Taking into account Proposition 4.1 and Remark 3.3, we may deduce that for every f
p ⇀ Du weakly * in the sense of measures,u p is bounded in L N N −1 (Ω), u p → u a.e. in Ω.These two last properties imply that u p ⇀ u weakly in L N N −1 (Ω), so that (4.27) becomes Ω f u p dx → Ω f u dx.
Remark 3.3.Let us point out that in the above proof, we have obtained ),(3.19)and(3.20)follow in a straightforward way from (3.2) and (3.3).