ON THE L p − THEORY OF ANISOTROPIC SINGULAR PERTURBATIONS OF ELLIPTIC PROBLEMS

. In this article we give an extension of the L 2 − theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlin- ear problems involving L p data (1 < p < 2). Convergences in pseudo Sobolev spaces are proved for weak and entropy solutions, and rate of convergence is given in cylindrical domains.


Introduction.
1.1. Preliminaries. In this article we shall give an extension of the L 2 −theory of the asymptotic behavior of elliptic, anisotropic singular perturbations problems. This kind of singular perturbations has been introduced by M. Chipot [6]. From the physical point of view, these problems can modelize diffusion phenomena when the diffusion coefficients in certain directions are going toward zero. The L 2 theory of the asymptotic behavior of these problems has been studied by M. Chipot and many co-authors. Before describing the problem, let us begin by a brief discussion on the uniqueness of the weak solution (by a weak solution we mean a solution in the sense of distributions) to the problem −∇ · (A∇u) = f u = 0 on ∂Ω, where Ω ⊂ R N , N ≥ 2, is a bounded Lipschitz domain, and f ∈ L p (Ω) with 1 < p < 2. The diffusion matrix A = (a ij ) is supposed to be bounded and satisfies the ellipticity assumption on Ω (see assumptions (3) and (4) in subsection 1.2). It is well known that (1) has at least a weak solution in W 1,p 0 (Ω). Moreover, if A is symmetric and continuous and ∂Ω ∈ C 2 [2] then (1) has a unique solution in W 1,p 0 (Ω). If A is discontinuous the uniqueness assertion is false, in [15] Serrin has given a counterexample when N ≥ 3. However, if N = 2 and if ∂Ω is sufficiently smooth, and without any continuity assumption on A, then (1) has a unique weak solution in W 1,p 0 (Ω). The proof is based on the Meyers regularity theorem (see for instance [13]). To treat this pathology, Benilin, Boccardo, Gallouet, et al. have introduced the concept of entropy solution [4], for problems involving L 1 data or more generally a Radon measure.

OGABI CHOKRI
For every k > 0 We define the function T k : R → R by: And we define the space T 1,2 0 introduced in [4]. is equivalent to the original one given in [4].In fact, this is a characterization of this space [4]. Now, more generally, for f ∈ L 1 (Ω) we have the following definition of entropy solution [4]. Definition 1.1. A function u ∈ T 1,2 0 (Ω) is said to be an entropy solution to (1) if We refer the reader to [4] for more details about the sense of this formulation. The main results of [4] show that (1) has a unique entropy solution which is also a weak solution of (1), moreover since Ω is bounded then this solution belongs to 1≤r< N N −1 W 1,r 0 (Ω).

1.2.
Description of the problem and functional setting. Throughout this article we will suppose that f ∈ L p (Ω), 1 < p < 2, (we can suppose that f / ∈ L 2 (Ω)). We give a description of the linear problem, some nonlinear problems will be studied later. Consider the following singular perturbations problem: where Ω is a bounded Lipschitz domain of R N (by Lipschitz we mean strongly Lipschitz). Let q ∈ N * , N − q ≥ 2. We denote by x = (x 1 , ..., x N ) = (X 1 , X 2 ) ∈ R q × R N −q i.e. we split the coordinates into two parts. With this notation we set where ∇ X1 = (∂ x1 , ..., ∂ xq ) T and ∇ X2 = (∂ xq+1 , ..., ∂ x N ) T Let A = (a ij (x)) be a N × N matrix which satisfies the ellipticity assumption ∃λ > 0 : Aξ · ξ ≥ λ |ξ| 2 ∀ξ ∈ R N for a.e x ∈ Ω, and a ij (x) ∈ L ∞ (Ω), ∀i, j = 1, 2, ...., N, (4) We have decomposed A into four blocks where A 11 , A 22 are respectively q × q and (N − q) × (N − q) matrices. For 0 < ≤ 1 we have set We denote Ω X1 = X 2 ∈ R N −q : (X 1 , X 2 ) ∈ Ω , and Ω 1 = P 1 Ω where P 1 : R N → R q is the usual projector. We introduce the space and for a.e X 1 ∈ Ω 1 , u(X 1 , ·) ∈ W 1,p 0 (Ω X1 ) We equip V p with the norm then one can show easily that (V p , · Vp ) is a separable reflexive Banach space. The passage to the limit (formally) in (2) gives the limit problem The L 2 -theory (when f ∈ L 2 ) of problem (2) has been treated in [8], convergence has been proved in V 2 and rate of convergence in the L 2 −norm has been given. For the L 2 −theory of several nonlinear problems we refer the reader to [9], [10], [14]. This article is mainly devoted to study the L p −theory of the asymptotic behavior of linear and nonlinear singularly perturbed problems. In other words, we shall study the convergence u → u 0 inV p . Notice that in [9], the authors have treated some problems involving L p data where some other data of the equations depend on p, one can check easily that it is not the L p theory which we expose in this manuscript. Let us briefly summarize the content of the paper: • In section 2: We study the linear problem, we prove convergences for weak and entropy solutions. • In section 3: We give the rate of convergence in cylindrical domains when the data is independent of X 1 . • In section 4: We treat some nonlinear problems.
Proof. This corollary follows immediately from Theorem 2.1 and uniqueness of the solutions of (2) and (5) as mentioned in subsection 1.1 (Notice that ∂Ω X1 ∈ C 2 ).
Proof. By density let (f n ) n∈N ⊂ L 2 (Ω) be a sequence such that f n → f in L p (Ω), we can suppose that ∀n ∈ N : f n L p ≤ M , M ≥ 0. Consider the regularized problem: Assumptions (3) and (4) show that u n exists and it is unique, by the Lax-Milgram theorem. Notice that u n belongs to W 1,p 0 (Ω). We introduce the function This kind of function has been used in [3]. We have θ (t) = (1 + |t|) p−2 ≤ 1 and θ(0) = 0, therefore we have θ(u) ∈ H 1 0 (Ω) for every u ∈ H 1 0 (Ω). Testing with θ(u n ) in (6) and using the ellipticity assumption we deduce In the other hand, by Hölder's inequality we have From the two previous integral inequalities we deduce By Hölder's inequality we get Using Minkowki inequality we get where the constant C depends on p, λ, mes(Ω), M and C Ω . Whence, we deduce Similarly we obtain where the constants C , C are independent of n and , so Fix , since W 1,p (Ω) is reflexive then (10) implies that there exist a subsequence (u n l ( ) ) l∈N and u ∈ W 1,p 0 (Ω) such that u n l ( ) u ∈ W 1,p 0 (Ω) (as l → ∞) in W 1,p (Ω)−weak. Now, passing to the limit in (6) as l → ∞ we deduce Whence u is a weak solution of (2) (u = 0 on ∂Ω in the trace sense of W 1,p −functions, indeed the trace operator is well defined since ∂Ω is Lipschitz). Now, from (8) we deduce and similarly from (9) we obtain Using reflexivity of L p (Ω) and continuity of the derivation operator on D (Ω), one can extract a subsequence (u k ) k∈N such that ∇ X2 u k ∇ X2 u 0 , k ∇ X1 u n k 0, u k u 0 in L p (Ω) − weak. Passing to the limit in (11) we get Now, we will prove that u 0 ∈ V p . Since ∇ X2 u k ∇ X2 u 0 and u k u 0 in L p (Ω) − weak then there exists a sequence (U n ) n∈N ⊂ conv({u k } k∈N ) such that ∇ X2 U n → ∇ X2 u 0 in L p (Ω) − strong, where conv({u k } k∈N ) is the convex hull of the set {u k } k∈N . Notice that we have U n ∈ W 1,p 0 (Ω) then -up to a subsequence-we have U n (X 1 , ·) ∈ W 1,p 0 (Ω X1 ), a.e X 1 ∈ Ω 1 . And we also have -up to a subsequence-∇ X2 U n (X 1 , ·) → ∇ X2 u 0 (X 1 , .) in L p (Ω X1 ) − strong a.e X 1 ∈ Ω 1 . Whence u 0 (X 1 , .) ∈ W 1,p 0 (Ω X1 ) for a.e X 1 ∈ Ω 1 , so u 0 ∈ V p . Finally, we will prove that u 0 is a solution of (5). Let E be a Banach space, a family of vectors {e n } n∈N in E is said to be a Banach basis or a Schauder basis of E if for every x ∈ E there exists a family of scalars (α n ) n∈N such that x = ∞ n=0 α n e n , where the series converges in the norm of E. Notice that Schauder basis does not always exist. In [11] P. Enflo has constructed a separable reflexive Banach space without Schauder basis!. However, the Sobolev space W 1,r 0 ( 1 < r < ∞) has a Schauder basis whenever the boundary of the domain is sufficiently smooth [12]. Now, we are ready to finish the proof.
where ∂V i is smooth (V i are Euclidian balls for example), such a covering always exists. Now, fix ψ ∈ D(V i ) then it follows from (12) that for every ϕ ∈ D(U i ) we have

OGABI CHOKRI
Whence for a.e X 1 ∈ U i we have Notice that by density we can take ψ ∈ W 1,p 0 (V i ) where p is the conjugate of p. Using the same techniques as in [8], where we use a Schauder basis of W 1,p 0 (V i ) and a partition of the unity, one can easily obtain for a.e X 1 ∈ Ω 1 . Finally, since u 0 (X 1 , ·) ∈ W 1,p 0 (Ω X1 ) (as proved above) then u 0 (X 1 , ·) is a solution of (5), notice that ∂Ω X1 is also Lipschitz so, the trace operator is well defined.
2.2. Strong convergence. Theorem 2.1 will be proved in three steps. the proof is based on the use of the approximated problem (6). First, we shall construct the solution of the limit problem.
Step1: Let u n ∈ H 1 0 (Ω) be the unique solution to (6), existence and uniqueness of u n follows from assumptions (3), (4) as mentioned previously. One have the following proposition.
in particular the two convergences holds in L p (Ω) and V p respectively, where u n 0 is the unique weak solution in V 2 to the problem Proof. This result follows from the L 2 −theory (Theorem 1 in [8]), The convergences in V p and L p (Ω) follow from the continuous embedding Now, we construct u 0 the solution of the limit problem (5). Testing with ϕ = θ(u n 0 (X 1 , ·)) in the weak formulation of (13) (θ is the function introduced in subsection 2.1) and estimating like in proof of Theorem 2.3 we obtain as in (7) Integrating over Ω 1 and using Cauchy-Schwaz's inequality in the right hand side we get and therefore where C is independent of n. Now, using the linearity of the problem and (13) with the test function θ(u n 0 (X 1 , ·) − u m 0 (X 1 , ·)), m, n ∈ N, one can obtain like in (14) .
Integrating over Ω 1 and using Cauchy-Schwarz inequality and (15) yields where C is independent of m and n. The Poincaré's inequality shows that . Since (f n ) n∈N is a converging sequence in L p (Ω) then this last inequality shows that (u n 0 ) n∈N is a Cauchy sequence in V p , consequently there exists u 0 ∈ V p such that u n 0 → u 0 in V p . Now, passing to the limit in (6) as → 0, we get according to Proposition 1 Passing to the limit as n → ∞ we deduce Then it follows, as proved in Theorem 2.3, that u 0 satisfies (5). Whence we have proved the following Step2: We shall construct the sequence (u ) 0< ≤1 solutions of (2), one can prove the following Proposition 3. There exists a sequence (u ) 0< ≤1 ⊂ W 1,p 0 (Ω) of weak solutions to (2) such that u n → u in W 1,p (Ω) as n → ∞, for every fixed. Moreover, u n → u in V p and ∇ X2 u n → ∇ X2 u , in L p (Ω) uniformly in .
Proof. Using the linearity of (6) testing with θ(u n − u m ), m, n ∈ N we obtain as in (7) .

OGABI CHOKRI
And (8) gives where C is independent of and n, whence Poincaré's inequality implies Similarly we obtain It follows that The last inequality implies that for every fixed (u n ) n∈N is a Cauchy sequence in W 1,p 0 (Ω), Then there exists u ∈ W 1,p 0 (Ω) such that u n → u in W 1,p (Ω), then the passage to the limit in (6) shows that u is a weak solution of (2). Finally (16) and (17) Step3: Now, we are ready to conclude. Proposition 1, 2 and 3 combined with the triangular inequality show that u → u 0 in V p and ∇ X2 u → 0 in L p (Ω), and the proof of Theorem 2.1 is finished.

Convergence of the entropy solutions.
In this subsection we prove Theorem 2.2. As mentioned in section 1 the entropy solution u of (2) exists and it is unique. We shall construct this entropy solution. Using the approximated problem (6), one has a W 1,p −strongly converging sequence u n → u ∈ W 1,p 0 (Ω) as shown in Proposition 3. We will show that u ∈ T 1,2 0 (Ω). Clearly we haveT k (u n ) ∈ H 1 0 (Ω) for every k > 0. Now testing with T k (u n ) in (6) we obtain Using the ellipticity assumption we get Fix , k, we have u n → u in L p (Ω) then there exists a subsequence (u n l ) l∈N such that u n l → u a.e x ∈ Ω and since T k is bounded then it follows that T k (u n l ) → T k (u ) a.e in Ω and strongly in L 2 (Ω) whence u ∈ T 1,2 0 (Ω). It follows by (18) that there exists a subsequence still labelled Whence u is the entropy solution of (2). Similarly the function u 0 (constructed in Proposition 2) is the entropy solution to (5) for a.e X 1 The uniqueness of u 0 in V p follows from the uniqueness of the entropy solution of problem (5). Finally, the convergences given in Theorem 2.2 follow from Theorem 2.1. Remark 1. Uniqueness of the entropy solution implies that it does not depend on the choice of the approximated sequence (f n ) n .

2.4.
A regularity result for the entropy solution of the limit problem. In this subsection we assume that Ω = ω 1 ×ω 2 where ω 1 , ω 2 are two bounded Lipschitz domains of R q , R N −q respectively. We introduce the space We suppose the following where u 0 is the entropy solution of (5).
Proof. Let (u n 0 ) be the sequence constructed in subsection 2.2, we have u n 0 → u 0 in V p , where u 0 is the entropy solution of (5).
Let ω 1 ⊂⊂ ω 1 be an open subset, for 0 < h < d(∂ω 1 , ω 1 ) and for X 1 ∈ ω 1 we set τ i h u n 0 = u n 0 (X 1 + he i , X 2 ) where e i = (0, .., 1, .., 0) then we have by (13) ω2 To make the notations less heavy we set Using the ellipticity assumption for the left hand side, and Hölder's inequality for the right hand side of the above integral equality we deduce Using Hölder's inequality we derive Then we deduce

OGABI CHOKRI
Now passing to the limit as δ → 0 using the Lebesgue theorem we deduce and Poincaré's inequality gives Now, integrating over ω 1 yields .
Passing to the limit as n → ∞ using the invariance of the Lebesgue measure under translations we get .
Whence, since f ∈ W p then where C is independent of h, therefore we have ∇ X1 u 0 ∈ L p (Ω). Combining this with u 0 ∈ V p we get the desired result.
3. The rate of convergence theorem. In this section we suppose that Ω = ω 1 × ω 2 where ω 1 , ω 2 are two bounded Lipschitz domains of R q and R N −q respectively. We suppose that A 12 , A 22 and f depend on X 2 only i.e A 12 (x) = A 12 (X 2 ), A 22 (x) = A 22 (X 2 ) and f (x) = f (X 2 ) ∈ L p (ω 2 ) (1 < p < 2), f / ∈ L 2 (ω 2 ). Let u , u 0 be the unique entropy solutions of (2), (5) respectively then under the above assumptions we have the following Theorem 3.1. For every ω 1 ⊂⊂ ω 1 and m ∈ N * there exists C ≥ 0 independent of such that Proof. Let u , u 0 be the entropy solutions of (2), (5) respectively, we use the approximated sequence (u n ) ,n , (u n 0 ) n introduced in section 2. Subtracting (13) from (6) we obtain where we have used that u n 0 is independent of X 1 (since f and A 22 are independent of X 1 ) and that A 12 is independent of X 1 .
Using the ellipticity assumption for the left hand side and assumption (4) for the right hand side of the previous integral equality we deduce Where C ≥ 0 depends on A and ρ. Using Young's inequality ab ≤ a 2 2c + c b 2 2 for the two terms in the right hand side of the previous inequality, we obtain where C is independent of and n Now, using Hölder's inequality and the above inequality we deduce Passing to the limit as δ → 0 using the Lebesgue theorem. Passing to the limit as n → ∞ we get Let m ∈ N * then there exist ω 1 ⊂⊂ ω 1 ⊂⊂ ...ω (m+1) 1 ⊂⊂ ω 1 . Iterating the above inequality m−time we deduce Now, from (20) (with ω 1 and ω 1 replaced by ω (m) 1 and ω (m+1) 1 respectively) we deduce is bounded and therefore we obtain And the proof of the theorem is finished.
Can one obtain a better convergence rate? In fact, the anisotropic singular perturbation problem (2) can be seen as a problem in a cylinder becoming unbounded. Indeed the two problems can be connected to each other via a scaling = 1 (see [5] for more details). So let us consider the problem: Ω = ω 1 × ω 2 a bounded domain where ω 1 , ω 2 are two bounded Lipschitz domain with ω 1 convex and containing 0.
We consider the limit problem: Then under the above assumptions we have Theorem 3.2. Let u , u ∞ be the unique entropy solutions to (21) and (24) then for every α ∈ (0, 1) there exists C ≥ 0, c > 0 independent of such that Proof. Let u , u ∞ the unique entropy solutions to (21) and (24) respectively, and let (u n ) and (u n ∞ ) the approximation sequences (as in section 2). we have u n → u in W 1,p 0 (Ω ) and u n ∞ → u ∞ in W 1,p 0 (ω 2 ).Subtracting the associated approximated problems to (21) and (24) and take the weak formulation we get Where we have used thatÃ 22 ,Ã 12 , u n ∞ are independent of X 1 . Now we will use the iteration technique introduced in [7], let 0 < 0 ≤ − 1, and let ρ ∈ D(R q ) a bump function such that where c 0 is the universal constant (see [5]). Testing with Using the ellipticity assumption (23) Notice that ∇ρ = 0 on Ω 0 , and Ω 0 ⊂ Ω 0+1 ( since ω 1 is convex and containing 0). Then by the Cauchy-Schwarz inequality we get where we have used (22). Whence we get ( since ρ = 1 on Ω 0 ) From Hölder's inequality it holds that Passing to the limit as δ → 0 (using the Lebesgue theorem) we get where we have used 0 ≤ ρ ≤ 1. Using Poincaré's inequality Let α ∈ (0, 1), iterating this formula starting from α we get where c, c > 0 are independent of and n. Now we have to estimate the right hand side of (26). Testing with θ(u n ) in the approximated problem associated to (21) one can obtain as in subsection 2.1 Similarly testing with θ(u n ∞ ) in the approximated problem associated to (24). we get We apply the triangular inequality to the right hand side of (26), and we use (27), (28). Passing to the limit as n → ∞ we obtain the desired result.

Corollary 2.
Under the above assumptions we have, for every α ∈ (0, 1) there exist C ≥ 0, c > 0 independent of such that where u , u 0 are the entropy solutions to (2) and (5) respectively Remark 2. It is very difficult to prove the rate convergence theorem for general data. When f (x) = f 1 (X 2 ) + f 2 (x) with f 1 ∈ L p (ω 2 ) and f 2 ∈ W 2 we only have the estimates This follows from the linearity of the equation, Theorem 3.1 and the L 2 −theory [8].
4. Some Extensions to nonlinear problems and applications.
First of all, suppose that f ∈ L 2 (Ω),then we have the following Proposition 4. Assume (3), (4) and a(0) = 0. Let u be the unique weak solution in H 1 0 (Ω) to (29) then ∇ X1 u → 0 in L 2 (Ω) and u → u 0 in V 2 where u 0 in the unique solution in V 2 to the limit problem (31).
Proof. Existence of u follows directly by a simple application of the Schauder fixed point theorem, for example. The uniqueness follows form monotonicity of a and the Poincaré's inequality.
Take u as a test function in (29) then one can obtain the estimates where C is independent of , we have used that Ω a(u )u dx ≤ 0 (thanks to monotonicity assumption and a(0) = 0). And we also have (thanks to assumption (30)) So there exist v ∈ L 2 (Ω), u 0 ∈ L 2 (Ω), ∇ X2 u 0 ∈ L 2 (Ω) and a subsequence (u k ) k∈N such that Passing to the limit in the weak formulation of (29) we get Take ϕ = u k in (33) and passing to the limit we get Let us compute the quantity (This quantity is positive thanks to the ellipticity and monotonicity assumptions).
Passing to the limit as k → ∞ using (32), (33), (34) we get lim I k = 0 And finally The ellipticity assumption and Poincaré's inequality show that Whence (33) becomes ∇ X2 (u k − u 0 ) L 2 (Ω) → 0 shows that u 0 ∈ V 2 , and therefore Hence u 0 (X 1 , ·) is a solution to (31). The uniqueness in H 1 0 (Ω X1 ) of the the solution of the limit problem (31) shows that u 0 is the unique function in V 2 which satisfies (36). Therefore the convergences (35) hold for the whole sequence (u ) 0< ≤1 . Now, we are ready to give the main result of this subsection Theorem 4.1. Suppose that f ∈ L p (Ω) where 1 < p < 2 (we can suppose that f / ∈ L 2 (Ω)), then there exists a unique u 0 ∈ V p such that u 0 (X 1 , ·) is the unique entropy solution to (31) for a.e X 1 ∈ Ω 1 , and we have u → u 0 in V p , ∇ X1 u → 0 in L p (Ω), where u is the unique entropy solution to (29).
Proof. We only give a sketch of the proof. Existence and uniqueness of the entropy solutions to (29) and (31) follows from the general result proved in [4]. As in proof of Theorem 2.2 we shall construct the entropy solution u . we consider the approximated problem: We follow the same arguments as in subsections 2.2 and 2.3 where we use the above proposition and the following inequality, which holds for every u, v ∈ L 2 (Ω) in fact this follows from the monotonicity of a and θ.
2) If (u ) 0< ≤1 is a sequence of entropy solutions to (37), then we have the above estimates.
Proof. 1) The existence of u is based on the Schauder fixed point theorem, we define the mapping Γ : where v is the entropy solution (which is also a weak solution) of the linearized problem: Since the entropy solution is unique then Γ is well defined, we can prove easily, by using the approximation method, that Γ is continuous. As in subsection 2.1 we can obtain the estimates where C 0 is independent of and v (thanks to (38)) For fixed we define the subset The subset K is convex and compact in L p (Ω) thanks to the Sobolev compact embedding W 1,p 0 (Ω) ⊂ L p (Ω). Since C 0 is independent of v then the subset K is stable under Γ. Whence Γ admits at least a fixed point u ∈ K , in other words u is an entropy solution to (37) which is also a weak solution.
2) Let (u ) 0< ≤1 be a sequence of entropy solutions to (37), u is the unique entropy solution (which is also a weak solution) to (39) with v replaced by u and therefore we obtain the desired estimates as proved in 1).

Remark 3.
In the general case the entropy solution u of (37) is not necessarily unique.
Now, assume that And assume that for every E ⊂ W p bounded in L p (Ω) we have where conv {B(E)} is the closed convex-hull of B(E) in L p (Ω). Assumption (41) has been introduced in our preprint [14]. We shall give later some concrete examples of operators which satisfy this assumption. Let us prove the following Proof. The proof is similar to the one given in our preprint [14]. Let (Ω i ) j∈N be an open covering of Ω such that Ω j ⊂ Ω j+1 . We equip the space Z = W 1,p loc (Ω) with the topology generated by the family of seminorms (p j ) j∈N defined by Equipped with this topology, Z is a separated locally convex topological vector space. We set Y = L p (Ω) equipped with its natural topology. We define the family of linear continuous mappings Λ : where v is the unique entropy solution to The continuity of Λ follows immediately if we observe Λ as a composition of Λ : Y → Y and the canonical injection Y → Z Now, we denote Z w , Y w the spaces Z, Y equipped with the weak topology respectively. then Λ : Y w → Z w is also continuous.
Consider the bounded (in Y ) subset where C 0 is the constant introduced in Proposition 5. Consider the subset G = f + conv {B(E 0 )} where the closure is taken in the L p −topology. G is convex closed in Y and it is bounded thanks to (38). Assume (41) then G ⊂ L p (ω 2 ) + W 2 , it follows from Remark 2 that for every g ∈ G the orbit {Λ g} is bounded in Z, and therefore {Λ g} is bounded in Z w . Clearly the set G is compact in Y w . Then it follows by the Banach-Steinhaus theorem (applied on the quadruple (Λ ) , G, Y w , Z w ) that there exists a bounded subset F in Z w such that ∀ : Λ (G) ⊂ F.
The boundedness of F in Z w implies its boundedness in Z.i.e For every j ∈ N there exists C j ≥ 0 independent of such that ∀ : p j (Λ (G)) ≤ C j .
Let (u ) be a sequence of entropy solutions to (37), then we have (u ) ⊂ E 0 as proved in Proposition 5 then Λ (f + B(u )) = u ∈ F for every , therefore Whence for every Ω ⊂⊂ Ω there exists C Ω ≥ 0 independent of such that ∀ : u W 1,p (Ω ) ≤ C Ω . Now, we are ready to prove the convergence theorem. Assume that where (L p (Ω), τ L p loc ) is the space L p (Ω) equipped with the L p loc (Ω)-topology. Notice that (42) implies that B : L p (Ω) → L p (Ω) is continuous. Then we have the following  (42), suppose that Ω is convex, then there exist u 0 ∈ V p and a sequence (u k ) k∈N of entropy solutions to (37) such that and u k → u 0 in L p loc (Ω) − strong Moreover u 0 satisfies in D (ω 2 ) the equation for a.e X 1 ∈ ω 1 .
Proof. The estimates given in Proposition 5 show that there exist u 0 ∈ L p (Ω) and a sequence (u k ) k∈N solutions to (37) such that As we have proved in Theorem 2.3 we have u 0 ∈ V p . The particular difficulty is the passage to the limit in the nonlinear term. This assertion is guaranteed by Theorem 4.2. Indeed, since Ω is convex and Lipschitz then there exists an open covering (Ω j ) j∈N , Ω j ⊂ Ω j+1 and Ω j ⊂ Ω such that each Ω j is a Lipschitz domain (Take an increasing sequence of numbers 0 < β j < 1 with lim β j = 1. Fix x 0 ∈ Ω and take Ω j = β j (Ω − x 0 ) + x 0 , since Ω is open and convex then Ω j ⊂ Ω. The Lipschitzness is conserved for Ω j , since homothecies and translations are C ∞ diffeomorphisms).
Theorem 4.2 shows that for every j ∈ N there exists C j ≥ 0 such that Since Ω j is Lipschitz and bounded then the embedding W 1,p (Ω j ) → L p (Ω j ) is compact [1] and therefore for each j there exists a subsequence (u j By the diagonal process one can construct a sequence still labeled (u k ) k such that u k → u 0 in L p (Ω j ) for every j, in other words we have Passing to the limit in the weak formulation of (37) we deduce where we have used (43) for the passage to the limit in the left hand side. For the passage to the limit in the nonlinear term we have used (44) and assumption (42).
We can prove similarly as in [14] that (41) holds for the above integral operator, therefore the assertion of the theorem is a simple application of Theorem 4.3.
Remark 4. Notice that the compacity of the operator given in this example is not sufficient to prove a such result as in the L 2 theory [10]. This shows the importance of assumption (41) which holds in this case.
Does operator whose assumption (41) holds admits necessarily an integral representation as in (45)?

Example 2.
We shall replace the integral by a general linear operator. Let us consider the following problem: Find u ∈ V p such that −∇ X2 · (A∇ X2 u) = f (X 2 ) + gP (ha(u)) u(X 1 , ·) = 0 on ∂ω 2 , where a, A and f are defined as in Example 1.
When P is not compact then the operator u → gP (ha(u)) is not necessarily compact, if this is the case then this operator cannot admit an integral representation.
Theorem 4.5. Under the assumptions of this example, there exists at least a solution u ∈ V p to (46) in the sense of D (ω 2 ) for a.e X 1 ∈ ω 1 Proof. Similarly, the proof is a simple application of Theorem 4.3.

Some open questions.
Problem.
1. Suppose that ∞ > p > 2. Given f ∈ L p (Ω) and consider (2), since f ∈ L 2 (Ω) then u → u 0 in V 2 . Assume that Ω and A are sufficiently regular. Can one prove that u → u 0 in V p ? 2. What happens when f ∈ L 1 (Ω)? As mentioned in the introduction there exists a unique entropy solution to (2) which belongs to 1≤r< N N −1 W 1,r 0 (Ω). Can one prove that u → u 0 in V r for some 1 ≤ r < N N −1 ? Can one prove at least weak convergence in L r for some 1 < r < N N −1 as given in Theorem 2.3?