On the L^p-theory of Anisotropic singular perturbations of elliptic problems

In this article we give an extention of the L^2-theory of anisotropic singular perturbations for elliptic problems. We study a linear and some nonlinear 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

1. 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. First of all, let us begin by a brief discussion on the uniqueness of the weak solution ( by weak a solution we mean a solution in the sense of distributions) to the problem − div(A∇u) = f u = 0 on ∂Ω (1) where Ω ⊂ R N , N ≥ 2 is a bounded Lipschitz domain, we suppose that f ∈ L p (Ω) (1 < p < 2). The diffusion matrix A = (a ij ) is supposed to be bounded and satisfies the ellipticity assumption on Ω ( see assumptions (2) and (3) 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, (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, and al have introduced the concept of the entropy solution [4] for problems involving L 1 data (or more generally a Radon measure).
For every k > 0 We define the function T k : R → R by T k (s) = s , |s| ≤ k ksgn(x) |s| ≥ k 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. 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 − div(A ǫ ∇u ǫ ) = f u ǫ = 0 on ∂Ω , where Ω is a bounded Lipschitz domain of R N . 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 , ..., ∂ xN ) 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, We have decomposed A into four blocks where A 11 , A 22 are respectively q × q and (N − q) × (N − q) matrices. For 0 < ǫ ≤ 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 p 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 ǫ → u 0 inV p (Notice that in [9], authors have treated some problems involving L p data where some others 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 a cylindrical domain when the data is independent of X 1 . • In section 4: We treat some nonlinear problems.
2.1. Weak convergence. Let us prove the following primary result Theorem 3. Assume (3), (4) then there exists a sequence (u ǫ k ) k∈N ⊂ W 1,p 0 (Ω) of weak solutions to (2) and u 0 satisfies (5) for a.e X 1 ∈ Ω 1 . 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 (2) and (3) shows that u n ǫ exists and it is unique by the Lax-Milgram theorem. (Notice that u n ǫ also 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 ǫ ) 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 Thanks to Poincaré's inequality u n ǫ L p (Ω) ≤ C Ω ∇ X2 u n ǫ L p (Ω) we obtain ∇ X2 u n ǫ 2 L p (Ω) ≤ C ′ (1 + ∇ X2 u n ǫ L p (Ω) ), 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 ǫ, so Fix ǫ, since W 1,p (Ω) is reflexive then (10) implies that there exists a subsequence (u n l (ǫ) ǫ k ) l∈N and u ǫ ∈ W 1,p 0 (Ω) such that u n l (ǫ) ǫ ⇀ u ǫ ∈ 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 ǫ is a weak solution of (2) (u ǫ = 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) and (9) we deduce and similarly we obtain Using reflexivity and continuity of the derivation operator on D ′ (Ω) one can extract a subsequence (u ǫ k ) k∈N such that ∇ X2 u ǫ k ⇀ ∇ X2 u 0 , ǫ k ∇ X1 u n ǫ k ⇀ 0, u ǫ 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 ǫ k ⇀ ∇ X2 u 0 and u ǫ k ⇀ u 0 in L p (Ω) − weak then there exists a sequence (U n ) n∈N ⊂ conv({u ǫ k } k∈N ) such that ∇ X2 U n → ∇ X2 u 0 in L p (Ω) − strong, where conv({u ǫ k } k∈N ) is the convex hull of the set {u ǫ k } k∈N . Notice that we have U n ∈ W 1,p 0 (Ω) then -up to a subsequencewe 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. Let (U i × V i ) i∈N be a countable covering of Ω 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 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 a Lipschitz domain so the trace operator is well defined).

2.2.
Strong convergence. Theorem 1 will be proved in three steps. the proof is based on the use of the approximated problem (6). In the first step, we shall construct the solution of the limit problem Step1 : Let u n ǫ ∈ H 1 0 (Ω) be the unique solution to (6), existence and uniqueness of u n ǫ follows from assumptions (3), (4) as mentioned previously. One have the following Proposition 1. Assume (3), (4) then there exists (u n 0 ) n∈N ⊂ V 2 such that ǫu n ǫ → 0 in L 2 (Ω), u n ǫ → u n 0 in V 2 for every n ∈ N, in particular the two convergences holds in L p (Ω) and V p respectively. And 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 the proof of Theorem 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 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 ǫ → 0 we get Passing to the limit as n → ∞ we deduce Then it follows as proved in Theorem 3 that u 0 satisfies (5). Whence we have proved the following Proposition 2. Under assumption of Proposition 1 there exists u 0 ∈ V p solution to (5) such that u n 0 → u 0 in V p where (u n 0 ) n∈N is the sequence given in Proposition 1 Step2 : In this second step we will construct the sequence (u ǫ ) 0<ǫ≤1 solutions of (2), one can prove the following Proposition 3. There exists a sequence (u ǫ ) 0<ǫ≤1 ⊂ W 1,p 0 (Ω) of weak solutions to (2) such that u n ǫ → u ǫ in W 1,p (Ω) for every ǫ fixed. Moreover, u n ǫ → u ǫ in V p and ǫ∇ X2 u n ǫ → ǫ∇ X2 u ǫ , uniformly in ǫ. Proof. Using the linearity of (6) testing with θ(u n ǫ − u m ǫ ), m, n ∈ N we obtain as in (7) And (8) gives where C is independent of ǫ and n, whence Poincaré's inequality implies Similarly we obtain its follows that The last inequality implies that for every ǫ fixed (u n ǫ ) n∈N is a Cauchy sequence in W 1,p 0 (Ω), Then there exists u ǫ ∈ W 1,p 0 (Ω) such that u n ǫ → u ǫ in W 1,p (Ω), then the passage to the limit in (6) shows that u ǫ is a weak solution of (2). Finally (16) and (17) show that u n ǫ → u ǫ (resp ǫ∇ X2 u n ǫ → ǫ∇ X2 u ǫ ) in V p ( resp in L p (Ω)) uniformly in ǫ.
Step3 : Now, we are ready to conclude. Proposition 1, 2 and 3 combined with the triangular inequality show that u ǫ → u 0 in V p and ǫ∇ X2 u ǫ → 0 in L p (Ω), and the proof of Theorem 1 is finished.

2.3.
Convergence of the entropy solutions. As mentioned in section 1 the entropy solution u ǫ 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 ǫ → u ǫ ∈ W 1,p 0 (Ω) as shown in Proposition 3. We will show that u ǫ ∈ T 1,2 0 (Ω). Clearly we haveT k (u n ǫ ) ∈ H 1 0 (Ω) for every k > 0. Now testing with Using the ellipticity assumption we get Fix ǫ, k, we have u n ǫ → u ǫ in L p (Ω) then there exists a subsequence (u n l ǫ ) l∈N such that u n l ǫ → u ǫ a.e x ∈ Ω and since T k is bounded then it follows that T k (u n l ǫ ) → T k (u ǫ ) a.e in Ω and strongly in L 2 (Ω) whence u ǫ ∈ T 1,2 0 (Ω). It follows by (18) that there exists a subsequence still labelled Whence u ǫ 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 follows from Theorem 1.
Remark 1. Uniqueness of the entropy solutions 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 ) 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) as mentioned in the above subsection.
We introduce the function θ δ (t) = t 0 (δ + |s|) p−2 ds, δ > 0, t ∈ R we have . 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 previous inequality we deduce Using Hölder's inequality we derive Then we deduce Now passing to the limit as δ → 0 using the Lebesgue theorem we deduce and Poincaré's inequality gives 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.

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 . Let u ǫ , u 0 be the unique entropy solutions of (2), (5) respectively then under the above assumptions we have the following Theorem 5. For every ω ′ 1 ⊂⊂ ω 1 and m ∈ N * there exists C ≥ 0 independent of ǫ such that Proof. Let u ǫ , u 0 be the entropy solutions of (2), (5) respectively, we use the approximated sequence (u n ǫ ) ǫ,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 previous 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 ǫ and n Now, using Hölder's inequality and the previous inequality we deduce Passing to the limit as δ → 0 using the Lebesgue theorem. Passing to the limit as n → ∞ we get ⊂⊂ ω 1 . Iterating the above inequality m−time we deduce Now, from (20) (with ω ′ 1 and ω ′′ 1 replaced by ω and ω (m+1) 1 respectively) we deduce Since u ǫ → u 0 in L p (Ω) then u ǫ − u 0 L p (Ω) is bounded and therefore we obtain And the proof of the theorem is finished.
We consider the limit problem Then under the above assumptions we have Theorem 6. Let u ℓ , u ∞ be the unique entropy solutions to (21) and (24) then for every α ∈ (0, 1) there exists C ≥ 0, c > 0 independent of ℓ such that Proof. Let u ℓ , u ∞ the unique entropy solutions to (21) and (24) respectively, and let (u n ℓ ) and (u n ∞ ) the approximation sequences (as in section 2). we have u n ℓ → u ℓ in W 1,p 0 (Ω ℓ ) 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 < ℓ 0 ≤ ℓ − 1, and let ρ ∈ D(R q ) a bump function such that 0 ≤ ρ ≤ 1, ρ = 1 on ℓ 0 ω 1 and ρ = 0 on R q (ℓ 0 + 1)ω 1 , |∇ X1 ρ| ≤ c 0 where c 0 is the universal constant (see [5]). Testing with Using the ellipticity assumption (23) Notice that ∇ρ = 0 on Ω ℓ0 , and Ω ℓ0 ⊂ Ω ℓ0+1 ( since ω 1 is convex and containing 0). Then by the Cauchy-Schwaz inequality we get where we have used (22). Whence we get ( since ρ = 1 on Ω ℓ0 ) 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 αℓ we get where c, c ′ > 0 are independent of ℓ and n. Now we have to estimate the right hand side of (26). Testing with θ(u n ℓ ) 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 Replace (28), (27) in (26) and passing to the limit as n → ∞ we obtain the desired result. Corollary 2. Under the above assumptions then for every α ∈ (0, 1) there exists C ≥ 0, c > 0 independent of ǫ such that where u ǫ , 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 Where the a : R → R is a continuous nonincreasing function which satisfies the growth condition ∀x ∈ R : |a(x)| ≤ K(1 + |x|), K ≥ 0 and f ∈ L p (Ω) where 1 < p < 2 , f / ∈ L 2 (Ω) and A is given as in Subsection 1.2. Clearly the Nemytskii operator u → a(u) maps L r (Ω) → L r (Ω) continuously for every 1 ≤ r < ∞. The passage to the limit (formally) gives the limit problem We can suppose that a(0) = 0. Indeed, in the general case the right hand side of (29) can be replaced by (a(0) a(0). Clearly b is continuous nonincreasing and satisfies |b(x)| ≤ (K + |a(0)|)(1 + |x|).
First of all, suppose that f ∈ L 2 (Ω),then we have the following Proof. Existence of u ǫ 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 ǫ as a test function in (29) then one can obtain the estimates where C is independent of ǫ, we have used that Ω a(u ǫ )u ǫ dx ≤ 0 (thanks to monotonicity assumption and a(0) = 0). And we also have (thanks to assumption (30)) a(u ǫ ) L 2 (Ω) ≤ K(|Ω| 1 2 + C) so there exists v ∈ L 2 (Ω), u 0 ∈ L 2 (Ω), ∇ X2 u 0 ∈ L 2 (Ω) and a subsequence (u ǫ k ) k∈N such that Passing to the in the weak formulation of (29) we get Take ϕ = u ǫ k in the previous equality and passing to the limit we get Let us computing 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 (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 ǫ ) 0<ǫ≤1 . Now, we are ready to give the main result of this subsection Theorem 7. Suppose that f ∈ L p (Ω) where 1 < p < 2 (we can suppose that f / ∈ L 2 (Ω)) then there exists u 0 ∈ V p such that u 0 (X 1 , ·) is the unique entropy solution to (31) and we have u ǫ → u 0 in V p , ǫ∇ X1 u ǫ → 0 in L p (Ω), where u ǫ 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 we shall construct the entropy solution u ǫ . we consider the approximated problem − div(A ǫ ∇u n ǫ ) = f n + a(u n ǫ ) u n ǫ = 0 on ∂Ω We follows the same arguments as in section 2, where we use the above proposition and the following Which holds for every u, v ∈ L 2 (Ω), in fact this follows from monotonicity of a and θ.
2) If (u ǫ ) 0<ǫ≤1 is a sequence of entropy and weak solutions to (37) then we have the above estimates.
Proof. 1) The existence of u ǫ is based on the Schauder fixed point theorem, we define the mapping Γ : 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 ǫ and v (thanks to (38)) Now, 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). The subset K is stable under Γ (since C 0 is independent of v as mentioned above). Whence Γ admits at least a fixed point u ǫ ∈ K, in other words u ǫ is a weak solution to (37) which is also an entropy solution, this last assertion follows from the definition of Γ.
2) Let (u ǫ ) 0<ǫ≤1 be a sequence of entropy and weak solutions to (37) u ǫ is the unique entropy solution to (39) with v replaced by u ǫ and therefore we obtain the desired estimates as proved above.
Remark 3. In the general case the entropy solution u ǫ 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) appears strange. We shall give later some concrete examples of operators which satisfy this assumption. Let us prove the following  (41). Let (u ǫ ) 0<ǫ≤1 ⊂ W 1,p 0 (Ω) be an entropy and weak solution to (37) then for every Ω ′ ⊂⊂ Ω there exists C Ω ′ ≥ 0 independent of ǫ such that The proof is similar the one given in our preprint [14]. Let (Ω i ) j∈N 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 p j (u) = u ǫ W 1,p (Ωj ) 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 the linear continuous mappings The continuity of Λ ǫ follows immediately if we observe Λ ǫ as a composition of Λ ǫ : 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 ǫ : 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. Thanks to assumption (41) and (38) G is closed convex and bounded in Y . Now for every g ∈ G the orbit {Λ ǫ g} ǫ is bounded in Z thanks to Remark 2. And therefore {Λ ǫ g} ǫ 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 ǫ , G, Y w , Z w ) that there exists a bounded subset F in Z w such that 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 ǫ such that ∀ǫ : p j (Λ ǫ (G)) ≤ C j Let u ǫ be an entropy and weak solution to (37) then we have u ǫ ∈ E 0 as proved in Proposition 5 then Λ ǫ (f + B(u ǫ )) = u ǫ ∈ F for every ǫ, therefore ∀ǫ : u ǫ W 1,p (Ωj ) ≤ C j Whence for every Ω ′ ⊂⊂ Ω there exists C Ω ′ ≥ 0 independent of ǫ such that ∀ǫ : u ǫ 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 Theorem 9. Under assumptions of Theorem 8, assume in addition (42), suppose that Ω is convex, then there exists u 0 ∈ V p and a sequence (u ǫ k ) k∈N of entropy and weak solution to (37) such that for a.e X 1 ∈ ω 1 Proof. The estimates given in Proposition 5 show that there exists u 0 ∈ L p (Ω) and a sequence (u ǫ k ) k∈N solutions to (37) such that As we have proved in Theorem 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 8. Indeed, since Ω is convex and Lipschitz then there 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 number 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 convex then Ω j ⊂ Ω. The Lipschitz character is conserved since the multiplication by β j and translations are C ∞ diffeomorphisms).
Theorem 8 shows that for every j ∈ N there exists C j ≥ 0 such that Since Ω j is Lipschitz then the embedding W 1,p (Ω j ) ֒→ L p (Ω j ) is compact [1] and therefore for each k there exists a subsequence (u ǫ j k ) k ⊂ L p (Ω j ) such that By the diagonal process one can construct a sequence (u ǫ k ) k such that u ǫ k → u 0 in L p (Ω j ) for every j, in other words we have Now 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). Example 1. We give a concrete example of application of the above abstract analysis. Let Ω = ω 1 × ω 2 be a Lispchitz convex domain of R q × R N −q and let A be a bounded (N − q) × (N − q) matrix defined on ω 2 which satisfies the ellipticity assumption. Let us consider the integro-differential problem where h ∈ L ∞ (ω 1 × Ω) and f ∈ L p (ω 2 ), 1 < p < 2, and a is a continuous real bounded function.
This equation is based on the Neutron transport equation (see for instance [10]) A solution to (45) is a function u ∈ V p Which satisfies (45) in D ′ (ω 2 ). suppose that ∇ X1 h(X ′ 1 , X 1 , X 2 ) ∈ L ∞ (ω 1 × Ω) Then we have Theorem 10. Under the assumptions of this example, (45) has at least a solution in V p in the sense of D ′ (ω 2 ) for a.e X 1 ∈ ω 1 Proof. We introduce the singular perturbation problem Clearly A ǫ satisfies the ellipticity assumption and it is Clear that the operator u → ω1 h(X ′ 1 , X 1 , X 2 )a(u(X ′ 1 , X 2 ))dX ′ 1 satisfies assumption (38).
We can prove easily that the above operator satisfies assumption (42). Indeed, let u n → u in L p loc (Ω) then there exists a subsequence (u n k ) (constructed by the diagonal process) such that u n k → u a.e in Ω. Since a is bounded then it follows by the Lebesgue theorem that ω1 h(X ′ 1 , X 1 , X 2 )a(u n k (X ′ 1 , X 2 ))dX ′ 1 → ω1 h(X ′ 1 , X 1 , X 2 )a(u(X ′ 1 , X 2 ))dX ′ 1 , in L p (Ω). Whence by a contradiction argument we get ω1 h(X ′ 1 , X 1 , X 2 )a(u n (X ′ 1 , X 2 ))dX ′ 1 → ω1 h(X ′ 1 , X 1 , X 2 )a(u(X ′ 1 , X 2 ))dX ′ 1 , in L p (Ω) We can prove similarly as in [14] that (41) holds, therefore the assertion of the theorem is a simple application of theorem 9 Remark 4. Notice that the compacity of the operator given in the previous example is not sufficient to prove a such result as in the L 2 theory [10]. This shows the importance of assumption (41) wich holds for the above operator.
Does operator whose assumption (41) holds admit necessarily an integral representation as in (45)?.
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 11. 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 9.

Some Open questions
Problem 1. Suppose that ∞ > p > 2. Given f ∈ L p and consider (2), since f ∈ L 2 then u ǫ → u 0 in V 2 . Assume that Ω and A are sufficiently regular .Can one prove that u ǫ → u 0 in V p ? Problem 2. What happens when f ∈ L 1 ? 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 ǫ → 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 4?