Linear diffusion with singular absorption potential and/or unbounded convective flow: the weighted space approach

In this paper we prove the existence and uniqueness of very weak solutions to linear diffusion equations involving a singular absorption potential and/or an unbounded convective flow on a bounded open set of $\mathbb R^N$. In most of the paper we consider homogeneous Dirichlet boundary conditions but we prove that when the potential function grows faster than the distance to the boundary to the power -2 then no boundary condition is required to get the uniqueness of very weak solutions. This result is new in the literature and must be distinguished from other previous results in which such uniqueness of solutions without any boundary condition was proved for degenerate diffusion operators (which is not our case). Our approach, based on the treatment on some distance to the boundary weighted spaces, uses a suitable regularity of the solution of the associated dual problem which is here established. We also consider the delicate question of the differentiability of the very weak solution and prove that some suitable additional hypothesis on the data is required since otherwise the gradient of the solution may not be integrable on the domain.


Introduction
In this paper we want to develop a weighted space approach to study the existence, uniqueness and regularity of linear diffusion equations involving singular and unbounded coefficients of the type − ∆ω + u · ∇ω + V ω = f on Ω, (1) where V is a very singular potential being in general non negative and locally integrable. To fix ideas, we shall consider mainly the case of Dirichlet boundary conditions ω = 0 on ∂Ω, but our weighted space approach can also be adapted to the case of Neumann boundary conditions and, what is more remarkable, to the case of no boundary conditions on ∂Ω (but still getting the uniqueness of solutions) for some specially singular potentials (see the subsection 4.2 in section 4).
Here Ω is an open bounded smooth (for instance with ∂Ω of class C 2,1 ) of IR N , N 2, (the case N = 1 and u =constant is considerably simpler) . The external forcing term f (x) will be assumed such that f ∈ L 1 (Ω; δ) where the weight in this space is given by (sharper results will require some slight restrictions to (3) (see for instance section 4.3). We recall that (3) is optimal in the cases V ≡ 0 and u = 0 as it can be shown by explicitly computing the Green kernel for special domains.
Although we shall indicate later the detailed assumptions on the data, we anticipate now that we shall always assume that the convective flow vector u satisfies u ∈ L N (Ω) N , div u = 0 in D ′ (Ω) and u · n = 0 on ∂Ω (5) where n denotes the unit exterior normal vector to ∂Ω. Notice that, due to (5), the weak solution notion adapted to equation (1) is equivalent to the one defined for the treatment of the equation in divergent form that is − ∆ω + div ( u ω) + V ω = f in Ω. (6) It is well-known that the mathematical treatment of diffusion equations such as (1) or (6) leads to quite satisfactory results (in view of some applications) when the data f , u and V are assumed to be bounded. Nevertheless, the main interest of this work concerns the limit cases in which V (x) is assumed to be a singular function (mainly with its singularity located on ∂Ω) and/or when u is an unbounded vector (satisfying (5)). Let us indicate some relevant applications leading to the consideration of such limit cases : 1. The vorticity equation in fluid mechanics. Equation (1) can be derived from the stationary Navier-Stokes in 2D − ∆ u + ( u · ∇) u + ∇p = F taking the curl of the equation and setting where k is the last element of the canonical basis in IR 3 (see e.g. [46]). Nevertheless, as far as we know no satisfactory theory is available in the literature under the general condition that F · k ∈ L 1 (Ω; δ).

Schrödinger equation with singular potentials.
It is well-known that the consideration of the bound states ψ(x, t) = e −iEt ω(x) leads to the stationary Schrödinger equation The Heisenberg uncertainty principle makes specially interesting the consideration of potentials which are critically singular on ∂Ω more precisely, such that a.e x ∈ Ω, (10) for some c > 0, which implies that ω = ∂ω ∂ n = 0 on ∂Ω, so that we can assume that ω ≡ 0 on IR N − Ω (see [15,16]). Here we shall not consider any eigenvalue problem like (9) but the study of (1) for potentials V (x) satisfying (10) will be very useful for later works in this direction.
3. Linearization of singular and/or degenerate nonlinear equations. For many different purposes, it is very convenient to "approximate" the solutions of quasilinear diffusion equations of the type − ∆ϕ(w) + div φ(w) + g(w) = f (x) in Ω (11) by the solutions of the associated linearized equation. This is what appears, for instance, in the study of the stability of the associated parabolic or hyperbolic equations and also in some control problems associated with (11). Usually, it is assumed that ϕ is a strictly increasing function. So by considering θ := ϕ(w) we get with Now, assume that θ ∞ (x) is a given solution of (12), satisfying, for instance, θ ∞ = 0 on ∂Ω. Then the "formal linearization" of equation (13) around the solution θ ∞ (x) coincides with equation (1) when we take u(x) := ψ θ ∞ (x) and What makes difficult the study of the corresponding problem (1) is the fact that in many cases relevant in the reaction-diffusion theory (see e.g. [26]) functions ψ ′ (r) and h ′ (r) present a singularity at r = 0 and so, at least on ∂Ω, the coefficients u and V are singular. A qualitative information on the behavior of θ ∞ (x) near ∂Ω allows us to get the precise information about the singularities of u and/or V near ∂Ω which, for instance, is of the type (10) . 4. Shape optimization in Chemical Engineering. When dealing with the problem of shape optimization for chemical reactors and applying technics of shape differentiation, it was shown that if g ∈ W 2,∞ (IR), then the solutions u 0 of the problem −∆u + g(u) = f, Ω, u = 1, ∂Ω, are differentiable with respect to the domain in the sense of Hadamard [25] and after developed in Murat and Simon [32,43] and the derivative u ′ in the direction of a deformation θ ∈ W 1,∞ (IR n , IR n ) is the solution of the problem −∆u ′ + g ′ (u 0 )u ′ = 0, u ′ + θ · ∇u ∈ H 1 0 (Ω).
Applying the theory developed for the general case (1), we can give a meaning to the shape derivative if the domain is not smooth as, for example, for root type kinetics (see [17,24]). These nonlinear terms g(u) are known in chemistry as Freundlich kinetics and have signifiant importance. Once again, taking V (x) ≡ g ′ u 0 (x) we arrive to problem (1).
We also mention that sometimes it is possible to get conclusions for the stationary problem (1) (with u = 0) through the consideration of the associated evolution equations (see e.g. [7], [8] and its references).
In this paper we shall work with the notion of "very weak solutions" (v.w.s.) of problem (1).  (1)). Let f be in L 1 (Ω; δ) and u ∈ L N,1 (Ω) N with div ( u) = 0 in D ′ (Ω), u · n = 0 on ∂Ω, V measurable and non negative function. A very weak solution ω of (1) is a function ω ∈ L N ′ ,∞ (Ω) satisfying Notice that we look for a function in the space L N ′ ,∞ (Ω) where N ′ = N N −1 instead of ω ∈ L 1 (Ω) as usual, in order to get more general assumptions on u and V .
We also also point out that our study will be concentrated in the case of "absorption" potentials V (x) 0 a.e. x ∈ Ω. In fact, as we shall see later, the study is also applicable to some general potentials such that e.g. V (x) −λ with 0 < λ < λ 1 (λ 1 being the first eigenvalue of the Laplacian on Ω with zero Dirichlet boundary condition). As we shall show, this does not induce a restriction on the growth of the singularity of such absorption potentials near ∂Ω (in contrast with the well-known results for negative potentials, see e.g. [7]).
The detailed definition of the Lorentz spaces L p,q (Ω) and some other spaces which we shall use in our study will be the object of Section 2 of this paper. Other preliminary results and the statement of some of our main conclusions will be also presented there.
The proof of the existence and uniqueness of a very weak solution (v.w.s.) for (1) needs a deep study of the dual problem associated with (1) Notice the change of sign in the convection term. We anticipate that in some cases no boundary condition will be assumed on φ.
In Section 3, we discuss, depending on V and u, the existence and the regularity of the solution of the dual problem. After this, we shall be concerned with the existence of the very weak solution in L N ′ ,∞ (Ω) ∩ L 1 (Ω; V δ), when V 0 is locally integrable. We will show that the very weak solution ω of equation (1) under zero Dirichlet or Neumann boundary condition has its gradient in the Sobolev-Lorentz weighted space W 1 L 1+ 1 N ,∞ (Ω; δ) in particular we shall get the estimate under the mere assumption u ∈ L N,1 (Ω) N . Thus, we can conclude that ∇ω ∈ L 1 loc (Ω).
The question of uniqueness of v.w.s. given by (16), when V is only in L 1 loc (Ω) is one of the major difficulties in this general framework. When V is sufficiently integrable, say V ∈ L N,1 (Ω), then we derive the uniqueness thanks to the regularity of the dual problem. If V is only locally integrable, but V is bounded from below by cδ −r , r > 2 near the boundary, then the v.w.s. is unique even when no boundary condition is specified on ∂Ω (but we additionally know that V ω ∈ L 1 (Ω; δ)). The uniqueness proof relies on the L 1 (Ω; δ)-accretiveness property of the operator (see [36]) . This is given through the following local version of the Kato's type inequality and a special approximation of test function ϕ in C 2 (Ω) by a sequence of functions of the type ϕ n (x) = δ(x) r h n (x) with h ∈ C 2 c (Ω) and r > 0 (see Lemma 4.4). We point out that, besides the concrete interest of (19) in itself; such an inequality has many consequences since it allows to apply the semigroup operators theory on suitable functional spaces. Concerning very weak solutions (where no differentiability is asked to the function ω), a natural question (originally set by H. Brézis in 1972 when u = 0) is then : when should we have |∇ω| in L 1 (Ω)? The answer to this question will require some suitable additional integrability conditions on f and u.
Note that for proving some additional integrability for the very weak solutions ω is a delicate task. Indeed, we shall show that for some special cases of u ∈ C 0,α (Ω), α > 0, there exists f ∈ L 1 + (Ω; δ) such that ||ω|| L N ′ = +∞ when N 3.This leads to an additional question : under what conditions could we improve the integrability of ω, to say ω ∈ L N ′ (Ω)? The answer to this question is also one of the main results of this paper.
Before stating the study of the main equation (1), we shall recall some notations and functional spaces that we shall use.

Notations, preliminary definitions and results
Before stating our main results concerning equation (1) we need to recall some notations and some functional spaces which are relevant for the study of the "dual problem" (17) under very general regularity assumptions on the coefficients u and T . Definition 2.1. ( bmo(IR N )) [23].
A locally integrable function f on IR N is said to be in bmo( where the supremum is taken over all cube Q ⊂ IR N the sides of which are parallel to the coordinates axes. Here Definition 2.2. ( bmo r (Ω) ) [11,12].
A locally integrable function f on a Lipschitz bounded domain Ω is said to be in bmo r (Ω) (r stands for restriction) if where the supremum is taken over all cube Q ⊂ Ω the sides of which are parallel to the coordinates axes.
In this case, there exists a function f ∈ bmo(IR N ) such that f Remark 1.
The above definition adapted to the case where the domain Ω is bounded, is equivalent to the definition given in [12,11]. The main property (21) is due to P.W Jones [27]. This extension result implies that bmo r (Ω) embeds continuously into L exp (Ω) (a space which we shall introduce below in Definition 2.5.) Here In fact the two above definitions are equivalent : For a Lipschitz bounded domain Ω one has L 2,N (Ω) = bmo r (Ω), with equivalent norms.
We set L 0 (Ω) = v : Ω → IR Lebesgue measurable and we denote by L p (Ω) the usual Lebesgue space 1 p +∞. Although it is not too standard, we shall use the notation W 1,p (Ω) = W 1 L p (Ω) for the associate Sobolev space. We shall need the following definitions: The generalized inverse u * of m is defined by and is called the decreasing rearrangement of u. We shall set Ω * =]0, |Ω| [.
For α > 0, we define When there is no possible confusion, we denote by the same symbol the space product V N and V .
We recall also that if v, u ∈ L 1 (Ω), then v * u= lim λց0 (u + λv) * − u * λ exists in a weak sense and it is called the relative rearrangement of v with respect to u. More precisely, we have the following result (see [31,35]).

Theorem 2.2.
Let Ω be a bounded measurable set in IR N , u and v two functions in L 1 (Ω) and let w : Ω * → IR be defined by: Moreover, dw ds L p (Ω * ) |v| L p (Ω) .
One property that we shall use for the relative rearrangement is the following one: Let v 0, and u be two functions in L 1 (Ω). Then There is a link between the derivative of u * and the relative rearrangement of the gradient of u as it was proved in [35,41]. We will use only the following result (see [35]) Note that u * is in W 1,1 loc (Ω * ) under statements (a) and (b) (see [35,41]).

Remark 3.
Here and along the paper u is at least in L N (Ω) N , div ( u) = 0 in D ′ (Ω) and u · n = 0 on ∂Ω, if N 3 and u ∈ L 2+ε (Ω), for some ε > 0 if N = 2. The value of u · n on ∂Ω is defined through the Green's formula (see [46]).
The following density result can be proved using the same argument as for the L p -case (see [46,13]) Proposition 2. (Density of smooth functions). Let 1 < p < +∞ and 1 q ∞. Then the closure of the set : div ( u) = 0 and u · n = 0 on ∂Ω .
We shall need the following classical result (see [28]) : Let X ֒→ c Y ֒→ Z be three Banach spaces each continuously embedded in the next one, the first inclusion is supposed to be compact. Then, for all ε > 0 there exists a constant c ε > 0 such that 3 Existence, uniqueness, regularity and results for the dual problem 3

.1 Case where the potential V is only measurable and bounded from below
We first study the solvability of the dual problem (17) (equivalent to (23) below and the regularity of its solutions.
The following result, consequence of the Lax-Milgram theorem, is a remarkable fact due to the low regularity assumed on the data u and V : Moreover, loc (Ω), then the equation (23) holds in the sense of distributions in D ′ (Ω) Proof. We endow W with the following norm Let us consider the bilinear form on W given by Then, by Lemmas 2.6 and 2.7 with α 0 > 0.
According to the above remark (22), since u ∈ L N (Ω) N , the bilinear form is continuous on W and we have Thus we may apply the Lax-Milgram theorem to derive the existence of a unique φ ∈ W , The estimate on φ follows from (24).
Moreover, since for any open set Ω ′ relatively compact in Ω. Writing the right hand is finite taking into account (25) and the fact that φ ∈ L 2 (Ω) . Thus, we have ∀ Ω ′ ⊂⊂ Ω, V φ ∈ L 1 (Ω ′ ). We conclude that V φ ∈ L 1 loc (Ω). As usual in some problems of Quantum Mechanics (see e.g. Lemma 2.1 of [15]) it is very useful to approximate the solution φ ∈ H 1 0 (Ω) of the dual problem (23) found in Proposition 3 by a sequence of solutions φ k corresponding to a sequence of bounded potentials V k approximating V . Let us define V k by V k = min(V, k).

Proposition 4. (Approximation by bounded potentials).
Let T ∈ H −1 (Ω), u and V as in Proposition 3. Then, the sequence φ k ∈ H 1 0 (Ω) of solutions of the problems converges to φ strongly in H 1 0 (Ω), where φ is the unique solution of (P) V,T found in Proposition 3.
Sketch of the proof of Proposition 4. One has, following the arguments of the Proposition 3, that Thus, φ k remains in a bounded set of H 1 0 (Ω). So we may assume that it converges to a function ϕ weakly in H 1 0 (Ω) and a.e. in Ω. The above relation (26) implies that: This shows that ϕ ∈ W (where W is the space defined in the proof of Proposition 3). Moreover, since for all ψ ∈ W we have uψ ∈ L 2 * ′ (Ω) (see the above remark), we deduce The sequence (V k + λ)φ k ψ satisfies Vitali's condition, since for any measurable subset B ⊂ Ω, we have and lim Thus lim k→+∞ Ω We then deduce that ϕ is solution of the problem (P) V,T and by uniqueness ϕ = φ. Therefore, the whole sequence φ k converges to φ weakly in W and strongly in L 2 (Ω).
To prove the strong convergence in H 1 0 (Ω), let us note, using the equations (P) V k ,T and (P) V,T , that For this reason, from now, we will assume that λ = 0.

Proposition 5.
Under the same assumptions as for Proposition For the treatment of (1) we shall need some additional regularity for the solutions of the dual problem (23) independent of u or V . We start by proving the boundedness of φ by means of some rearrangement technics ( [35] p.126 of Th 5.5.1, see also [45]).
We point out that L Let φ be the solution of (23) when T ∈ L N 2 ,1 (Ω), V 0. Then φ ∈ L ∞ (Ω) and there exists a constant K N (Ω) independent of u and V such that Proof. We shall argue in a way similar to the proof of Theorem 5.3.1 in [35]. According to Proposition 4 , it is enough to prove the proposition for V ∈ L ∞ + (Ω), and and for T 0, since the equation (23) is linear. Thus φ 0, therefore, in this proof v = |φ| = φ, but we shall keep the notation v because in the general case we cannot use anymore this maximum principle.
Differentiating this relation with respect to s, we find where T * is the monotone rearrangement of T (we use the fact that V 0). Therefore, we arrive at Since |∇φ| = |∇v|, and − v ′ * (s) the PSR property (see Theorem 3 of [35]) and |∇v| * v |∇v| 2 Thus, integrating (33) between 0 to |Ω|, we find .

Proposition 7.
Let N 2, and let φ be a solution of (23) when Then φ ∈ L ∞ (Ω) and there exists a constant K N (Ω) > 0 independent of u and V such that Proof. For convenience, we write F for F . Thanks to Proposition 4, we can use the same test function G s (φ) as in the proof of Proposition 6. Then We differentiate this equation with respect to s as before, for a.e. s ∈ Ω * , and find Since, V 0 and v ′ * (s) 0, we obtain We have as before: We infer that for a.e. s − v ′ * (s) Integrating this relation between 0 and |Ω| and using the Hardy-Littlewood inequality (see [35] p.118-121) we obtain We conclude as in [35] p. 118-120, Proposition 5.2.2.
Remark 5. The problem considered in this Section 3.1 was previously considered by other authors in the special case of − → u ≡ − → 0 (see, e.g. [14] and its references), nevertheless we emphasize that the results of this section must be understood as preliminary results with respect the study we shall present in the following sections of this paper. In particular, what is specially important for us is to obtain a continuous dependence estimate with respect to the data (namely the velocity − → u , the potential V, and the right hand side f ) since we need to carry out several perturbations of those data in the next sections. As far as we know, such estimates are new in the literature (and, of course, they were not given in the above mentioned reference).

Some regularity results with an integrable potential V and bounded from below
As a first consequence of Proposition 3 and Proposition 7 we can deduce Meyer's type regularity giving a better information on the gradient of the solution of (23).
Then, the unique solution φ of the equation (23) belongs to W 1 L p,q (Ω). Moreover, there exists a constant K pq > 0 independent of u such that : Proof. (We shall simply write F, F 0 , F 1 for F , F 0 , F 1 ). We first assume that u ∈ V. We know from Proposition 7 that φ ∈ L ∞ (Ω) and that there exists a constant independent of u, V and F and V such that Therefore, there exists a vector field F 0 ∈ L p,q (Ω) N such that But, we have uφ ∈ L p,q (Ω) N since φ ∈ L ∞ (Ω) according to the above Proposition 7. Hence We may apply the W 1 L p,q result to (40) (see [42,9,2,36]) to deduce that For the general case, we consider u k ∈ V such that u k → u strongly in L p,q (Ω) N . Let φ k be the solution of equation (23) The sequence (φ k ) k satisfies and then (φ k ) k converges weakly in H 1 0 (Ω) to φ the solution of (23) . Since φ k satisfies (41) , we deduce that φ also satisfies (41) and (23) .
As an immediate consequence of the above result.

Proposition 9.
Let u and F be in L p,∞ (Ω) N for some p > N . Then, the solution of (23) satisfies Proof. According to the Sobolev embedding (see [35]), we have Now we shall consider the case of more general data u and V .

Proposition 10.
Assume that u and F are in bmo r (Ω) N and V is in bmo r (V ). Then the solution φ of the equation (23) satisfies Proof. Since bmo r (Ω) ֒→ L p,q (Ω) for all p > N and q ∈ [1, +∞], we deduce from Proposition 8 and Proposition 9 that : where F 1 was defined in the proof of Proposition 8 (see equation (40)). From Stegenga multiplier's result, uφ ∈ bmo r (Ω) N whenever u is in bmo r (Ω) N [44,47]. Therefore F 1 − uφ ∈ bmo r (Ω) N . We may appeal to Campanato's result [10] to derive then that ∇φ ∈ bmo r (Ω) N and We shall end this paragraph by proving a W 2 L p,q (Ω)-regularity result for the solutions of the dual problem (23) which will lead to interesting conclusions for the direct problem (1). For this, we shall use the following ADN constant which is finite due to the well-known Agmon-Douglis-Nirenberg's regularity result combined with the Marcinkiewicz interpolation Theorem. We shall improve now the regularity obtained in Proposition 10. We consider ε 0 > 0 (fixed) so that K s pq ε 0 || u|| L p,q (Ω) 1 2 . Proposition 11. (W 2 L p,q (Ω) regularity for p > N ) Let φ be the solution of (23) when T ∈ L p,q (Ω), p > N, q ∈ [1, +∞]. Assume, furthermore, that u ∈ L p,q (Ω) N and V ∈ L p,q (Ω). Then φ ∈ W 2 L p,q (Ω).
Moreover, there exist constants c ε0 , K pqN > 0 such that Proof. We assume first that u ∈ V. Arguing as in Proposition 8, since we can assume that T = div F for suitable F we get that the solution φ of (23) is in W 1 L p,q (Ω) and then −∆φ = u ∇φ + T − V φ ∈ L p,q (Ω).
By the Agmon-Douglis-Nirenberg regularity results and the Marcinkiewicz interpolation theorem we deduce that φ ∈ W 2 L p,q (Ω). Moreover, since p > N and q ∈ [ 1, +∞], we have the following continuous embeddings : The first inclusion is compact so we may appeal to Lemma 2.8 to derive that ∀ ε > 0, there exists From the equation satisfied by φ, we have and using the ADN constant We combine those last three equations and derive that for any ε > 0 Next, we consider u k ∈ V such that u k → u ∈ V. Then, choosing ε = ε 0 > 0 such that , we deduce from relation (46) that φ k corresponding to the solution of (23) , that is −∆φ k − u k · ∇φ k + V φ k = T ∈ L p,q (Ω), belongs to a bounded set of W 2 L p,q (Ω) when k varies. Therefore, the strong limit φ in C 1 (Ω) is the solution of (23) and it satisfies also the relation (46) for all ε ∈]0, ε 0 ]. From Proposition 6, we have Combining relations (46) and (47) with ε = ε 0 , we derive the result.
The case where p = N can also be treated in the same way provided that the norm of u in L N,1 (Ω) is small enough in the sense that Proposition 12. (Regularity in W 2 L N,1 (Ω)). Let φ be the solution of (23) when T ∈ L N,1 (Ω), V ∈ L N,1 (Ω). Assume that u satisfies relation (48). Then φ ∈ W 2 L N,1 (Ω). Moreover, there exists a constant K ′ N (Ω) (independent of u) such that Proof. The proof follows the same argument as for the proof of Proposition 11. Nevertheless, the embedding W 1 L N,1 ⊂ C(Ω) is not compact and this explains the condition (48).
There are many other spaces between the space L p,1 (Ω) and L N,1 (Ω) for which we can obtain a regularity result for the second derivatives of φ.
Here we want only to consider the space Λ = (L N (Log L) β N ) N for β > N − 1. Indeed this space is included in L N,1 (Ω) and contains L p (Ω) for all p > N .
Theorem 3.1. (Regularity in W 2 L N (Ω)). Let T and V be in L N (Ω), u ∈ Λ, div ( u) = 0 and u · n = 0 on ∂Ω. Then the unique solution φ of (23) belongs to W 2 L N (Ω) and choosing ε > 0 such that ε|| u|| Λ 1 2 , there exists a constant K ε > 0 such that The proof firstly depends on the following Trudinger's type embedding : Proof. According to the pointwise Sobolev inequality for the relative rearrangement, we have for u = |v| (see Theorem 2.3) − u ′ * (s) We integrate this formula from t to |Ω| knowing that u * (|Ω|) = 0, and using the Hölder inequality, we get Therefore from (51), implies using Theorem 2.2 The key result for the proof of Theorem 3.1 is the following compactness inclusion : Proof. Let (u n ) n be a bounded sequence in W 1 0 L N (Ω). We may assume that u n ⇀ u in W 1 0 L N (Ω)-weakly and almost everywhere in Ω. Let c = Max n ||u n − u|| For ε > 0, there exists δ > 0 such that Therefore, we have : if t δ if t > δ then, since |u n − u| * is nonincreasing The right hand side of this inequality tends to zero as n goes to infinity. Hence, for n n ε with n ε large enough sup 0<t<|Ω| |u n − u| * (t) As a corollary of the above theorem, since W 2 L N ∩ W 1 0 L N ֒→ W 1 0 L α exp ֒→ L N , we have: Proof. We use the equivalence of norms ||v|| W 2 L N (Ω)∩H 1 0 ≡||∆v|| L N +||v|| L N and apply Lemma 2.8 with Proof of Theorem 3.1. We first assume that u ∈ V, and T ∈ L ∞ (Ω). Then, the unique solution φ of (23) satisfies We have Let ε > 0 be fixed. There exists c ε > 0 such that (see Corollary 1 of Theorem 3.2). Combining this with relation (52), we have ∀ ε > 0, ∃ c 1 ε > 0 Secondly, we consider T ∈ L N (Ω) and u ∈ V. There exist u k ∈ V such that u k → u strongly in Λ and T k ∈ L ∞ (Ω) with ||T k || L N ||T || L N .
Then from relation (54), the solution φ k of (23) satisfies We choose ε 0 > 0 such that Then φ k remains in a bounded set of W 2 L N (Ω) ∩ H 1 0 (Ω). So it converges to φ weakly in W 2 L N (Ω) ∩ H 1 0 (Ω) and we have and (according to Proposition 6). This gives the results.

Very weak solutions of problem (1) with and without the Dirichlet boundary condition.
We now want to apply all those regularity results to the study of equation (1). We first start with some definitions of the weak solution associated with (1).

Existence and regularity of the very weak solution for a locally integrable potential V 0
We start by considering the existence of very weak solutions of equation (1) with the Dirichlet boundary condition (23) when the potential V is a nonnegative locally integrable function. We can use the definition of very weak solution (see Definition 1.1).

Remark 6.
In section 4.2, we shall discuss the uniqueness of the v.w.s when V / ∈ L N,1 (Ω).
Proof. First, we assume that f 0. Let u j ∈ V be such that u j → u strongly in L p,1 (Ω) N and f j ∈ L ∞ (Ω) such that 0 f j (x) f (x) a.e and f j (x) → f (x) a.e. According to Proposition 4, Proposition 11 or Proposition 12, there exists a unique function ω j 0 such that which is equivalent to saying that We argue as in [20,18,36]. Let E be a measurable subset of Ω and χ E its characteristic function. Then, there exists a non negative function φ j ∈ W 2 L m (Ω), ∀ m < +∞, satisfying We consider a small number ε > 0 such ε sup j || u j || L N,1 1 2 . Therefore, we have By the Hardy-Littlewood property we conclude that Moreover, choosing φ = ϕ 1 as the test function with −∆ϕ 1 = λ 1 ϕ 1 , and ϕ 1 = 0 on ∂Ω, we have for a suitable constant c > 0. Thus V j ω j remains in a bounded set of L 1 (Ω; δ) and If f has a constant sign, we write Denoting by ω + j the v.w.s. associated to f j+ and by ω − j the one associated to f j− ,we see that ω j = ω + j − ω − j satisfies (58) and we have also the estimates (61) and (62). In particular, since We conclude that (ω j ) j converges weak-* to ω in L N ′ ,∞ (Ω) = L N,1 (Ω) * . To obtain a strong convergence, we need a local estimate of the gradient. For that purpose, we shall prove the boundedness of ω j in the Lorentz-Sobolev weighted space W 1 L 1+ 1 N ,∞ (Ω; δ). For this, we shall need the following result due to Philippe Bénilan and co-authors whose proof can be found in [5] Lemma 4.2, with generalization in [40].

Proposition 13.
Let v ∈ L 1 (Ω, δ α ), and α ∈ [ 0, 1 ]. Assume that there exists a constant c 0 > 0 such that for all and Then, there exists a constant c, depending continuously on c 0 > 0, such that for all λ > 0 In particular, if v j is a sequence converging weakly in L 1 (Ω) to a function v, satisfying the inequality (64) then v j converges to v weakly in W 1,q (Ω ′ ) for all q ∈ 1, N + α N + α − 1 and all Ω ′ ⊂⊂ Ω, with a subsequence, v j (x) → v(x) a.e. in Ω.
We first need to prove the following a priori estimate : Let ω j be the solution of (57), ω its weak limit in L N ′ ,∞ (Ω). Under the same assumptions as for Theorem 3.1, there exists a constant c 0 > 0 such that: Proof. Let ϕ 1 be the first eigenvalue of the Dirichlet problem −∆ϕ 1 = λ 1 ϕ 1 in Ω, ϕ 1 = 0 on ∂Ω. Then, there exist constants such that c 1 δ(x) ϕ 1 (x) c 2 δ(x) ∀ x ∈ Ω. We consider the approximate problem given in equation (57) say 1 (Ω) N −strongly and ω j → ω weakly-* in L N ′ ,∞ (Ω). For k > 1, we choose T k (ω j )ϕ 1 as a test function; then V j ω j T k (ω j )ϕ 1 0 and we derive after some integrations by parts : This relation implies: By the Hölder inequality From relation (66) and (67), we then have : Letting j → +∞, we deduce from (68) and Proposition 13 : Then the L N ′ ,∞ -regularity of ω implies

Corollary 2 (of Propositions 13 and 14).
Let ω be as in the proof of the previous proposition. Then, there exists a constant c 6 > 0 such that ||∇ω|| In particular, we have, for all q < 1 + 1 N , To pass to the limit in (57), we argue as in [19] p. 1041. We emphasize the main differences due to the additional term u · ∇ω.
Let us note that by the above Proposition 11, we have (for a subsequence still denoted as (ω j ) j ) that 1. ω j (x) → ω(x) a.e. (and thus V j ω j → V ω a.e. in Ω).
In particular, we deduce from the above statement 1., relation (63) and Fatou's lemma  Proof. Since u j → u in L N,1 (Ω), and a.e. in Ω, we have It is enough to show that ( u j ω j ) j satisfies Vitali's condition : But from Hölder's inequality we have we derive that it satisfies the Vitali condition. Therefore, we have proved the lemma.
Then we have the following result analogous to Lemma 2.3 of [19].
Taking ϕ 1 γ m (ω j ) as a test function in relation (57) we get We As m → +∞, treating the remaining terms in (69) as in [19], we derive |ωj |>t This relation proves that V j ω j δ remains in a bounded set of L 1 (Ω) but also that the set V j |ω j |δ, j ∈ IN is x compact for the σ(L 1 ; L ∞ )-topology, so we may appeal to the Dunford-Pettis to conclude. Indeed, let us set we deduce that for any ε > 0, there exists t ε > 0 such that, for all j ∈ IN, Let Ω 0 ⊂ Ω such that V δ ∈ L 1 (Ω 0 ) (thus Ω 0 = Ω if V is only locally integrable). Then by the Lebesgue convergence dominate theorem for a.e. t, lim j→+∞ Ω0 Therefore there exists η > 0 such that if A ⊂ Ω 0 , |A| η, then for all j ∈ IN, This conclude the proof of Lemma 7.
The passage to the limit, we will distinguish two different cases : (since φ δ ∈ L ∞ (Ω) and V j ω j δ converges to V ωδ for σ(L 1 ; L ∞ ) topology). Therefore, since we let j → +∞ to deduce that ω is a v.w.s. using Lemma 4.2 and the convergences of ω j .

A result of uniqueness of solution when the potential is bounded
from below by c δ −r , r > 2 The purpose of this section is to show the following uniqueness result.

Theorem 4.2.
Assume that V is locally integrable V 0, and such that ∃ c > 0, V (x) cδ(x) −r , in a neighborhood U of the boundary, with r > 2.
Then, the v.w.s. ω found in Theorem 4.1 is unique.
This theorem relies on the following general result which does not require any information about the boundary condition, since the required additional information is written in another way :
As an immediate corollary of the above theorem we have

Corollary 3. of Theorem 4.3
Assume the hypotheses of Theorem 4.3 hold and let f ∈ L 1 loc (Ω). Then there exists at most one function ω ∈ L 1 (Ω; δ −r ) ∩ W 1,1 loc (Ω), r > 1 solution of Lω = f in D ′ (Ω). For the proof of Theorem 4.3, we need the following extension of the Kato's inequality whose proof is similar to the one given in [30] :
Thus, the conclusion 1. will be proved if we show that For this purpose, we may assume that ω ∈ W 1,1 (Ω) with compact support and Lω ∈ L 1 (Ω). Moreover, if ρ j ∈ C ∞ c (IR N ) is a sequence of mollifiers, and ω ⋆ ρ j ∈ C ∞ c (Ω) we have So, it is sufficient to show the inequality number for ω ∈ C ∞ c (Ω). From here, we argue as for the case where L is replaced by the Laplacian operator (see Proposition 1.5.4 p.21 in [30] for more details). We approximate the functions sign + by a sequence of convex, non-decreasing functions h ε such that lim Thus, for all ψ ∈ C ∞ c (Ω), ψ 0, we have where L * ψ = −∆ψ − u · ∇ψ.
Since we have, by Leibniz's formula (c γβ are constant depending only on γ, β) and for γ = 0.

Remark 7.
In Theorem 4.3 and Theorem 4.4, if u ≡ 0 (or u ∈ C 1 (Ω) N ) then we can weaken the conditions on ω reducing it to ω belongs to L 1 (Ω; δ −r ), r > 1. Then the above conclusions hold true.

Remark 8.
In fact, in Corollary 3, we can state that the unique solution of (1) (without any indication of the boundary condition) must satisfy that ω = 0 on ∂Ω at least if ω is differentiable. Indeed, a consequence of Lemma 7 we have

Remark 9.
There is a large amount of works in the literature in which the uniqueness of solutions of suitable elliptic problems is established without indicating any boundary condition but these previous papers deal with degenerate elliptic operators (see, e.g. [3], [4], [21] and the references therein). We point out that the main reason to get this type of results in our case (in which the diffusion operator is the simplest one and is not degenerate) is the presence of a very singular coefficient of the zero order term (the potential V (x)) which is "pathological" since it is more singular on the boundary of the domain than what the Hardy inequality may allow. Let f be in L 1 Ω; δ 1 + |Log δ|
We recall the Proof of Theorem 4.5 (boundedness in L N ′ (Ω)). Let ω be the very weak solution found in Theorem 4.1 and assume that f ∈ L 1 Ω; δ(1 + |Log δ|) We know that there exists a sequence u j ∈ V such that the corresponding sequence (ω j ) j satisfying relation (58) verifies ω j ⇀ ω weak-* in L N ′ ,∞ and that Here u j converges in (L N (Log L) β N ) N = Λ to u strongly where β > N − 1. Let g ∈ L N (Ω)and let φ j be the solution of φ j ∈ W 2 L N (Ω) such that − ∆φ j − u j ∇φ j = g in Ω, φ j = 0 on ∂Ω.
Then according to Theorem 3.1, we have Thus By the Trudinger's type inclusion (see Lemma 3.1) Therefore, considering equation (81), we have with the help of Lemma 4.5 with α = 1 N ′ and estimate (83), this relation gives: Hence sup which shows that : proving the result.
For the case V ≡ 0, we can always obtain the W 1,q (Ω)-regularity, for q 1, provided some integrability on f but also on u. Here is a first result in that direction : Theorem 4.6. Let f be in L 1 (Ω; δ(1 + |Log δ|)), V = 0, and u in bmo r (Ω) N . Then, the very weak solution found in Theorem 4.1 belongs to W 1,1 0 (Ω).
By standard argument, we derive the existence of ω satisfying (92) as a weak limit of ω k in W 1 L p ′ ,q ′ (Ω).
For the explosion of the norm of ω in L N ′ , we can adopt the same proof as for the explosion of the gradient in L 1 (Ω). We have N ′ ) such that the very weak solution ω found in Theorem 4.1 satisfies that ω does not belong to L N ′ (Ω)).

Remark 10.
We can give the more precise information that the function f in Theorem 4.9 is not in L 1 (Ω; δ(1+ |Log δ|) 1 N ′ ) (due to Theorem 4.5).

Some final conclusion
In the opinion of the authors, the results of this paper open many different further applications in different directions. Besides the consideration of the list of concrete problems mentioned in the Introduction other studies can be carried out. For instance, following the arguments of [19], it is not complicated to extend many of the results of this paper to the study of semilinear problems for which equation (1) is replaced by the equation −∆ω + u · ∇ω + V ω + β(x, u, ∇u) = f (x) on Ω, when β is nondecreasing in u. Moreover the consideration of parabolic problems of the type ω t − ∆ω + u · ∇ω + V ω + β(x, u, ∇u) = f (t, x) on Ω × (0, T ), can be carried out with the help of the results of this paper (mainly the L 1 (Ω; δ)-accretiveness property of the associates operator). The details will be given in some separate work by the authors.