Morrey estimates for a class of elliptic equations with drift term

where Ω is a bounded open subset of R , with N > 2, M : Ω → R 2 is a matrix with measurable bounded entries M ij (x) satisfying the standard ellipticity condition, E(x) and f (x) are respectively a vector eld and function both belonging to suitable Morrey spaces to be speci ed later on. The study of the above problem goes back to the papers [53, 54] by G. Stampacchia and it presents a di culty due to thenoncoercivity of thedi erential operator u → −div[M(x)∇u − E(x)u]. In thequotedpapers the existence and uniqueness of a weak solution u ∈ W1,2 0 (Ω) have been proved assuming that


Introduction
This paper is devoted to the study of the regularity of a weak solution u of the following homogeneous Dirichlet problem −div[M(x)∇u − E(x)u] = f (x) in Ω u = on ∂Ω, where Ω is a bounded open subset of R N , with N > , M : Ω → R N is a matrix with measurable bounded entries M ij (x) satisfying the standard ellipticity condition, E(x) and f (x) are respectively a vector eld and function both belonging to suitable Morrey spaces to be speci ed later on.
The study of the above problem goes back to the papers [53,54] by G. Stampacchia and it presents a di culty due to the noncoercivity of the di erential operator u → −div[M(x)∇u − E(x)u]. In the quoted papers the existence and uniqueness of a weak solution u ∈ W , (Ω) have been proved assuming that Later, using a nonlinear approach and exploiting techniques issued from those of G. Stampacchia, L. Boccardo in [3] retrieved the previous results without the smallness condition in (1.2). The aforementioned results are stated in a global fashion and they have been obtained when the data belong to usual Lebesgue spaces. On the other hand, G. Stampacchia in [54] also studies local properties of the solution of the problem (1.1). In particular, under the hypotheses stated in (ii), he proves that a W ,solution of the problem (1.1) is locally bounded and locally Hölder continuous (¹).
So that, naturally it raises the question of studying the problem in the setting of Morrey spaces, in order to try to obtain results similar to those obtained by S. Campanato in [6], for example. As far as we know the only available results in this framework are contained in the paper [21], where Hölder continuity and a Morrey estimate of a weak solution u have been proved. The technique used is essentially based on the representation formula of the solution but it doesn't allow to obtain any Morrey estimate of the gradient of the solution.
In this paper we suppose that the right-hand side f and the vector eld E belong to suitable Morrey spaces and we recover the gradient estimate of a solution u in the corresponding Morrey space, so to retrieve the regularity theory at the "gradient level" as in Campanato's work.
Moreover, we also weaken the assumptions on the data when dealing with the existence of bounded weak solutions.
Moreover, if f has an higher summability exponent and a lower Morrey exponent, that is then any weak solution u of problem (1.1) satis es |Du| ∈ L ,λ (Ω) (see Theorem 3.7 below). At last, Morrey estimate obtained for |Du| allows us to extend to the problem (1.1) the Calderon-Zygmund theory introduced in the paper [50] by G. Mingione (see also [2]).

Main notations, functions spaces and auxiliary lemmas
In this section, for reader's convenience, we recall some useful properties of functions spaces and some lemmas that we are going to exploit.
In the sequel, Ω is a bounded open subset of R N , N > , with a su ciently smooth boundary ∂Ω. 1 Further local properties of solutions depending on local properties of the data are object of study in the forthcoming paper [11].
We will then need the following technical lemma.  for every u ∈ W ,p (Ω).
Next lemma concerns the product of two functions belonging to Morrey spaces. for some C > independent of u and E.
and moreover there exists a positive constant C depending on N, ν such that .

Statement of the main results
Let M : Ω → R N be a matrix with measurable entries M ij such that We will initially consider the problem (1.1) under the following assumptions on tha data f ∈ L (Ω), |E| ∈ L (Ω).
Our rst result concerns the existence and uniqueness of a bounded solution of the problem (1.1) with data E and f belonging to Morrey spaces having a large Morrey exponent but a lower summability exponent. with corresponding norms estimates Du L ,λ (Ω) ≤ c , (3.8) [u] where c , c are two positive constants depending on c and u W , (Ω) .
It is worth noticing that, with respect to the results in [3,20,21], the boundedness of the solution u and its Hölder's continuity have been obtained under weaker assumptions on data (see Remark 2.2). Namely, in [3] Theorem 5.6 or in [20] while in [21] Theorem 5.2, Hölder continuity of u is achieved under the hypotheses We stress that, for some constants c , c > , one has so that our lower order term lies in the framework of "controlli limite" as described in Campanato's book [6] pages 122 and 125 (see, in particular, Osservazione 4.I).
For this problem, also in the nonlinear setting, it is proven an L p -estimate for |Du| under the stronger assumption |E| N/ ∈ L p (Ω), p > (see Theorem 4.III, pag. 125 in [6]).  [6], where it assumed that Indeed, by embedding properties of Morrey spaces (see Remark 2.2) one has Consequently our result improves the previous one, at least in the case when λ > N − as stated in the following Then there exists a unique weak solution u of the problem (1.1) satisfying (3.8) and (3.9).
If we assume and we cannot expect any bounded solution u of the problem (1.1). However, a regularity result similar to Campanato's one (see [6] pag. 91) can be proved for weak solutions of problem (1.1). We point out that, in this case, since the datum f has higher integrability (i.e. equal to the duality exponent N N+ ) by a weak solution of problem (1.1) we mean a function u such that (3.11) T 3.7. Assume that hypotheses (3.4), (3.10) and hold and let u be a weak solution of the problem (1.1). Then, We point out that the existence and uniqueness of a weak solution of the problem (1.1) is ensured by the additional assumption |E| ∈ L N (Ω) (see [20], [3] and [4]). We recall that, being N ≥ , in general it is not true that L ,µ (Ω) → L N (Ω). R 3.9. The above theorem improves Theorem 5.1 from [21] where it is proved that u ∈ L p,λ (Ω) for any p ∈ [ , λ [. Moreover our result provides as well information on the gradient of the solution. Finally we state a Theorem on the fractional di erentiability of Du. For the sake of brevity we will focus only on the case of lower Morrey exponent. The same calculations can be repeated also in the previous case λ > N − (see [50], Theorem 1.10 for the case |E| = ). T 3.11. Assume that hypotheses (3.4), (3.10), (3.12) and Finally we notice that, at least formally, if λ = then no fractional di erentiability property of Du seems to be achievable.

Proof of Theorem 3.2.
The proof will be performed in several steps.
Step 1 (Global boundedness) For every n ∈ N and for a.e. x ∈ Ω, let us introduce the bounded functions .
We consider the following approximating problems  for every φ ∈ W , (Ω).
Moreover, due to the boundedness of the functions En(x) un + n |un| and fn, every un is bounded (see [53], [54]).
Next, we prove that the sequence {un} is uniformly bounded in L ∞ (Ω). Let (see [3]) and take φ = ψ(un) as test function in the weak formulation (4.18). Using Young's inequality and taking into account that |ψ(un)| ≤ , we have Due to Lemma 2.5, applied with σ = λ and q = λ N − note that q > since λ > N − , and by virtue of (4.20), (4.21) we deduce the inequality where A(k) = x ∈ Ω : vn(x) > k , which in turn implies for every h > k.
Thanks to the inequality (4.22) and a well-known Stampacchia's Lemma we obtain that log( + |un(x)|) ≤ L a.e. in Ω, with L independent of n, and so where the constant M is independent of n. Now, choosing φ = un in (4.18) and taking into account (3.4), we deduce where M is a positive constant independent of n ∈ N.
Step 2 (Local Morrey regularity) In a xed ball B R ⊂⊂ Ω we write and consequently . Now we choose wn as test function in the weak formulation of the problem (4.26) and we use Young's inequality, (3.4) and the boundedness of un and wn we obtain On the other hand, it is well known that vn satis es the so-called Saint Venaint's principle (see [6] pag. 91, Theorem 8.IV), that is, there exist two constants c = c (α, β, N) > and γ = γ α β , N ∈] , [ such that From (4.29) and (4.30) we deduce Finally, we remark that the uniqueness of u is a consequence of Theorem 6.1 of [3], where only the assumption |E| ∈ L (Ω) was used.

Step 3 (Boundary Morrey regularity)
Now we prove the regularity of |Du| up to the boundary of Ω. For this purpose, following the idea of G. M. Troianiello [55] we will deduce boundary and then global regularity from the previous interior result through an extension technique and successive standard " attening and covering" arguments.
This technique is illustrated in Lemma 2.18 and in Theorem 2.19 of the cited book. We will reproduce here the main steps for reader's convenience.
We denote a vector of R N by x = (x , · · · , x N− , x N ) ≡ (x , x N ). If y = (y , ) we de ne Fixed R > , let M be a simmetrix matrix with bounded coe cients M ij , E be a vector eld and f be a function de ned in Ω = B + R (y). We begin by investigating a solution of the problem (4.34) We state the following Proof. We extend the functions M ij (x), E i (x), f and u a.e. to B R (y) by setting ij , E and f satisfy the assumption (3.4), (3.7) and (3.6) in B R (y), respectively and u ∈ W , B R (y) .
Fixed a function v ∈ C B R (y) , we note that v −ṽ ∈ C B + R (y) . Therefore, simple calculations show that and by density argument function u is solution of the problem (1.1) in Ω = B R (y) with M , E and f replaced by M, E and f . Therefore u veri es (4.33) and the Lemma follows by changing back the coordinates.
Step 4 (Global Morrey regularity) Now we can prove the global Morrey regularity.
We extend Du a.e. to B R setting Du (x , As a consequence, since for some r > Br(ȳ) ∩ Ω ⊂ Λ − (B + R ), the vector-function Du (Λ(y)), y ∈ Br(ȳ) belongs to L ,λ (Br(ȳ)) that is, by the chain rule, Du belongs to L ,λ (Ω ∩ Br(ȳ)); moreover, by virtue of (4.37) and (4.35), Let u ∈ W , (Ω) be a weak solution of the problem (1.1) in the weak sense (3.11). As we have done in the proof of the previous theorem we x B R ⊂⊂ Ω and we set set and v ∈ W , (B R ) are respectively solutions of the following boundary problems . (4.42) Choosing w as test function in the weak formulation of the problem (4.41) and using hypothesis (3.4), by standard calculations, we obtain where S is the Sobolev's constant and σ is a positive constant chosen su ciently small in order to get As in (4.30) the solution v of the problem (4.27) satis es the Saint Venaint's principle; therefore, by (4.44), we deduce for every < ρ ≤ R We point out that u ∈ L * (Ω) ⊂ L , (Ω). Consequently, by virtue of Lemma 2.6 Eu ∈ L ,µ (Ω) with µ = µ − N + and from (4.45) we obtain where µ = min µ , λ < N − . for some c > independent of u and E Using in (4.45) the improved norm estimate (4.50) of |Eu|, we have where c > is a constant independent of u, E and f .
Iterating the previous procedure and setting for every k = , , . . .
it follows i) E u ∈ L ,µ +µ k (Ω), with the norm estimate for some constant C k > independent of u and E; ii) for some constant c > independent of u, E and f .
We will proceed as in the proof of Theorem 4 of [16]. Let B ⊂⊂ Ω be a ball of radius R and letB be the enlarged ball of radius R. We shall denote by Q inn (B) and Q out (B) the largest and the smallest cubes, concentric to B and with sides parallel to the coordinate axes, contained in B and containing B respectively. If we put Let Ω and Ω be a couple of open subset such that Ω ⊂⊂ Ω ⊂⊂ Ω and x ∈ Ω . For any τ ∈] , [ (that will be chosen later) we x h ∈ R with < |h| << min , d(Ω , ∂Ω ) such that, denoted with B = B(x , |h| τ ) the ball centered in x and with radius |h| τ , the outer cube of B,Q out is included in Ω .  The rst term on the right-hand side of (4.57) can be estimated as for all z ∈ R n (see [50]). The third term on the right-hand side of (4.57) can be estimated as |Du| dx (4.59) (see [50]). Finally, we estimate From this point on, we gather together inequalities (4.57), (4.58), (4.59) and (4.62) and we can argue as in the proof of Theorem 4 in [16], exploiting the method introduced in [50].