Gradient estimates and comparison principlefor some nonlinear elliptic equations

We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. 
We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also 
prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators.


1.
Introduction. Let us consider the class of the homogeneous Dirichlet problems −div (a (x, ∇u)) = H (x, ∇u) + f in Ω u = 0 on ∂Ω, where Ω is a bounded open subset of R N , N ≥ 2. We assume that a : Ω × R N → R N and H : Ω × R N → R are Carathéodory functions which satisfy the ellipticity condition a (x, z) · z ≥ |z| p , the monotonicity condition and the growth conditions |H (x, z)| ≤ h |z| q , h > 0 (5) with 1 < p < N , p − 1 < q ≤ p, for almost every x ∈ R N , for every z, z ∈ R N , and f is in a suitable Lorentz space.
Existence of solutions to problem (1) have been extensively studied in literature. Among them we quote only some contributes and refer to the references therein: [1,4,22,23,29]. Existence results have been proved under suitable assumptions on summability of the datum f and all these results require a smallness condition on its norm. Usually one has to distinguish three intervals for q, i.e.
Depending on these intervals, the notion of solution to problem (1) has to be specified (see Section 2). Indeed u is the standard weak solution when the datum f is an element of the dual space W −1,p (Ω), as for example when q satisfies (8), but this notion does not fit the cases when q satisfies (6) or (7). In these cases we adopt the notion of "solution obtained as limit of approximations" ( [17], see also [18]) which is based on a delicate procedure of passage to the limit. Other equivalent notion of solutions are available in letterature, such as renormalized solution ( [28,30]) or entropy solution ( [10]). The purpose of this article is twofold: we consider solutions to (1) and we firstly study how the summability of the gradient of a solution increases when the summability of f increases. Then we prove comparison principles, and therefore uniqueness results, under more restrictive assumptions on the structure of the operator.
The study of the summability of the gradient of a solution to (1) is faced in Section 3 and the main results are stated in Theorems 3.6 -3.7. We assume that the datum f belongs to Lorentz spaces, whose properties are recalled in Section 2. The main ingredients in proving such results are a pointwise estimate of the gradient of a solution (see Lemma 3.2), a priori estimates proved in [4] and an Hardy-type inequality (see Lemma 2.1).
As far as the uniqueness concerns we consider elliptic operators which satisfy further standard structural conditions. We change the monotonicity condition (3) in the following "strong monotonicity" condition (a (x, z) − a (x, z )) · (z − z ) ≥ α(ε + |z| + |z |) p−2 |z − z | 2 , for some α > 0, with ε nonnegative and strictly positive if p > 2. Moreover we assume the following locally Lipschitz condition on H where β > 0, η is nonnegative and strictly positive if 1 < p ≤ 2.
In Section 4 we prove comparison principles, and in turn uniqueness results, for weak solutions or "solution obtained as limit of approximations" to problem (1) when p − 1 < q ≤ p N + p − 1; the continuous dependence on the data is proved in [15]. Actually classical counter-examples show that uniqueness of weak solutions fails when q > p − 1 + p N (see e.g. [31]. In order to avoid technicalities we assume that the datum f belongs to some Lebesgue space (and not more to the Lorentz spaces as in Section 3). Our main results, Theorems 4.1, 4.3, 4.4, concern the case where 1 < p ≤ 2. The first one gives uniqueness for weak solutions to (1) when q = p − 1 + p N , while Theorems 4.3, 4.4 give uniqueness for "solution obtained as limit of approximations" whose gradient has a suitable summability when q satisfies (7). These results improve the known results in literature (see [31]). Finally in Theorem 4.2 a uniqueness result is stated for "solution obtained as limit of approximations" when q satisfies (6). Actually this result is already proved (for renormalized solution) in [14] (see also [31]). Uniqueness results when p > 2 are contained in [15].

2.
Preliminary results and notion of solutions. In this section we recall a few properties of Lorentz spaces and the notion of solutions for problem (1) which we use in the following.
Let us begin by introducing the definition of Lorentz spaces. If u is a measurable function on Ω we denote by its distribution function and by the decreasing rearrangement of u (see, e.g. [26]). Lorentz spaces are rearrangements invariant spaces. Given r, t ∈ ]0, ∞] , the Lorentz space L(r, t) is the set of all measurable functions on Ω such that u * r,t is finite, where we use the notation if t = ∞. These spaces give in some sense a refinement of the usual Lebesgue spaces. Indeed L(r, r) = L r (Ω) for any r ≥ 1 and L(r, ∞) is the Marcinkiewicz space L r -weak. Moreover the following embeddings hold (see [25])) and In general · * r,t is not a norm. However it leads to a topology in L(r, t) as follows We can introduce a metric in the spaces L(r, t) in the following way (see [25]). If τ ∈]0, 1] with τ < r and τ ≤ t, we set Then By Hardy inequality (see, for example, [11], Chapter 2, Proposition 3.6), it results So the convergence induced by the distance d is the same as (15). Finally these metric spaces are complete and when τ = 1 they are Banach spaces. We explicitely remark that our discussion about Lorentz spaces has mainly the goal to give a meaning to the notion of convergence in L(r, t).
We finally recall an Hardy-type inequality proved in [5] .
where c is a positive constant depending only on θ, γ and δ.
Now let us explain what we mean for solution to problem (1). If the datum f belongs to the dual space W −1,p (Ω), a solution to problem (1) is a standard weak solution. A function u ∈ W 1,p 0 (Ω) is a weak solution to (1) if The definition of weak solution does not fit the case when p − 1 < q < p − 1 + p N , since in general the right-hand side of (1) is not more an element of the dual space W −1,p (Ω). A different notion of solution has to be adopted and we refer to solutions obtained as a limit of weak solutions to approximated problems whose data are smooth enough, the so-called "solution obtained as limit of approximations" .
Here T n denotes the usual truncation at level n > 0, defined, for a given n > 0, as We explicitely remark that the existence of a weak solution to (19) is assured by classical results (see [27]). Moreover the gradient ∇u is the generalized gradient of u defined according to Lemma 2.1 in [10] which states the existence of a measurable function v : Ω → R N such that in Ω, for every k > 0.
We define the gradient ∇u as this function v. The gradient defined in (23) is not the gradient used in the definition of Sobolev space, since it is possible that the function u does not belong to L 1 loc (Ω) or v does not belong to L 1 loc (Ω) N . However, if v belongs to L 1 loc (Ω) N , then u belongs to W 1,1 loc (Ω) and v is the distributional gradient of u (see [19]).
The same definition has a meaning if the datum f belongs to a Lorentz space L(m, k) with k < ∞, when we substitute the convergence in (18) with the convergence in Lorentz space recalled above. If f belongs to Marcinkiewicz space L(m, ∞), the definition of "solution obtained as limit of approximations" is adapted by taking into account approximated source terms f n ∈ L ∞ (Ω) ∩ W −1,p (Ω); one can take for example f n = T n (f ).

3.
A priori estimates for the gradient. The main results of this section is stated in Theorems 3.6 and 3.7 below. Their proofs are obtained in various steps. The first step consists in proving Lemma 3.2 below. It gives a pointwise estimate for the decreasing rearrangement of |∇u n |, the gradient of a weak solution u n to the approximated problem (19). Then we prove Propositions 3.3 -3.5 below from which one can deduce the apriori estimates for |∇u n | in certain Lorentz spaces depending on the summability of the datum f . Finally, since ∇u n converge to ∇u according to (21), we obtain the corresponding estimate for |∇u| by passing to the limit in a standard way.
The summability in Lebesgue spaces of the gradient of a solution to (1) have been studied in [23]. Theorems 3.6 and 3.7 give results proved in [5] when the lower order term H does not appear.
We begin by recalling the following result proved in [21, Lemma 4.1].
Lemma 3.1. Let us suppose that (2)-(5) hold true with 1 < p < N and Let u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) be a weak solution to problem (1) with f ∈ L ∞ (Ω). We have, a.e. in (0, |Ω|), Now we prove a pointwise estimate for the gradient of weak solutions to problem (1) with bounded data. This pointwise estimate is not known in literature; the novelty is due to the presence of the first order term H. Our result overlaps with the result in [5] when H = 0; further estimates for gradient when H = 0 with Neumann or Dirichlet boundary conditions are proved in [3] and [16] respectively (cf. [20] for local inequality in terms of nonlinear potentials).
Our first result concerns the case where p − 1 < q < N (p−1) N −1 . In the following we denote by t * , for 1 < t < N , the Sobolev exponent of t, i. where (36) Then, for any m, k such that where C is a positive constant depending on N, p, q, h, |Ω|, m, k and on f m,k .
Moreover C depends on f m,k in such a way that it is bounded when f varies in sets which are bounded and equi-integrable in L(m, k  (35), then the following pointwise estimate for (−u * (s)) holds true for every s ∈ (0, |Ω|]. Finally observe that the first index of summability of |∇u| in (37) can be also less than 1. Indeed m * (p − 1) < 1 when 1 ≤ m < N N p−N +1 and 1 < p ≤ 2 − 1 N . Proof. In order to prove (37) we apply Lemma 3.2. We begin by evaluating ψ(r, σ) and we prove that where ψ is defined by (26).
Here and in the following c will denote a positive constant which can vary from line to line and depends only on the data of the problem but not on f m,k .
To this aim we use the pointwise estimate (38). Since q < N (p−1) which implies (39). By using (25) and (39), we obtain Now we proceed by distinguishing the case where 0 < k < ∞ and the case where k = ∞.
Remark 2. If m = k = 1, by Lemma 3.2 and (39), it is easy to verify that This a priori estimate overlaps with the well-known a priori estimate for |∇u| proved in [4] and in [23]. Moreover, observe that, since we assume m > 1, it results m * (p − 1) > N (p−1) N −1 , i.e. (37) says that the summability of the gradient increases when the summability of f increases.
Our second result concerns the case where N (p−1) Let u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) be a weak solution to (1) with f ∈ L ∞ (Ω) such that with Then, for any m, k such that we have where C is a positive constant depending on N, p, q, h, m, k, |Ω| and f m,k .
Moreover C depends on f m,k in such a way that it is bounded when f varies in sets which are bounded and equi-integrable in L(m, k).

Remark 3.
In order to prove Proposition 3.4 we need to apply a pointwise estimate proved in [4,Theorem 4.2] which says that, if q satisfies (42) and f ∈ L ∞ (Ω) satisfies the sharp smallness assumption (43), the following pointwise estimate for (−u * (s)) holds true where Y 0 ≥ 0 is the smallest nonnegative solution to the equation Proof. In order to prove (46) we apply Lemma 3.2. We begin by evaluating ψ(r, σ).
Since f ∈ L ∞ (Ω) and we assume (43), the pointwise estimate (47) holds true (see Remark 3 above). Therefore, we have where Now we proceed by distinguishing the case where 0 < k < ∞ and the case where k = ∞.
The case 0 < k < ∞. We prove that under these bounds on k, we have where ψ is defined by (26).
The case k = ∞. If f ∈ L(m, ∞), by (49) and (16), we get Therefore by using Lemma 3.2, we deduce This completes the proof. , k) small enough we get also an a priori estimate of |∇u| in L(t, k(p − 1)) with t < N (q − p + 1).

Our third result concerns the limit case
Let u ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) be a weak solution to (1) with f ∈ L ∞ (Ω) such that for a constant M > |Ω|, satisfies with Then for any m, k such that we have where C is a positive constant depending on N, p, q, h, m, k, |Ω| and f m,k .
Moreover C depends on f m,k in such a way that it is bounded when f varies in sets which are bounded and equi-integrable in L(m, k).

Remark 5.
In order to prove Proposition 3.5 we need to apply a pointwise estimate proved in [4,Theorem 4.3] which says that, if q satisfies (57) and f ∈ L ∞ (Ω) satisfies (58), then the following pointwise estimate for (−u * (s)) holds true where X 0 is the smallest positive solution to the equation h Proof. As in the previous proofs we get the assert by evaluating ψ(r, σ). Since f ∈ L ∞ (Ω) and it satisfies (58), the pointwise estimate (62) holds true (see Remark 5 above). Therefore we have where, by (63), Now we proceed by distinguishing the case where 0 < k < ∞ and the case where k = ∞.
The case 0 < k < ∞. We prove that under these bounds on k, we have where ψ is defined by (26). Let us begin by assuming k > 1. By Hölder inequality, inequalities (16) and estimate (64), we get Moreover it is easy to verify that the following estimate holds Combining (66) and (67), we get which coincides with (53) of the proof of Proposition 3.4. Therefore (54) holds true for 0 < k < ∞ and the conclusion follows as in the previous proof.
The case k = ∞. If f ∈ L(m, ∞), by (64) and (16), we get Therefore by Lemma 3.2 we deduce This completes the proof. The main results of this section are stated in Theorems 3.6 -3.7 which say as the summability of the gradient of a "solution obtained as limit of approximations" to problem (1) varies with the summability of the datum.

Moreover assume that either
. Let u be a "solution obtained as limit of approximations" to (1). Then where C is a positive constant depending on N, p, q, h, m, k, |Ω| and f m,k .
Moreover C depends on f m,k in such a way that it is bounded when f varies in sets which are bounded and equi-integrable in L(m, k).
Moreover assume that f ∈ L(m, k) with and where K 3 is defined by (59). Let u be a "solution obtained as limit of approximations" to (1). Then (69) holds true.
Proof of Theorem 3.6. Assume that (i) holds true, for k < ∞. Since u is a "solution obtained as limit of approximations" , we can consider a sequence (u n ) n of weak solutions to the approximated problem (19) with f n ∈ C ∞ 0 (Ω) such that f n → f in L(m, k) , and |∇u n | → |∇u| a.e. in Ω, according to the definition given in Section 2. Since L(m, k) ⊂ L 1 (Ω) for m > 1, we get f n → f in L 1 (Ω), then f n 1 → f 1 . Therefore, since f satisfies (35), for n large enough, we have f n L 1 < K 1 . Therefore by Proposition 3.3, we obtain ∇u n m * (p−1),k(p−1) ≤ C , where C depends on f n m,k in such a way that it is bounded when f n vary in sets which are bounded and equi-integrable in L(m, k). Since which, together with (72) and the fact that f n m,k ≤ c f m,k , gives (69) for 0 < k < ∞. Finally we obtain (69) for k = ∞ in analogous way. Assume now that (ii) holds for k < ∞. Since u is a "solution obtained as limit of approximations" , we can consider a sequence (u n ) n of weak solutions to the approximated problem (19) Therefore, since f satisfies (43), for n large enough, we have f n N (q−p+1) q ,∞ < K 2 . Therefore by Proposition 3.4, we obtain ∇u n m * (p−1),k(p−1) ≤ C , where C depends on f n m,k in such a way that it is bounded when f n vary in sets which are bounded and equi-integrable in L(m, k). Then we get the conclusion arguing as in the previous case.
Proof of Theorem 3.7. Assume that k < ∞. As in the previous proof, we can consider a sequence (u n ) n of weak solutions to the approximated problem (19) with f n → f in L(m, k) , and |∇u n | → |∇u| a.e. in Ω.
Then f n m,k → f m,k and, since f satisfies (70), for n large enough, we have Therefore by Proposition 3.5, we obtain ∇u n m * (p−1),k(p−1) ≤ C , where C depends on f n m,k in such a way that it is bounded when f n vary in sets which are bounded and equi-integrable in L(m, k). The conclusion follows arguing as in the previous proof.
Remark 8. Let us make some remarks on the bounds of q and m. Observe that in Proposition 3.4 and in Theorem 3.6 we assume q < p − 1 + p N . This is due to the fact that when p − 1 + p N ≤ q ≤ p, weak solutions to problem (1) have to be considered (see Section 2) and therefore it is not natural to ask larger summability of the gradient.
As far as the bounds on m concerns, if m = (p * ) , i.e. f ∈ L((p * ) , k), Propositions 3.3 -3.5, and therefore Thorems 3.6 -3.7, can not be proved since we can not apply Lemma 2.1. However if we assume that f ∈ L((p * ) , p ), under the assumptions of Propositions 3.3 -3.5, one can prove that |∇u| ∈ L p (Ω) by using (25) and the fact ψ(r, σ) ≤ c. Moreover for m = (p * ) , Propositions 3.3 -3.5, and therefore Thorems 3.6 -3.7, are not sharp since we can not obtain gradient estimates depending on k. Indeed if we assume f ∈ L((p * ) , k) with 0 < k ≤ p is just possible to prove that |∇u| ∈ L p (Ω) (and not in L(p, k(p − 1)) as in the case m < (p * ) ).
Finally when m = max 1, N (q−p+1) q , by Remarks 2, 4 e 6 we can prove estimates of |∇u| which overlaps with the estimates proved in [4] for the existence results.
Remark 9. A remark on the summability of u is in order. Under the assumptions of Theorem 3.6, we have When N (p−1) N −1 ≤ q < p this is an easy consequence of (54), Hardy inequality (see, for example, [11], Chapter 2, Proposition 3.6) and integration on (s, |Ω|). When q < N (p−1) N −1 , it is easy to prove firstly the analogous of (54) and then to conclude by integration on (s, |Ω|). Analogous estimates for u have been proved in [23] for data in Lebesgue spaces.
We explicitly observe that Theorems 3.6 -3.7 and this Remark hold true also for operators a which depends on u and satisfy usual growth conditions on u.

Comparison principles.
In this section we present comparison principles which in turn imply uniqueness of a solution to problem (1), under the assumptions (9) and (10). In order to avoid technicalities we assume that the datum f belongs to Lebesgue spaces (and not more to Lorentz spaces). In this section we assume Indeed, as pointed out in Introduction a well-known example shows that uniqueness of a weak solution to (1) does not hold if q > p − 1 + p N . Uniqueness results for weak solutions when q = p − 1 are well-known (cf. [2,8,12]), while the uniqueness of renormalized solution or "solution obtained as limit of approximations" are proved, for example, in [6,7,9,13,14,24,31]. In the following we distinguish the comparison principle for weak solutions from comparison principle for "solution obtained as limit of approximations" .

4.1.
Comparison principle for weak solutions. As pointed out we have to consider only the case where q = p − 1 + p N .
Moreover the sharp smallness assumptions on the data (82) are made just to assure the existence of a weak solution. Our result improves the uniqueness result proved in [31], since we have uniqueness of weak solutions for a larger interval of values of p.
Proof. Let us denote Assume that D has positive measure. Let us fix t ∈ [0, sup w[ . We denote Using w t as test function in (79) we obtain By assumptions (9) and (10), we have Let us estimate the integral on the right-hand side of (84) by using Hölder inequality and Sobolev inequality. Since p ≥ 2N N +2 , we obtain where C p is the best constant in Sobolev embedding W 1,p 0 (Ω) ⊂ L p * (Ω). From (84) and (85) we get On the other hand, since p ≤ 2, if N ≥ 3 or p < 2 if N = 2, Hölder inequality gives Then, by using (86) we obtain .
Letting t → sup w the left-hand side goes to zero; this gives a contradiction. Therefore we conclude that |D| = 0 and we get the assert.
Remark 11. We explicitly observe that we can not prove a comparison result when p > 2 and q satisfies (81) since our approach would require a larger summability of |∇u| which is not natural for a weak solution u. The same occurs also in [31].

4.2.
Comparison principle for "solution obtained as limit of approximations" . A comparison principle, and uniqueness result for (1), holds true also for the range of q p − 1 < q < p − 1 + p N .
In this case by comparison principle we mean that if u, v are "solution obtained as limit of approximations" to the following Dirichlet problems respectively with Q defined in (74), f, g ∈ L r (Ω), r ≥ 1 and When q belongs to the interval a comparison principle is proved in [14] (see also [31]) for renormalized solution (which are equivalent to "solution obtained as limit of approximations" ). Here we just recall its statement by expliciting the sharp assumptions which give existence.

Assume now
We make assumptions on f which assure both existence of a "solution obtained as limit of approximations" and a larger summability of its gradient, (cfr. Theorems 3.6-3.7). We begin with the case q > N (p−1) N −1 and we prove the following comparison principle.
Remark 12. We explicitly observe that our result improves the uniqueness result proved in [31] where a comparison principle for renormalized solutions which satisfy a further regularity condition is proved for the smaller interval of values of p, that is 2N N +1 < p ≤ 2.
Assume that D has positive measure. Let us fix t ∈ [0, sup w[. Denote Since u, v are "solution obtained as limit of approximations" to problems (87), and (88) respectively, and since assumptions of Theorem 3.6 are satisfied, we have |∇u|, |∇v| ∈ L m * (p−1) (Ω). Moreover by definition recalled in Section 2, two sequences of functions f n , g n ∈ C ∞ 0 (Ω) exist such that f n → f strongly in L m (Ω) and g n → g strongly in L m (Ω) and sequences of weak solutions u n , v n ∈ W 1,p 0 (Ω) ∩ L ∞ (Ω) to the approximated problems Q n u n ≡ −div (a (x, ∇u n )) − T n (H (x, ∇u n )) = f n in Ω u n = 0 on ∂Ω, Q n v n ≡ −div (a (x, ∇v n )) − T n (H (x, ∇v n )) = g n in Ω v n = 0 on ∂Ω, respectively, exist such that u n → u a.e. in Ω and v n → v a.e. in Ω .
and ∇u n → ∇u a.e. in Ω and ∇v n → ∇v a.e. in Ω .
For n large enough D n has positive measure. Consider the function w n,t = w n − t if w n > t 0 otherwise and denote E n,t = {x ∈ D n : w n (x) > t} .
We begin by estimating the integral on the left-hand side of (102). To this aim let us observe that, by using Sobolev embedding theorem for the function [(w t + ϑ) where C s is the best constant in Sobolev embedding W 1,s 0 (Ω) ⊂ L s * (Ω).