Failure of the Hopf-Oleinik lemma for a linear elliptic problem with singular convection of non-negative divergence

In this paper we study existence, uniqueness, and integrability of solutions to the Dirichlet problem $-\mathrm{div}( M(x) \nabla u ) = -\mathrm{div} (E(x) u) + f$ in a bounded domain of $\mathbb R^N$ with $N \ge 3$. We are particularly interested in singular $E$ with $\mathrm{div} E \ge 0$. We start by recalling known existence results when $|E| \in L^N$ that do not rely on the sign of $\mathrm{div} E $. Then, under the assumption that $\mathrm{div} E \ge 0$ distributionally, we extend the existence theory to $|E| \in L^2$. For the uniqueness, we prove a comparison principle in this setting. Lastly, we discuss the particular cases of $E$ singular at one point as $Ax /|x|^2$, or towards the boundary as $\mathrm{div} E \sim \mathrm{dist}(x, \partial \Omega)^{-2-\alpha}$. In these cases the singularity of $E$ leads to $u$ vanishing to a certain order. In particular, this shows that the Hopf-Oleinik lemma, i.e. $\partial u / \partial n<0$, fails in the presence of such singular drift terms $E$.


Introduction
It is well known that many relevant applications lead to the presence of a convection term in the correspondent model which, in its simplest formulation, leads to a boundary value problem for linear elliptic second order equation of the following type: − div(M (x)∇u) = − div(uE(x)) + f (x) in Ω u = 0 on ∂Ω.
Here Ω ⊂ R N , N ≥ 3, is an open, bounded set, and we assume that M ∈ L ∞ (Ω) N ×N is elliptic ∀ξ ∈ R N and a.e.x ∈ Ω.
In the mentioned references it assumed that the convection term is regular (for instance E ∈ W 1,∞ (Ω)) and that it satisfies an additional condition which helps to have a maximum principle: div E ≥ 0 a.e. on Ω. ( More recently, some effort has been devoted to get an existence and regularity theory under more general conditions on the convection term E by different authors (see, e.g.[1], [5] and their references).For instance, solutions in the energy space can be considered under the conditions |E| ∈ L N (Ω) and f ∈ L 2N N +2 (Ω).In [13] and [12] the authors study the case in which |E| ∈ L N (Ω) and div E = 0 in Ω and E • n = 0 on ∂Ω, f ∈ L 1 (Ω, δ).See also [15,20].
In this paper, we will show that (2) makes div E behave like a non-negative potential in the Schrödinger case, and we can apply techniques from that setting.See, for example, [12,13,14,17].We focus on the case where (2) holds in distributional sense.
The paper is structured as follows.First, in Section 2 we review known results for the case |E| ∈ L N and f ∈ L 2N N +2 (Ω) which were published in [1], were shown there is a unique weak solution of (1) that can be constructed by approximation.In Section 3 we show that if |E| ∈ L 2 (Ω), div E ≥ 0, and f ∈ L m (Ω) for some m > 1 then the same approximation procedure converges to a weak solution of (1), and we give some a priori bounds for this solution.In Section 4 we show that, if we also assume f ∈ L 2N N +2 (Ω), then this constructed solution is the unique weak solution of (1).
Then we move to discussing interesting examples that fall in this setting.In Section 5 we focus on the case which is somehow in the limit of theory since it is not in In [5] the authors examined the more general class The authors show existence of solutions u, where the summability is reduced as |A| is increased.
Their results indicate that the sign of A should play a role, but the application of Hardy's inequality (which they use in a crucial way) is not able to detect this fact.In Theorem 16 we show that if N > 1, f ∈ L m (Ω) for suitable m, and A > 0 then we can use the sign of div E to deduce that the solution u A of ( 1) with By the contrary, when A < 0 we cannot improve the result in [5].Notice that this is similar to the equation Lastly, in Section 6, we discuss the case where E is suitably singular only on the boundary.We present an example showing that if div E behaves like d(x, ∂Ω) −2−γ for some γ > 0 and f is bounded, then the solutions are flat on the boundary, i.e.
In particular, this shows that the Hopf-Oleinik lemma, i.e. ∂u ∂n < 0 on ∂Ω, fails in the presence of such singular drift terms E. Our example can be easily extended to a more general class of E, as we comment in Section 7. Again, we use the fact that div E acts as a potential.However, in the Schrödinger equation it is sufficient that V (x) ≥ Cδ −2 to get flat solutions, whereas for E we need a strictly larger exponent (see Remark 22).Questions of this type are quite relevant in the framework of linear Schrödinger equations associated to singular potential since they can be understood as complements to the Heisenberg Incertitude Principle (see, e.g.[10,11,12,13,18,14]).
We conclude with some further comments and open problems in Section 7.

Known results when |E| ∈ L N
We define the Sobolev conjugate exponent We have that m Notice that, since m ≥ 1 we have m * ≥ m.In order to compute explicit a priori estimates, we use the Sobolev embedding constant S p such that, for 1 < p < +∞ S p u L p * (Ω) ≤ ∇u L p (Ω) . ( We point out the relevance of the constants, for N > 2 of (2 * ) ′ = 2N N +2 .This constant depends only of N .Since we are going to require the Sobolev embedding for p = 2, we assume that N ≥ 3.In [1] the author proves the following existence theorem with a priori estimates.
Then, there exists a unique weak solution u of (1) in the sense that and it satisfies: 1. Logarithmic estimate:

Stampacchia-type summability: For
4. Stampacchia-type boudedness: Let r > N and m > N 2 .There exists C such that Remark 2. The natural theory for this problem in energy space is precisely |E| ∈ L N (Ω), since in the weak formulation we need to justify a term of the form Eu∇v, where u, v ∈ W 1,2 0 (Ω).This means that u ∈ L 2 * whereas ∇v ∈ L 2 .So we always have that uE ∈ L 2 (Ω).
In [1] the main tool to study the linear problem (1) are the auxiliary non-linear Dirichlet problems where the author take f n = T n (f ) a truncation of f through the family and n |E| .We will take advantage of a similar approximation.Remark 3. Since the problem is linear, for t ∈ R we have that tu is solution of and that E does not change.Thus, using (6) .
Dividing by t −2 and taking the limit as t → ∞ gives Notice that in Theorem 11 we will prove this fact for the case div E ≥ 0.
3 Existence theory when |E| ∈ L 2 and div E ≥ 0 The structural assumption in this section is the following: Theorem 4. Assume (11) and and let p = min{2, m * }.Then, there exists a weak solution u of (1) in the sense that Remark 5. Due to the gradient estimates, we can extend ( 13) to all v ∈ W 1,q 0 (Ω) by approximation, where q = p ′ .
Since the construction of solutions in the proof of Theorem 4 is achieved by approximation, we have that Corollary 6.The solutions constructed in Theorem 4 satisfy (7) and (8).
We say that u n is a weak solution of (9 The existence of a weak solution if N is a consequence of the Schauder theorem.The proof of Theorem 4 is based on the following approximation lemma Lemma 7. Let u n be any weak solution of ( 15) with E n = E, (11), (12), and Then, for any weak solution u n of (15) we have that where C m does not depend on E.
Hence, up to a subsequence, {u n } converges weakly in L m * * .
Proof.Our proof is the same of [4], since we will see that the contribution of new term on E is a negative number.We use 2 * ; we repeat it is possible since every T k (u n ) has exponential summability.Note that 2γ − 1 > 0 since m > 1.Thus, we have To study the second integral, we define the function It is easy to check that H γ (s) ≥ 0 for all s ∈ R. Thus, using the sign condition on div E we have that Hence, we have that which is the starting point of [4], and we get the estimates Letting k → ∞ we recover (16).
With this lemma, we can pass to the limit to prove Theorem 4.
Proof of Theorem 4. Up to subsequences, the sequence {T k (u n )} constructed above, weakly converges (in ) and it is possible to pass to some u (note that u ∈ L m * * (Ω)).Recall that E ∈ (L 2 ) N .In order to pass to the limit in That is equivalent to m ≥ 2N N +2 .Thus we pass also to the limit in (16).Remark 8.Note that, once more it is possible to develop an approximate method in order to prove the existence when E ∈ L r .Indeed, let E 0 ∈ L r , r > 1 and E n ∈ L 2 converging to E 0 in L r .Define now u n in the corresponding way, we can use the statement of (17), so that we can say that estimates (14) still hold for this new sequence {u n } and once more we can pass to the limit, and we prove the existence if We can provide further a priori estimates when div E ≥ 0 Proposition 9.The solutions constructed in Theorem 4 satisfy the following additional estimates: We will later take advantage of (18) and present several extensions.See, e.g., Lemma 19 where we extend the result to div E ∈ L 1 loc .
Remark 10.Notice that (19) blows up as m → 1.In fact, it is known that the case m = 1 does not satisfy such an estimate.
We prove a priori estimates under the assumption of div E ≥ 0 for bounded (or even smooth) E, which we now know will hold for approximations.
Proof of Proposition 9. Assume first that E ∈ (L N ) N , and f ∈ L m for m ≥ 2N N +2 .Then, we can deal with the unique solution u ∈ W 1,2 0 (Ω) that exists by Theorem 1. Due to the construction by approximation in Theorem 4 the estimates pass to the limit in the construction.Take h ∈ W 1,∞ (R) such that h(0) = 0. We take v = h(u) as a test function we can write We can write where

Now we prove both items
• Item 1.Since div E ∈ L 1 (Ω) we can integrate by parts again to deduce Let us consider It is clear that F ε (s) → |s| a.e. as ε → 0.Then, Hence, going back to (20) Hence, we simplify

Comparison principle and uniqueness
To show uniqueness of solutions we prove a weak maximum principle.11).Then, if u ∈ W 1,2 0 (Ω) is a solution of (13) then .
Hence, there is, at most, one solution of (13) in W 1,2 0 (Ω).Furthermore, if f ≥ 0 then u ≥ 0. We first prove the following lemma Proof.By definition of having a sign in distributional sense, for 0 ≤ ϕ ∈ C ∞ c (Ω), we have that In particular, ∇ϕ → ∇v in L r (Ω).We can pass to the limit in the estimate.
Proof of Theorem 11.Let u be a solution.Take ρ n a family of non-negative mollifiers, and use v n = ρ n * u + as a test function.Passing to the limit in n and applying the previous lemma We recover the estimate.

Finally, define E
(1) n → E (i) in L r (Ω) N for i = 1, 2, and the proof is complete.
Then, taking q = min{2, m * } (using formally m * = ∞ for m ≥ N ) there exists a solution of Furthermore, if m ≥ 2N N +2 and r ≥ N it is the unique solution of (13).Proof.Let f k = T k (f ) where T k is the cut-off function.We consider E k constructed in Lemma 13.By Proposition 9 there exists a unique weak u k solution of (13).Since the • * operation is monotone, then q * = min{2 * , m * * }.The sequence u k is uniformly bounded in W 1,q 0 (Ω).Therefore, by the Sobolev embedding theorem, it is uniformly bounded on L q * (Ω).Up to a subsequence, there exists u ∈ W 1,q 0 (Ω) such that (22) we have that Therefore, we can pass to the limit in the weak formulation for v ∈ W 1,∞ 0 (Ω).If m ≥ 2N N +2 and r ≥ N , then uE ∈ L 2 (Ω), and it is a solution of (13) by approximation.

Convection with singularity at one point
With the approach developed in this paper we are able to study the special situation where A > 0 (24) which is somehow in the limit of theory since it is not in In [5] the authors examined the framework of drifts such that The authors show existence of solutions u under (25), where the summability is reduced as |A| is increased.They prove Theorem 15 ([5]).Let f ∈ L m (Ω) and |E| ≤ |A|/|x|.Then, there exists a solution u the solution of (1) and Above, M N N −1 denotes the Marcinkiewicz space (see [5] for the definition and some properties).The argument in [5] is based on Hardy's inequality We are able to extend this result to distinguish depending on the sign of A. Our result is the following for some m > 1 and (24).Then, there exists a solution u A of (23), and it satisfies the estimates in Proposition 9. Furthermore, u A → 0 as A → ∞ in the sense that We point out that, if m > 2N N +2 , we have furthermore u A E ∈ L 2 (Ω).
Proof.Since N ≥ 3 we know that |E| ∈ L 2 (Ω) and that is non-negative, and it is in L 1 (Ω).Then, we have satisfied the existence theory of Theorem 4. Due to Proposition 9 and (27) the estimate follows.

Convection with singularity on the boundary
The aim of this section is to understand the case where E is regular inside Ω but blows up towards ∂Ω.For the sake of simplicity we present an example, which as mentioned in Section 7 can be generalized, but the computations become quite technical.Let us consider ϕ 1 the first eigenfunction of −∆ with Dirichlet boundary conditions, i.e., We normalize it so that ∇ϕ 1 L ∞ = 1.It is known that there exists C > 0 such that and near ∂Ω we have that We focus our efforts on the particular case and f ∈ L ∞ c (Ω), the space of bounded functions with compact support in Ω.The aim of this section is to prove Theorem 17.Let E be given by (28), M = I and f ∈ L ∞ c (Ω).Then, there exists a unique u ∈ H 1 0 (Ω) ∩ L ∞ (Ω) such that uE ∈ L ∞ (Ω) and u is a weak solution in the sense that (13) holds.Furthermore, u is flat on the boundary in the sense that for all α > 1 we have that We will give the proof below.First, we prove positivity in the interior.
Proposition 18.In the assumptions of Theorem 17 if f ≥ 0 and Ω f > 0, then u > 0 in Ω.
It is immediate to compute that Hence div E(x) ≥ c dist(x, ∂Ω) −2−γ near the boundary.Notice that E and div E are not in L 1 (Ω).We start the proof with a lemma.
Proof.We consider the approximating sequence for Theorem 14.For the approximation we know that Let us fix K ⋐ Ω.We have that Since we know that div E n → div E in L 1 (K), we have that, up to a further subsequence, div E n converges a.e. in K. Hence, applying Fatou's lemma Since this estimate is uniform in K, we can take K h = {x ∈ Ω : dist(x, ∂Ω) ≥ h} and deduce, as h → 0, that (18) holds.
The solution found in Theorem 17 is unique in a certain class.We provide a uniqueness result extending Theorem 11, which can itself be generalised to a larger framework In particular, there is at most one weak solution in H 1 0 (Ω) of the (1).
Proof.We want to repeat the argument in Theorem 11, i.e., taking v = u + in the weak formulation and using that We prove this formula by approximation.Take η ∈ C ∞ c (Ω).There exists K ⋐ Ω and φ m ∈ C ∞ 0 (K) such that φ m → u + η in H 1 0 (Ω).We have that Since E ∈ L ∞ (K), we pass to the limit to deduce Now we take η m ր 1.In particular η m (x) = η 0 (mϕ 1 (x)) where η 0 is non-decreasing, η 0 (s) = 0 if s ≤ 1 and η 0 (s) = 1 if s > 2. Clearly ∇η m L ∞ ≤ Cm.Since u + ∈ H 1 0 (Ω), then u + (x)/ϕ 1 (x) ∈ L 2 (Ω) by Hardy's inequality.And we compute and the size of the domain tends to zero.We conclude, by Dominated Convergence that We are finally ready to prove the result.
Proof of Theorem 17.The uniqueness claim is proven in Lemma 20.We now prove the existence and bounds by approximation.We can assume, without loss of generality, that f ≥ 0, and construct approximations of E given by . These satisfy the assumptions of Theorem 4. Hence, there exists a weak solution u ℓ ∈ H 1 0 (Ω) of ( 1) where E = E ℓ .We compute This is non-negative.Hence, due to Theorem 11 we have that Splitting the behaviour near the boundary and away from the boundary, it is easy to see that div E ℓ ≥ c 0 > 0 uniformly.Therefore, due to Proposition 9 we have that Now we must construct barrier functions.Select a single α > 1 and the barrier We drop the dependence on ℓ and α to make the presentation below more readable.Plugging it into the equation we get There exists η α > 0 small enough such that We will use the neighbourhood of the boundary A α = {x ∈ Ω : dist(x, ∂Ω) < η α }.Also, we consider the candidate super-solution We denote the constant on the right-hand side as By the previous computations, if ℓ ≥ 2( α−1 γ ) 1 γ , we have f ≥ 0 = f in A α , and clearly f ∈ L ∞ (A α ).Hence, due to Theorem 11 we have that 0 ≤ u ℓ (x) ≤ u(x), x ∈ A α .
Also, due to (30) and the second part of C α , we have that Eventually, we deduce that for any α > 1 we that In particular, picking α = γ + 1 we deduce that We deduce that, up to a subsequence, u ℓ → u a.e. and strongly in L 2 and u ℓ ⇀ u weakly in H 1 0 (Ω).
This implies that u ℓ E ℓ → uE a.e.And hence uE is bounded.Passing to the limit in the weak formulation by the Dominated Convergence Theorem, the result is proven.
Remark 21.Notice that the construction of the super-solution above can be done in any dimension N ≥ 1.However, most of the results in the rest of the paper are only available for N ≥ 3.
Remark 22.For Schrödinger-type equations −∆u + V u = f , it is known that if the potential V is greater than dist(x, ∂Ω) −2 and f is compactly supported, then u is flat on the boundary, in the sense that |u| ≤ C dist(x, ∂Ω) 1+ε .This means that ∂ n u = 0 on ∂Ω.This means that it satisfies Dirichlet and Neumann homogeneous boundary conditions.And it can be extended by 0 outside Ω with higher regularity than H 1 .In contrast, the exponent γ in the above result can not be taken as γ = 0 in order to get flat solutions.Indeed, the convection term E • ∇ϕ 1 , in the above computations, is more singular than the term ϕ 1 div E. Proof.We maintain the notation of the proof of Theorem 17.We have already shown that, on a neighbourhood of the boundary, For α in the range (1, γ + 2 − ω), we can take as a supersolutions for the approximating sequence And the rest of the proof remains as in Theorem 17.
7 Further remarks, extensions, and open problems when a ≥ 0. As above, our approach allows for less regularity in a than most previous literature, e.g. a ∈ L 1 loc (Ω).Furthermore, one will then obtain Hence, one can reduce the hypothesis to a + div E ≥ 0 in the whole analysis.
4. The study of a ≡ 1 is useful in the study of the evolution problem u t − div(M (x)∇u) + div(uE(x)) = 0.
For the study of this problem one can write u t + Au = 0 where Au = − div(M (x)∇u) + div(uE(x)).
In order to obtain solutions in semigroup form in L p (where 1 ≤ p ≤ +∞), following the theory of accretive operators, it is sufficient that, Letting f = u + λAu, this is precisely what we have proven above, where M = λI and a ≡ 1. See also [6].
5. We point out that when |E| ≤ |A|/|x|, we have that, if m > 2N N +2 then u|E| ∈ L 2 (Ω).It seems possible to extend the uniqueness result (20) to this setting.