Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 13, pp. 1–16. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.13 FAILURE OF THE HOPF-OLEINIK LEMMA FOR A LINEAR ELLIPTIC PROBLEM WITH SINGULAR CONVECTION OF NON-NEGATIVE DIVERGENCE LUCIO BOCCARDO, JESÚS ILDEFONSO DÍAZ, DAVID GÓMEZ-CASTRO Abstract. In this article we study the existence, uniqueness, and integrability of solutions to the Dirichlet problem − div(M(x)∇u) = − div(E(x)u) + f in a bounded domain of RN with N ≥ 3. We are particularly interested in singular E with divE ≥ 0. We start by recalling known existence results when |E| ∈ LN that do not rely on the sign of divE. Then, under the assumption that divE ≥ 0 distributionally, we extend the existence theory to |E| ∈ L2. 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 divE ∼ dist(x, ∂Ω)−2−α. 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. ∂u/∂n < 0, fails in the presence of such singular drift terms E. 1. Introduction It is well known that many relevant applications lead to the presence of a con- vection term in the correspondent model which, in its simplest formulation, leads to a boundary value problem for linear elliptic second order equation of the type −div(M(x)∇u) = −div(uE(x)) + f(x) in Ω u = 0 on ∂Ω. (1.1) Here Ω ⊂ RN , N ≥ 3, is an open, bounded set, and we assume that M ∈ L∞(Ω)N×N is elliptic M(x)ξ · ξ ≥ α|ξ|2, ∀ξ ∈ RN and a.e. x ∈ Ω. According to the regularity of the right-hand side datum f(x) it is natural to search the solution in the energy space W 1,2 0 (Ω) (case of f ∈ H−1(Ω): see, e.g. [21, 16, 1]), or in a larger Sobolev space if f is singular (see [1]); when f ∈ L1(Ω), see, for instance, [8], or when L1(Ω, δ) with δ(x) = d(x, ∂Ω), see, e.g., [7, 13]. 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 2020 Mathematics Subject Classification. 35J25, 35J75, 35B50, 35B60. Key words and phrases. Linear elliptic equation; convection with singularity on the boundary; strong maximum principle; flat solutions. ©2024. This work is licensed under a CC BY 4.0 license. Submitted October 31, 2023. Published January 31, 2024. 1 2 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 have a maximum principle: divE ≥ 0 a.e. on Ω. (1.2) 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| ∈ LN (Ω) and f ∈ L 2N N+2 (Ω). In [13] and [12] the authors study the case in which |E| ∈ LN (Ω) and divE = 0 in Ω and E ·n = 0 on ∂Ω, f ∈ L1(Ω, δ). See also [15, 20]. In this article, we show that (1.2) makes divE behave like a non-negative poten- tial 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 (1.2) holds in distributional sense. The article is structured as follows. First, in Section 2 we review known results for the case |E| ∈ LN and f ∈ L 2N N+2 (Ω) which were published in [1], were shown there is a unique weak solution of (1.1) that can be constructed by approximation. In Section 3 we show that if |E| ∈ L2(Ω), divE ≥ 0, and f ∈ Lm(Ω) for some m > 1 then the same approximation procedure converges to a weak solution of (1.1), 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 (1.1). Then we move to discussing interesting examples that fall in this setting. In Section 5 we focus on the case E(x) = A x |x|2 , (1.3) which is somehow in the limit of theory since it is not in LN (Ω) but it is in Lr(Ω) for r ∈ [1, N). In [5] the authors examined the more general class |E| ≤ |A| |x| . (1.4) 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 5.2 we show that if N > 1, f ∈ Lm(Ω) for suitable m, and A > 0 then we can use the sign of divE to deduce that the solution uA of (1.1) with E = A x |x|2 satisfies uA → 0 in L1(Ω) as A→ +∞ . By the contrary, when A < 0 we cannot improve the result in [5]. Notice that this is similar to the equation L(uB) +B uB = f , whereas B →∞ we have uB → 0. Lastly, in Section 6, we discuss the case where E is suitably singular only on the boundary. We present an example showing that if divE 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. |u(x)| ≤ C dist(x, ∂Ω)α for some α > 1 . 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 divE acts as a potential. However, in the Schrödinger equation it is EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 3 sufficient that V (x) ≥ Cδ−2 to get flat solutions, whereas for E we need a strictly larger exponent (see Remark 6.6). 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. 2. Known results when |E| ∈ LN We define the Sobolev conjugate exponent m∗ = mN N −m if m ≤ N, m∗∗ = (m∗)∗ = mN N − 2m if m ≤ N 2 . We have that m∗∗ ∈ [1,∞] for N N+2 ≤ m ≤ N 2 . Notice that m∗ ≥ 2 if and only if m ≥ 2N N+2 = (2∗)′. Notice that, since m ≥ 1 we have m∗ ≥ m. In order to compute explicit a priori estimates, we use the Sobolev embedding constant Sp such that, for 1 < p < +∞ Sp‖u‖Lp∗ (Ω) ≤ ‖∇u‖Lp(Ω). (2.1) 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. Theorem 2.1 ([1]). Let f ∈ L 2N N+2 (Ω) and |E| ∈ LN (Ω). Then, there exists a unique weak solution u of (1.1) in the sense that u ∈W 1,2 0 (Ω) is such that ∫ Ω M(x)∇u∇v = ∫ Ω uE(x) · ∇v + ∫ Ω f(x) v(x), for all v ∈W 1,2 0 (Ω). and it satisfies (1) Logarithmic estimate:(∫ Ω | log(1 + |u|)|2 )2/2∗ ≤ 1 S2 2α 2 ∫ Ω |E|2 + 2 S2 2α ∫ Ω |f |, (2) Gradient estimate: there exists C = C(α,N) such that∫ Ω |∇u|2 ≤ C ( ‖E‖2LN + ‖f‖2 L 2N N+2 ) . (2.2) (3) Stampacchia-type summability: For m ∈ [ 2N N+2 , N 2 ) there exists a constant C = C(m,α, N, ‖E‖LN ) such that ‖u‖m∗∗ ≤ C‖f‖m. (2.3) (4) Stampacchia-type boudedness: Let r > N and m > N 2 . There exists C such that ‖u‖L∞ ≤ C(m, r, α, ‖f‖Lm , ‖E‖Lr ). (2.4) Remark 2.2. The natural theory for this problem in energy space is precisely |E| ∈ LN (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 ∈ L2∗ whereas ∇v ∈ L2. So we always have that uE ∈ L2(Ω). 4 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 In [1] the main tool to study the linear problem (1.1) are the auxiliary non-linear Dirichlet problems −div(M(x)∇un) = −div ( un 1 + 1 nun En(x) ) + fn(x) Ω u = 0 ∂Ω, (2.5) where the author take fn = Tn(f) a truncation of f through the family Tn(s) = { s |s| ≤ k, k sign(s) |s| > k, and En = E 1+ 1 n |E| . We will take advantage of a similar approximation. Remark 2.3. Since the problem is linear, for t ∈ R we have that tu is solution of −div(M(x)∇[t u]) = −div([t u]E(x)) + t f(x), and that E does not change. Thus, using (2.2) t2 ∫ Ω |∇u|2 ≤ C ( ‖E‖2LN + t2‖f‖2 L 2N N+2 ) . Dividing by t−2 and taking the limit as t→∞ gives∫ Ω |∇u|2 ≤ C‖f‖2 L 2N N+2 . (2.6) Notice that in Theorem 4.1 we will prove this fact for the case divE ≥ 0. 3. Existence theory when |E| ∈ L2 and divE ≥ 0 The structural assumption in this section is the following: E belongs to the Lebesgue space (L2(Ω))N , divE ≥ 0 in D′(Ω), that is ∫ Ω E · ∇φ ≤ 0, ∀0 ≤ φ ∈W 1,2 0 (Ω). (3.1) Theorem 3.1. Assume (3.1) and f ∈ Lm(Ω), 1 < m < N 2 , (3.2) and let p = min{2,m∗}. Then, there exists a weak solution u of (1.1) in the sense that u ∈W 1,p 0 (Ω) is such that∫ Ω M(x)∇u∇v = ∫ Ω uE(x) · ∇v + ∫ Ω f(x) v(x), ∀v ∈W 1,∞ 0 (Ω). (3.3) Furthermore, it satisfies ‖u‖ W 1,m∗ 0 (Ω) ≤ Cm‖f‖m, if 1 < m < 2N N + 2 ; ‖u‖W 1,2 0 (Ω) + ‖u‖m∗∗ ≤ C̃m ‖f‖m, if 2N N + 2 ≤ m 0 since m > 1. Thus, we have∫ Ω M(x)∇un∇(Tk(un)|Tk(un)|2γ−2) = ∫ Ω un 1 + 1 n |un| E(x) · ∇(Tk(un)|Tk(un)|2γ−2) + ∫ Ω fn(x)Tk(un)|Tk(un)|2γ−2. To study the second integral, we define the function Hγ(s) = ∫ s 0 t |t|2γ−2 1 + 1 n |t| dt. It is easy to check that Hγ(s) ≥ 0 for all s ∈ R. Thus, using the sign condition on divE we have that∫ Ω un 1 + 1 n |un| E(x) · ∇(Tk(un)|Tk(un)|2γ−2) = ∫ Ω (2γ − 1) Tk(un) |Tk(un)|2γ−2 1 + 1 n |Tk(un)| E(x) · ∇Tk(un) = ∫ Ω Hγ(Tk(un))E(x) · ∇Tk(un) = ∫ Ω E(x) · ∇[Hγ(Tk(un))] ≤ 0. Hence, we have that∫ Ω M(x)∇un∇(Tk(un)|Tk(un)|2γ−2) ≤ ∫ Ω fn(x)Tk(un)|Tk(un)|2γ−2, which is the starting point of [4], and we obtain the estimates ‖Tk(un)‖ W 1,m∗ 0 (Ω) ≤ Cm ‖f‖m, if 1 < m < 2N N + 2 ; 6 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 ‖Tk(un)‖W 1,2 0 (Ω) + ‖Tk(un)‖m∗∗ ≤ C̃m ‖f‖, if 2N N + 2 ≤ m 1 and En ∈ L2 converging to E0 in Lr. Define now un in the corresponding way, we can use the statement of (3.7), so that we can say that estimates (3.4) still hold for this new sequence {un} and once more we can pass to the limit, and we prove the existence if 1 ≥ 1 r + 1 m − 2 N We can provide further a priori estimates when divE ≥ 0. Proposition 3.6. The solutions constructed in Theorem 3.1 satisfy the following additional estimates: (1) (L1 estimate) If divE ∈ L1(Ω) then we have that∫ Ω |u|divE ≤ ∫ Ω |f |. (3.8) (2) (Lm estimate) If divE ≥ c0 > 0 and m > 1 then ‖u‖Lm ≤ m m− 1 ‖f‖Lm c0 . (3.9) We will later take advantage of (3.8) and present several extensions. See, e.g., Lemma 6.3 where we extend the result to divE ∈ L1 loc. Remark 3.7. Notice that (3.9) 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 divE ≥ 0 for bounded (or even smooth) E, which we now know will hold for approximations. Proof of Proposition 3.6. Assume first that E ∈ (LN )N , and f ∈ Lm for m ≥ 2N N+2 . Then, we can deal with the unique solution u ∈ W 1,2 0 (Ω) that exists by Theorem 2.1. Because of the construction by approximation in Theorem 3.1 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 α ∫ Ω h′(u)|∇u|2 ≤ ∫ Ω M(x)∇u · ∇h(u) = ∫ Ω uE · ∇h(u) + ∫ Ω fh(u). EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 7 We can write u∇h(u) = uh′(u)∇u = ∇F (u) where F (s) = ∫ s 0 τ h′(τ)dτ . Hence, α ∫ Ω h′(u)|∇u|2 ≤ ∫ Ω E · ∇F (u) + ∫ Ω fh(u). Now we prove both items (1) Since divE ∈ L1(Ω) we can integrate by parts again to deduce α ∫ Ω h′(u)|∇u|2 + ∫ Ω F (u) divE ≤ ∫ Ω fh(u). (3.10) Let us consider hε(s) = Tε(s)/ε. Then h′ε ≥ 0 and |hε| ≤ 1 and, hence, in (3.10)∫ Ω Fε(u) divE ≤ ∫ Ω |f |. It is clear that Fε(s)→ |s| a.e. as ε→ 0. Then∫ Ω |u|divE ≤ ∫ Ω |f |. (2) Let us take, for m > 1, h(s) = |s|m−1 then F (s) = (m− 1) ∫ s 0 |τ |m−2 sign(τ)τdτ = m− 1 m sm. Hence, going back to (3.10), c0 m− 1 m ‖u‖mLm ≤ m− 1 m ∫ Ω |u|m divE ≤ ∫ Ω f |u|m−1 ≤ ‖f‖Lm‖u‖m−1 Lm . Hence, we simplify ‖u‖Lm ≤ m m− 1 ‖f‖Lm c0 . � 4. Comparison principle and uniqueness To show uniqueness of solutions we prove a weak maximum principle. Theorem 4.1. Let f ∈ L 2N N+2 (Ω) and (3.1). Then, if u ∈W 1,2 0 (Ω) is a solution of (3.3) then ‖∇u+‖2 ≤ 1 αS2 ‖f+‖ 2N N+2 . Hence, there is, at most, one solution of (3.3) in W 1,2 0 (Ω). Furthermore, if f ≥ 0 then u ≥ 0. We first prove the following lemma. Lemma 4.2. Let m, r > 1, E ∈ Lr′(Ω) with 0 ≤ divE ∈ D′(Ω). Then, we have that − ∫ Ω E∇v ≥ 0 ∀ 0 ≤ v ∈W 1,r 0 (Ω). (4.1) 8 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 Proof. By definition of having a sign in distributional sense, for 0 ≤ ϕ ∈ C∞c (Ω), we have that − ∫ Ω E∇ϕ = 〈divE,ϕ〉 ≥ 0. For 0 ≤ v ∈ W 1,r 0 (Ω), we can find a sequence 0 ≤ ϕn ∈ C∞c (Ω), such that ϕn → v in W 1,r 0 (Ω). In particular, ∇ϕ → ∇v in Lr(Ω). We can pass to the limit in the estimate. � Proof of Theorem 4.1. Let u be a solution. Take ρn a family of non-negative molli- fiers, and use vn = ρn∗u+ as a test function. Passing to the limit in n and applying the previous lemma α ∫ Ω |∇u+|2 ≤ ∫ Ω E∇ u2 + 2 + ∫ Ω fu+ ≤ ‖f‖(2∗)′‖u+‖2∗ ≤ 1 S ‖f‖(2∗)′‖∇u+‖2. We recover the estimate. � Lemma 4.3. Let E ∈ Lr(Ω)N for r > 1 with divE ≥ 0 in D′(Ω). Then, there exists a sequence En ∈W 1,∞(Ω) with divEn ≥ 0 such that E → E in Lr(Ω)N . Proof. We use a similar decomposition to [22, Theorem 1.5] (done there for r = 2). First, we define −∆p(1) = divE in Ω, p(1) = 0 on ∂Ω. By well-known results we obtain a unique solution p(1) ∈ W 1,r 0 (Ω). Take E(1) = ∇p(1). Lastly, take E(2) = E − E(1) ∈ Lr(Ω). Notice that divE(2) = 0. Due to [19], E(2) admits a divergence-free extension to Lr(Rd), which we denote Ẽ(2). We can take a family of mollifiers ρn, and E (2) n = Ẽ(2) ∗ ρn ∈ W 1,∞(Ω). Now let 0 ≤ g(1) = divE(1) ∈ W−1,r′(Ω). Let g (1) n ≥ 0 be a sequence of C∞c (Ω) functions with g (1) n → g(1) in Lr ′ (Ω). Take p (1) n the unique solution to −∆p(1) n = g(1) n in Ω, p(1) n = 0 in ∂Ω. Finally, define E (1) n = ∇p(1) n ∈ W 1,∞(Ω). It is now easy to see that E (i) n → E(i) in Lr(Ω)N for i = 1, 2, and the proof is complete. � Theorem 4.4. Let f ∈ Lm(Ω) and E ∈ Lr(Ω) such that 0 ≤ divE ∈ D′(Ω) and 1 min{2∗,m∗∗} + 1 r ≤ 1 if 2N N + 2 ≤ m ≤ N 2 1 2∗ + 1 r ≤ 1 otherwise. (4.2) Then, taking q = min{2,m∗} (using formally m∗ = ∞ for m ≥ N) there exists a solution of: u ∈W 1,q 0 (Ω) such that∫ Ω M(x)∇u∇v = ∫ Ω uE∇v + ∫ Ω fv, ∀v ∈W 1,q′ 0 (Ω). (4.3) Furthermore, if m ≥ 2N N+2 and r ≥ N it is the unique solution of (3.3). EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 9 Proof. Let fk = Tk(f) where Tk is the cut-off function. We consider Ek constructed in Lemma 4.3. By Proposition 3.6 there exists a unique weak uk solution of (3.3). Since the ·∗ operation is monotone, then q∗ = min{2∗,m∗∗}. The sequence uk is uniformly bounded in W 1,q 0 (Ω). Therefore, by the Sobolev embedding theorem, it is uniformly bounded on Lq ∗ (Ω). Up to a subsequence, there exists u ∈ W 1,q 0 (Ω) such that ∇uk ⇀ ∇u in Lq(Ω) uk ⇀ u in Lq ∗ (Ω). Since M ∈ L∞(Ω)N×N , Ek → E ∈ Lr(Ω)N strongly and (4.2) we have that M(x)∇uk ⇀M(x)∇u in Lq(Ω) ukEk ⇀ uE in L1(Ω). 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 ∈ L2(Ω), and it is a solution of (3.3) by approximation. � 5. Convection with singularity at one point With the approach developed in this paper we are able to study the special situation E = A x |x|2 where A > 0 (5.1) which is somehow in the limit of theory since it is not in LN (Ω), but it is in Lr(Ω) for r ∈ [1, N). In [5] the authors examined the framework of drifts such that |E| ≤ |A| |x| , (5.2) The authors show existence of solutions u under (5.2), where the summability is reduced as |A| is increased. They proved the following result. Theorem 5.1 ([5]). Let f ∈ Lm(Ω) and |E| ≤ |A|/|x|. Then, there exists a solution u the solution of (1.1) and (1) If |A| < α(N−2m) m and m ∈ [ 2N N+2 , N 2 ) then u ∈W 1,2 0 (Ω) ∩ Lm∗∗(Ω). (2) If |A| < α(N−2m) m and m ∈ (1, 2N N+2 ) then u ∈W 1,m∗ 0 (Ω). (3) If |A| < α(N − 2) and m = 1 then ∇u ∈ (M N N−1 (Ω))N and u ∈ W 1,q 0 (Ω), for every q < N N−1 . 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(N − 2 N )2 ∫ RN |u|2 |x|2 ≤ ∫ RN |∇u|2. (5.3) We are able to extend this result to distinguish depending on the sign of A. Our result is the following theorem. 10 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 Theorem 5.2. Let f ∈ Lm(Ω) for some m > 1 and (5.1). Then, there exists a solution uA of (4.3), and it satisfies the estimates in Proposition 3.6. Furthermore, uA → 0 as A→∞ in the sense that∫ Ω |uA(x)| |x|2 ≤ 1 A(N − 2) ∫ Ω |f |. We point out that, if m > 2N N+2 , we have furthermore uAE ∈ L2(Ω). Proof. Since N ≥ 3 we know that |E| ∈ L2(Ω) and that divE = r1−N ∂ ∂r (rN−1Ar−1) = A(N − 2) |x|2 (5.4) is non-negative, and it is in L1(Ω). Then, we have satisfied the existence theory of Theorem 3.1. Because of Proposition 3.6 and (5.4) the estimate follows. � 6. 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 ϕ1 the first eigenfunction of −∆ with Dirichlet boundary conditions, i.e., −∆ϕ1 = λ1ϕ1in Ω, ϕ1 = 0on ∂Ω. We normalize it so that ‖∇ϕ1‖L∞ = 1. It is known that there exists C > 0 such that 0 < C dist(x, ∂Ω) ≤ ϕ1(x) ≤ C−1 dist(x, ∂Ω), ∀x ∈ Ω, and near ∂Ω we have that |∇ϕ1(x)| ≥ C > 0. We focus our efforts on the particular case E = −ϕ−1−γ 1 ∇ϕ1, for some γ > 0, (6.1) and f ∈ L∞c (Ω), the space of bounded functions with compact support in Ω. The aim of this section is to prove the following theorem. Theorem 6.1. Let E be given by (6.1), M = I and f ∈ L∞c (Ω). Then there exists a unique u ∈ H1 0 (Ω) ∩ L∞(Ω) such that uE ∈ L∞(Ω) and u is a weak solution in the sense that (3.3) holds. Furthermore, u is flat on the boundary in the sense that for all α > 1 we have that |u(x)| ≤ Cα dist(x, ∂Ω)α, for a.e. x ∈ Ω. (6.2) We will give the proof below. First, we prove positivity in the interior. Proposition 6.2. In the assumptions of Theorem 6.1 if f ≥ 0 and ∫ Ω f > 0, then u > 0 in Ω. Proof. Let Ωη = {x ∈ Ω : d(x,Ω) > η}. Consider uη the solution of (1.1) with E given by (6.1) and uη = 0 in ∂Ωη. Notice that E is smooth on Ωη for η > 0. Since we already know from Theorem 4.1 that u ≥ 0 in Ω, the classical comparison principle in Ωη ensures that uη ≤ u for any η ≥ 0. Take η > 0 small enough so that∫ Ωη f > 0. Then, by the “classical” strong maximum principle we obtain uη > 0 in Ωη, and the proof is complete. � EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 11 It is immediate to compute that divE = (1 + γ)ϕ−2−γ 1 |∇ϕ1|2 − ϕ−1−γ 1 ∆ϕ1 = (1 + γ)ϕ−2−γ 1 |∇ϕ1|2 + λ1ϕ −γ 1 . Hence divE(x) ≥ cdist(x, ∂Ω)−2−γ near the boundary. Notice that E and divE are not in L1(Ω). We start the proof with a lemma. Lemma 6.3. In the assumptions of Theorem 4.4, assume furthermore that divE ∈ L1 loc(Ω). Then udivE ∈ L1(Ω), still satisfying estimate (3.8). Proof. We consider the approximating sequence for Theorem 4.4. For the approxi- mation we know that ∫ Ω |un|divEn ≤ ∫ Ω |f |. Let us fix K b Ω. We have that∫ K |un|divEn ≤ ∫ Ω |f |. Since we know that divEn → divE in L1(K), we have that, up to a further subsequence, divEn converges a.e. in K. Hence, applying Fatou’s lemma∫ K |u|divE ≤ ∫ Ω |f |. Since this estimate is uniform in K, we can take Kh = {x ∈ Ω : dist(x, ∂Ω) ≥ h} and deduce, as h→ 0, that (3.8) holds. � The solution found in Theorem 6.1 is unique in a certain class. We provide a uniqueness result extending Theorem 4.1, which can itself be generalized to a larger framework. Lemma 6.4. Assume that u ∈ H1 0 (Ω), E ∈ L∞loc(Ω), u|E| ∈ L2(Ω), divE ≥ 0 distributionally, and f ∈ L 2N N+2 (Ω). Then ‖∇u+‖2 ≤ 1 αS2 ‖f+‖ 2N N+2 . In particular, there is at most one weak solution in H1 0 (Ω) of (1.1). Proof. We want to repeat the argument in Theorem 4.1, i.e., taking v = u+ in the weak formulation and using that − ∫ Ω uE · ∇u+ ≥ 0. We prove this formula by approximation. Take η ∈ C∞c (Ω). There exists K b Ω and φm ∈ C∞0 (K) such that φm → u+η in H1 0 (Ω). We have that − ∫ Ω φmE · ∇φm = 〈divE, φ2 m 2 〉 ≥ 0. Since E ∈ L∞(K), we pass to the limit to deduce − ∫ Ω (u+η) · E∇(u+η) ≥ 0. 12 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 Now we expand∫ Ω (u+η)E · ∇(u+η) = ∫ Ω u2 +ηE · ∇η + ∫ Ω u+η 2E · ∇u+ Now we take ηm ↗ 1. In particular ηm(x) = η0(mϕ1(x)) where η0 is non- decreasing, η0(s) = 0 if s ≤ 1 and η0(s) = 1 if s > 2. Clearly ‖∇ηm‖L∞ ≤ Cm. Since u+ ∈ H1 0 (Ω), then u+(x)/ϕ1(x) ∈ L2(Ω) by Hardy’s inequality. And we compute∣∣ ∫ Ω u2 +ηm · E∇ηm ∣∣ ≤ ∫ ϕ1(x)≤ 1 m u+ ϕ1 ϕ1 m |uE|Cm ≤ C ∫ ϕ1(x)≤ 1 m u+ ϕ1 |uE| → 0 since u+ ϕ1 |uE| ∈ L1(Ω) and the size of the domain tends to zero. We conclude, by Dominated Convergence that 0 ≥ ∫ Ω (u+ηm) · E∇(u+ηm)→ ∫ Ω u+E · ∇u+ = ∫ Ω uE · u+. � We are finally ready to prove the main result. Proof of Theorem 6.1. The uniqueness claim is proven in Lemma 6.4. 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 E` = −(ϕ1 + 1 ` )−1−γ∇ϕ1. Clearly E` ∈ L∞(Ω). These satisfy the assumptions of Theorem 3.1. Hence, there exists a weak solution u` ∈ H1 0 (Ω) of (1.1) where E = E`. We compute divE` = (1 + γ)(ϕ1 + 1 ` )−2−γ |∇ϕ1|2 + λ1(ϕ1 + 1 ` )−1−γϕ1. This is non-negative. Hence, from Theorem 4.1 we have that ‖∇u`‖L2 ≤ C‖f‖L∞ . Splitting the behaviour near the boundary and away from the boundary, it is easy to see that divE` ≥ c0 > 0 uniformly. Therefore, by Proposition 3.6 we have that ‖u`‖L∞ ≤ ‖f‖L∞ c0 . (6.3) Now we must construct barrier functions. Select a single α > 1 and the barrier U = 1 α (ϕ1 + 1 ` )α. We drop the dependence on ` and α to make the presentation below more readable. Plugging it into the equation we obtain −∆U + div(UE`) = −∆U +∇U · E` + U divE` = −(α− 1)(ϕ1 + 1 ` )α−2|∇ϕ1|2 + λ1(ϕ1 + 1 ` )α−1ϕ1 − (ϕ1 + 1 ` )α−2−γ |∇ϕ1|2 + (1 + γ)(ϕ1 + 1 ` )α−2−γ |∇ϕ1|2 + λ1(ϕ1 + 1 ` )α−1−γϕ1 ≥ ( γ(ϕ1 + 1 ` )−γ − (α− 1) ) (ϕ1 + 1 ` )α−2|∇ϕ1|2. EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 13 This is non-negative if ϕ1(x) + 1 m ≤ (α−1 γ )−1/γ . There exists ηα > 0 small enough such that f(x) = 0 and ϕ1(x) ≤ 1 2 ( α− 1 γ )−1/γ , ∀x such that dist(x, ∂Ω) ≤ ηα. We will use the neighborhood of the boundary Aα = {x ∈ Ω : dist(x, ∂Ω) < ηα}. Also, we consider the candidate super-solution u(x) = U(x) ( α mindist(x,∂Ω)=ηα ϕ1(x)α + α c0 ‖f‖L∞ mindist(x,∂Ω)≥ηα ϕ1(x)α ) . We denote the constant on the right-hand side as Cα. Using the first term of Cα, u ≥ u when dist(x, ∂Ω) = ηα. Also, u = 1 mα ≥ 0 = u on ∂Ω. Let us call f = −∆u+ div(uE`). By the previous computations, if ` ≥ 2(α−1 γ ) 1 γ , we have f ≥ 0 = f in Aα, and clearly f ∈ L∞(Aα). Hence, by Theorem 4.1 we have that 0 ≤ u`(x) ≤ u(x), x ∈ Aα. Also, by (6.3) and the second part of Cα, we have that 0 ≤ u`(x) ≤ u(x), x ∈ Ω \Aα. Eventually, we deduce that for any α > 1 we have 0 ≤ u`(x) ≤ Cα α (ϕ1 + 1 ` )α, ∀x ∈ Ω and ` ≥ 2( α− 1 γ ) 1 γ . In particular, picking α = γ + 1 we deduce that |u`E`| ≤ Cγ+1 γ + 1 ‖∇ϕ1‖L∞ = Cγ+1 γ + 1 . We deduce that, up to a subsequence, u` → u a.e. and strongly in L2 and u` ⇀ u weakly in H1 0 (Ω). This implies that u`E` → 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 6.5. 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 6.6. 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 ∂nu = 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 H1. 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 · ∇ϕ1, in the above computations, is more singular than the term ϕ1 divE. A very explicit example can be done in one dimension: if we consider E = −Cx−1 then this drift does not generate flat solutions since if we take U(x) = xα then −U ′′ + (EU)′ = (−αxα−1 − Cxα−1)′ = −(α+ C)(α− 1)xα−2, 14 L. BOCCARDO, J. I. DÍAZ, D. GÓMEZ-CASTRO EJDE-2024/13 and this is a supersolution only if α ≤ 1. Corollary 6.7. In the hypothesis of Theorem 6.1 replace f ∈ L∞c (Ω) by |f(x)| ≤ C dist(x, ∂Ω)ω for 0 ≤ ω ≤ γ + 1. Then |u(x)| ≤ dist(x, ∂Ω)α for all α ∈ (1, γ + 2− ω) . Proof. We maintain the notation of the proof of Theorem 6.1. We have already shown that, on a neighborhood of the boundary, −∆U + div(UEm) ≥ γ 2 (ϕ1 + 1 m )α−2−γ |∇ϕ1|2 ≥ c1(ϕ1 + 1 m )α−2−γ ≥ c2|f |. For α in the range (1, γ + 2− ω), we can take as a supersolutions for the approxi- mating sequence u(x) = U(x) ( 1 c2 + α mindist(x,∂Ω)=ηα ϕ1(x)α + α c0 ‖f‖L∞ mindist(x,∂Ω)≥ηα ϕ1(x)α ) . And the rest of the proof remains as in Theorem 6.1. � 7. Further remarks, extensions, and open problems (1) We point that the proofs of our estimates can be extended to many non-linear settings. (2) Theorem 6.1 admits many generalizations. For instance, one can consider the case |E| ≤ c0 dist(x, ∂Ω)−γ−1 with divE ≥ c1 dist(x, ∂Ω)−γ−2 up to suitable conditions on the constants. Also, the techniques in this paper could be extended to the situation where dist(x, ∂Ω) is replaced by dist(x,Γ) with a suitable part Γ ⊂ ∂Ω. The case Γ an interior manifold can also be studied. (3) Including a non-negative potential. The same analysis can be performed on the equation −div(M(x)∇u) + a(x)u = −div(uE(x)) + f(x) when a ≥ 0. As above, our approach allows for less regularity in a than most previous literature, e.g. a ∈ L1 loc(Ω). Furthermore, one will then obtain∫ Ω |u|(a+ divE) ≤ ∫ Ω |f |. Hence, one can reduce the hypothesis to a+ divE ≥ 0 in the whole analysis. (4) The study of a ≡ 1 is useful in the study of the evolution problem ut − div(M(x)∇u) + div(uE(x)) = 0. For the study of this problem one can write ut +Au = 0 where Au = − div(M(x)∇u) + div(uE(x)). To obtain solutions in semigroup form in Lp (where 1 ≤ p ≤ +∞), following the theory of accretive operators, it is sufficient that, ‖u‖Lp ≤ ‖u+ λAu‖Lp . 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| ∈ L2(Ω). It seems possible to extend the uniqueness result (6.4) to this setting. EJDE-2024/13 FAILURE OF THE HOPF-OLEINIK LEMMA 15 Acknowledgements. This article was started during the doctoral course of LB in Madrid in 2019, funded by the Instituto de Matemática Interdisciplinar. The research of J. I. Dı́az was partially supported by the project PID2020-112517GB- I00 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) of the UCM. The research of D. Gomez-Castro was supported by the Advanced Grant Nonlocal-CPD (Nonlocal PDEs for Complex Particle Dynamics: Phase Transi- tions, Patterns and Synchronization) of the European Research Council Executive Agency (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 883363) and RYC2022-037317-I from the Spanish Government. References [1] L. Boccardo; Some developments on Dirichlet problems with discontinuous coefficients. Bol- letino dell Unione Matematica Italiana, 2(1):285–297, 2009. [2] L. Boccardo; Dirichlet problems with singular convection term and applications; J. Differen- tial Equations, 258 (2015), 2290-2314. [3] L. Boccardo; The impact of the zero order term in the study of Dirichlet problems with convection or drift terms. Revista Matemática Complutense https://doi.org/10.1007/s13163- 022-00434-1 [4] L. Boccardo, T. Gallouët; Nonlinear elliptic equations with right hand side measures; Comm. 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Stampacchia: Le probléme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus. Ann. Inst. Fourier (Grenoble), 15 (1965) 189-258. [22] R. Temam. Navier-Stokes equations: theory and numerical analysis (Vol. 343). American Mathematical Soc., 1979. Lucio Boccardo Istituto Lombardo & Sapienza Università di Roma, Italy Email address: boccardo@mat.uniroma1.it Jesús Ildefonso D́ıaz Inst. de Matemática Interdisciplinar, Fac. de Matemáticas, Univ. Complutense de Madrid, Spain Email address: jidiaz@ucm.es David Gómez-Castro Inst. de Matemática Interdisciplinar, Fac. de Matemáticas, Univ. Complutense de Madrid, Spain Email address: dgcastro@ucm.es 1. Introduction 2. Known results when |E| LN 3. Existence theory when |E| L2 and divE 0 4. Comparison principle and uniqueness 5. Convection with singularity at one point 6. Convection with singularity on the boundary 7. Further remarks, extensions, and open problems Acknowledgements References