THE WEAK MAXIMUM PRINCIPLE FOR SECOND-ORDER ELLIPTIC AND PARABOLIC CONORMAL DERIVATIVE PROBLEMS

. We prove the weak maximum principle for second-order elliptic and parabolic equations in divergence form with the conormal derivative boundary conditions when the lower-order coeﬃcients are unbounded and domains are beyond Lipschitz boundary regularity. In the elliptic case we consider John domains and lower-order coeﬃcients in L n spaces ( a i ,b i ∈ L q , c ∈ L q/ 2 , q = n if n ≥ 3 and q > 2 if n = 2). For the parabolic case, the lower-order coeﬃcients a i , b i , and c belong to L q,r spaces ( a i ,b i , | c | 1 / 2 ∈ L q,r with n/q + 2 /r ≤ 1), q ∈ ( n, ∞ ], r ∈ [2 , ∞ ], n ≥ 2. We also consider coeﬃcients in L n, ∞ with a smallness condition for parabolic equations.


1.
Introduction. The classical properties of solutions to elliptic and parabolic differential equations, such as the maximum principle, Harnack's inequality, Hölder estimates, and L p estimates, have been studied vastly over the last 70 years. Among them, the maximum principle plays an important role in the study of second-order linear and nonlinear elliptic and parabolic equations. For instance, it is an essential ingredient in proving a priori estimates and existence (see, for instance, [7,8]). In this paper, we consider the weak maximum principle for second-order linear elliptic and parabolic equations in divergence form with the conormal derivative boundary conditions (Neumann boundary conditions) on bounded non-smooth domains.
We mainly study the conormal derivative problem for parabolic operators in divergence form Lu = −u t + D i (a ij (x, t)D j u + a i (x, t)u) + b i (x, t)D i u + c(x, t)u whose coefficients a ij , a i , b i , c, i, j = 1, 2, . . . , n, are measurable functions on a bounded cylindrical domain in R n ×R. The leading coefficients a ij (x, t) are assumed to be uniformly elliptic and bounded; that is, there exists δ ∈ (0, 1) such that δ|ξ| 2 ≤ a ij (x, t)ξ j ξ i and |a ij (x, t)| ≤ 1/δ for all (x, t) ∈ R n × R and ξ ∈ R n . Throughout the paper, we adopt the summation convention over repeated indices.
We focus on two objectives: unbounded lower-order coefficients and non-smooth domains (beyond Lipschitz category). To get the weak maximum principle for equations with such coefficients and domains, we first suggest minimal integrability assumptions on the lower-order coefficients without any extra structural conditions on b := (b 1 , . . . , b n ), such as div b ≤ 0. We suppose that a i , b i ∈ L q,r , n q + 2 r ≤ 1. Notably, in the case n q + 2 r = 1, we establish the weak maximum principle for equations defined on non-smooth domains without smallness assumptions on a i Lq,r or b i Lq,r , provided that 2 ≤ n < q ≤ ∞. In the case q = n ≥ 3 with r = ∞, we obtain the same result under a smallness assumption on a i Ln,∞ and b i Ln,∞ . For the case n = 1, see Remark 2.5. In contrast, for the elliptic conormal derivative problem, if n ≥ 3, we do not require the smallness assumption on a i Ln or b i Ln . The choice of a i , b i ∈ L n , n ≥ 3, is optimal in the sense that, for a i , b i ∈ L n−ε , weak solutions may not be well-defined unless they have sufficient smoothness. Concerning this for the parabolic case, see Remark 2.2. We note that there is a research activity in the case 1 ≤ n q + 2 r < 2 for parabolic problems. Indeed, a version of the strong maximum principle is proved for any Lipschitz solutions in [21]. In this case the additional assumption div b ≤ 0 in the sense of distributions is imposed.
Secondly, regarding the maximum principle on non-smooth domains, we consider John domains. They can be defined in many equivalent ways and we refer the reader to [16,17] for various definitions of John domains. Definition 1.1. A bounded domain Ω in R n is a John domain with center z 0 ∈ Ω and constant K ≥ 1 if for each z ∈ Ω \ {z 0 }, there is a rectifiable curve γ(z, z 0 ) ⊂ Ω connecting z and z 0 such that where d(x) := dist(x, ∂Ω), and |γ(z, x)| denotes the length of the subcurve γ(z, x) ⊂ γ(z, z 0 ) connecting z and x.
Note that it follows from (1. If there is no confusion, hereafter, we shall simply say "John domain" instead of saying "John domain with center z 0 and constant K". See [1,17]. In general, the Hausdorff dimension of the boundary of a John domain in R n can be strictly larger than n − 1. The boundary of a (planar) John domain may contain an interior cusp, while exterior cusps are ruled out. Moreover, John domains satisfy an optimal geometric condition for the Sobolev-Poincaré inequality to hold. See Proposition 3.1 and Remark 3.2. We point out that if all lower-order coefficients are bounded, then the weak maximum principle for elliptic and parabolic conormal derivative problems is established, for instance, on Lipschitz domains. We refer to [19,Chapter VI] for the parabolic case and [20,Chapter 5] for the elliptic case. In [3], the authors proved a generalized maximum principle for degenerate parabolic operators with discontinuous coefficients on a Lipschitz domain, when n ≥ 2. Due to the degeneracy, the indices for the integrability of lower-order coefficients are implicitly given. In this paper, the case n = 1 is also mentioned and, for n ≥ 3, the smallness of a i Ln,∞ and b i Ln,∞ required for the maximum principle is explicitly estimated. We believe that some of our results belong to the folklore and are more or less known to the experts. However, it has been very hard to find a specific reference and we anticipate that our results fill a gap in the literature.
Finally, a few remarks are in order regarding the proofs of our main results. To prove the critical case a i , b i ∈ L q,r with n q + 2 r = 1, r = ∞, of the main results, we use a decomposition on a i and b i , that is, for instance such that b i 1 with its small norm and b i 2 ∈ L ∞ . See the proof of Theorem 2.1 and Lemma 6.2. In addition, we prove a multiplicative version of the Gagliardo-Nirenberg-Sobolev inequality on John domains (Lemma 3.4), which is the main tool of the proof. Lemma 3.4 clearly holds on W 1 p -extension domains. According to Jones' results [15], uniform domains (as mentioned earlier, a strict subclass of John domains) are W 1 p -extension domains and these two domains are equivalent only for bounded and finitely connected domains in the plane. We refer to [10,[12][13][14] for more details.
A brief outline of the paper is as follows. In Section 2, we state the main results (Theorems 2.1 and 2.6). Section 3 is devoted to some embeddings for parabolic function spaces based on multiplicative inequalities. We prove Theorem 2.1 for the parabolic problem and further discuss the critical case in Section 4. The proof of Theorem 2.1 for elliptic problems is in Section 5. Finally, in Section 6, we give proofs for subtle issues.
2. Main results. The parabolic function spaces considered here are following. Let Ω T = Ω × (0, T ) be a parabolic cylinder. Let q, r ≥ 1 and consider the Banach space with the norm v Lq,r(Ω T ) := v(·, t) Lq(Ω) Lr(0,T ) . L q,q (Ω T ) will be denoted by L q (Ω T ). W 1,0 2 (Ω T ) is the Hilbert space with the scalar product Let p ≥ 1 and consider the Banach space The continuity in t of a function v(x, t) in the norm of L 2 (Ω) means that v(·, t + ∆t) − v(·, t) L2(Ω) → 0 as ∆t → 0.
We now present the main results of this paper, the first of which is the weak maximum principle for parabolic equations with the conormal derivative boundary condition. and for all nonnegative ϕ defined on Ω T such that where u + := max{u, 0}.
Remark 2.2. The condition (2.1) is unavoidable in terms of well-posedness of the integrals. In fact, we have (Ω T ) < +∞ by Proposition 3.5.

Remark 2.3.
As is seen in the proof of Theorem 2.1, the inequality (2.2) is required to hold only for uv in place of ϕ, where u, v ∈ V 1,0 2 (Ω T ). In this case, from the definition of V 1,0 2 (Ω T ) and Proposition 3.5, we see that u, v ∈ L 2q Also note that by Hölder's inequality, (2.2) is well-defined.
Remark 2.4. In the case q = n ≥ 3 with r = ∞, we obtain the same result under an additional smallness assumption on |a i | + |b i | Lq,r(Ω T ) . See Section 4.
Remark 2.5. Note that John domains are irrelevant when n = 1. In this case, Theorem 2.1 still holds, provided that q ∈ [2, ∞] and r ∈ [2, ∞] satisfy (2.1) and that Ω is a bounded open interval. The proof is the same as that of Theorem 2.1 for n ≥ 2 using the embedding inequality (3.3) for n = 1. One can find a proof of the inequality (3.3) in [18,Chapter II] when Ω is a bounded open interval or has a piecewise smooth ∂Ω in the case n ≥ 2.
We next state the weak maximum principle for second-order elliptic equations in divergence form with the conormal derivative boundary condition.
Theorem 2.6. Let Ω be a John domain in R n , n ≥ 2. Suppose that a i , b i , and for all nonnegative ϕ ∈ W 1 2 (Ω). Then either u is a constant or else u ≤ 0 in Ω.
As mentioned in the introduction, the result concerning elliptic conormal derivative problems does not require any smallness assumptions on the L n norms of lower-order coefficients in case q = n ≥ 3. See Section 5. We finally remark that Theorem 2.6 is a natural extension of [20,Theorem 5.15] to both unbounded coefficients and non-smooth domains.
3. Multiplicative inequalities. In this section, we provide some multiplicative inequalities on John domains. To do this, we first introduce the Sobolev-Poincaré inequality on John domains.
Remark 3.2. John domain satisfies an optimal geometric condition (in some sense) for the Sobolev-Poincaré inequality (3.1) to hold. If Ω is a simply connected plane domain of finite area, for instance, the necessary and sufficient condition for the validity of (3.1) is that Ω is a John domain. Similarly if Ω satisfies an additional assumption -the so-called separation property (see [4]), then John domain is optimal in the higher dimension case. We refer to [4,5,11,17] for more details.
with average value zero on a set E ⊂ Ω of positive volume. For N > p ≥ 1 and N ≥ n, there is a constant C, depending only n, p, N , c 1 , |Ω|, and |Ω|/|E|, such that Proof. Find p 1 such that This is possible because p 1 = p if p ∈ [1, n). Otherwise, one can find p 1 such that 1 < p 1 < n ≤ p because N np N p + N n − np < n.
We next state embedding inequalities for the parabolic spaces V p (Ω T ). One can prove Proposition 3.5 in the same way as in the proof of Proposition 3.4 in [6, Chapter 1] with Theorem 2.1 replaced by Lemma 3.4. For the reader's convenience, we give a short proof.

Proposition 3.5.
Let Ω be a John domain in R n , n ≥ 2. Let N and p be constants such that N > p ≥ 1 and N ≥ n. There exists a constant C, depending only on n, p, N , c 1 , |Ω|, and T , such that, for every In particular, we have that

3)
where the constants q ∈ (n, ∞] and r ∈ [2, ∞) are linked by n q + 2 r = 1. Moreover, the inequality (3.3) holds for q = n and r = ∞ if n ≥ 3. This along with Young's inequality implies the desired inequalities. In particular, the inequality (3.3) for q = ∞ and r = 2 follows directly from the definition of V 2 (Ω T ).

Proof of Theorem 2.1.
There is a well-recognized inconvenience when dealing with weak solutions of parabolic problems: weak solutions may not be differentiable in the time variable. Regarding this, we adopt, whenever needed, the so-called Steklov average of weak solutions. More precisely, due to the Steklov average and a proper smooth cut-off function depending only on the time variable, it may be possible to take a weak solution, even any function in V 1,0 2 (Ω T ), as a test function. We refer to [18,Chapter III] It follows from the definitions that Γ k ⊂ {u > k}, Du = Dv in Γ k , and Dv = 0 If v can be taken as a test function, the inequality (2.3) can be written as Since (2.2) is valid for uv, in place of ϕ, we have that From this inequality we derive There are a few comments. The inequality (4.1) can be derived by a standard argument based on the Steklov average or a proper mollification technique. In fact, (4.1) is obtained by the method presented in [18, pages 142-143 and 182-183] for the problems subject to the Dirichlet boundary condition. Since we are dealing with the Neumann boundary condition, for the reader's convenience we give a proof of (4.1) in Appendix. See Section 6.1.
We only deal with the case n q + 2 r = 1 with q ∈ (n, ∞] and r ∈ [2, ∞). Otherwise, that is, if n q + 2 r < 1, one can find Q ∈ (n, q) and R ∈ (2, r) such that n Q + 2 R = 1. Then a i , b i , |c| 1/2 ∈ L Q,R (Ω T ). Moreover, the sign condition (2.2) is satisfied for all nonnegative ϕ on Ω T such that Note that even if n/q + 2/r < 1 with r = ∞, we have n/Q + 2/R = 1 with R < ∞ because Q is to be found so that Q > n.

DOYOON KIM AND SEUNGJIN RYU
We decompose a i and b i as follows (for a detailed proof of the decompositions, see Lemma 6.2). For ε > 0 (Ω T ) ≤ Λ 2 for some Λ 1 and Λ 2 . Then it follows from (4.1) that (4.2) By Proposition 3.5 with n q + 2 r = 1, we have that v L 2q Hence, I 1 is estimated as follows: By Young's inequality, we see that for any γ > 0 where Λ * = max{Λ 1 , Λ 2 }. Now, we combine (4.2), (4.4), and (4.5) to get Now we select ε and γ satisfying Thus we have Combining (4.3) and (4.6) yields v L 2q where by Hölder's inequality, for some C > 0 and α > 0, that is, Therefore, it follows that |Γ k | is bounded from below by a positive constant, independent of k, so is {u > k} because Γ k ⊂ {u > k}. This shows that the supremum of u on Ω T is finite (otherwise, u is not, for instance, in L 2 (Ω T )). We now see that Γ k ⊂ Γ l whenever sup BΩ T u + ≤ l < k < sup Ω T u. Hence has a positive measure. Set Since u = sup Ω T u on Γ * , Γ ⊂ Γ * , and Du = 0 on Γ , we have that u = sup Ω T u on Γ as well as Du = 0 on Γ , which is a contradiction. Thus, sup Ω T u ≤ sup BΩ T u + . The theorem is proved.
In the remaining part of this section, we briefly discuss the critical case q = n ≥ 3 with r = ∞. As mentioned in Remark 2.4, we obtain the same result under a smallness assumption on |a i | + |b i | Lq,r(Ω T ) .
To show this, let us assume that there exists a constant k satisfying sup BΩ T u + ≤ k < sup Ω T u. We again derive (4.1), from which we get (4.7)

DOYOON KIM AND SEUNGJIN RYU
Then by Proposition 3.5 with q = n ≥ 3 and r = ∞, and the definition of · V2(Ω T ) , we obtain from (4.7) that where we used the fact that δ < 1. Now, we additionally assume that Then we have 5. Proof of Theorem 2.6. In this section, we prove the weak maximum principle for second-order elliptic equations in divergence form with the conormal derivative boundary condition.
Proof of Theorem 2.6. It suffices to prove that u is constant if sup Ω u > 0. Thus, to get a contradiction, assume that sup Ω u > 0, but u is not constant. In this case there has to be a constant k 1 such that 0 < k 1 < sup Ω u and |{u < k 1 }| > 0. Now we write (2.5) as follows.
Also define Then we have that Dv = w a.e. in Ω. We write It follows by the definitions that Γ k ⊂ {u > k}, Du = Dv a.e. in Γ k , and Dv = 0 a.e. in Ω \ Γ k . Hence we have where v = 0 on a set of positive volume because of the choice of k. Now, we decompose a i and b i as follows. For ε > 0 for some L 1 and L 2 . From these decompositions and the ellipticity of a ij , we have that δ Using Hölder's inequality, Combining (5.4), (5.5), and (5.6) yields for all γ > 0. Now, we take ε and γ such that 0 < ε < δ 8C * and 0 < γ < δ 4(L 1 + L 2 ) .
Thus, we obtain that Combining (5.1) and (5.7) yields v L 2q Hence we have that v L 2q It follows that |Γ k | is bounded below by a positive constant, independent of k, so is {u > k} because Γ k ⊂ {u > k}. This shows that the supremum of u on Ω is finite. Now we see that Γ k ⊂ Γ l whenever l < k and l, k ∈ [k 1 , sup Ω u). Hence has a positive measure. Set Since u = sup Ω u on Γ * , Γ ⊂ Γ * , and Du = 0 on Γ , we have that u = sup Ω u on Γ as well as Du = 0 on Γ , which is a contradiction. Therefore, u is constant.
By the Lebesgue dominated convergence theorem v − v k Lq,r(Ω T ) → 0 as k → ∞.
Then we take v 1 = v − v k and v 2 = v k for a sufficiently large k.