Conormal derivative problems for stationary Stokes system in Sobolev spaces

We prove the solvability in Sobolev spaces of the conormal derivative problem for the stationary Stokes system with irregular coefficients on bounded Reifenberg flat domains. The coefficients are assumed to be merely measurable in one direction, which may differ depending on the local coordinate systems, and have small mean oscillations in the other directions. In the course of the proof, we use a local version of the Poincar\'e inequality on Reifenberg flat domains, the proof of which is of independent interest.


Introduction
We study L q theory of the conormal derivative problem for the stationary Stokes system with variable coefficients: Lu + ∇p = f + D α f α in Ω, Bu + np = f α n α on ∂Ω, where Ω is a bounded domain in R d and n = (n 1 , . . . , n d ) T is the outward unit normal to ∂Ω. The differential operator L is in divergence form acting on column vector valued functions u = (u 1 , . . . , u d ) T as follows: We denote by Bu = A αβ D β un α the conormal derivative of u on the boundary of Ω associated with the operator L. Throughout the paper, the coefficients A αβ = A αβ (x) are d × d matrix valued functions on R d with the entries A αβ ij satisfying the strong ellipticity condition; see (2.1). We assume that the coefficients A αβ are merely measurable in one direction and have small mean oscillations in the other directions (partially BMO). For more a precise definition of partially BMO coefficients, see Assumption 2.1. Stokes systems with this type of variable coefficients may be used to describe the motion of inhomogeneous fluids with density dependent viscosity and two fluids with interfacial boundaries; see [11,12] and the references therein. We note that Stokes systems with variable coefficients can also occur when performing a change of coordinates or when flattening the boundary. See [9].
is available, where the constant N depends only on d, q, and the flatness of ∂Ω. This result is an easy consequence of a Poincaré type inequality on the ball B r (x 0 ) because u can be extended to a function on B r (x 0 ) by setting u ≡ 0 on B r (x 0 ) \ Ω. For the conormal derivative problem, one may consider a boundary Sobolev-Poincaré inequality u − c L dq/(d−q) (Ω∩Br (x0)) ≤ N Du Lq(Ω∩BR(x0)) , 1 ≤ q < d, (1.2) where c = - It is well known that the inequality (1.2) holds if Ω ∩ B r (x 0 ) and Ω ∩ B R (x 0 ) are replaced by a Lipschtiz domain Ω or a Reifenberg flat domain Ω since these domains are in the category of extension domains. It is also known that if Ω is a Lipschitz domain, the above inequality holds with a constant N depending only on d, q, and the Lipschitz constant of Ω; see, for instance, [19,Lemma 8.1]. However, if Ω is a Reifenberg flat domain, it is not quite obvious that the inequality (1.2) follows from the same type of inequality for Ω because the intersection may not retain the same nice properties as those of Ω. We were unable to find any literature dealing with the inequality (1.2) on a Reifenberg flat domain Ω intersected with a ball. To show the exact information on the parameters that the constant N depends on, we provide a proof of (1.2) in Appendix. On the other hand, in the proof of [4,Corollary 3], Byun-Wang used such type of inequality without a proof, referring to the Sobolev inequality on extension domains. See also [5,6], in which conormal derivative problems for parabolic equations are considered. The inequality (1.2) is the key ingredient in establishing reverse Hölder's inequality of conormal derivative problems for the Stokes system; see Section 3.2.
In a subsequent paper, we will study Green functions for the Stokes system with the conormal derivative boundary condition. We note that L q -estimates for boundary value problems play an essential role in the study of Green functions. For instance, in [7], the authors obtained global pointwise estimates of Green functions for elliptic systems with conormal boundary conditions by using L q -estimates for the system.
The remainder of this paper is organized as follows. In Section 2, we state our main result along with some notation and assumptions. In Section 3, we provide some auxiliary results, and in Section 4, we establish interior and boundary Lipschitz estimates for solutions. Finally, in Section 5 we prove the main theorem using a level set argument. In Appendix, we provide the proof of a local version of the Poincaré inequality on a Reifenberg flat domain.

Main results
Throughout this paper, we denote by Ω a domain in the Euclidean space R d , where d ≥ 2. For any x ∈ Ω and r > 0, we write Ω r (x) = Ω∩B r (x), where B r (x) is a usual Euclidean ball of radius r centered at x. We also denote B + We use the abbreviations B r := B r (0) and B + r := B + r (0), where 0 ∈ R d , and Let L be a strongly elliptic operator of the form The coefficients A αβ = A αβ (x) are d × d matrix valued functions on R d with the entries A αβ ij satisfying the strong ellipticity condition, i.e., there is a constant δ ∈ (0, 1] such that for any x ∈ R d and ξ α ∈ R d , α ∈ {1, . . . , d}. We denote by Bu = A αβ D β un α the conormal derivative of u on the boundary of Ω associated with the elliptic operator L. The i-th component of Bu is given by where n = (n 1 , . . . , n d ) T is the outward unit normal to ∂Ω. Let q, q 1 ∈ (1, ∞), q 1 ≥ qd/(q + d), and Ω be a bounded domain in R d . For f ∈L q1 (Ω) d and f α ∈ L q (Ω) d , we say that (u, p) ∈ W 1 q (Ω) d × L q (Ω) is a weak solution of the problem holds for any φ ∈ W 1 q/(q−1) (Ω) d . Assumption 2.1 (γ). There exists R 0 ∈ (0, 1] such that the following hold. (i) For x 0 ∈ Ω and 0 < R ≤ min{R 0 , dist(x 0 , ∂Ω)}, there exists a coordinate system depending on x 0 and R such that in this new coordinate system, we have that For any x 0 ∈ ∂Ω and 0 < R ≤ R 0 , there is a coordinate system depending on x 0 and R such that in the new coordinate system we have that (2.2) holds, and {y : where x 01 is the first coordinate of x 0 in the new coordinate system.
The main result of the paper reads as follows. and

Auxiliary results
In this section, we derive some auxiliary results. We impose no regularity assumptions on the coefficients A αβ of the operator L.
The lemma below shows that the divergence equation is solvable inW 1 q (Ω) d provided that Ω is bounded.
where the constant N depends only on d and q.
Proof. Assume that Ω ⊂ B R for some R ≥ 1. We denoteḡ = gχ Ω , where χ Ω is the characteristic function. By the well-known result on the existence of solutions of the divergence equation in a ball, there exists v ∈W 1 It then follows that div u = g in Ω. Moreover, we get where I is the d × d identity matrix.

Lemma 3.2.
Let Ω be a bounded domain in R d . Assume that there exists a constant K 0 > 0 such that Then, for any f α ∈ L 2 (Ω) d and g ∈ L 2 (Ω), there exists a unique (u, p) Bu + np = f α n α on ∂Ω. (3.2)

Moreover, we have
Proof. By (3.1),W 1 2 (Ω) can be understood as a Hilbert space with the inner product The proof of the lemma is then nearly the same as that of [  There exists a positive constant R 0 such that the following holds: for any x 0 ∈ ∂Ω and R ∈ (0, R 0 ], there is a coordinate system depending on x 0 and R such that in this new coordinate system (called the coordinate system associated with (x 0 , R)), we have where x 01 is the first coordinate of x 0 in the new coordinate system.
If Ω is a bounded Reifenberg flat domain, then the Poincaré inequality holds over Ω. However, the domain of the Poincaré inequality presented in the theorem below is Ω ∩ B R (x 0 ), x 0 ∈ ∂Ω, which is not a Reifenberg flat domain with the same flatness as that of Ω. Moreover, we need correct information on the parameters on which the constant of the Poincaré inequality depends. Thus, for the reader's convenience, we provide a proof of the theorem in Appendix. Based on the L 2 -estimate and Poincaré inequality in Theorem 3.3, we obtain the following estimates for Du and p.
Proof. We prove only the case x 0 ∈ ∂Ω because the other case is the same with obvious modifications. Without loss of generality, we assume that x 0 = 0. Let R ∈ (0, R 0 /8] and η be a smooth function on R d satisfying By applying η 2 (u − (u) Ω2R ) as a test function to (3.2), and using both Hölder's and Young's inequalities, we obtain for θ ∈ (0, 1) that We extend p by zero on B 2R \Ω. From the existence of solutions to the divergence equation in a ball, there exists w ∈W 1 We extend w to be zero on Ω \ Ω R and apply w as a test function to (3.2) to get Using (3.7) with the fact that and thus, we get from (3.6) that for any θ ∈ (0, 1), where N = N (d, δ, q, θ). This together with (3.6) yields From Hölder's inequality and the Poincaré inequality in Theorem 3.3, it follows that 1 Lq(Ω4R) . Combining (3.8) and the above inequality, we conclude the desired estimate. The lemma is proved.
Using Lemma 3.4 and Gehring's lemma, we get the following reverse Hölder's inequality.
(Ω). Then there exist constants q 0 ∈ (2, q 1 ) and N > 0, depending only on d, δ, and q 1 such that for any x 0 ∈ R d and R ∈ (0, R 0 ], where Dū,p,f α , andḡ are the extensions of Du, p, f α , and g to R d so that they are zero on R d \ Ω. Proof. We fix a constant q ∈ (2d/(d + 2), 2), and set Then, by Lemma 3.4, it follows that for any Using Lemma 3.4 and the fact that 3R ≤ R 0 /8, we obtain (3.9) with B 3R (y 0 ) and B 12R (y 0 ) in place of B R (x 0 ) and B 14R (x 0 ), respectively. Hence, we get the inequality (3.9). If B 2R (x 0 ) ⊂ R d \ Ω, by the definition of Dū andp, (3.9) trivially holds. For x 0 ∈ R d and R ∈ (0, R 0 ], using a covering argument and (3.9) with y ∈ B R (x 0 ) and R/24 in place of x 0 and R, respectively, and taking a sufficiently small where N = N (d, δ). Therefore, by Gehring's lemma (see, for instance, [17, Ch. V]), we get the desired estimate. The lemma is proved.

L ∞ and Hölder estimates
In this section, we prove L ∞ -estimates of Du and p. We set We start with the following boundary estimates. For the corresponding interior estimates, see [11,Section 3]. (4.1) where N = N (d, δ). Moreover, for any 0 < r < R, we have Remark 4.1. In the above lemma and throughout the paper, (u, . By the existence of solutions to the divergence equation in a ball, there exists w ∈W 1 . Applying w as a test function to (4.1), we have and thus, we get (4.2). The inequality (4.3) is deduced from (4.2) in the same way as [11,Lemma 3.7] is deduced from [11, Lemmas 3.4 and 3.6]. The lemma is proved.
Using the standard finite difference argument, we obtain the following estimates for DD x ′ u and D x ′ p. The corresponding interior estimates can be found in [11,Lemmas 4.1 and 4.2].
Since the coefficients are functions of only x 1 , we obtain that δ). This, along with the standard finite difference argument (see, for instance, [14, Section 5.8.2]), implies the desired estimate.
In the lemma below, we obtain L ∞ -estimates for Du and p when (u, p) is a weak solution of L 0 u + ∇p = 0. We also prove Hölder semi-norm estimates for linear combinations of derivatives of u. Indeed, we do not use Hölder semi-norm estimates in this paper, but we present here the results for later use of the estimates in the study of weighted L q -estimates. For the Dirichlet counterpart of the estimates and their application to L q -estimates with Muckenhoupt weights, see [12].
As usual, the Hölder semi-norm of u is defined by Using the fact that A αβ 0 are independent of x ′ ∈ R d−1 , we observe that if u and p are sufficiently smooth, where e α is the α-th unit vector in R d and In other words, we have

then we have
. Proof. For the proof of the assertion (a), we refer to [12, Lemma 4.1 (a)]. To prove the assertion (b), we let r 1 ∈ (1, 2) and i = 2, . . . , d. By Lemma 4.2, we have We then use Lemma 4.2 again as above with r 2 in place of r 1 and with r 1 in place of 2, where 1 < r 2 < r 1 . By repeating this process, we obtain that for any r ∈ [1, 2) and k ∈ {1, 2, . . .}, where N = N (d, δ, r, k). Since the above inequality holds for i = 2, . . . , d, we have for any r ∈ [1, 2) and k ∈ {0, 1, 2, . . .}, where we used Lemma 4.1 for the case when k = 0. Then, using (4.5) and an anisotropic Sobolev embedding theorem with k > (d − 1)/2 (see, for instance, the proof of [13, Lemma 3.5]), we get . Using the relation div u = 0, we get from the above inequality that Now we are ready to prove the assertion (b). From the definition of U and (4.5), it follows that for any k ∈ {0, 1, 2, . . .}, where N = N (d, δ, k). Since L 0 u + ∇p = 0, we obtain by (4.4) that This together with (4.5) yields that D 1 U has sufficiently many derivatives in x ′ with the estimates for any k ∈ {0, 1, 2, . . .}. Combining (4.7) and (4.8), and using the anisotropic Sobolev embedding as above with k > (d − 1)/2, we have Notice from the definition of U that d j=2 By the ellipticity condition (2.1), . Taking · L∞(B + 1 ) of both sides of the above inequality and using (4.6) and (4.9), we conclude that . From this, (4.9), and the fact that . The lemma is proved.  ∞). Moreover, the L 2 norms on the right-hand side of the estimates can be replaced by the corresponding L 1 norms.
Proof of Proposition 5.1. Without loss of generality, we assume that x 0 = 0. Let µ, ν be constants satisfying 1/µ + 1/ν = 1 and 2µ = q 0 , where q 0 ∈ (2, ∞) is a number from Lemma 3.5 that depends only on d, δ, and q. Case 1. B R ⊂ Ω. By Assumption 2.1 (γ) (i), there exists a coordinate system such that where we set Let L 0 be the elliptic operator with the coefficients A αβ 0 and let B 0 be the conormal derivative operator associated with L 0 .
Note that in the ball B R , the hypothesis of Lemma 3.2 is satisfied. Hence, there exists a unique (w, where N = N (d, δ). By Hölder's inequality, the boundedness of A αβ , (5.4), and Lemma 3.5, we have N (d, δ, q). Using this together with (5.5), we obtain (5.2) with W = Dw.
Since the hypothesis of Lemma 3.2 holds onB + R , there exists a unique (ŵ,p 1 ) ∈ . By Hölder's inequality, Lemma 3.5, and the fact that N (d, δ, q). Using this and following the same arguments in the proof of (5.2) in Case 1, one can easily show that Hence, defining (W, p 1 ) in Ω R by We write y 0 = (γR, 0, . . . , 0) ∈ R d . We then have
The proof of Theorem 2.1 relies on the following estimate of level sets.
By Proposition 5.1, (Du, p) admits a decomposition with the estimates . From this together with Chebyshev's inequality, it follows that which contradicts (5.7) if we choose a sufficiently large κ.

By
where N 2 = N 2 (d, δ, q). We then obtain that which contradicts (5.7) if we choose a sufficiently large κ.

Appendix: Poincaré inequality
In this section, we provide a detailed proof of a local version of the Poincaré inequality on a Reifenberg flat domain. Throughout the appendix, we denote the line segment connecting x and y by xy. such that and (6.1) is satisfied for all z ∈ xx k0 with k 0 in place of k. Here, ℓ(x x k0 ) is the length of x x k0 .
Proof. Set ρ k = R/2 k . By Assumption 3.1, there exists a coordinate system associated with (x 0 , ρ k ) satisfying (3.4) with ρ k in place of R. Then, let x k be the intersection of the boundary of the ball B ρ k /2 (x 0 ) and the positive y 1 -axis of the coordinate system associated with (x 0 , ρ k ). Note that, as k changes, the y 1 -direction of the coordinate system may differ because, for each k ∈ {0, 1, . . .}, a coordinate system is chosen depending on (x 0 , ρ k ). Since it holds that By repeating the above argument we choose In particular, x k and x k+1 are located in the half space where the coordinate system (y 1 , y ′ ) is that associated with (x 0 , ρ k ). Indeed, it is clear that x k ∈ H + k . If x k+1 ∈ R d \ H + k , then by Assumption 3.1, it belongs to {y : x 01 − γρ k < y 1 ≤ x 01 + 5γρ k } , which implies that dist(x k+1 , ∂Ω) ≤ 6γρ k .
By applying Lemma 6.1 to Ω R (x 0 ) we obtain and, in case k 0 ≥ 1, where k ∈ {0, . . . , k 0 − 1}. To construct a curve joining x and z 0 , we connect x to z 0 by the line segment x x k0 and x k x k+1 , k = 0, 1, . . . , k 0 , up to x 0 , and then connect x 0 to z 0 by the line segment x 0 z 0 . Precisely, we connect x and z 0 by the curve where η 1 is a curve defined by Since x 0 ∈ B 5R/4 and |z − x 0 | ≤ R/2 for z ∈ η 1 , we have We also obtain by (6.12) and (6.14) that Note that the line segment x 0 z 0 satisfies for z ∈ x 0 z 0 . Indeed, since x 0 ∈ B 5R/4 and x 0 ∈ ∂B R/2 (x 0 ), it follows that x 0 ∈ B 7R/4 . This together with the fact that z 0 ∈ B R/2 shows that Moreover, by the choice of z 0 and (6.3) with k = 0, we have Using this together with the fact that x 0 , z 0 ∈ Ω 2R , we have This along with (6.16) proves (6.15). From (6.15) and the fact that η 1 ⊂ Ω 7R/4 and ℓ(η 1 ) ≤ 2R, it follows that η ⊂ Ω 7R/4 and ℓ(η) ≤ 5R.
The proposition is proved.
By setting y =η(s; x, z) in the above identity, and using the fact that y ∈ Ω 2R and φ L1(R d ) ≥ N (d, h) Using the estimate for fractional integrations, we see that u −ū L q * (ΩR) ≤ N ∇u Lq(Ω2R) , where q * = dq/(d − q) and N = N (d, q). Therefore, by using the fact that u − (u) ΩR L q * (ΩR) ≤ u −ū L q * (ΩR) , we obtain the desired estimate. The theorem is proved.