Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces

We prove the unique solvability in weighted Sobolev spaces of non-divergence form elliptic and parabolic equations on a half space with the homogeneous Neumann boundary condition. All the leading coefficients are assumed to be only measurable in the time variable and have small mean oscillations in the spatial variables. Our results can be applied to Neumann boundary value problems for {\em stochastic} partial differential equations with BMO$_x$ coefficients.


Introduction
In this paper, we study L p estimates for elliptic and parabolic equations in non-divergence form: with the homogeneous Neumann boundary condition, where λ is a nonnegative constant and R d + is a half space defined by R d + = {x = (x 1 , · · · , x d ) = (x 1 , x ′ ) : x 1 > 0}. We consider the equations in weighted Sobolev spaces with measures µ d (dx) = x θ−d 1 dx and µ(dx dt) = x θ−d 1 dx dt in the elliptic and parabolic cases, respectively, for some θ ∈ (d−1, d−1+p).
Krylov [17] first studied Laplace's equation and the heat equation in weighted Sobolev spaces H γ p,θ and H γ p,θ ; see Section 2 for precise definitions. After [17], there has been quite a few work on the solvability theory for elliptic and parabolic equations in weighted Sobolev spaces, for instance, see [12,15,13,10]. In particular, the authors of [13,10] studied secondorder parabolic equations with the Dirichlet boundary condition in weighted Sobolev spaces with leading coefficients having small mean oscillations. The motivation of such theory came from stochastic partial differential equations (SPDEs) and is well explained in [16].
Recently, Dong and Kim [4] studied both divergence and non-divergence type elliptic and parabolic equations on a half space in weighted Sobolev spaces with the Dirichlet boundary condition. The coefficients in [4] are contained in a larger class than those in [13,10]. Namely, the leading coefficients are assumed to be only measurable in t and x 1 except a 11 , which is measurable in either t or x 1 , where x 1 is the normal direction. Kozlov and Nazarov [14] considered an oblique derivative problem for non-divergence type parabolic equations on a half space with coefficients discontinuous in t (and continuous in x) in a weighted Sobolev space. Their proof is based on a careful investigation of Green's functions. In this paper, we extend the result in [14] to a more general setting. Namely, the coefficients considered in this paper are measurable in the time variable and have small mean oscillations with respect to a weighted measure in the spatial variables. We call this class of coefficients BMO x . The weight, for instance, for the parabolic case is µ(dx dt) = x θ−d 1 dx dt, where θ ∈ (d − 1, d − 1 + p). The condition θ ∈ (d − 1, d − 1 + p) is sharp even for the heat equation; see [17]. We note that the coefficients a ij in [4] also have small mean oscillations with respect to a weighted measure as functions of x ′ ∈ R d−1 (whereas, in this paper as functions of x ∈ R d ), but the size of the modulus of regularity of a ij is proportional to the distance to the boundary. See Assumption 2.1 and Remark 2.2.
Since the counterexamples of Ural'ceva [26] and Nadirashvili [25], particular types of discontinuous coefficients have been considered for the solvability of equations. One type of discontinuous coefficients, which has been widely considered, is the class of vanishing mean oscillation (VMO) coefficients. The study of equations with VMO coefficients was initiated by Chiarenza, Frasca, and Longo [1,2]. In [18] Krylov gave a unified approach to investigating parabolic and elliptic equations in unweighted Sobolev spaces with coefficients that are measurable in the time variable and have small mean oscillations with respect to the usual Lebesgue measure in the spatial variables (BMO x with respect to the Lebesgue measure); see also [19]. In fact, the coefficients in [18] are called VMO x coefficients, but their mean oscillations in x do not have to vanish as the radii of cylinders go to zero. For more related work about L p theory with BMO x or partially BMO x coefficients for parabolic systems and higher-order parabolic systems, we refer the reader to [7,8,6,5] and the references therein.
Our proof is in the spirit of the approach by Krylov [18]. The key point of such approach is to establish mean oscillation type estimates for equations with simple coefficients, i.e., coefficients are only measurable as functions of t. Then we apply a perturbation argument, which is well suited to the mean oscillation estimates, to deal with BMO x coefficients. Finally we obtain the desired L p estimates by applying the Fefferman-Stein theorem on sharp functions and the Hardy-Littlewood maximal function theorem in weighted L p spaces.
Here one of the main steps is to get the mean oscillation estimates of D 2 u. For a simple equation the Neumann boundary condition D 1 u = 0 on {x 1 = 0}, we treat DD 1 u and D 2 x ′ u separately. We estimate DD 1 u as follows. Differentiating the equation above with respect to x 1 , it is easily seen that D 1 u satisfies the divergence type parabolic equation . Therefore, we can apply a result in [4] to obtain the mean oscillation estimates of DD 1 u. On the other hand, the estimates of D 2 x ′ u are much involved. We treat the mean oscillations of D 2 x ′ u in the x 1 variable and x ′ variables differently. By integrating by parts and the Poincaré inequality in weighted spaces, we manage to bound the mean oscillations of D 2 x ′ u in the x 1 variable by the maximal functions of DD 1 u. For the mean oscillations in x ′ variables, we write the equation in the following form which can be regarded as a non-divergence type parabolic equation in R × R d−1 . Then by applying an interior estimate result without weights for D 2 u, where u is, as a function of x ′ ∈ R d−1 , a solution of a non-divergence type equation in the whole space R d−1 (see, for instance, [18]), we bound the mean oscillations of D 2 x ′ u in the x ′ variables by the maximal functions of f , DD 1 u and D 2 x ′ u. As an application of our results, in a forthcoming paper we are going to study non-divergence form SPDEs in weighted or unweighted Sobolev spaces with the Neumann boundary condition. A particular case is the solvability of SPDEs in the form where w k t are independent one-dimensional Wiener processes, a ij , b i , and c satisfy the same conditions as in the current paper, and f , Dg k , g k ∈ L p,θ (−∞, T ); see the definition of the L p,θ space at the beginning of Section 2. We note that SPDEs in weighted Sobolev spaces with the Dirichlet boundary condition have been studied extensively in the past fifteen years. We refer the reader to [22,21,20,11] and the references therein.
This paper is organized as follows. In the next section, we introduce some notation and state our main results. In Section 3, we obtain the mean oscillation estimates for D 2 x ′ u and DD 1 u separately for a parabolic equation with simple coefficients. In Section 4, we prove our main theorem (Theorem 2.3).

preliminaries and main results
Throughout the paper we use, for example, the following Einstein summation convention: a ij D ij u = i,j a ij D ij u. We introduce some notation used in the paper. As hinted in the introduction, a point in R d is denoted by x = (x 1 , · · · , x d ), and also by x = ( , and Q r = Q r (0). Similarly, we define Q + r (a) and Q + r . By a + we mean max{a, 0}.
Throughout the paper, we assume that the leading coefficients a ij are bounded, measurable, and satisfy the ellipticity condition: To introduce the function spaces used in this paper, we first recall the weighted Sobolev spaces H γ p,θ introduced in [17]. If γ is a non-negative integer For a general real number γ, H γ p,θ is defined as follows. Take and fix a nonnegative function ζ ∈ C ∞ 0 (0, ∞) such that for all x 1 ∈ R. For any γ, θ ∈ R and p ∈ (1, ∞), let H γ p,θ be the set of all functions u on R d + such that where · γ,p is the norm in the Bessel potential space H γ p (R d ). For any a ∈ R, let M α be the operator of multiplication by (x 1 ) α and M := M 1 .
Our solution spaces are defined as follows. For the elliptic case, we set For the parabolic case, We also use the following Hölder spaces. For a function f on D ⊂ R d+1 , define where a, b ∈ (0, 1]. For a ∈ (0, 1], we set The space corresponding to · a/2,a,D is denoted by C a/2,a (D).
Throughout the paper, we use the weighted measures: . Now we state our regularity assumption on the leading coefficients. For a function g on R d+1 Then we define the mean oscillation of g in Q + r (s, y) with respect to x as (2.1) Using the notation above with a ij in place of g, we state the following regularity assumption on a ij with a sufficiently small parameter ρ > 0 to be specified later.
Note that under this assumption, the coefficients a ij may not have any regularity with respect to t. Remark 2.2. While we have a fixed size of the modulus of regularity R 0 above, in [4] the size of the modulus is proportional to the distance from the boundary to the location where the mean oscillations of a ij are measured. To express this, one can replace R in (2.1) by y 1 R. This means, in particular, that the coefficients a ij in [4] are allowed to be much rougher near the boundary than those in this paper.
For lower-order terms, we assume that the coefficients b i and c are only measurable (without any regularity assumptions) and bounded so that The following theorems are our main results, the first of which is the unique solvability of parabolic equations.
In particular, when b i = c = 0 and a ij = a ij (t), we can take λ 0 = 0.
(ii) For any λ > λ 0 and f ∈ L p,θ (−∞, T ), there is a unique solution u ∈ W 1,2 p,θ (−∞, T ) to the equation (2.2). We now present our results for elliptic equations, where the coefficients are independent of t. Since Assumption 2.1 (ρ) does not concern the regularity of coefficients in t, we still require the coefficients to satisfy Assumption 2.1 (ρ). By adapting, for example, the proof of [18, Theorem 2.6] to the results above for parabolic equations, i.e., by regarding elliptic equations as steady state parabolic equations, we obtain the following theorem for elliptic equations.

equations with coefficients independent of x
In this section, we deal with equations in the form Note that now the coefficients a ij depend only on t. Let us state several technical lemmas. The first one is Hardy's inequality, which can be found in [24].
We summarize Lemmas 4.2 and 4.3 in [4] as the following results, which were proved by localizing the results in [9,3] and using the Sobolev embedding theorem.
Using Lemma 3.2, we obtain the following mean oscillation type estimate.
with the Dirichlet boundary condition u = 0 on {x 1 = 0}. Then there exists a constant C = C(d, δ, p, θ, α) such that for any r < 1, Proof. From Lemma 3.2, we obtain Combining the two inequalities above, we prove the corollary.
Before we state the next theorem, we introduce a function space. For Now we state a special case (a ij = a ij (t)) of [4, Theorem 3.9], where a ij are allowed to be merely measurable in (t, x 1 ) except that a 11 = a 11 (t) or a 11 = a 11 (x 1 ). Note that in the theorem below there is no specification of the boundary condition, but functions in the solution space H 1,λ p,θ (−∞, T ) necessarily satisfies u = 0 on the boundary. Hence the theorem is about the Dirichlet boundary value problem for divergence type equations in weighted Sobolev spaces.
κr (y 1 ) with the Dirichlet boundary condition v = 0 on {x 1 = 0}. By a scaling argument, it is sufficient to set κr = 8. We consider two cases.

(3.6)
For the same reason as before, we may assume that w is smooth and so isû = D 1 u − w. It is easily seen that u t − a ij D ijû = 0 in Q 1/2 (y 1 ).
It remains to estimate D ij u with i, j > 1. Let us first state a Poincaré inequality in weighted L p spaces.
Proof. When α ∈ [0, ∞), the inequality is proved in [13, Lemma 4.1] with a missing constant depending on d. For the sake of completeness, we here present a proof when α ∈ (−1, ∞). Since the weight is with respect to x 1 , we only prove (3.8) for d = 1. In fact, to prove (3.8) for d > 1 we just need to combine the case when d = 1 with the unweighted Poincaré inequality. Due to scaling, it suffices to prove (3.8) with r = 1. We further assume a ∈ (0, 2). Indeed, if a ≥ 2, the inequality (3.8) is equivalent to the usual Poincaré inequality without weights. For each x, y ∈ ((a − 1) + , a + 1), by Hölder's inequality, where 1/p + 1/q = 1 and C = C(p, α). Then, to conclude (3.8), we integrate the above inequalities with respect to ν(dx) and ν(dy), and use the fact that a ∈ (0, 2).
In the sequel, we denote the standard parabolic cylinder in R d+1 aŝ Q d+1 r (X) = (t − r 2 , t) ×B r (x), whereB r (x) is the Euclidean ball in R d with radius r and center x. For a function f on R d+1 , we define the average of f inQ d+1 r (X) without weight as Then there exists a constant C = C(d, δ, p) such that for any κ ≥ 4, r > 0, we have Proof. See [23, Theorem 5.1].
To estimate D 2 x ′ u, we introduce the following notation. For in Q + κr (y 1 ) with D 1 u = 0 on {x 1 = 0}. Then there exist constants C = C(d, δ, p, θ) and α = α(d, p, θ) > 0 such that . Proof. By scaling, we may assume that κr = 8. In this case r ≤ 1/4 because κ ≥ 32. Let η = η(t, We consider two cases. Case (i): y 1 ∈ [0, 1]. By Hölder's inequality and the triangle inequality, it is easily seen that x ′ u(s, y)| p µ(dx dt) µ(dy ds). (3.11) Since, by the triangle inequality, the right-hand side of (3.11) is bounded by C(I + II), where Let us now estimate I and II separately and first consider II. By integrating by parts and Hölder's inequality, where 1/p + 1/q = 1. From (3.10), we have We plug the two inequalities above into II to achieve Applying Lemma 3.6 with d = 1, we get because |Q + r (y 1 )| ≥ Cr d+θ+2 . Next, we estimate I, which can be written as By Hölder's inequality, we have We now estimate I(x 1 ) by writing the equation (3.9) as Here, for each fixed x 1 ∈ (0, ∞), we regard u(t, x 1 , x ′ ) as a solution to the above equation defined in R × R d−1 . Then thanks to Theorem 3.7 with d in place of d + 1 and the triangle inequality, we get is the unweighted average of g with respect to (t, x ′ ) in the d dimensional parabolic cylinder with radius κr. We plug the inequality above into (3.12) to obtain Bearing in mind that µ 1 (B 1+ r (y 1 )) ≥ Cr ζ+1−d , where ζ = max{d, θ}, and r = 8/κ, we obtain from (3.13) that . By the definition of ζ, p + d − ζ − 1 > 0. Combining the estimates of I and II, we complete the proof of Case (i) with α = p + d − ζ − 1.
Case (ii): y 1 > 1. Set v = D k u, k = 1, . . . , d, and note that v satisfies the divergence type equation in B κr (y 1 ). Since this is an interior estimate, we do not care about the boundary value of v on {x 1 = 0}. Then we repeat the second part of the proof of Lemma 3.5 with D k in place of D 1 . The lemma is proved.

proof of theorem 2.3
In this section, we deal with operators with coefficients depending on both x and t. We denote L = a ij D ij and assume p > 1, λ ≥ 0, and θ ∈ (d − 1, d − 1 + p).
First we need the following lemma.
The lemma is proved.
The following two lemmas are mean oscillation estimates for the operator a ij (t, x)D ij . We prove them by using the mean oscillation estimates for a ij (t)D ij proved in Section 3 combined with a perturbation argument.
Proof. Throughout the proof, we assume Q + R (X 1 ) ∩ Q + κr (Y ) = ∅. Otherwise, (4.2) holds trivially. Fix a z ∈ R d + and set Then we have It follows from Lemma 3.5 with α = 1/2 and a translation of the coordinates that where C 0 = C 0 (d, δ, p, θ). By the definition off , the triangle inequality, and the fact that u vanishes outside Q + R (X 1 ), we have where is the indicator function of Q + R (X 1 ). Denote B + to be B + κr (y) if κr < R, or to be B + R (x) otherwise. Define Q + in the same fashion. We note that It is obvious if κr < R, i.e., Q + = Q + κr (Y ). If κr ≥ R, the inequality is proved in Lemma 4.1. Combining (4.3) and (4.4) and taking the average of each term with respect to z in B + , we reach Since u vanishes outside Q + R (X 1 ), by Hölder's inequality, we get By the boundedness of a ij , Hölder's inequality, and the definition of osc x , we have where C 1 = C 1 (d, δ, p, β). From (4.7), (4.8), and (4.5), we obtain Combining (4.6) and (4.9), we get (4.2). The lemma is proved.
Next we recall the Hardy-Littlewood maximal function theorem and the Fefferman-Stein theorem on sharp functions. Let For a function g defined on R d+1 + , the weighted (parabolic) maximal and sharp functions of g are given by For any θ > d − 1 and g ∈ L p,θ (R d+1 + ), we have where p ∈ (1, ∞) and C = C(d, p, θ). The first inequality above is known as the Fefferman-Stein theorem on sharp functions and the second one is the Hardy-Littlewood maximal function theorem, for instance see [19,Chapter 3] Now we are ready to prove our main theorem.
Proof of Theorem 2.3. By the method of continuity, it is enough to prove the a priori estimate (2.3). Moreover, since the set of functions in p,θ (−∞, T ), we only need to prove (2.3) for infinitely differentiable functions with compact support. In this case, the proof of (2.3) can be divided into several steps.
Hence it is sufficient to show that for large λ λ u p,θ ≤ C 0 f p,θ .
Step 4: We remove the assumption that b i = c = 0 by moving the terms of b i and c to the right-hand side By the conclusion in Step 3 with b i = c = 0, there exists λ 0 = λ 0 (d, δ, p, θ, R 0 ) such that for any λ ≥ λ 0 , u t p,θ + λ u p,θ + √ λ Du p,θ + M −1 D 1 u p,θ + D 2 u p,θ ≤ C 0 f p,θ + C 0 K Du p,θ + C 0 K u p,θ .
Finally, in the case when b i = c = 0 and a ij = a ij (t), by using a scaling argument we can take R 0 = 0. The theorem is proved.