Neumann functions for second order elliptic systems with measurable coefficients

We study Neumann functions for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domain or a Lipschitz graph domain. We establish existence, uniqueness, and various estimates for the Neumann functions under the assumption that weak solutions of the system enjoy interior H\"older continuity. Also, we establish global pointwise bounds for the Neumann functions under the assumption that weak solutions of the system satisfy a certain natural local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is in fact equivalent to the global pointwise bound for the Neumann function. We present a unified approach valid for both the scalar and the vectorial cases.


Introduction
In this article, we are concerned with Neumann functions (or sometimes called Neumann Green's function) for divergence form, second order elliptic systems with bounded measurable coefficients in a bounded Lipschitz domains or a Lipschitz graph domain. More precisely, we consider Neumann functions for the m × m elliptic systems Here, we assume that the coefficients are measurable functions defined in the whole space R d satisfying the strong ellipticity and the uniform boundedness condition; see Section 2 for their precise definitions. We do not assume that the coefficients of the system (1.1) are symmetric. We will later impose some further assumptions on the system (1.1) in the case when m > 1 but not explicitly on its coefficients. Analogous to the role of Green's functions in the study of Dirichlet boundary value problem of elliptic equations, Neumann functions play a significant role in the study of Neumann boundary value problem. By this reason, the Neumann functions are discussed in many papers, but however, with only a few exceptions, it is assumed that the coefficients and the domains are sufficiently regular. In the case when m = 1, Kenig and Pipher [16] constructed Neumann functions for the divergence form elliptic equations with L ∞ coefficients and derived various estimates for the Neumann functions in the unit ball B. Those estimates are the same sorts of estimates known for the Green's functions as appear in [11,17] and are nicely summarized in [14,Theorem 1.6.3]. Their methods of proof are general enough to allow B to be a bounded star-like Lipschitz domain but however, as pointed out in [19], it is not immediately clear whether they also work for general bounded Lipschitz domains. Also, their methods do not seem to work for unbounded domains such as the half space. On the other hand, Hofmann and Kim [12] recently proved existence and various (interior) estimates for the Green's function of the system (1.1) in arbitrary domains under the assumption that weak solutions of the system (1.1) satisfy an interior Hölder continuity estimate. Their result has been complemented by a very recent article by Kang and Kim [13], where global estimates of Green's functions for the system (1.1) are established under some other (but similar) assumptions. In the case when m = 1, the De Giorgi-Moser-Nash theory for weak solutions implies such estimates and thus, in particular, they were able to reproduce the related classical results of [11,17].
The goal of this article is to present a unified approach for the construction and estimates of Neumann functions of the elliptic systems (1.1) in a bounded Lipschitz domain as well as in an unbounded domain above a Lipschitz graph. As a matter of fact, it is exactly where the strength of our paper lies. By using our unified method, we reproduce the estimates for Neumann functions of scalar equations with L ∞ coefficients in the unit ball presented in [16] as well as those for systems with C α coefficients in C 1,α domains appearing in a recent article [15]. Recently, there have been some interest in studying boundary value problems for divergence form elliptic equations with complex L ∞ coefficients above a Lipschitz graph; see e.g., [1,2,3]. In this context, it is natural to consider Green's functions and Neumann functions for elliptic systems with L ∞ coefficients in a Lipschitz graph domain. In fact, properties of Green's function investigated in [12], for elliptic equations whose coefficients are complex perturbations of real L ∞ coefficients, were used in [1]. However, we are not even able to find a literature dealing with Neumann functions in the half space for scalar elliptic equations with L ∞ coefficients. As we have already mentioned, our method also goes through in that case, and in particular, we derive the estimates for the Neumann function of the scalar elliptic equations with L ∞ coefficients in a Lipschitz graph domain that corresponds to the estimates in [14,Theorem 1.6.3]. We hope this article may serve as a reference for the Neumann functions and their properties, so that it may become a useful tool for other authors.
We shall now describe our main result briefly. Let Ω ⊂ R d with d ≥ 3 be a bounded Lipschitz domain or a Lipschitz graph domain. We first construct the Neumann function of the system (1.1) under the assumption that its weak solutions are locally Hölder continuous. In doing so, we also derive various interior estimates for the Neumann function; see Theorem 3.1 and 5.1. We then show that if the system (1.1) has such a property that weak solutions of Neumann problems with nice data are locally bounded and satisfy a certain natural estimate (see the conditions (LB) and (LB ′ ) in Section 3 and 5), then its Neumann function N(x, y) has the following global pointwise bound; see Theorem 3.6 and 5.2.
Conversely, if the Neumann function has the above pointwise bound, then we prove that the system should satisfy the aforementioned local boundedness property; see Theorem 3.9 and 5.
for some µ ∈ (0, 1]; see Remark 3.8. The organization of the paper is as follows. In Section 2, we introduce some notation and definitions including weak formulations of Neumann problems and the precise definition of Neumann functions of the system (1.1). In Section 3, we state our main theorems including existence and global pointwise estimates for Neumann functions in bounded Lipschitz domains, and their proofs are presented in Section 4. Section 5 is devoted to the study of Neumann function in a Lipschitz graph domain. In the appendix we provide the proofs of some technical lemmas.
Finally, a few remarks are in order. This article is, in spirit, very similar to [12,13], where corresponding results for Green's functions have been established. However, the technical details are very different since Neumann boundary condition is more difficult to handle than the Dirichlet condition. For instance, in [12], the Green's functions are constructed in arbitrary domains but here Neumann functions are constructed only in domains with Lipschitz boundary. We do not treat the case d = 2 in our paper. In dimension two, the Neumann functions should have logarithmic growth and requires some other methods. As a matter of fact, our method breaks down and is not applicable in two dimensional case. One way to overcome this difficulty is to utilize so-called Neumann heat kernel of the elliptic operator defined in a Lipschitz cylinder Ω × (0, ∞) ⊂ R 3 . However, this approach requires first establishing a pointwise bound for Neumann heat kernel that is sharp enough to be integrable in t-variable; see [6,7] for the treatment of Green's functions of elliptic systems in two dimensional domains. This topic will be discussed elsewhere because Neumann heat kernel is an interesting subject in its own right. After submission of the first version of this paper, Taylor et al. [20] constructed the Green's function for the mixed problem for elliptic systems in two dimensions.

Preliminaries
2.1. Basic Notation. We mainly follow the notation used in [12,13]. Let d ≥ 3 be an integer. We recall that a function ϕ : where ϕ : R d−1 → R is a Lipschitz function. Throughout the entire article, we let Ω be a Lipschitz domain or Lipschitz graph domain in R d . For p ≥ 1 and k a nonnegative integer, we denote by W k,p (Ω) the usual Sobolev space. When Ω is a Lipschitz domain, we define the spaceW 1,2 (Ω) as the family of all functions u ∈ W 1,2 (Ω) satisfying ∂Ω u = 0 in the sense of trace. We warn the reader that the spacẽ W 1,2 (Ω) is different from the spaceW 2 1 (B) used in [16]. By using Rellich-Kondrachov compactness theorem, one can easily show that there is a constant C = C(d, Ω) such that (2.1) u L 2 (Ω) ≤ C Du L 2 (Ω) , ∀u ∈W 1,2 (Ω).
The space Y 1,2 (Ω) is defined as the family of all weakly differentiable functions u ∈ L 2d/(d−2) (Ω), whose weak derivatives are functions in L 2 (Ω). The space Y 1,2 (Ω) is endowed with the norm For a Lipschitz graph domain Ω with Lipschitz constant K, it is easy to show (see Appendix) that the following Sobolev inequality holds:

Elliptic systems.
Let L be an elliptic operator acting on column vector valued functions u = (u 1 , . . . , u m ) T defined on a subset of R d , in the following way: where we use the usual summation convention over repeated indices α, β = 1, . . . , d, and A αβ = A αβ (x) are m×m matrix valued functions defined on the whole space R d with entries A αβ i j that satisfy the strong ellipticity condition for some constant λ > 0 and also the uniform boundedness condition for some constant M > 0. Notice that the i-th component of the column vector Lu coincides with L i j u j in (1.1). The adjoint operator t L is defined by 2.3. Neumann boundary value problem. We denote by ADu · n the conormal derivative of u associated with the operator L; i.e., i-th component of ADu · n is defined by where n = (n 1 , . . . , n d ) T is the outward unit normal to ∂Ω. Let Σ be an open subset of ∂Ω and f ∈ L 1 loc (Ω) m and g ∈ L 1 loc (Σ) m . We shall say that u ∈ W 1,1 loc (Ω) m is a weak solution of Lu = f in Ω, ADu · n = g on Σ if the following identity holds: Observe that (2.5) makes sense if f is a vector-valued measure in Ω; see part ii) in the definition of Neumann function below. We are mostly interested in the case when Σ = ∂Ω.

Neumann problem in a bounded Lipschitz domain.
Let Ω ⊂ R d be a bounded Lipschitz domain. Notice that the inequality (2.1) implies that H :=W 1,2 (Ω) m becomes a Hilbert space with the inner product If we define the bilinear form associated to the operator L as Let f ∈ L 2d/(d+2) (Ω) m and g ∈ L 2 (∂Ω) m satisfy the "compatibility" condition Then, by the inequality (2.8) and the trace theorem combined with (2.1), we find that is a bounded linear functional on H. Therefore, the Lax-Milgram theorem implies that there exists a unique u in H such that Observe that any function v ∈ W 1,2 (Ω) m is represented as a sum of a function in H and a constant vector in R m as follows.

Neumann problem in a Lipschitz graph domain.
Let Ω ⊂ R d be a Lipschitz graph domain and recall the inequality (2.2). Similar to the bounded Lipschitz domain case, H := Y 1,2 (Ω) m becomes a Hilbert space with the inner product (2.6). Also, the bilinear form B in (2.7) is coercive and bounded in Y 1,2 (Ω) m . For any f ∈ L 2d/(d+2) (Ω) m , the inequality (2.2) implies that is a bounded linear functional on Y 1,2 (Ω) m . Therefore, the Lax-Milgram theorem implies that there exists a unique u in Y 1,2 (Ω) m such that B(u, v) = F(v) for all v ∈ Y 1,2 (Ω) m ; i.e., we have (2.11) Hereafter, we shall say that u is a unique solution in Y 1,2 (Ω) m of the Neumann problem (Ω) m and satisfies the identity (2.11).

Neumann function.
In the definitions below, N = N(x, y) will be an m × m matrix valued function with measurable entries N i j : Ω × Ω → R.

Neumann function in a bounded Lipschitz domain.
We say that N is a Neumann function of L in a bounded Lipschitz domain Ω if it satisfies the following properties: loc (Ω) and N(·, y) ∈ W 1,2 (Ω \ B r (y)) for all y ∈ Ω and r > 0. Moreover, ∂Ω N(·, y) = 0 in the sense of trace. ii) LN(·, y) = δ y I in Ω and ADN(·, y) · n = − 1 |∂Ω| I on ∂Ω for all y ∈ Ω in the sense is a unique solution inW 1,2 (Ω) m of the problem (2.14)

Neumann function in a Lipschitz graph domain.
We say that N is a Neumann function of L in a Lipschitz graph domain Ω if it satisfies the following properties: i) N(·, y) ∈ W 1,1 loc (Ω) and N(·, y) ∈ Y 1,2 (Ω \ B r (y)) for all y ∈ Ω and r > 0. ii) LN(·, y) = δ y I in Ω and ADN(·, y) · n = 0 on ∂Ω for all y ∈ Ω in the sense iii) For any f ∈ C ∞ c (Ω) m , the function u given by (2.13) is a unique solution in Y 1,2 (Ω) m of the problem We point out that part iii) in the above definitions give the uniqueness of a Neumann function. Indeed, letÑ(x, y) is another function satisfying the above properties. Then by the uniqueness, we have and thus we conclude that N =Ñ a.e. in Ω × Ω.

Main results
The following "interior Hölder continuity" condition (IH) means that weak solutions of Lu = 0 and t Lu = 0 enjoy interior Hölder continuity. In the case m = 1, it is a consequence of the celebrated De Giorgi-Moser-Nash theorem. If m > 1 and d > 2, it is not true in general, but however, if the coefficients of the system (1.1) belong to the class of VMO and if Ω is bounded, then it is known that the condition (IH) holds in that case; see e.g., [12,Lemma 5.3].

Condition (IH).
There exist µ 0 ∈ (0, 1] and C 0 > 0 such that for all x ∈ Ω and 0 , then u is Hölder continuous in B R with the following estimate: for all y ∈ Ω and the following the identity holds: Moreover, for any f ∈ L q (Ω) m with q > d/2 and g ∈ L 2 (∂Ω) m satisfying Ω f + ∂Ω g = 0, the function u given by is a unique solution inW 1,2 (Ω) m of the problem Furthermore, the following estimates hold for all y ∈ Ω: Similarly, if g ∈ L 2 (∂Ω) m satisfies ∂Ω g = 0, then we find that

and C depends on p as well in ii) and iv). The estimates i) -vii) are also valid forÑ(x, y).
is a unique solution inW 1,2 (Ω) m of the problem Also, the following estimates are easy consequences of the identity (3.2) and the estimates i) -vii) forÑ(·, x): . The following "local boundedness" condition (LB) is used to obtain global pointwise bounds for the Neumann function N(x, y) of L in Ω. Again, in the case m = 1, it is well known that the condition (LB) holds in bounded Lipschitz domains; see e.g., [18]. In the case when m > 1, this condition does not hold in general and requires certain restrictions on the coefficients and domains. It can be shown, for example, that if the coefficients belong to the VMO class and the domain is bounded and has C 1 boundary, then the condition (LB) holds via W 1,p estimates; see Appendix.

Condition (LB).
There exists a constant C 1 > 0 such that the following holds: For any f ∈ C ∞ c (Ω) m and g ∈ C ∞ (∂Ω) satisfying Ω f + ∂Ω g = 0, let u ∈W 1,2 (Ω) m be a unique weak solution of the problem Then for all x ∈ Ω and 0 < R < diam(Ω), we have In the above, C = C(d, m, λ, M, Ω, C 1 ) > 0 and C depends on p as well in ii) and iv). The estimates i) -v) are also valid for the Neumann functionÑ of the adjoint t L. Remark 3.8. As we have pointed out, the condition (LB) is satisfied, for example, in the scalar case and also in the case when the system has VMO coefficients and the domain is of class C 1 . In fact, in those cases, we also have the following "local Hölder continuity" condition: There exist constants µ 0 ∈ (0, 1] and C 1 > 0 such that for all x ∈ Ω and 0 < R < diam(Ω), the following holds: Let u ∈ W 1,2 (Ω R (x)) m be a weak solution of either By using (LH) and modifying the proof for the estimate vii) in Theorem 3.1 (c.f. the proof for (3.7) in Section 4.2), we have the following global version of the estimate vii): The same estimate is also valid forÑ.
Finally, the following theorem says that the converse of Theorem 3.6 is also true, and thus that condition (LB) is equivalent to a global bound (3.7) for the Neumann function.
We now turn to pointwise bound for N(x, y). For any x ∈ Ω such that 0 < |x − y| < d y /2, set R := 2|x − y|/3. Notice that (4.14) implies that N(·, y) ∈ W 1,2 (B R (x)) and satisfies LN(·, y) = 0 weakly in B R (x). Then, by [12, Lemma 2.4] and the estimate ii) in the theorem, we have We have thus shown that the estimate vi) in the theorem holds. Then, it is routine to see that the estimate vii) in the theorem follows from the condition (IH) and the above estimate.
Next, let x ∈ Ω \ {y} be fixed but arbitrary, and letÑ ε ′ (·, x) ∈W 1,2 (Ω) m 2 be the mollified Neumann function of the adjoint operator t L in Ω, where ε ′ > 0. By setting φ in (4.4) to be the l-th column ofÑ ε ′ (·, x) and utilizing an integral identity forÑ ε ′ (·, x) similar to (4.4), we obtain the following identity: LetÑ(·, x) be a Neumann function of t L in Ω obtained by a sequence {ε ′ ν } ∞ ν=1 tending to 0. Then, by following the same steps as in [12, p. 151], we conclude N lk (x, y) =Ñ kl (y, x), ∀k, l = 1, . . . , m, which obviously implies the identity (3.2). In fact, by following a similar line of reasoning as in [12, p. 151], we find Now, let u be a unique solution inW 1,2 (Ω) m of the problem (2.14) with f ∈ C ∞ c (Ω) m . We remark that the condition (IH) implies that u is continuous in Ω; see [12,Eq. (3.14)]. By setting w to be the k-th column of N ε (·, y) in (4.6) and setting φ = u in (4.4), we get We take the limit ε → 0 above to get which is equivalent to (2.13). We have shown that N(x, y) satisfies the property iii) in Section 2.4.1, and thus that N(x, y) is a unique Neumann function of the operator L in Ω.
Finally, let f ∈ L q (Ω) m with q > d/2 and g ∈ L 2 (∂Ω) m satisfy the compatibility condition (2.9), and let u be a unique weak solution inW 1,2 (Ω) m of the problem (3.4); see Section 2.3. Then u satisfies the identity (2.10). By setting v to be the k-th column of N ε (·, x) in (2.10) and utilizing an integral identity forÑ ε (·, x) similar to (4.4), we get We remark that the condition (IH) together with the assumption that f ∈ L q (Ω) m with q > d/2 implies that u is Hölder continuous in Ω; see e.g., [12,Section 3.2]. Then by proceeding similarly as above and using (3.2), we obtain which is the formula (3.3). The proof is complete.
To derive the estimates i) -v) in the theorem, we need to repeat some steps in the proof of Theorem 3.1 with a little modification. Let v ε be the k-th column of N ε (·, y), where k = 1, . . . , m, 0 < ε < min(d y , r)/6, and 0 < r < diam(Ω). Let η be a smooth function on R d satisfying the conditions (4.10). We set φ = η 2 v ε in (4.3) to get (4.29) where we used the fact ∂Ω v ε = 0 and η 2 Φ ε ≡ 0. We then use (2.3), (2.4), Cauchy's inequality to get where C = C(λ, M). By using the conditions in (4.10) and the pointwise bound for N ε (x, y) obtained above, we get Therefore, by taking the limit ε → 0, we get Observe that the pointwise bound (3.7) together with the above estimate yields where C = C(d, m, λ, M, Ω, C 1 ). By following literally the same step used in deriving (4.15) -(4.18) from (4.14), and using the fact that |Ω| < ∞, we obtain the estimates i) -v) from (4.30). The proof is complete.
By a standard covering argument and the fact that R < diam(Ω) < ∞, we obtain (LB) from the above inequality. The proof is complete.

Neumann functions in Lipschitz graph domain
This separate section is devoted to the study of Neumann functions in an unbounded domain above a Lipschitz graph.

Main results.
Since Lipschitz graph domains are necessarily unbounded domains, it is more practical to replace the condition (IH) in Section 3 by the following condition (IH ′ ). In the case when the domain is bounded, it is equivalent to the condition (IH), but it is weaker if the domain is unbounded. By the well known De Giorgi-Moser-Nash theorem, we have the condition (IH ′ ) with R c = ∞ in the scalar case and thus, it reduces to the condition (IH).

Condition (IH ′ ).
There exist µ 0 ∈ (0, 1], R c ∈ (0, ∞], and C 0 > 0 such that for all x ∈ Ω and R ∈ (0, d ′ x ), where d ′ x := min(d x , R c ), the following holds: If u ∈ W 1,2 (B R (x)) is a weak solution of either Lu = 0 or t Lu = 0 in B R = B R (x), then u is Hölder continuous in B R with the following estimate:

2). Furthermore, the estimates i) -vii) in Theorem 3.1 are valid for N(·, y) and N(·, y) for all y ∈ Ω provided d x is replaced by d
. We also replace the condition (LB) in Section 3 by the following condition (LB ′ ). In the scalar case, it is well known that the condition (LB ′ ) holds in Lipschitz graph domains.

Condition (LB ′ ).
There exists a constant C 1 > 0 such that the following holds: For any f ∈ C ∞ c (Ω) m , let u ∈ Y 1,2 (Ω) m be a unique weak solution of the problem Then for all x ∈ Ω and R > 0, we have ADv · n = 0 on ∂Ω, and define N ε (·, y) by (4.2). Then N ε (·, y) satisfies the identity By the same argument as in the proof of Theorem 3.1, we then obtain (4.5). Let R ∈ (0, d ′ y ) be arbitrary, but fixed. Assume that f ∈ C ∞ c (Ω) m is supported in B R = B R (y) ⊂ Ω and let u be a unique weak solution in Y 1,2 (Ω) m of the problem (2.16). Then, we get the identity (4.7) and also the estimate (4.8). By literally the same steps in the proof of Theorem 3.1, we get (4.9) and (4.12) -(4.20) with d y replaced by d ′ y . Therefore, by the same reasoning as in the proof of Theorem 3.1, we find that N(·, y) satisfies the properties i) and ii) in Section 2.4.2, and also the estimates i) -vii) in Theorem 3.1 with d y replaced by d ′ y ; see [12,Section 4.1]. Let u be a unique solution in Y 1,2 (Ω) m of the problem (2.16) with f ∈ C ∞ c (Ω) m . Then as in the proof of Theorem 3.1 again, we get (2.13). Therefore, N(x, y) satisfies the property iii) in Section 2.4. 2, and thus that N(x, y) is a unique Neumann function of the operator L in Ω. The proof is complete.

Proof of Theorem 5.2.
We follow the proofs of Theorem 3.6 and 3.9 with a few adjustment. Let y ∈ Ω and R > 0 be arbitrary, but fixed. Assume that f ∈ C ∞ c (Ω) m is supported in Ω R (y) and let u be a unique weak solution in Y 1,2 (Ω) m of the problem Then we have (4.7) and (4.8) as in the proof of Theorem 3.6, and thus by (2.2) we get (4.21). By (LB ′ ), (4.21), and Hölder's inequality, we obtain (4.23). Then by following the same steps as in the proof of Theorem 3.6 we get (4.25) with C = C(d, m, λ, M, K, C 1 ). Let x ∈ Ω, r > 0, and ε > 0 be such that B ε (y) ∩ B r (x) = ∅. Since the k-th column of N ε (·, y) is a unique weak solution in Y 1,2 (Ω) m of the problem (5.3), the condition (LB ′ ) implies that N ε (·, y) L ∞ (Ω r/2 (x)) ≤ C 1 r −d/2 N ε (·, y) L 2 (Ω r (x)) . Then by following literally the same steps as in the proof of Theorem 3.6, we obtain the desired pointwise bound (3.7).

Appendix
Let v = ζu, where ζ : R d → R is a smooth function to be chosen later, and observe that v is a weak solution in W 1,2 (Ω) m of the problem Then, by [9, Corollary 9.3] together with the embedding theorems of Sobolev and Besov spaces (see e.g., [4]), we have the following estimate for Dw i provided p > d/(d − 1): Notice that if we set h i α = D α w i + F i α , then v becomes a weak solution of the problem We then apply [5, Theorem 1] to conclude that v ∈ W 1,p (Ω) m with the estimate By choosing ζ ≡ 1, we find that u ∈ W 1,p (Ω) m and and thus, via Morrey's imbedding theorem, we find that u ∈ C µ (Ω) for any µ ∈ (0, 1), which particularly implies that u is globally bounded in Ω.
Then by using the assumptions on ζ, we estimate terms in (6.2) as follows.
Therefore, by using Hölder's inequality and Cauchy's inequality, we estimate Similarly, by Hölder's inequality and Cauchy's inequality, we obtain . By combining (6.7) and the above inequality, we get . The above inequality obviously yields (6.6).
To prove (6.11), we follow the same steps in the usual proof of extension theorem for Sobolev functions in Lipschitz domain (see, e.g., [8]). For u ∈ C ∞ c (Ω), set u + (y) = u(y) if y ∈ Ω, Note u − = u + = u on ∂Ω. Then, it is routine to check (see [8,Section 4