THE GREEN FUNCTION ESTIMATES FOR STRONGLY
ELLIPTIC SYSTEMS OF SECOND ORDER
STEVE HOFMANN AND SEICK KIM
Abstract. We establish existence and pointwise estimates of fundamental so-
lutions and Green’s matrices for divergence form, second order strongly elliptic
systems in a domain Ω ⊆ Rn, n ≥ 3, under the assumption that solutions of
the system satisfy De Giorgi-Nash type local Hölder continuity estimates. In
particular, our results apply to perturbations of diagonal systems, and thus
especially to complex perturbations of a single real equation.
1. Introduction
In this article, we study Green’s functions (or Green’s matrices) of second order,
strongly elliptic systems of divergence type in a domain Ω ⊂ Rn with n ≥ 3.
In particular, we treat the Green matrix in the entire space, usually called the
fundamental solution. We shall prove that if a given elliptic system has the property
that all weak solutions of the system are locally Hölder continuous, then it has the
Green’s matrix in Ω. (For example, if coefficients of the system belong to the
space of VMO introduced by Sarason [19], then it will enjoy such a property). For
such elliptic systems, we study standard properties of the Green’s matrix including
pointwise bounds, Lp and weak Lp estimates for Green’s matrix and its derivatives,
For the scalar case, i.e., a single elliptic equation, the existence and properties
of Green’s function was studied by Littman, Stampacchia, and Weinberger [14]
and Grüter and Widman [11]. In this article, we follow the approach of Grüter
and Widman in constructing Green’s matrix. The main technical difficulties arise
from lack of Harnack type inequalities and the maximum principle for the systems.
The key observation on which this article is based is that even in the scalar case,
one can get around Moser’s Harnack inequality [18] or maximum principle but
instead rely solely on De Giorgi-Nash type oscillation estimates [5] in constructing
and studying properties of Green’s functions. From this point of view, this article
provides a unified approach in studying Green’s function for both scalar and systems
of equations. We should point out that there has been some study of Green’s matrix
for systems with continuous coefficients, notably by Fuchs [7] and Dolzmann-Müller
[6]. Our existence results and interior estimates of Green’s function will include
theirs, since as is well known, weak solutions of systems with uniformly continuous
(or VMO) coefficients enjoy local Hölder estimates. On the other hand, we have
not attempted to replicate their boundary estimates, which depend in particular on
having a C1 boundary. Our method does not require boundedness of the domain
nor regularity of the boundary in constructing Green’s matrices, while the methods
2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35J45.
Key words and phrases. Green’s function, fundamental solution, second order elliptic system.
http://arxiv.org/abs/0704.1352v2
2 S. HOFMANN AND S. KIM
of Fuchs [7] and Dolzmann-Müller [6] require both boundedness and regularity of
the domain at the very beginning. We note that a scalar elliptic equation with
complex coefficients can be identified as an elliptic system with real coefficients
satisfying a special structure, and thus our results apply in particular to complex
perturbations of a scalar real equation. In the complex coefficients setting, the main
results of Section 3 in our paper can be also obtained by following the method of
Auscher [2]. The estimates of the present paper will be applied to the development
of the layer potential method for equations with complex coefficients in [1].
The organization of this paper is as follows. In Section 2, we define the prop-
erty (H), which is essentially equivalent to De Giorgi’s oscillation estimates in the
scalar case, and introduce a function space Y
0 (Ω) which substitutes W
0 (Ω) in
constructing Green’s functions; they are identical if Ω is bounded but in general,
0 (Ω) is a larger space and is more suitable for our purpose. In Section 3, we
study Green’s functions defined in the entire space, which are usually referred to
as the fundamental solutions. The main result is that for a system whose coeffi-
cients are close to those of a diagonal system, the fundamental solution behaves
very much like that of a single equation. In Section 4, we study Green’s matrices in
general domains, including unbounded ones. We also study the boundary behav-
ior of Green’s matrices when the boundary of domain satisfies a measure theoretic
exterior cone condition, called the condition (S). We prove in particular that if the
coefficients of the system are close to those of a diagonal system, then again the
boundary behavior of its Green’s function is much like that of a single equation.
In section 5, we discuss the Green’s matrices of the strongly elliptic systems with
VMO coefficients. By following the same techniques already developed in the pre-
vious two sections, we construct the Green’s matrix in general domains including
the entire space. One subtle difference is that in this VMO coefficients case, one
should play with a localized version of property (H) since basically, the regularity of
weak solutions of the systems with VMO coefficients is inherited from the systems
with constant coefficients when the scale is made small enough. Therefore, all the
estimates for the Green’s matrix stated in this section are only meaningful near a
pole.
Finally, we would like to mention that when n = 2, the method used in this
article breaks down in several places and for that reason we plan to treat the two
dimensional case in a separate paper.
2. Preliminaries
2.1. Strongly elliptic systems. Throughout this article, the summation con-
vention over repeated indices shall be assumed. Let L be a second order elliptic
operator of divergence type acting on vector valued functions u = (u1, . . . , uN )T
defined on Rn (n ≥ 3) in the following way:
(2.1) Lu = −Dα(A
αβ Dβu),
where Aαβ = Aαβ(x) (α, β = 1, . . . , n) are N by N matrices satisfying the strong
ellipticity condition, i.e., there is a number λ > 0 such that
(2.2) A
ij (x)ξ
α ≥ λ |ξ|
|ξiα|
2, ∀x ∈ Rn
GREEN FUNCTION ESTIMATES 3
We also assume that A
ij are bounded, i.e., there is a number Λ > 0 such that
(2.3)
i,j=1
α,β=1
ij (x)|
2 ≤ Λ2, ∀x ∈ Rn.
If we write (2.1) component-wise, then we have
(2.4) (Lu)i = −Dα(A
ij Dβu
j), ∀i = 1, . . . , N.
The transpose operator of tL of L is defined by
(2.5) tLu = −Dα(
αβDβu),
where tAαβ = (Aβα)T (i.e., tA
ij = A
ji ). Note that the coefficients
ij satisfy
(2.2), (2.3) with the same constants λ,Λ.
In the sequel, we shall use the notation −
f := 1
f (assuming 0 < |S| < ∞),
where S is a measurable subset of Rn and |S| denotes the Lebesgue measure of
measurable S.
Definition 2.1. We say that the operator L satisfies the property (H) if there exist
µ0, H0 > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) satisfy
(2.6)
)n−2+2µ0
, 0 < r < s ≤ R.
Similarly, we say that the transpose operator tL satisfies the property (H) if corre-
sponding estimates hold for all weak solutions u of tLu = 0 in BR.
Lemma 2.2. Let (aαβ(x))nα,β=1 be coefficients satisfying the following conditions:
There are constants λ0,Λ0 > 0 such that for all x ∈ R
(2.7) aαβ(x)ξβξα ≥ λ0 |ξ|
, ∀ξ ∈ Rn;
∣aαβ(x)
≤ Λ20.
Then, there exists ǫ0 = ǫ0(n, λ0,Λ0) such that if
(2.8) ǫ2(x) :=
ij (x) − a
αβ(x)δij
< ǫ20, ∀x ∈ R
then the operator L associated with the coefficients A
ij satisfies the condition (H)
with µ0 = µ0(n, λ0,Λ0), H0 = H0(n,N, λ0,Λ0) > 0.
Proof. See e.g., [12, Proposition 2.1]. �
Lemma 2.3. Suppose that the operator L satisfies the following Hölder property
for weak solutions: There are constants µ0, C0 > 0 such that all weak solutions u
of Lu = 0 in B2R = B2R(x0) satisfy the estimate
(2.9) [u]Cµ0 (BR) ≤ C0R
where [f ]Cµ(Ω) denotes the usual C
µ(Ω) semi-norm of f ; see [10] for the definition.
Then, the operator L satisfies the property (H) with µ0 and H0 = H0(n,N, λ,Λ, C0).
4 S. HOFMANN AND S. KIM
Proof. We may assume that r < s/4; otherwise, (2.6) is trivial. Denote ur = −
We may assume, by replacing u by u − us, if necessary, that us = 0. From the
Caccioppoli inequality, (2.9), and then the Poincaré inequality, it follows
≤ Cr−2
|u− u2r|
≤ Cr−2
|u(x) − u(y)|
dy dx
≤ Cr−2[u]2Cµ0 (B2r)(2r)
2µ0 |B2r| ≤ Cr
n−2+2µ0 [u]2Cµ0(Bs/2)
≤ C(r/s)n−2+2µ0s−2
≤ C(r/s)n−2+2µ0
The proof is complete. �
Lemma 2.4. Assume that the operator L satisfies the property (H). Then, the
operator L satisfies the Hölder property (2.9). Moreover, for any p > 0, there
exists Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0 such that all weak solutions u of Lu = 0
in BR = BR(x0) satisfy
(2.10) ‖u‖L∞(Br) ≤
(R− r)n/p
‖u‖Lp(BR) , ∀r ∈ (0, R).
Proof. From a theorem of Morrey [17, Thoerem 3.5.2], the property (H), and the
Caccioppoli inequality, it follows that
(2.11) [u]2Cµ0(BR) ≤ CR
2−n−2µ0 ‖Du‖
L2(B3R/2)
≤ CR−n−2µ0 ‖u‖
L2(B2R)
Then, by a well known averaging argument (see e.g., [12]) we derive
(2.12) ‖u‖L∞(BR/2) ≤ C
where C = C(n,N, λ,Λ, µ0, H0) > 0. For the proof that (2.12) implies (2.10), we
refer to [9, pp. 80–82]. �
2.2. Function spaces Y 1,2(Ω) and Y
0 (Ω).
Definition 2.5. For an open set Ω ⊂ Rn (n ≥ 3), the space Y 1,2(Ω) is defined as
the family of all weakly differentiable functions u ∈ L2
(Ω), where 2∗ = 2n
, whose
weak derivatives are functions in L2(Ω). The space Y 1,2(Ω) is endowed with the
‖u‖Y 1,2(Ω) := ‖u‖L2∗ (Ω) + ‖Du‖L2(Ω) .
We define Y
0 (Ω) as the closure of C
c (Ω) in Y
1,2(Ω), where C∞c (Ω) is the set of
all infinitely differentiable functions with compact supports in Ω.
We note that in the case Ω = Rn, it is well known that Y 1,2(Rn) = Y
see e.g., [15, p. 46]. By the Sobolev inequality, it follows that
(2.13) ‖u‖L2∗ (Ω) ≤ C(n) ‖Du‖L2(Ω) , ∀u ∈ Y
0 (Ω).
Therefore, we have W
0 (Ω) ⊂ Y
0 (Ω) and W
0 (Ω) = Y
0 (Ω) when Ω has finite
Lebesgue measure.
From (2.13), it follows that the bilinear form
(2.14) 〈u,v〉
GREEN FUNCTION ESTIMATES 5
defines an inner product on H := Y
0 (Ω)
N . Also, it is routine to check that H
equipped with the inner product (2.14) is a Hilbert space.
Definition 2.6. We shall denote by H the Hilbert space Y
0 (Ω)
N with the inner
product (2.14). We denote
:= 〈u,u〉
= ‖Du‖L2(Ω) .
We also define the bilinear form associated to the operator L as
B(u,v) :=
ij Dβu
By the strong ellipticity (2.2), it follows that the bilinear form B is coercive; i.e,
(2.15) B(u,u) ≥ λ 〈u,u〉
3. Fundamental matrix in Rn
Throughout this section, we assume that the operators L and tL satisfy the
property (H). The main goal of this section is to construct the fundamental matrix
of the the operator L in the entire Rn, where n ≥ 3. Since Y 1,2(Rn) = Y
we have, as in Definition 2.5,
‖u‖L2∗ (Rn) ≤ C(n) ‖Du‖L2(Rn) , ∀u ∈ Y
1,2(Rn).
We note that W 1,2(Rn) ⊂ Y 1,2(Rn) ⊂ W
loc (R
n). Unless otherwise stated, we
employ the letter C to denote a constant depending on n, N , λ, Λ, µ0, H0, and
sometimes on an exponent p characterizing Lebesgue classes. It should be under-
stood that C may vary from line to line.
3.1. Averaged fundamental matrix. Our approach here is based on that in
[11]. Let y ∈ Rn and 1 ≤ k ≤ N be fixed. For ρ > 0, consider the linear functional
u 7→ −
Bρ(y)
uk. Since
(3.1)
Bρ(y)
≤ Cρ(2−n)/2 ‖u‖L2∗(Rn) ≤ Cρ
(2−n)/2 ‖u‖
Lax-Milgram lemma implies that there exists a unique vρ = vρ;y,k ∈ H such that
(3.2)
ij Dβv
ρ Dαu
i = −
Bρ(y)
uk, ∀u ∈ H.
Note that (2.15), (3.2), and (3.1) imply that
λ ‖vρ‖
≤ B(vρ,vρ) ≤ Cρ
(2−n)/2 ‖vρ‖H ,
and thus we have
(3.3) ‖Dvρ‖L2(Rn) = ‖vρ‖H ≤ Cρ
(2−n)/2.
We define the “averaged fundamental matrix” Γρ( · , y) = (Γ
jk( · , y))
j,k=1 by
(3.4) Γ
jk( · , y) = v
ρ = v
ρ;y,k.
Note that we have
(3.5)
ij DβΓ
jk( · , y)Dαu
i = −
Bρ(y)
uk, ∀u ∈ H,
6 S. HOFMANN AND S. KIM
and equivalently (α ↔ β , i ↔ j).
(3.6)
ij Dβu
ik( · , y) = −
Bρ(y)
uk, ∀u ∈ H.
In the sequel, we shall denote by L∞c (Ω) the family of all L
∞ functions with
compact supports in Ω. For a given f ∈ L∞c (R
n)N consider a linear functional
(3.7) w 7→
f ·w,
which is bounded on H since
(3.8)
≤ ‖f‖
n+2 (Rn)
n−2 (Rn)
≤ C ‖f‖
n+2 (Rn)
Therefore, by Lax-Milgram lemma, there exists u ∈ H such that
(3.9)
ij Dβu
f iwi, ∀w ∈ H.
In particular, if we set w = vρ in (3.9), then by (3.6), we have
(3.10)
ik( · , y)f
i = −
Bρ(y)
Moreover, by setting w = u in (3.9), it follows from (3.8) that
(3.11) ‖Du‖L2(Rn) ≤ C ‖f‖L2n/(n+2)(Rn) .
3.2. L∞ estimates for averaged fundamental matrix. Let u ∈ H be given as
in (3.9). We will obtain local L∞ estimates for u in BR(x0), where x0 ∈ R
n and
R > 0 are fixed but arbitrary.
Fix x ∈ BR(x0) and 0 < s ≤ R. We decompose u as u = u1 + u2, where
u1 ∈ W
1,2(Bs(x))
N is the weak solution of tLu1 = 0 in Bs(x) satisfying u1 = u on
∂Bs(x); i.e., u1 − u ∈ W
0 (Bs(x)). Then, for 0 < r < s, we have
Br(x)
Br(x)
|Du1|
Br(x)
|Du2|
)n−2+2µ0
Bs(x)
|Du1|
Bs(x)
|Du2|
)n−2+2µ0
Bs(x)
Bs(x)
|Du2|
Since u2 ∈ W
0 (Bs(x))
N is a weak solution of tLu2 = f in Bs(x), we have
Bs(x)
|Du2|
≤ C ‖f‖
L2n/(n+2)(Bs(x))
For given p > n/2, choose p0 ∈ (n/2, p) such that µ1 := 2− n/p0 < µ0. Then
(3.12) ‖f‖
n+2 (Bs(x))
≤ ‖f‖
Lp0(Bs(x))
1+2/n−2/p0 ≤ C ‖f‖
Lp0(Rn) s
n−2+2µ1 .
Therefore, after combining the above inequalities, we have for all r < s ≤ R
Br(x)
)n−2+2µ0
Bs(x)
+ Csn−2+2µ1 ‖f‖
Lp0(Rn) .
GREEN FUNCTION ESTIMATES 7
By a well known iteration argument (see e.g., [8, Lemma 2.1, p. 86]), we have
Br(x)
)n−2+2µ1
BR(x)
+ Crn−2+2µ1 ‖f‖
Lp0(Rn)
)n−2+2µ1
+ Crn−2+2µ1 ‖f‖
Lp0(Rn) ,
(3.13)
for all 0 < r < R and x ∈ BR(x0). From (3.13) it follows (see, e.g. [12])
(3.14) [u]2Cµ1 (BR(x0)) ≤ C
R−(n−2+2µ1) ‖Du‖
L2(Rn) + ‖f‖
Lp0(Rn)
Note that since u ∈ H, we have
L2(BR(x0))
≤ ‖u‖
(BR(x0))
≤ CR2 ‖Du‖
L2(Rn) .
Consequently, we have
L∞(BR/2(x0))
≤ CR2µ1 [u]2Cµ1 (BR(x0)) + CR
−n ‖u‖
L2(BR(x0))
R2−n ‖Du‖
L2(Rn) +R
2µ1 ‖f‖
Lp0(Rn)
+ CR2−n ‖Du‖
L2(Rn)
≤ CR2−n ‖f‖
L2n/(n+2)(Rn) + CR
2µ1 ‖f‖
Lp0(Rn) ,
where we used the inequality (3.11) in the last step.
Therefore, if f is supported in BR(x0), then (3.12) yields (recall µ1 = 2− n/p0)
(3.15) ‖u‖L∞(BR/2(x0)) ≤ CR
2−n/p0 ‖f‖Lp0(BR(x0)) ≤ CR
2−n/p ‖f‖Lp(BR(x0)) .
Now, (3.10) implies that for ρ < R/2, we have, by setting x0 = y in (3.15),
(3.16)
BR(y)
ik( · , y)f
Bρ(y)
|u| ≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀p > n/2
provided that f is supported in BR(y). Therefore, by duality, we see that
(3.17) ‖vρ‖Lq(BR(y)) ≤ CR
2−n+n/q, ∀q ∈ [1, n
), ∀ρ ∈ (0, R/2),
where vρ = vρ;y,k is as in (3.4).
Fix x 6= y and let r := 2
|x− y|. If ρ < r/2, then since vρ ∈ W
1,2(Br(x))
N and
satisfies Lvρ = 0 weakly in Br(x), it follows from Lemma 2.4 that
(3.18) |vρ(x)| ≤ Cr
−n ‖vρ‖L1(Br(x)) ≤ Cr
−n ‖vρ‖L1(B3r(y)) ≤ Cr
Since ρ, y, k are arbitrary, we have obtained the following estimates.
(3.19) |Γρ(x, y)| ≤ C |x− y|
, ∀ρ < |x− y| /3.
3.3. Uniform weak-L
n−2 estimates for Γ
ρ( · , y). We claim that the following
estimate holds:
(3.20)
Rn\BR(y)
|Γρ( · , y)|
n−2 ≤ CR−n, ∀R > 0, ∀ρ > 0.
If R > 3ρ, then by (3.19) we have
Rn\BR(y)
|Γρ(x, y)|
n−2 dx ≤ C
Rn\BR(y)
|x− y|
dx ≤ CR−n.
8 S. HOFMANN AND S. KIM
Next, we consider the case R ≤ 3ρ. Let vTρ be the k-th column of the averaged
fundamental matrix Γρ( · , y) as in (3.4). From (3.3), we see that
‖vρ‖L2∗ (Rn\BR(y)) ≤ ‖vρ‖L2∗ (Rn) ≤ ‖Dvρ‖L2(Rn) ≤ Cρ
(2−n)/2.
and thus (3.20) also follows in the case when R ≤ 3ρ.
Now, let At = {x ∈ R
n : |Γρ(x, y)| > t} and choose R = t−1/(n−2). Then,
|At \BR(y)| ≤ t
At\BR(y)
|Γρ( · , y)|
n−2 ≤ Ct−
n−2 t
n−2 = Ct−
n−2 .
Obviously, |At ∩BR(y)| ≤ CR
n = Ct−
n−2 . Therefore, we obtained that for all
t > 0, we have
(3.21) |{x ∈ Rn : |Γρ(x, y)| > t}| ≤ Ct−
n−2 , ∀ρ > 0.
3.4. Uniform weak-L
n−1 estimates for DΓρ( · y). Let vρ be as before. Fix a
cut-off function η ∈ C∞(Rn) such that η ≡ 0 on BR/2(y), η ≡ 1 outside BR(y),
and |Dη| ≤ C/R. If we set u := η2vρ, then by (3.2)
ij Dβv
ij Dβv
ρDαη,
which together with (3.19) implies that if R > 6ρ, then
Rn\BR(y)
|Dvρ|
≤ CR−2
BR(y)\BR/2(y)
≤ CR2−n.
On the other hand, if R ≤ 6ρ, then (3.3) again implies
Rn\BR(y)
|Dvρ|
|Dvρ|
≤ Cρ2−n ≤ CR2−n.
Therefore, we have
(3.22)
Rn\BR(y)
|DΓρ( · , y)|
≤ CR2−n, ∀R > 0, ∀ρ > 0.
Next, let At = {x ∈ R
n : |DxΓ
ρ(x, y)| > t} and choose R = t−1/(n−1). Then
|At \BR(y)| ≤ t
At\BR(y)
|DΓρ( · , y)|
≤ Ct−
and |At ∩BR(y)| ≤ CR
n = Ct−
n−1 . We have thus find that for all t > 0, we have
(3.23) |{x ∈ Rn : |DxΓ
ρ(x, y)| > t}| ≤ Ct−
n−1 , ∀ρ > 0.
3.5. Construction of the fundamental matrix. First, we claim
(3.24) ‖DΓρ( · , y)‖Lp(BR(y)) ≤ CpR
1−n+n/p, ∀ρ > 0, ∀p ∈ (0, n
Let vρ be as before. Note that
BR(y)
|Dvρ|
BR(y)∩{|Dvρ|≤τ}
|Dvρ|
BR(y)∩{|Dvρ|>τ}
|Dvρ|
≤ τp |BR|+
{|Dvρ|>τ}
|Dvρ|
GREEN FUNCTION ESTIMATES 9
By using (3.23), we estimate
{|Dvρ|>τ}
|Dvρ|
ptp−1 |{|Dvρ| > max(t, τ)}| dt
≤ Cτ−
ptp−1 dt+ C
ptp−1−n/(n−1) dt
1− p/(p− n
τp−n/(n−1).
By optimizing over τ , we get
(3.25)
BR(y)
|Dvρ|
≤ CR(1−n)p+n,
from which (3.24) follows.
If we utilize (3.21) instead of (3.23), we obtain a similar estimates for Γρ( · , y)
(3.26) ‖Γρ( · , y)‖Lp(BR(y)) ≤ CpR
2−n+n/p, ∀ρ > 0, ∀p ∈ (0, n
Let us fix q ∈ (1, n
). We have seen that for all R > 0, there exists some
C(R) < ∞ such that
‖Γρ( · , y)‖W 1,q(BR(y)) ≤ C(R), ∀ρ > 0.
Therefore, by a diagonalization process, we obtain a sequence {ρµ}
and Γ( · , y)
loc (R
n)N×N such that limµ→∞ ρµ = 0 and that
(3.27) Γρµ( · , y) ⇀ Γ( · , y) in W 1,q(BR(y))
N×N , ∀R > 0,
where we recall that ⇀ denotes weak convergence. Then, for any φ ∈ C∞c (R
n)N ,
it follows from (3.5)
ij DβΓjk( · , y)Dαφ
i = lim
ij DβΓ
jk ( · , y)Dαφ
= lim
Bρµ (y)
φk = φk(y).
(3.28)
Let vTρ be the k-th column of Γ
ρ( · , y) as before, and let vT be the corresponding
k-th column of Γ( · , y). Then, for any g ∈ L∞c (BR(y))
N , (3.26) yields
(3.29)
v · g
= lim
vρµ · g
≤ CpR
2−n+n/p ‖g‖Lp′(BR(y)) ,
where p′ denotes the conjugate exponent of p ∈ [1, n
). Therefore, we obtain
(3.30) ‖Γ( · , y)‖Lp(BR(y)) ≤ Cp R
2−n+n/p, ∀p ∈ [1, n
By a similar reasoning, we also have by (3.24)
(3.31) ‖DΓ( · , y)‖Lp(BR(y)) ≤ Cp R
1−n+n/p, ∀p ∈ [1, n
Also, with the aid of (3.20) and (3.22), we obtain
Rn\BR(y)
|Γ( · , y)|
≤ CR−n,(3.32)
Rn\BR(y)
|DΓ( · , y)|
≤ CR2−n.(3.33)
10 S. HOFMANN AND S. KIM
In particular, (3.32), (3.33) imply that
(3.34) ‖Γ( · , y)‖Y 1,2(Rn\Br(y)) ≤ Cr
1−n/2, ∀r > 0.
Moreover, arguing as before, we see that the estimates (3.32) and (3.33) imply
|{x ∈ Rn : |Γ(x, y)| > t}| ≤ Ct−
n−2 , ∀t > 0(3.35)
|{x ∈ Rn : |DxΓ(x, y)| > t}| ≤ Ct
n−1 ∀t > 0.(3.36)
Next, we turn to pointwise bounds for Γ( · , y). Let vT be the k-th column of
Γ( · , y). For each x 6= y, denote r = 2
|x− y|. Then, it follows from (3.34) and
(3.28) that v is a weak solution of Lv = 0 in Br(x). Therefore, by Lemma 2.4 and
(3.30) we find
(3.37) |v(x)| ≤ Cr−n ‖v‖L1(Br(x)) ≤ Cr
−n ‖v‖L1(B3r(y)) ≤ Cr
from which it follows
(3.38) |Γ(x, y)| ≤ C |x− y|
, ∀x 6= y.
3.6. Continuity of the fundamental matrix. From the property (H), it follows
that Γ( · , y) is Hölder continuous in Rn \ {y}. In fact, (2.11) together with (3.28)
and (3.33) implies
(3.39) |Γ(x, y)− Γ(z, y)| ≤ C |x− z|
µ0 |x− y|
2−n−µ0 if |x− z| < |x− y| /2.
Moreover, by the same reasoning, it follows from (2.11) and (3.22) that for any
given compact set K ⋐ Rn \ {y}, the sequence {Γρµ( · , y)}
µ=1 is equicontinuous on
K. Also, by Lemma 2.4 and (3.20), we find that there are CK < ∞ and ρK > 0
such that
(3.40) ‖Γρ( · , y)‖L∞(K) ≤ CK ∀ρ < ρK for any compact K ⋐ R
n \ {y} .
Therefore, we may assume, by passing if necessary to a subsequence, that
(3.41) Γρµ( · , y) → Γ( · , y) uniformly on K, for any compact K ⋐ Rn \ {y} .
We will now show that Γ(x, · ) is also Hölder continuous in Rn \ {x}. Denote
by tΓσ( · , x) the averaged fundamental matrix associated to tL, the transpose of L.
Since each column of Γρ( · , y) and tΓσ( · , x) belongs to H, we have by (3.5),
Bρ(y)
tΓσkl( · , x) =
ij DβΓ
jk( · , y)Dα
tΓσil( · , x)
ji Dα
tΓσil( · , x)DβΓ
jk( · , y) = −
Bσ(x)
lk( · , y).
(3.42)
By the same argument as appears in Sec. 3.5, we obtain a sequence {σν}
ν=1 tending
to 0 such that tΓσν ( · , x) converges to tΓ( · , x) uniformly on any compact subset of
n \ {x}, where tΓ( · , x) is a fundamental matrix for tL satisfying all properties
stated in Sec. 3.5. By (3.42), we find that
gklµν := −
Bρµ (y)
tΓσνkl ( · , x) = −
Bσν (x)
lk ( · , y).
From the continuity of Γ
lk ( · , y), it follows that for x, y ∈ R
n with x 6= y, we have
gklµν = lim
Bσν (x)
lk ( · , y) = Γ
lk (x, y)
GREEN FUNCTION ESTIMATES 11
and thus by (3.41) we obtain
gklµν = lim
lk (x, y) = Γlk(x, y).
On the other hand, (3.27) yields
gklµν = lim
Bρµ (y)
tΓσνkl ( · , x) = −
Bρµ (y)
tΓkl( · , x)
and thus it follows from the continuity of tΓkl( · , x) that
gklµν = lim
Bρµ (y)
tΓkl( · , x) =
tΓkl(y, x).
We have thus shown that
Γlk(x, y) =
tΓkl(y, x), ∀k, l = 1, . . . , N, ∀x 6= y,
which is equivalent to say
(3.43) Γ(x, y) = tΓ(y, x)T , ∀x 6= y.
Therefore, we have proved the claim that Γ(x, · ) is Hölder continuous in Rn \ {x}.
So far, we have seen that there is a sequence {ρµ}
tending to 0 such that
ρµ( · , y) → Γ( · , y) in Rn \ {y}. However, by (3.42), we obtain
lk(x, y) = limν→∞
Bσν (x)
lk( · , y) = limν→∞
Bρ(y)
tΓσνkl ( · , x)
Bρ(y)
tΓkl( · , x) = −
Bρ(y)
Γlk(x, · ),
(3.44)
i.e., we have the following representation for the averaged fundamental matrix:
(3.45) Γρ(x, y) = −
Bρ(y)
Γ(x, z) dz.
Therefore, by the continuity, we obtain
(3.46) lim
ρ(x, y) = Γ(x, y), x 6= y.
3.7. Properties of fundamental matrix. We record what we obtained so far in
the following theorem:
Theorem 3.1. Assume that operators L and tL satisfy the property (H). Then,
there exists a unique fundamental matrix Γ(x, y) = (Γij(x, y))
i,j=1 (x 6= y) which is
continuous in {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ(x, · ) is locally integrable
in Rn for all x ∈ Rn and that for all f = (f1, . . . , fN)T ∈ C∞c (R
n)N , the function
u = (u1, . . . , uN )T given by
(3.47) u(x) :=
Γ(x, y)f (y) dy
belongs to Y 1,2(Rn)N and satisfies Lu = f in the sense
(3.48)
ij Dβu
f iφi, ∀φ ∈ C∞c (R
n)N .
Moreover, Γ(x, y) has the property
(3.49)
ij DβΓjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (R
n)N .
12 S. HOFMANN AND S. KIM
Furthermore, Γ(x, y) satisfies the following estimates:
‖Γ( · , y)‖Y 1,2(Rn\Br(y)) + ‖Γ(x, · )‖Y 1,2(Rn\Br(x)) ≤ Cr
2 , ∀r > 0,(3.50)
‖Γ( · , y)‖Lp(Br(y)) + ‖Γ(x, · )‖Lp(Br(x)) ≤ Cpr
2−n+ n
p , ∀p ∈ [1, n
),(3.51)
‖DΓ( · , y)‖Lp(Br(y)) + ‖DΓ(x, · )‖Lp(Br(x)) ≤ Cpr
1−n+ n
p , ∀p ∈ [1, n
),(3.52)
|{x ∈ Rn : |Γ(x, y)| > t}|+ |{y ∈ Rn : |Γ(x, y)| > t}| ≤ Ct−
n−2 ,(3.53)
|{x ∈ Rn : |DxΓ(x, y)| > t}|+ |{y ∈ R
n : |DyΓ(x, y)| > t}| ≤ Ct
n−1 ,(3.54)
|Γ(x, y)| ≤ C |x− y|
, ∀x 6= y,(3.55)
|Γ(x, y)− Γ(z, y)| ≤ C |x− z|
µ0 |x− y|
2−n−µ0 if |x− z| < |x− y| /2,(3.56)
|Γ(x, y)− Γ(x, z)| ≤ C |y − z|
µ0 |x− y|
2−n−µ0 if |y − z| < |x− y| /2,(3.57)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Proof. Let Γρ(x, y) and Γ(x, y) be constructed as above. We have already seen
that Γ is continuous in {(x, y) ∈ Rn × Rn : x 6= y} and satisfies all the properties
(3.49) – (3.57). By using the Lax-Milgram lemma as in Sec. 3.1, we find that for
all f ∈ C∞c (R
n)N , there is a unique u ∈ Y 1,2(Rn)N satisfying
ij Dβu
f ivi, ∀v ∈ Y 1,2(Rn)N .
If we set vi = Γ
ki(x, · ) above, then (3.5) together with (3.43) implies that
(3.58)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Bρ(x)
Assume that f is supported in BR(x) for some R > 0. Then, by (3.27) and (3.43)
we have
ki(x, · )f
i = lim
BR(x)
ki(x, · )f
Γki(x, · )f
By the same argument which lead to (3.14) in Section 3.2, we find that u is Hölder
continuous. Therefore, (3.47) follows by taking the limits in (3.58).
Now, it only remains to prove the uniqueness. Assume that Γ̃(x, y) is another ma-
trix such that Γ̃ is continuous on {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ̃(x, · )
is locally integrable in Rn for all x ∈ Rn and that for all f ∈ C∞c (R
n)N ,
ũ(x) :=
Γ̃(x, y)f (y) dy
belongs to Y 1,2(Rn) and satisfies Lu = f in the sense of (3.48). Then by the
uniqueness in H = Y 1,2(Rn)N , we must have u = ũ. Therefore, for all x ∈ Rn we
(Γ− Γ̃)(x, ·)f = 0, ∀f ∈ C∞c (R
n)N ,
and thus we have Γ ≡ Γ̃ in {(x, y) ∈ Rn × Rn : x 6= y}. �
Theorem 3.2. Assume that the operators L and tL satisfy the property (H). If
f ∈ (L
n+2 (Rn) ∩ L
loc(R
n))N for some p > n/2, then there exists a unique u in
GREEN FUNCTION ESTIMATES 13
Y 1,2(Rn)N such that
(3.59)
ij Dβu
f ivi, ∀v ∈ Y 1,2(Rn)N .
Moreover, u is continuous and has the following representation:
(3.60) uk(x) =
Γki(x, y)f
i(y) dy, k = 1, . . . , N,
where (Γki(x, y))
k,i=1 is the fundamental matrix of L.
Proof. Since f ∈ L
n+2 (Rn)N , the same argument as appears in Sec. 3.1 implies
that there is u ∈ Y 1,2(Rn)N satisfying (3.59). If we set vi = Γ
ki(x, · ) in (3.59),
then (3.2) implies that
(3.61)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Bρ(x)
Next, note that (3.20), (3.26), and the assumption f i ∈ L
n+2 (Rn) ∩ L
loc(R
n) for
some p > n/2, imply that
ki(x, · )f
i = lim
B1(x)
ki(x, · )f
Rn\B1(x)
ki(x, · )f
B1(x)
Γki(x, · )f
Rn\B1(x)
Γki(x, · )f
Γki(x, · )f
(3.62)
Finally, by the same argument which lead to (3.14) in Sec. 3.2, we find that u is
Hölder continuous, and thus (3.60) follows from (3.61) and (3.62). �
Corollary 3.3. Suppose that f = (f1, . . . , fN )T has a bound
(3.63) |f(x)| ≤ C(1 + |x|)−(1+n/2+ǫ) ∀x ∈ Rn for some ǫ > 0.
Then, u = (u1, . . . , uN)T given by (3.60) is a unique Y 1,2(Rn)N solution of Lu = f
in Rn in the sense of (3.59).
Proof. Note that (3.63) implies f ∈ (L
n+2 (Rn) ∩ Lp(Rn))N . �
Theorem 3.4. Assume that L and tL satisfy the property (H). If f ∈ Y 1,2(Rn)N
satisfies Df ∈ L
loc(R
n)N×n for some p > n, then
(3.64) fk(x) =
DαΓki(x, · )A
ij Dβf
j , k = 1, . . . , N,
where (Γki(x, y))
k,i=1 is the fundamental matrix of L.
Proof. We denote by tΓρ the averaged fundamental matrix of tL. Recall that
columns of tΓρ belong to H. Then, by (3.5) we have
ji Dα
ik( · , x)Dβf
j = −
Bρ(x)
14 S. HOFMANN AND S. KIM
As in (3.62), the assumption Df ∈ L
loc(R
n)N for p > n, together with (3.22) and
(3.24) yields
Bρ(x)
fk = lim
B1(x)
Rn\B1(x)
ji Dα
ik( · , x)Dβf
B1(x)
Rn\B1(x)
ji Dα
tΓik( · , x)Dβf
ij DαΓki(x, · )Dβf
(3.65)
where we used (3.43) in the last step. By the Morrey’s inequality [17], f is contin-
uous and thus (3.64) follows from (3.65). �
Corollary 3.5. Assume that L, tL, L̃, and tL̃ satisfy the property (H). Denote by
Γ and Γ̃ the fundamental matrices of L and L̃, respectively. If the coefficients A
of L and Ã
ij of L̃ are Hölder continuous, then
(3.66) Γ̃lm(x, y) = Γlm(x, y) +
DαΓli(x, · )(A
ij − Ã
ij )DβΓ̃jm( · , y), x 6= y.
Proof. We denote by Γρ and Γ̃ρ (ρ < |x− y| /4) the averaged fundamental matrices
of L and L̃ respectively. Recall that columns of Γρ and Γ̃ρ belong to H. Moreover,
since we assume that the coefficients are Hölder continuous, the standard elliptic
theory, (3.38), and (3.45) implies thatDΓρ(x, · ) andDΓ̃ρ( · , y) are locally bounded.
Therefore, by setting f j = Γ̃
jm( · , y) in (3.64) we have
(3.67) Γ̃
lm(x, y) =
DαΓli(x, · )A
ij DβΓ̃
jm( · , y),
Next, set f j = Γ
lj(x, · ) and apply (3.64) with L replaced by
tL̃ to get
lm(x, y) =
t̃Γmi(y, · )
ij DβΓ
lj(x, · ).
By using (3.43) and interchanging indices (α ↔ β, i ↔ j), we obtain
(3.68) Γ
lm(x, y) =
li(x, · )Ã
ij DβΓ̃jm( · , y).
Now, set r = |x− y| /4 and split the integral (3.67) into three pieces (recall ρ < r)
Br(x)
Br(y)
Rn\(Br(x)∪Br(x))
DαΓli(x, · )A
ij DβΓ̃
jm( · , y).
Since we assume that the coefficients are Hölder continuous, it follows from the stan-
dard elliptic theory that DΓ(x, · ) and DΓ̃( · , y) are continuous (and thus bounded)
on Br(y) and Br(x) respectively. Moreover, (3.45) implies
DΓ̃ρ( · , y) → DΓ̃( · , y) uniformly on Br(x) as ρ → 0.
Therefore, as in (3.65), we may take the limit ρ → 0 in (3.67) to get
Γ̃lm(x, y) =
DαΓli(x, · )A
ij DβΓ̃jm( · , y),
GREEN FUNCTION ESTIMATES 15
Similarly, by taking the limit ρ → 0 in (3.68), we obtain
Γlm(x, y) =
DαΓli(x, · )Ã
ij DβΓ̃jm( · , y).
The proof is complete. �
Remark 3.6. We note that in terms of matrix multiplication (3.60) is written as
u(x) =
Γ(x, y)f (y) dy,
where both u,f are understood as column vectors. Also, (3.66) reads
Γ̃(x, y) = Γ(x, y) +
DαΓ(x, · )(A
αβ − Ãαβ)DβΓ( · , y).
4. Green’s matrix in general domains
4.1. Construction of Green’s matrix. In this section, we shall construct the
Green’s matrix in any open, connected set Ω ⊂ Rn, where n ≥ 3. To construct the
Green’s matrix in Ω, we need to adjust arguments in Section 3.
Henceforth, we shall denote Ωr(y) := Ω ∩Br(y) and dy := dist(y, ∂Ω). Also, as
in Section 3, we use the letter C to denote a constant depending on n, N , λ, Λ, µ0,
H0, and sometimes on an exponent p characterizing Lebesgue classes.
It is routine to check that for any given y ∈ Ω and 1 ≤ k ≤ N , the linear
functional u 7→ −
Ωρ(y)
uk is bounded on H = Y
0 (Ω)
N . Therefore, by Lax-Milgram
lemma, there exists a unique vρ = vρ;y,k ∈ H such that
(4.1)
ij Dβv
ρ Dαu
i = −
Ωρ(y)
uk, ∀u ∈ H.
Note that as in (3.3), we have
(4.2) ‖Dvρ‖L2(Ω) = ‖vρ‖H ≤ C |Ωρ(y)|
We define the “averaged Green’s matrix” Gρ( · , y) = (G
jk( · , y))
j,k=1 by
jk( · , y) = v
ρ = v
ρ;y,k.
Note that as in (3.5), we have
(4.3)
ij DβG
jk( · , y)Dαu
i = −
Ωρ(y)
uk, ∀u ∈ H.
Next, observe that as in (3.7)–(3.10), for any given f ∈ L∞c (Ω)
N , there exists a
unique u ∈ H such that
ik( · , y)f
i = −
Ωρ(y)
Moreover, as in (3.11), we have
‖Du‖L2(Ω) ≤ C ‖f‖L2n/(n+2)(Ω) .
Also, by following the argument as appears in Section 3.2, we find that if f is
supported in BR(y), then we have
‖u‖L∞(BR/4(y)) ≤ CR
2−n/p ‖f‖Lp(BR(y)) , ∀R < dy, ∀p > n/2.
16 S. HOFMANN AND S. KIM
Therefore, as in (3.16), for any f ∈ L∞c (BR(y)), R < dy, we have
BR(y)
ik( · , y)f
≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀ρ < R/4, ∀p > n/2.
Therefore, as in (3.17), we see that if R < dy, then
‖Gρ( · , y)‖Lq(BR(y)) ≤ CR
2−n+n/q, ∀ρ < R/4, ∀q ∈ [1, n
Then, by following the lines in (3.18)–(3.19), we obtain
|Gρ(x, y)| ≤ C |x− y|
if |x− y| < dy/2, ∀ρ < |x− y| /3.
Next, we shall derive an estimate corresponding to (3.22). Let η ∈ C∞(Rn) be
a cut-off function such that 0 ≤ η ≤ 1, η ≡ 1 outside BR/2(y), η ≡ 0 on BR/4(y),
and |Dη| ≤ C/R, where R ≤ dy . By setting u = η
2vρ ∈ H in (4.1), we obtain
η2 |Dvρ|
≤ CR−2
BR/2(y)\BR/4(y)
≤ CR−2
BR/2(y)\BR/4(y)
|x− y|
2(2−n)
= CR−2R4−n = CR2−n, ∀ρ < R/12.
(4.4)
Therefore, we have (r = R/2)
(4.5)
Ω\Br(y)
|DGρ( · , y)|
≤ Cr2−n, ∀ρ < r/6, ∀r < dy/2.
On the other hand, (4.2) implies that if ρ ≥ r/6, then
(4.6)
Ω\Br(y)
|DGρ( · , y)|
|DGρ( · , y)|
≤ C |Ωρ(y)|
n ≤ Cr2−n.
Therefore, by combining (4.5) and (4.6), we obtain
(4.7)
Ω\Br(y)
|DGρ( · , y)|
≤ Cr2−n, ∀r < dy/2, ∀ρ > 0.
From the estimate (4.7), which corresponds to (3.22), we can derive an estimate
corresponding to (3.24) as follows. By following the lines between (3.22) and (3.23),
we obtain
(4.8) |{x ∈ Ω : |DxG
ρ(x, y)| > t}| ≤ Ct−
n−1 , ∀ρ > 0 if t > (dy/2)
Then, by following lines (3.24)–(3.25), we find (set τ = (R/2)1−n)
(4.9)
BR(y)
|DGρ( · , y)|
≤ CRp(1−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0,
Now, we will derive estimates corresponding (3.20) and (3.26). Let η be the
same as in (4.4). Note that (4.4) and (4.7) implies that for R < dy,
(4.10)
|D(ηvρ)|
η2 |Dvρ|
≤ CR2−n, ∀ρ < R/12.
Since ηvρ ∈ H = Y
0 (Ω), it follows from (4.10) and (2.13) that
(4.11)
Ω\Br(y)
2∗ ≤ Cr−n, ∀r < dy/2, ∀ρ < r/6.
GREEN FUNCTION ESTIMATES 17
On the other hand, if ρ ≥ r/6, then (4.2) implies
Ω\Br(y)
2∗ ≤ C
|Dvρ|
)2∗/2
≤ C |Ωρ|
≤ Cr−n.
(4.12)
Therefore, by combining (4.11) and (4.12), we obtain
(4.13)
Ω\Br(y)
|Gρ( · , y)| 2
≤ Cr−n, ∀r < dy/2, ∀ρ > 0.
As in Section 3.3, the above estimate (4.13) yields
(4.14) |{x ∈ Ω : |Gρ(x, y)| > t}| ≤ Ct−
n−2 , ∀ρ > 0 if t > (dy/2)
Then, as we argued in (4.9), we find (set τ = (R/2)2−n)
(4.15)
BR(y)
|Gρ( · , y)|
≤ CRp(2−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0,
Now, observe that (4.9) and (4.15) in particular imply that
(4.16) ‖Gρ( · , y)‖W 1,p(Bdy (y))
≤ C(dy) for some p ∈ (1,
), uniformly in ρ.
Therefore, from (4.16) together with (4.7) and (4.13), it follows that there exist a
sequence {ρµ}
tending to 0 and functions G( · , y) and G̃( · , y) such that
ρµ( · , y) ⇀ G( · , y) in W 1,p(Bdy (y))
N×N and(4.17)
ρµ( · , y) ⇀ G̃( · , y) in Y 1,2(Ω \Bdy/2(y))
N×N as µ → ∞.(4.18)
Since G( · , y) ≡ G̃( · , y) on Bdy(y) \Bdy/2(y), we shall extend G( · , y) to entire Ω
by setting G( · , y) = G̃( · , y) on Ω \ Bdy(y) but still call it G( · , y) in the sequel.
Moreover, by applying a diagonalization process and passing to a subsequence, if
necessary, we may assume that
(4.19) Gρµ( · , y) ⇀ G( · , y) in Y 1,2(Ω \Br(y))
N×N as µ → ∞, ∀r < dy.
We claim that the following holds:
(4.20)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
To see (4.20), write φ = ηφ+(1− η)φ, where η ∈ C∞c (Bdy (y)) is a cut-off function
satisfying η ≡ 1 on Bdy/2(y). Then, (4.3), (4.17), and (4.19) yield
φk(y) = lim
Ωρµ (y)
ηφk + lim
Ωρµ (y)
(1 − η)φk
= lim
ij DβG
jk (·, y)Dα(ηφ
i) + lim
ij DβG
jk (·, y)Dα((1− η)φ
ij DβGjk( · , y)Dα(ηφ
ij DβGjk( · , y)Dα((1 − η)φ
ij DβGjk( · , y)Dαφ
i as desired.
Next, we claim thatG( · , y) = 0 on ∂Ω in the sense that for all η ∈ C∞c (Ω) satisfying
η ≡ 1 on Br(y) for some r < dy, we have
(1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
18 S. HOFMANN AND S. KIM
To see this, it is enough to show that
(4.21) (1 − η)Gρµ( · , y) ⇀ (1− η)G( · , y) in Y 1,2(Ω)N×N as µ → ∞,
for (1 − η)Gρµ( · , y) ∈ Y
0 (Ω)
N×N for all µ ≥ 1 and Y
0 (Ω) is weakly closed in
Y 1,2(Ω) by Mazur’s theorem. To show (4.21), we note that (4.19) yields
(1− η)Gkl( · , y)φ =
Gkl( · , y)(1− η)φ = lim
kl ( · , y)(1− η)φ
= lim
(1− η)G
kl ( · , y)φ, ∀φ ∈ L
n+2 (Ω),
D((1 − η)Gkl( · , y)) · ψ = −
Gkl( · , y)Dη ·ψ +
DGkl( · , y) · (1 − η)ψ
= − lim
kl ( · , y)Dη · ψ + limµ→∞
kl ( · , y) · (1− η)ψ
= lim
D((1 − η)G
kl ( · , y)) · ψ, ∀ψ ∈ L
2(Ω)N .
By using the same duality argument as in (3.29), we derive the following esti-
mates that correspond to (3.30)–(3.36):
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.22)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.23)
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < dy/2,(4.24)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dy/2)
2−n,(4.25)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dy/2)
1−n.(4.26)
Also, we obtain pointwise bound and Hölder continuity estimate for G( · , y)
corresponding to (3.38) and (3.39), respectively, as follows. Denote by vT the k-th
column of G( · , y) and set R := d̄x,y/2, where
(4.27) d̄x,y := min(dx, dy, |x− y|).
Since v is a weak solution of Lu = 0 in B3R/2(x) ⊂ Ω \ BR/2(y), it follows from
(2.10) and (4.24) that
|v(x)| ≤ CR(2−n)/2 ‖v‖L2∗(Ω\BR/2(y)) ≤ CR
which in turn implies that
(4.28) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|).
In particular, we have
(4.29) |G(x, y)| ≤ C |x− y|
if |x− y| < dx/2 or |x− y| < dy/2.
Similarly, it follows from (2.11) and (4.24) that
(4.30) [v]2Cµ0(BR(x)) ≤ CR
2−n−2µ0
B3R/2(x)
≤ CR2(2−n−µ0).
Therefore, we find that
(4.31) |G(x, y)−G(z, y)| ≤ C |x− z|
µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2,
where d̄x,y is given by (4.27).
GREEN FUNCTION ESTIMATES 19
Denote by tGσ( · , x) the averaged Green’s matrix of tL in Ω with a pole at x ∈ Ω.
Observe that we have an identity corresponding to (3.42).
(4.32) −
Ωρ(y)
tGσkl( · , x) = −
Ωσ(x)
lk( · , y).
Let tG( · , x) be a Green’s matrix of tL in Ω with a pole at x ∈ Ω that is obtained
by a sequence {σν}
ν=1 tending to 0. Then, by a similar argument as appears in
Section 3.6, we obtain
(4.33) Glk(x, y) =
tGkl(y, x), ∀k, l = 1, . . . , N, ∀x, y ∈ Ω, x 6= y,
which is equivalent to say
(4.34) G(x, y) = tG(y, x)T , ∀x, y ∈ Ω, x 6= y.
Using (4.34), we find that G(x, · ) satisfies the estimates corresponding to (4.22)–
(4.26) and (4.31). Moreover, by following the lines (3.44)–(3.45) and using (4.32)
we obtain
(4.35) Gρ(x, y) = −
Ωρ(y)
G(x, z) dz.
Therefore, by the continuity, we find
(4.36) lim
ρ(x, y) = G(x, y), ∀x, y ∈ Ω, x 6= y.
Finally, we summarize what we obtained so far in the following theorem.
Theorem 4.1. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω)
for x ∈ Ω; we set dx = ∞ if Ω = R
n. Assume that operators L and tL satisfy the
property (H). Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y))
i,j=1
(x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that
G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈
C∞c (Ω)
N , the function u = (u1, . . . , uN )T given by
(4.37) u(x) :=
G(x, y)f (y) dy
belongs to Y
0 (Ω)
N and satisfies Lu = f in the sense
(4.38)
ij Dβu
f iφi, ∀φ ∈ C∞c (Ω)
Moreover, G(x, y) has the properties that
(4.39)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy,
(4.40) (1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
Furthermore, G(x, y) satisfies the following estimates:
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.41)
‖G(x, · )‖Lp(Br(x)) ≤ Cp r
2−n+n/p, ∀r < dx, ∀p ∈ [1,
),(4.42)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.43)
‖DG(x, · )‖Lp(Br(x)) ≤ Cp r
1−n+n/p, ∀r < dx, ∀p ∈ [1,
),(4.44)
20 S. HOFMANN AND S. KIM
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < dy/2,(4.45)
‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr
1−n/2, ∀r < dx/2,(4.46)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dy/2)
2−n,(4.47)
|{y ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dx/2)
2−n,(4.48)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dy/2)
1−n.(4.49)
|{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dx/2)
1−n,(4.50)
(4.51) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|),
|G(x, y)−G(z, y)| ≤ C |x− z|
µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2,(4.52)
|G(x, y)−G(x, z)| ≤ C |y − z|
µ0 d̄2−n−µ0x,y if |y − z| < d̄x,y/2,(4.53)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Proof. LetGρ(x, y) andG(x, y) be constructed as above. We have already seen that
G is continuous on {(x, y) ∈ Ω× Ω : x 6= y} and satisfies all the properties (4.39)
– (4.53). Also, as in the proof of Theorem 3.1, we find that for all f ∈ C∞c (Ω)
there is a unique u ∈ (Y
0 (Ω) ∩ C(Ω))
N satisfying
ij Dβu
f ivi, ∀v ∈ Y
0 (Ω)
If we set vi = G
ki(x, · ) above, then by (4.3) and (4.34), we find
(4.54)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Ωρ(x)
Fix r < dx/2. By (4.17), (4.18), and (4.34), we have
ki (x, · )f
i = lim
Br(x)
ki (x, · )f
Ω\Br(x)
ki (x, · )f
B1(x)
Gki(x, · )f
Ω\B1(x)
Gki(x, · )f
Gki(x, · )f
Therefore, (4.37) follows by taking the limits in (4.54). By proceeding as in the
proof of Theorem 3.1, we also derive the uniqueness of Green’s matrix in Ω. �
4.2. Boundary regularity. Let Σ be any subset of Ω and u be aW 1,2(Ω) function.
Then we shall say u = 0 on Σ (in the sense of W 1,2(Ω)) if u is a limit in W 1,2(Ω)
of a sequence of functions in C∞c (Ω \ Σ).
We shall denote ΣR(x) := ∂Ω ∩ BR(x) for any R > 0. We shall abbreviate
ΩR = ΩR(x) and ΣR = ΣR(x) if the point x is well understood in the context.
Lemma 4.2 (Boundary Poincaré inequality). Assume that |BR \ Ω| ≥ θ |BR| for
some θ > 0. Then, for any u ∈ W 1,2(ΩR) satisfying u = 0 on ΣR, we have the
following estimate:
(4.55) ‖u‖L2(ΩR) ≤
R ‖Du‖L2(ΩR) .
GREEN FUNCTION ESTIMATES 21
Proof. Since u = 0 in ΣR, we may extend u to a W
1,2(BR) function by setting
u = 0 in S := BR \Ω. Note that Du = 0 in S. Then the lemma follows from (7.45)
in [10, p. 164]. �
Lemma 4.3 (Boundary Caccioppoli inequality). Let the operator L satisfy condi-
tions (2.2), (2.3). Suppose u is a W 1,2(ΩR)
N solutions of Lu = 0 in ΩR satisfying
u = 0 on ΣR. Then, we have
(4.56) ‖Du‖L2(Ωr) ≤
‖u‖L2(ΩR) , ∀0 < r < R,
where C = C(n,N, λ,Λ) > 0.
Proof. It is well known. �
Definition 4.4. We say that Ω satisfies the condition (S) at a point x̄ ∈ ∂Ω if
there exist θ > 0 and Ra ∈ (0,∞] such that
(4.57) |BR(x̄) \ Ω| ≥ θ |BR(x̄)| , ∀R < Ra.
We say that Ω satisfies the condition (S) uniformly on Σ ⊂ ∂Ω if there exist θ > 0
and Ra such that (4.57) holds for all x̄ ∈ Σ.
Definition 4.5. Let Ω satisfy the condition (S) at x̄ ∈ ∂Ω. We shall say that
an operator L satisfies the property (BH) if there exist µ1, H1 > 0 such that if
u ∈ W 1,2(ΩR(x̄))
N is a weak solution of the problem, Lu = 0 in ΩR(x̄) and u = 0
on ΣR(x̄), where R < Ra, then u satisfies the following estimates:
(4.58)
Ωr(x̄)
)n−2+2µ1
Ωs(x̄)
, ∀0 < r < s ≤ R.
Lemma 4.6. There exists ǫ0 = ǫ0(n, λ0,Λ0) > 0 such that if the coefficients of the
operator L in (2.1) satisfies (2.8) in Lemma 2.2, then L satisfies the property (BH)
with µ1 = µ1(n, λ0,Λ0, θ) > 0 and H1 = H1(n,N, λ0,Λ0, θ) > 0.
Proof. Throughout the proof, we shall abbreviate Ωr = Ωr(x̄) for any r > 0,
Σr = Σr(x̄), the point x̄ ∈ ∂Ω to be understood. For any s ≤ R < Ra, let
vi (i = 1, . . . , N) be a unique W 1,2(Ωs) solution of L0v
i = 0 in Ωs satisfying
vi − ui ∈ W
0 (Ωs), where L0v
i = −Dα(a
αβDβv
We claim that there exist µ2(n, λ0,Λ0, θ) > 0 and C(n, λ0,Λ0, θ) > 0 such that
the following estimate holds:
(4.59)
)n−2+2µ2
, ∀0 < r < s.
We first note that we may assume that r ≤ s/8; otherwise (4.59) becomes trivial.
Since each vi satisfies vi = 0 on Σs, it follows from Theorem 8.27 [10, pp. 203–204]
and Theorem 8.25 [10, pp. 202–203] that there is µ2 = µ2(n, λ0,Λ0, θ) > 0 and
C = C(n, λ0,Λ0, θ) > 0 such that
(4.60) osc
vi ≤ Crµ2s−µ2 sup
|vi| ≤ Crµ2s−µ2−n/2‖vi‖L2(Ωs/2).
22 S. HOFMANN AND S. KIM
In particular, the estimate (4.60) implies vi(x̄) = lim
vi(x) = 0. Then, Lemma 4.3
and Lemma 4.2 imply that for all i = 1, . . . , N (recall r < s/8)
≤ Cr−2
|vi|2 = Cr−2
|vi − vi(x̄)|2
≤ Crn−2
)n−2+2µ2
|vi|2
)n−2+2µ2
|Dvi|2,
and thus we have proved the claim.
Next, note that w := u− v belongs to W
0 (Ωs)
N and thus it satisfies
aαβDβw
(aαβδij −A
ij )Dβu
Therefore, we have
(4.61)
≤ (λ−1 ‖ǫ‖L∞)
where ǫ(x) is as defined in (2.8). By combining (4.59) and (4.61), we obtain
)n−2+2µ2
+ C0 ‖ǫ‖
, ∀0 < r < s.
Now, choose a µ1 ∈ (0, µ2). Then, from a well known iteration argument (see, e.g.,
[8, Lemma 2.1, p. 86]), it follows that there is ǫ0 such that if ‖ǫ‖L∞ < ǫ0, then
(4.58) holds. �
Theorem 4.7. Let the operator L satisfy the properties (H) and (BH). Assume
that Ω satisfies the condition (S) at x̄ ∈ ∂Ω with parameters θ,Ra. Let x ∈ Ω such
that |x− x̄| = dx ≤ R/2, where R < Ra is given. Then, any weak solution u of
Lu = 0 in ΩR(x̄) satisfying u = 0 on ΣR(x̄), we have
(4.62) |u(x)| ≤ CdµxR
1−n/2−µ ‖Du‖L2(ΩR(x̄)) , dx := dist(x, ∂Ω),
where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1).
Proof. The proof is an adaptation of a technique due to Campanato [4]. In this
proof, we shall use the notation ux,r := −
Ωr(x)
u. Also, we shall abbreviate d = dx.
Observe that
(4.63) Ωd(x) = Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄).
We may assume that R > 3d so that Ω2d(x) ⊂ ΩR(x̄); otherwise 2d ≤ R ≤ 3d and
(4.62) follows from Lemma 2.4. We estimate u(x) by
|u(x)| ≤ |u(x)− ux,2d|+ |ux,2d − ux̄,2d|+ |ux̄,2d| := I + II + III.
We shall estimate I first. For any r1 < r2 ≤ 2d, we estimate
(4.64) |ux,r1 − ux,r2 |
≤ 2 |u(z)− ux,r1|
+ 2 |u(z)− ux,r2 |
Note that since Bd(x) ⊂ Ω, we have
|Ωr(x)| ≥ Cr
n, ∀r ≤ 2d.
GREEN FUNCTION ESTIMATES 23
Therefore, by integrating (4.64) over Ωr1(x) with respect to z, we estimates
(4.65) |ux,r1 − ux,r2 |
≤ Cr−n1
|u− ux,r1 |
|u− ux,r2 |
Since u = 0 on ΣR(x̄), we may extend u to BR(x̄) as a W
1,2 function by setting
u = 0 on BR(x̄) \Ω. Therefore, by a version of Poincaré inequality (see, e.g. (7.45)
in [10, p. 164]), we have for all r ≤ 2d,
(4.66)
|u− ux,r|
|u− ux,r|
≤ Cr2
= Cr2
Therefore, by (4.65) and (4.66), we obtain
(4.67) |ux,r1 − ux,r2 |
≤ Cr−n1
Ωr1(x)
+ r22
Ωr2 (x)
Next, we claim that the following estimate holds:
(4.68)
Ωr(x)
)n−2+2µ
ΩR(x̄)
, ∀r ≤ 2d.
We first consider the case when r ≤ d. Note that in this case, we have Ωr(x) = Br(x)
and Ωd(x) = Bd(x). Since L satisfies (H), it follows from (4.63) that
(4.69)
Ωr(x)
)n−2+2µ
Ωd(x)
)n−2+2µ
Ω2d(x̄)
On the other hand, since L satisfies (BH), it follows from (4.58) that
(4.70)
Ω2d(x̄)
)n−2+2µ ∫
ΩR(x̄)
By combining (4.69) and (4.70), we obtain (4.68). Next, consider the case when
d < r. In this case, we have Ωr(x) ⊂ Ω2r(x̄), and thus it follows from (4.58)
Ωr(x)
Ω2r(x̄)
)n−2+2µ
ΩR(x̄)
We proved the claim (4.68).
Now, by using (4.68), we estimates (4.67) as follows (recall r1 < r2 ≤ 2d):
(4.71) |ux,r1 − ux,r2 |
≤ Cr−n1 (r
1 + r
2−n−2µ
ΩR(x̄)
For any r ≤ 2d, set r1 = r2
−(i+1) and r2 = r2
−i in (4.71) to get
∣ux,r2−(i+1) − ux,r2−i
≤ Cr2µ2−2µ(i+1)R2−n−2µ
ΩR(x̄)
Therefore, for 0 ≤ j < k, we obtain
∣ux,r2−k − ux,r2−j
∣ux,r2−(i+1) − ux,r2−i
≤ Crµ
2−µ(i+1)
R1−n/2−µ ‖Du‖L2(ΩR(x̄))
= C2−jµrµR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
(4.72)
24 S. HOFMANN AND S. KIM
By setting r = 2d, j = 0, and letting k → ∞ in (4.72), we obtain
(4.73) I = |u(x) − ux,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
Next, we estimate III. Since |Br(x̄) ∩Bd(x)| ≥ Cr
n for r ≤ 2d, we have
(4.74) |Ωr(x̄)| ≥ Cr
n, ∀r ≤ 2d.
Also, as in (4.66), we have for all r ≤ 2d (recall u ≡ 0 on BR(x̄) \ Ω)
(4.75)
|u− ux̄,r|
|u− ux̄,r|
≤ Cr2
= Cr2
Therefore, as in (4.67) we have for r1 < r2 ≤ 2d,
|ux̄,r1 − ux̄,r2 |
≤ Cr−n1
Ωr1(x̄)
+ r22
Ωr2 (x̄)
Then, by using the property (BH), we obtain (c.f. (4.72), (4.73))
(4.76) |û(x̄)− ux̄,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) ,
where û(x̄) := limk→∞ ux̄,2−kr. (note that (4.72) implies û(x̄) exists). It follows
from (4.74), (4.55), and (4.58) that for any r ≤ 2d, we have
|ux̄,r|
Ωr(x̄)
≤ Cr−n
Ωr(x̄)
≤ Cr2−n
Ωr(x̄)
≤ Cr2−n
)n−2+2µ
ΩR(x̄)
= Cr2µR2−n−2µ
ΩR(x̄)
and thus that û(x̄) = 0. Therefore, by (4.76) we obtain
(4.77) III = |ux̄,2d| = |û(x̄)− ux̄,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
Finally, we estimate II.
(4.78) |ux,2d − ux̄,2d|
≤ 2 |u(z)− ux,2d|
+ 2 |u(z)− ux̄,2d|
By integrating (4.78) over Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄) with respect to z, we estimate
|ux,2d − ux̄,2d|
≤ Cd−n
Ω2d(x)
|u− ux,2d|
Ω2d(x̄)
|u− ux̄,2d|
≤ Cd2−n
Ω2d(x)
Ω2d(x̄)
≤ Cd2µR2−n−2µ
ΩR(x̄)
(4.79)
where we have used (4.66), (4.75), (4.68), and (4.58). Therefore, by combining
(4.73), (4.77), and (4.79), we obtain (4.62). �
Theorem 4.8. Let the operators L, tL satisfy the properties (H) and (BH). Assume
that Ω satisfies the condition (S) uniformly on ∂Ω with parameters θ,Ra. Denote
Rx,y := min(|x− y| , 4Ra).
GREEN FUNCTION ESTIMATES 25
Then the Green matrix G(x, y) satisfies
|G(x, y)| ≤ CdµxR
1−n/2−µ
x,y d
1−n/2
y if dx ≤ Rx,y/8,(4.80)
|G(x, y)| ≤ CdµyR
1−n/2−µ
x,y d
1−n/2
x if dy ≤ Rx,y/8,(4.81)
where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1). As a conse-
quence, we have G( · , y) = 0, G(x, · ) = 0 on ∂Ω in the usual sense.
Proof. We only need to prove (4.80), for (4.81) will then follow from (4.34). Set
R = Rx,y/4, r = dy/2, and choose x̄ ∈ ∂Ω such that |x− x̄| = dx. Then, since
dy ≤ |x− y|+ dx ≤
|x− y| ,
we have
|y − x̄| ≥ |x− y| − dx ≥
|x− y| ≥ R+ r,
and thus, ΩR(x̄) ⊂ Ω\Br(y). Now, we apply Theorem 4.7 with u = G( · , y). Then,
by (4.62) and (4.24), we obtain
|G(x, y)| ≤ CdµxR
1−n/2−µ ‖DG( · , y)‖L2(Ω\Br(y)) ≤ Cd
1−n/2−µ
x,y d
1−n/2
The proof is complete. �
Remark 4.9. We note that in the scalar case, the maximum principle yields (see
[11, Theorem 1.1])
(4.82) G(x, y) ≤ C |x− y|
, ∀x 6= y ∈ Ω.
Then, by the boundary Caccioppoli inequality, we have (c.f. (4.4)–(4.7))
Ω\Br(y)
|DG( · , y)|
≤ Cr2−n, ∀r > 0.
Therefore, in the scalar case we don’t need to require that r < dy/2 (or r < dx/2)
in the proof of Theorem 4.8 and we may as well set r = |x− y| /2 to get
G(x, y) ≤ CdµxR
1−n/2−µ
x,y |x− y|
1−n/2
if dx ≤ Rx,y/8,
G(x, y) ≤ CdµyR
1−n/2−µ
x,y |x− y|
1−n/2
if dy ≤ Rx,y/8.
In particular, if G(x, y) is the Green’s function on Rn+, then we obtain
G(x, y) ≤ Cdµx |x− y|
2−n−µ
if dx ≤ |x− y| /8,
G(x, y) ≤ Cdµy |x− y|
2−n−µ
if dy ≤ |x− y| /8,
for ∂Rn+ satisfies the condition (S) with θ = 1/2 and Ra = ∞.
5. Remarks on VMO coefficients case
Definition 5.1 (Sarason [19]). For a measurable function f defined on Rn, we
shall denote fx,r = −
Br(x)
f and for 0 < δ < ∞ we define
(5.1) Mδ(f) := sup
Br(x)
∣f − fx,r
∣ ; M0(f) := lim
Mδ(f).
We shall say that f belongs to VMO if M0(f) = 0.
26 S. HOFMANN AND S. KIM
Definition 5.2. We say that the operator L satisfies the property (H)loc if there
exist µ0, H0, Rc > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) with
R ≤ Rc satisfy
(5.2)
)n−2+2µ0
, 0 < r < s ≤ R.
Similarly, we say that the transpose operator tL satisfies the property (H)loc if
corresponding estimates hold for all weak solutions u of tLu = 0 in BR with R ≤ Rc.
Lemma 5.3. Let the coefficients of the operator L in (2.1) satisfy the conditions
(2.2) and (2.3). If the coefficients belong to VMO in addition, then L satisfies the
property (H)loc.
Proof. It is well known that if the coefficients are uniformly continuous, then L
satisfies the property (H)loc; see e.g. [8, pp. 87–89]. Essentially, the same proof
carries over to the VMO coefficients case. One only needs to make a note of the
following two facts. First, a theorem of Meyers [16] implies that there is some
p = p(n,N, λ,Λ) > 2 such that if u is a weak solution of Lu = 0 in BR(x), then
Br(x)
B2r(x)
, ∀r < R/2.
Secondly, note that the John-Nirenberg theorem [13] implies that
Br(x)
∣f − fr,x
≤ C(n, q)Mδ(f), ∀r < c(n)δ, ∀q ∈ (0,∞),
where Mδ(f) is defined as in (5.1). For the details, we refer to [3, pp. 47–48]. �
In the rest of this section, we shall assume that the operators L and tL satisfy
the property (H)loc with parameters µ0, H0, Rc. We shall denote
(5.3) rx := min(dx, Rc), r̄x,y := min(d̄x,y, Rc),
where dx = dist(x, ∂Ω) and d̄x,y is as in (4.28). It is routine to check that all
estimates appearing in Section 4.1 remain valid if dx, d̄x,y are replaced by rx, r̄x,y,
respectively. Therefore, we have the following theorem:
Theorem 5.4. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω) for
x ∈ Ω; we set dx = ∞ if Ω = R
n. Assume that operators L and tL satisfy the prop-
erty (H)loc. Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y))
i,j=1
(x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that
G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈
C∞c (Ω)
N , the function u = (u1, . . . , uN )T given by
(5.4) u(x) :=
G(x, y)f (y) dy
belongs to Y
0 (Ω)
N and satisfies Lu = f in the sense
(5.5)
ij Dβu
f iφi, ∀φ ∈ C∞c (Ω)
Moreover, G(x, y) has the properties that
(5.6)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
GREEN FUNCTION ESTIMATES 27
and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy,
(5.7) (1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
Furthermore, G(x, y) satisfies the following estimates: For rx, ry, r̄x,y as in (5.3),
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < ry, ∀p ∈ [1,
),(5.8)
‖G(x, · )‖Lp(Br(x)) ≤ Cp r
2−n+n/p, ∀r < rx, ∀p ∈ [1,
),(5.9)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < ry , ∀p ∈ [1,
),(5.10)
‖DG(x, · )‖Lp(Br(x)) ≤ Cp r
1−n+n/p, ∀r < rx, ∀p ∈ [1,
),(5.11)
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < ry/2,(5.12)
‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr
1−n/2, ∀r < rx/2,(5.13)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (ry/2)
2−n,(5.14)
|{y ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (rx/2)
2−n,(5.15)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (ry/2)
1−n,(5.16)
|{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (rx/2)
1−n,(5.17)
(5.18) |G(x, y)| ≤ Cr̄2−nx,y , ∀x, y ∈ Ω,
|G(x, y)−G(z, y)| ≤ C |x− z|
µ0 r̄2−n−µ0x,y if |x− z| < r̄x,y/2,(5.19)
|G(x, y)−G(x, z)| ≤ C |y − z|
µ0 r̄2−n−µ0x,y if |y − z| < r̄x,y/2,(5.20)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Remark 5.5. Dolzmann-Müller [6] derived a global estimate
(5.21) |G(x, y)| ≤ C |x− y|
∀x, y ∈ Ω, x 6= y,
assuming that Ω is a bounded C1 domain. We have not attempted to derive the
corresponding estimate here. However, we would like to point out that the constant
C in their estimate depends on the domain (e.g., the diameter of the domain and
also some characteristics of ∂Ω) while our interior estimate (5.18) does not.
References
[1] Alfonseca, A.; Auscher, P.; Axelsson, A.; Hofmann, S.; Kim, S. Analyticity of layer
potentials and L2 Solvability of boundary value problems for divergence form elliptic equations
with complex L∞ coefficients. preprint.
[2] Auscher, P. Regularity theorems and heat kernel for elliptic operators. J. London Math. Soc.
(2) 54 (1996), no. 2, 284–296.
[3] Auscher, P.; Tchamitchian, Ph. Square root problem for divergence operators and related
topics. Astérisque No. 249 (1998)
[4] Campanato, S. Equazioni ellittiche del II◦ ordine espazi L(2,λ). (Italian) Ann. Mat. Pura
Appl. (4) 69 (1965) 321–381.
[5] De Giorgi, E. Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli re-
golari. (Italian) Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43.
[6] Dolzmann, G.; Müller, S. Estimates for Green’s matrices of elliptic systems by Lp theory.
Manuscripta Math. 88 (1995), no. 2, 261–273.
28 S. HOFMANN AND S. KIM
[7] Fuchs, M. The Green matrix for strongly elliptic systems of second order with continuous
coefficients. Z. Anal. Anwendungen 5 (1986), no. 6, 507–531.
[8] Giaquinta, M. Multiple integrals in the calculus of variations and nonlinear elliptic systems.
Princeton University Press, Princeton, NJ, 1983.
[9] Giaquinta, M. Introduction to regularity theory for nonlinear elliptic systems. Birkhäuser
Verlag, Basel, 1993.
[10] Gilbarg, D.; Trudinger, N. S. Elliptic partial differential equations of second order, Reprint
of the 1998 ed. Springer-Verlag, Berlin, 2001.
[11] Grüter, M.; Widman, K.-O. The Green function for uniformly elliptic equations.
Manuscripta Math. 37 (1982), no. 3, 303–342.
[12] Hofmann, S.; Kim, S. Gaussian estimates for fundamental solutions to certain parabolic
systems. Publ. Mat. Vol. 48 (2004), pp. 481-496.
[13] John, F.; Nirenberg, L. On functions of bounded mean oscillation. Comm. Pure Appl.
Math. 14 (1961) 415–426.
[14] Littman, W.; Stampacchia, G.; Weinberger, H. F. Regular points for elliptic equations
with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 43–77.
[15] Malý, J.; Ziemer, W. P. Fine regularity of solutions of elliptic partial differential equations.
American Mathematical Society, Providence, RI, 1997.
[16] Meyers, N. G. An Lp-estimate for the gradient of solutions of second order elliptic diver-
gence equations, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963) 189–206.
[17] Morrey, C. B., Jr. Multiple integrals in the calculus of variations. Springer-Verlag New
York, Inc., New York 1966
[18] Moser, J. On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math.
14 (1961) 577–591.
[19] Sarason, D. Functions of vanishing mean oscillation. Trans. Amer. Math. Soc. 207 (1975),
391–405.
(S. Hofmann) Mathematics Department, University of Missouri, Columbia, Missouri
65211, United States of America
E-mail address: hofmann@math.missouri.edu
(S. Kim) Centre for Mathematics and its Applications, The Australian National
University, ACT 0200, Australia
E-mail address: seick.kim@maths.anu.edu.au
	1. Introduction
	2. Preliminaries
	2.1. Strongly elliptic systems
	2.2. Function spaces Y1,2() and Y1,20()
	3. Fundamental matrix in Rn
	3.1. Averaged fundamental matrix
	3.2. L estimates for averaged fundamental matrix
	3.3. Uniform weak-Lnn-2 estimates for bold0mu mumu (,y)
	3.4. Uniform weak-Lnn-1 estimates for Dbold0mu mumu (y)
	3.5. Construction of the fundamental matrix
	3.6. Continuity of the fundamental matrix
	3.7. Properties of fundamental matrix
	4. Green's matrix in general domains
	4.1. Construction of Green's matrix
	4.2. Boundary regularity
	5. Remarks on VMO coefficients case
	References

  We establish existence and pointwise estimates of fundamental solutions and
Green's matrices for divergence form, second order strongly elliptic systems in
a domain $\Omega \subseteq \mathbb{R}^n$, $n \geq 3$, under the assumption that
solutions of the system satisfy De Giorgi-Nash type local H\"{o}lder continuity
estimates. In particular, our results apply to perturbations of diagonal
systems, and thus especially to complex perturbations of a single real
equation.

Introduction
In this article, we study Green’s functions (or Green’s matrices) of second order,
strongly elliptic systems of divergence type in a domain Ω ⊂ Rn with n ≥ 3.
In particular, we treat the Green matrix in the entire space, usually called the
fundamental solution. We shall prove that if a given elliptic system has the property
that all weak solutions of the system are locally Hölder continuous, then it has the
Green’s matrix in Ω. (For example, if coefficients of the system belong to the
space of VMO introduced by Sarason [19], then it will enjoy such a property). For
such elliptic systems, we study standard properties of the Green’s matrix including
pointwise bounds, Lp and weak Lp estimates for Green’s matrix and its derivatives,
For the scalar case, i.e., a single elliptic equation, the existence and properties
of Green’s function was studied by Littman, Stampacchia, and Weinberger [14]
and Grüter and Widman [11]. In this article, we follow the approach of Grüter
and Widman in constructing Green’s matrix. The main technical difficulties arise
from lack of Harnack type inequalities and the maximum principle for the systems.
The key observation on which this article is based is that even in the scalar case,
one can get around Moser’s Harnack inequality [18] or maximum principle but
instead rely solely on De Giorgi-Nash type oscillation estimates [5] in constructing
and studying properties of Green’s functions. From this point of view, this article
provides a unified approach in studying Green’s function for both scalar and systems
of equations. We should point out that there has been some study of Green’s matrix
for systems with continuous coefficients, notably by Fuchs [7] and Dolzmann-Müller
[6]. Our existence results and interior estimates of Green’s function will include
theirs, since as is well known, weak solutions of systems with uniformly continuous
(or VMO) coefficients enjoy local Hölder estimates. On the other hand, we have
not attempted to replicate their boundary estimates, which depend in particular on
having a C1 boundary. Our method does not require boundedness of the domain
nor regularity of the boundary in constructing Green’s matrices, while the methods
2000 Mathematics Subject Classification. Primary 35A08, 35B45; Secondary 35J45.
Key words and phrases. Green’s function, fundamental solution, second order elliptic system.
http://arxiv.org/abs/0704.1352v2
2 S. HOFMANN AND S. KIM
of Fuchs [7] and Dolzmann-Müller [6] require both boundedness and regularity of
the domain at the very beginning. We note that a scalar elliptic equation with
complex coefficients can be identified as an elliptic system with real coefficients
satisfying a special structure, and thus our results apply in particular to complex
perturbations of a scalar real equation. In the complex coefficients setting, the main
results of Section 3 in our paper can be also obtained by following the method of
Auscher [2]. The estimates of the present paper will be applied to the development
of the layer potential method for equations with complex coefficients in [1].
The organization of this paper is as follows. In Section 2, we define the prop-
erty (H), which is essentially equivalent to De Giorgi’s oscillation estimates in the
scalar case, and introduce a function space Y
0 (Ω) which substitutes W
0 (Ω) in
constructing Green’s functions; they are identical if Ω is bounded but in general,
0 (Ω) is a larger space and is more suitable for our purpose. In Section 3, we
study Green’s functions defined in the entire space, which are usually referred to
as the fundamental solutions. The main result is that for a system whose coeffi-
cients are close to those of a diagonal system, the fundamental solution behaves
very much like that of a single equation. In Section 4, we study Green’s matrices in
general domains, including unbounded ones. We also study the boundary behav-
ior of Green’s matrices when the boundary of domain satisfies a measure theoretic
exterior cone condition, called the condition (S). We prove in particular that if the
coefficients of the system are close to those of a diagonal system, then again the
boundary behavior of its Green’s function is much like that of a single equation.
In section 5, we discuss the Green’s matrices of the strongly elliptic systems with
VMO coefficients. By following the same techniques already developed in the pre-
vious two sections, we construct the Green’s matrix in general domains including
the entire space. One subtle difference is that in this VMO coefficients case, one
should play with a localized version of property (H) since basically, the regularity of
weak solutions of the systems with VMO coefficients is inherited from the systems
with constant coefficients when the scale is made small enough. Therefore, all the
estimates for the Green’s matrix stated in this section are only meaningful near a
pole.
Finally, we would like to mention that when n = 2, the method used in this
article breaks down in several places and for that reason we plan to treat the two
dimensional case in a separate paper.
2. Preliminaries
2.1. Strongly elliptic systems. Throughout this article, the summation con-
vention over repeated indices shall be assumed. Let L be a second order elliptic
operator of divergence type acting on vector valued functions u = (u1, . . . , uN )T
defined on Rn (n ≥ 3) in the following way:
(2.1) Lu = −Dα(A
αβ Dβu),
where Aαβ = Aαβ(x) (α, β = 1, . . . , n) are N by N matrices satisfying the strong
ellipticity condition, i.e., there is a number λ > 0 such that
(2.2) A
ij (x)ξ
α ≥ λ |ξ|
|ξiα|
2, ∀x ∈ Rn
GREEN FUNCTION ESTIMATES 3
We also assume that A
ij are bounded, i.e., there is a number Λ > 0 such that
(2.3)
i,j=1
α,β=1
ij (x)|
2 ≤ Λ2, ∀x ∈ Rn.
If we write (2.1) component-wise, then we have
(2.4) (Lu)i = −Dα(A
ij Dβu
j), ∀i = 1, . . . , N.
The transpose operator of tL of L is defined by
(2.5) tLu = −Dα(
αβDβu),
where tAαβ = (Aβα)T (i.e., tA
ij = A
ji ). Note that the coefficients
ij satisfy
(2.2), (2.3) with the same constants λ,Λ.
In the sequel, we shall use the notation −
f := 1
f (assuming 0 < |S| < ∞),
where S is a measurable subset of Rn and |S| denotes the Lebesgue measure of
measurable S.
Definition 2.1. We say that the operator L satisfies the property (H) if there exist
µ0, H0 > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) satisfy
(2.6)
)n−2+2µ0
, 0 < r < s ≤ R.
Similarly, we say that the transpose operator tL satisfies the property (H) if corre-
sponding estimates hold for all weak solutions u of tLu = 0 in BR.
Lemma 2.2. Let (aαβ(x))nα,β=1 be coefficients satisfying the following conditions:
There are constants λ0,Λ0 > 0 such that for all x ∈ R
(2.7) aαβ(x)ξβξα ≥ λ0 |ξ|
, ∀ξ ∈ Rn;
∣aαβ(x)
≤ Λ20.
Then, there exists ǫ0 = ǫ0(n, λ0,Λ0) such that if
(2.8) ǫ2(x) :=
ij (x) − a
αβ(x)δij
< ǫ20, ∀x ∈ R
then the operator L associated with the coefficients A
ij satisfies the condition (H)
with µ0 = µ0(n, λ0,Λ0), H0 = H0(n,N, λ0,Λ0) > 0.
Proof. See e.g., [12, Proposition 2.1]. �
Lemma 2.3. Suppose that the operator L satisfies the following Hölder property
for weak solutions: There are constants µ0, C0 > 0 such that all weak solutions u
of Lu = 0 in B2R = B2R(x0) satisfy the estimate
(2.9) [u]Cµ0 (BR) ≤ C0R
where [f ]Cµ(Ω) denotes the usual C
µ(Ω) semi-norm of f ; see [10] for the definition.
Then, the operator L satisfies the property (H) with µ0 and H0 = H0(n,N, λ,Λ, C0).
4 S. HOFMANN AND S. KIM
Proof. We may assume that r < s/4; otherwise, (2.6) is trivial. Denote ur = −
We may assume, by replacing u by u − us, if necessary, that us = 0. From the
Caccioppoli inequality, (2.9), and then the Poincaré inequality, it follows
≤ Cr−2
|u− u2r|
≤ Cr−2
|u(x) − u(y)|
dy dx
≤ Cr−2[u]2Cµ0 (B2r)(2r)
2µ0 |B2r| ≤ Cr
n−2+2µ0 [u]2Cµ0(Bs/2)
≤ C(r/s)n−2+2µ0s−2
≤ C(r/s)n−2+2µ0
The proof is complete. �
Lemma 2.4. Assume that the operator L satisfies the property (H). Then, the
operator L satisfies the Hölder property (2.9). Moreover, for any p > 0, there
exists Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0 such that all weak solutions u of Lu = 0
in BR = BR(x0) satisfy
(2.10) ‖u‖L∞(Br) ≤
(R− r)n/p
‖u‖Lp(BR) , ∀r ∈ (0, R).
Proof. From a theorem of Morrey [17, Thoerem 3.5.2], the property (H), and the
Caccioppoli inequality, it follows that
(2.11) [u]2Cµ0(BR) ≤ CR
2−n−2µ0 ‖Du‖
L2(B3R/2)
≤ CR−n−2µ0 ‖u‖
L2(B2R)
Then, by a well known averaging argument (see e.g., [12]) we derive
(2.12) ‖u‖L∞(BR/2) ≤ C
where C = C(n,N, λ,Λ, µ0, H0) > 0. For the proof that (2.12) implies (2.10), we
refer to [9, pp. 80–82]. �
2.2. Function spaces Y 1,2(Ω) and Y
0 (Ω).
Definition 2.5. For an open set Ω ⊂ Rn (n ≥ 3), the space Y 1,2(Ω) is defined as
the family of all weakly differentiable functions u ∈ L2
(Ω), where 2∗ = 2n
, whose
weak derivatives are functions in L2(Ω). The space Y 1,2(Ω) is endowed with the
‖u‖Y 1,2(Ω) := ‖u‖L2∗ (Ω) + ‖Du‖L2(Ω) .
We define Y
0 (Ω) as the closure of C
c (Ω) in Y
1,2(Ω), where C∞c (Ω) is the set of
all infinitely differentiable functions with compact supports in Ω.
We note that in the case Ω = Rn, it is well known that Y 1,2(Rn) = Y
see e.g., [15, p. 46]. By the Sobolev inequality, it follows that
(2.13) ‖u‖L2∗ (Ω) ≤ C(n) ‖Du‖L2(Ω) , ∀u ∈ Y
0 (Ω).
Therefore, we have W
0 (Ω) ⊂ Y
0 (Ω) and W
0 (Ω) = Y
0 (Ω) when Ω has finite
Lebesgue measure.
From (2.13), it follows that the bilinear form
(2.14) 〈u,v〉
GREEN FUNCTION ESTIMATES 5
defines an inner product on H := Y
0 (Ω)
N . Also, it is routine to check that H
equipped with the inner product (2.14) is a Hilbert space.
Definition 2.6. We shall denote by H the Hilbert space Y
0 (Ω)
N with the inner
product (2.14). We denote
:= 〈u,u〉
= ‖Du‖L2(Ω) .
We also define the bilinear form associated to the operator L as
B(u,v) :=
ij Dβu
By the strong ellipticity (2.2), it follows that the bilinear form B is coercive; i.e,
(2.15) B(u,u) ≥ λ 〈u,u〉
3. Fundamental matrix in Rn
Throughout this section, we assume that the operators L and tL satisfy the
property (H). The main goal of this section is to construct the fundamental matrix
of the the operator L in the entire Rn, where n ≥ 3. Since Y 1,2(Rn) = Y
we have, as in Definition 2.5,
‖u‖L2∗ (Rn) ≤ C(n) ‖Du‖L2(Rn) , ∀u ∈ Y
1,2(Rn).
We note that W 1,2(Rn) ⊂ Y 1,2(Rn) ⊂ W
loc (R
n). Unless otherwise stated, we
employ the letter C to denote a constant depending on n, N , λ, Λ, µ0, H0, and
sometimes on an exponent p characterizing Lebesgue classes. It should be under-
stood that C may vary from line to line.
3.1. Averaged fundamental matrix. Our approach here is based on that in
[11]. Let y ∈ Rn and 1 ≤ k ≤ N be fixed. For ρ > 0, consider the linear functional
u 7→ −
Bρ(y)
uk. Since
(3.1)
Bρ(y)
≤ Cρ(2−n)/2 ‖u‖L2∗(Rn) ≤ Cρ
(2−n)/2 ‖u‖
Lax-Milgram lemma implies that there exists a unique vρ = vρ;y,k ∈ H such that
(3.2)
ij Dβv
ρ Dαu
i = −
Bρ(y)
uk, ∀u ∈ H.
Note that (2.15), (3.2), and (3.1) imply that
λ ‖vρ‖
≤ B(vρ,vρ) ≤ Cρ
(2−n)/2 ‖vρ‖H ,
and thus we have
(3.3) ‖Dvρ‖L2(Rn) = ‖vρ‖H ≤ Cρ
(2−n)/2.
We define the “averaged fundamental matrix” Γρ( · , y) = (Γ
jk( · , y))
j,k=1 by
(3.4) Γ
jk( · , y) = v
ρ = v
ρ;y,k.
Note that we have
(3.5)
ij DβΓ
jk( · , y)Dαu
i = −
Bρ(y)
uk, ∀u ∈ H,
6 S. HOFMANN AND S. KIM
and equivalently (α ↔ β , i ↔ j).
(3.6)
ij Dβu
ik( · , y) = −
Bρ(y)
uk, ∀u ∈ H.
In the sequel, we shall denote by L∞c (Ω) the family of all L
∞ functions with
compact supports in Ω. For a given f ∈ L∞c (R
n)N consider a linear functional
(3.7) w 7→
f ·w,
which is bounded on H since
(3.8)
≤ ‖f‖
n+2 (Rn)
n−2 (Rn)
≤ C ‖f‖
n+2 (Rn)
Therefore, by Lax-Milgram lemma, there exists u ∈ H such that
(3.9)
ij Dβu
f iwi, ∀w ∈ H.
In particular, if we set w = vρ in (3.9), then by (3.6), we have
(3.10)
ik( · , y)f
i = −
Bρ(y)
Moreover, by setting w = u in (3.9), it follows from (3.8) that
(3.11) ‖Du‖L2(Rn) ≤ C ‖f‖L2n/(n+2)(Rn) .
3.2. L∞ estimates for averaged fundamental matrix. Let u ∈ H be given as
in (3.9). We will obtain local L∞ estimates for u in BR(x0), where x0 ∈ R
n and
R > 0 are fixed but arbitrary.
Fix x ∈ BR(x0) and 0 < s ≤ R. We decompose u as u = u1 + u2, where
u1 ∈ W
1,2(Bs(x))
N is the weak solution of tLu1 = 0 in Bs(x) satisfying u1 = u on
∂Bs(x); i.e., u1 − u ∈ W
0 (Bs(x)). Then, for 0 < r < s, we have
Br(x)
Br(x)
|Du1|
Br(x)
|Du2|
)n−2+2µ0
Bs(x)
|Du1|
Bs(x)
|Du2|
)n−2+2µ0
Bs(x)
Bs(x)
|Du2|
Since u2 ∈ W
0 (Bs(x))
N is a weak solution of tLu2 = f in Bs(x), we have
Bs(x)
|Du2|
≤ C ‖f‖
L2n/(n+2)(Bs(x))
For given p > n/2, choose p0 ∈ (n/2, p) such that µ1 := 2− n/p0 < µ0. Then
(3.12) ‖f‖
n+2 (Bs(x))
≤ ‖f‖
Lp0(Bs(x))
1+2/n−2/p0 ≤ C ‖f‖
Lp0(Rn) s
n−2+2µ1 .
Therefore, after combining the above inequalities, we have for all r < s ≤ R
Br(x)
)n−2+2µ0
Bs(x)
+ Csn−2+2µ1 ‖f‖
Lp0(Rn) .
GREEN FUNCTION ESTIMATES 7
By a well known iteration argument (see e.g., [8, Lemma 2.1, p. 86]), we have
Br(x)
)n−2+2µ1
BR(x)
+ Crn−2+2µ1 ‖f‖
Lp0(Rn)
)n−2+2µ1
+ Crn−2+2µ1 ‖f‖
Lp0(Rn) ,
(3.13)
for all 0 < r < R and x ∈ BR(x0). From (3.13) it follows (see, e.g. [12])
(3.14) [u]2Cµ1 (BR(x0)) ≤ C
R−(n−2+2µ1) ‖Du‖
L2(Rn) + ‖f‖
Lp0(Rn)
Note that since u ∈ H, we have
L2(BR(x0))
≤ ‖u‖
(BR(x0))
≤ CR2 ‖Du‖
L2(Rn) .
Consequently, we have
L∞(BR/2(x0))
≤ CR2µ1 [u]2Cµ1 (BR(x0)) + CR
−n ‖u‖
L2(BR(x0))
R2−n ‖Du‖
L2(Rn) +R
2µ1 ‖f‖
Lp0(Rn)
+ CR2−n ‖Du‖
L2(Rn)
≤ CR2−n ‖f‖
L2n/(n+2)(Rn) + CR
2µ1 ‖f‖
Lp0(Rn) ,
where we used the inequality (3.11) in the last step.
Therefore, if f is supported in BR(x0), then (3.12) yields (recall µ1 = 2− n/p0)
(3.15) ‖u‖L∞(BR/2(x0)) ≤ CR
2−n/p0 ‖f‖Lp0(BR(x0)) ≤ CR
2−n/p ‖f‖Lp(BR(x0)) .
Now, (3.10) implies that for ρ < R/2, we have, by setting x0 = y in (3.15),
(3.16)
BR(y)
ik( · , y)f
Bρ(y)
|u| ≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀p > n/2
provided that f is supported in BR(y). Therefore, by duality, we see that
(3.17) ‖vρ‖Lq(BR(y)) ≤ CR
2−n+n/q, ∀q ∈ [1, n
), ∀ρ ∈ (0, R/2),
where vρ = vρ;y,k is as in (3.4).
Fix x 6= y and let r := 2
|x− y|. If ρ < r/2, then since vρ ∈ W
1,2(Br(x))
N and
satisfies Lvρ = 0 weakly in Br(x), it follows from Lemma 2.4 that
(3.18) |vρ(x)| ≤ Cr
−n ‖vρ‖L1(Br(x)) ≤ Cr
−n ‖vρ‖L1(B3r(y)) ≤ Cr
Since ρ, y, k are arbitrary, we have obtained the following estimates.
(3.19) |Γρ(x, y)| ≤ C |x− y|
, ∀ρ < |x− y| /3.
3.3. Uniform weak-L
n−2 estimates for Γ
ρ( · , y). We claim that the following
estimate holds:
(3.20)
Rn\BR(y)
|Γρ( · , y)|
n−2 ≤ CR−n, ∀R > 0, ∀ρ > 0.
If R > 3ρ, then by (3.19) we have
Rn\BR(y)
|Γρ(x, y)|
n−2 dx ≤ C
Rn\BR(y)
|x− y|
dx ≤ CR−n.
8 S. HOFMANN AND S. KIM
Next, we consider the case R ≤ 3ρ. Let vTρ be the k-th column of the averaged
fundamental matrix Γρ( · , y) as in (3.4). From (3.3), we see that
‖vρ‖L2∗ (Rn\BR(y)) ≤ ‖vρ‖L2∗ (Rn) ≤ ‖Dvρ‖L2(Rn) ≤ Cρ
(2−n)/2.
and thus (3.20) also follows in the case when R ≤ 3ρ.
Now, let At = {x ∈ R
n : |Γρ(x, y)| > t} and choose R = t−1/(n−2). Then,
|At \BR(y)| ≤ t
At\BR(y)
|Γρ( · , y)|
n−2 ≤ Ct−
n−2 t
n−2 = Ct−
n−2 .
Obviously, |At ∩BR(y)| ≤ CR
n = Ct−
n−2 . Therefore, we obtained that for all
t > 0, we have
(3.21) |{x ∈ Rn : |Γρ(x, y)| > t}| ≤ Ct−
n−2 , ∀ρ > 0.
3.4. Uniform weak-L
n−1 estimates for DΓρ( · y). Let vρ be as before. Fix a
cut-off function η ∈ C∞(Rn) such that η ≡ 0 on BR/2(y), η ≡ 1 outside BR(y),
and |Dη| ≤ C/R. If we set u := η2vρ, then by (3.2)
ij Dβv
ij Dβv
ρDαη,
which together with (3.19) implies that if R > 6ρ, then
Rn\BR(y)
|Dvρ|
≤ CR−2
BR(y)\BR/2(y)
≤ CR2−n.
On the other hand, if R ≤ 6ρ, then (3.3) again implies
Rn\BR(y)
|Dvρ|
|Dvρ|
≤ Cρ2−n ≤ CR2−n.
Therefore, we have
(3.22)
Rn\BR(y)
|DΓρ( · , y)|
≤ CR2−n, ∀R > 0, ∀ρ > 0.
Next, let At = {x ∈ R
n : |DxΓ
ρ(x, y)| > t} and choose R = t−1/(n−1). Then
|At \BR(y)| ≤ t
At\BR(y)
|DΓρ( · , y)|
≤ Ct−
and |At ∩BR(y)| ≤ CR
n = Ct−
n−1 . We have thus find that for all t > 0, we have
(3.23) |{x ∈ Rn : |DxΓ
ρ(x, y)| > t}| ≤ Ct−
n−1 , ∀ρ > 0.
3.5. Construction of the fundamental matrix. First, we claim
(3.24) ‖DΓρ( · , y)‖Lp(BR(y)) ≤ CpR
1−n+n/p, ∀ρ > 0, ∀p ∈ (0, n
Let vρ be as before. Note that
BR(y)
|Dvρ|
BR(y)∩{|Dvρ|≤τ}
|Dvρ|
BR(y)∩{|Dvρ|>τ}
|Dvρ|
≤ τp |BR|+
{|Dvρ|>τ}
|Dvρ|
GREEN FUNCTION ESTIMATES 9
By using (3.23), we estimate
{|Dvρ|>τ}
|Dvρ|
ptp−1 |{|Dvρ| > max(t, τ)}| dt
≤ Cτ−
ptp−1 dt+ C
ptp−1−n/(n−1) dt
1− p/(p− n
τp−n/(n−1).
By optimizing over τ , we get
(3.25)
BR(y)
|Dvρ|
≤ CR(1−n)p+n,
from which (3.24) follows.
If we utilize (3.21) instead of (3.23), we obtain a similar estimates for Γρ( · , y)
(3.26) ‖Γρ( · , y)‖Lp(BR(y)) ≤ CpR
2−n+n/p, ∀ρ > 0, ∀p ∈ (0, n
Let us fix q ∈ (1, n
). We have seen that for all R > 0, there exists some
C(R) < ∞ such that
‖Γρ( · , y)‖W 1,q(BR(y)) ≤ C(R), ∀ρ > 0.
Therefore, by a diagonalization process, we obtain a sequence {ρµ}
and Γ( · , y)
loc (R
n)N×N such that limµ→∞ ρµ = 0 and that
(3.27) Γρµ( · , y) ⇀ Γ( · , y) in W 1,q(BR(y))
N×N , ∀R > 0,
where we recall that ⇀ denotes weak convergence. Then, for any φ ∈ C∞c (R
n)N ,
it follows from (3.5)
ij DβΓjk( · , y)Dαφ
i = lim
ij DβΓ
jk ( · , y)Dαφ
= lim
Bρµ (y)
φk = φk(y).
(3.28)
Let vTρ be the k-th column of Γ
ρ( · , y) as before, and let vT be the corresponding
k-th column of Γ( · , y). Then, for any g ∈ L∞c (BR(y))
N , (3.26) yields
(3.29)
v · g
= lim
vρµ · g
≤ CpR
2−n+n/p ‖g‖Lp′(BR(y)) ,
where p′ denotes the conjugate exponent of p ∈ [1, n
). Therefore, we obtain
(3.30) ‖Γ( · , y)‖Lp(BR(y)) ≤ Cp R
2−n+n/p, ∀p ∈ [1, n
By a similar reasoning, we also have by (3.24)
(3.31) ‖DΓ( · , y)‖Lp(BR(y)) ≤ Cp R
1−n+n/p, ∀p ∈ [1, n
Also, with the aid of (3.20) and (3.22), we obtain
Rn\BR(y)
|Γ( · , y)|
≤ CR−n,(3.32)
Rn\BR(y)
|DΓ( · , y)|
≤ CR2−n.(3.33)
10 S. HOFMANN AND S. KIM
In particular, (3.32), (3.33) imply that
(3.34) ‖Γ( · , y)‖Y 1,2(Rn\Br(y)) ≤ Cr
1−n/2, ∀r > 0.
Moreover, arguing as before, we see that the estimates (3.32) and (3.33) imply
|{x ∈ Rn : |Γ(x, y)| > t}| ≤ Ct−
n−2 , ∀t > 0(3.35)
|{x ∈ Rn : |DxΓ(x, y)| > t}| ≤ Ct
n−1 ∀t > 0.(3.36)
Next, we turn to pointwise bounds for Γ( · , y). Let vT be the k-th column of
Γ( · , y). For each x 6= y, denote r = 2
|x− y|. Then, it follows from (3.34) and
(3.28) that v is a weak solution of Lv = 0 in Br(x). Therefore, by Lemma 2.4 and
(3.30) we find
(3.37) |v(x)| ≤ Cr−n ‖v‖L1(Br(x)) ≤ Cr
−n ‖v‖L1(B3r(y)) ≤ Cr
from which it follows
(3.38) |Γ(x, y)| ≤ C |x− y|
, ∀x 6= y.
3.6. Continuity of the fundamental matrix. From the property (H), it follows
that Γ( · , y) is Hölder continuous in Rn \ {y}. In fact, (2.11) together with (3.28)
and (3.33) implies
(3.39) |Γ(x, y)− Γ(z, y)| ≤ C |x− z|
µ0 |x− y|
2−n−µ0 if |x− z| < |x− y| /2.
Moreover, by the same reasoning, it follows from (2.11) and (3.22) that for any
given compact set K ⋐ Rn \ {y}, the sequence {Γρµ( · , y)}
µ=1 is equicontinuous on
K. Also, by Lemma 2.4 and (3.20), we find that there are CK < ∞ and ρK > 0
such that
(3.40) ‖Γρ( · , y)‖L∞(K) ≤ CK ∀ρ < ρK for any compact K ⋐ R
n \ {y} .
Therefore, we may assume, by passing if necessary to a subsequence, that
(3.41) Γρµ( · , y) → Γ( · , y) uniformly on K, for any compact K ⋐ Rn \ {y} .
We will now show that Γ(x, · ) is also Hölder continuous in Rn \ {x}. Denote
by tΓσ( · , x) the averaged fundamental matrix associated to tL, the transpose of L.
Since each column of Γρ( · , y) and tΓσ( · , x) belongs to H, we have by (3.5),
Bρ(y)
tΓσkl( · , x) =
ij DβΓ
jk( · , y)Dα
tΓσil( · , x)
ji Dα
tΓσil( · , x)DβΓ
jk( · , y) = −
Bσ(x)
lk( · , y).
(3.42)
By the same argument as appears in Sec. 3.5, we obtain a sequence {σν}
ν=1 tending
to 0 such that tΓσν ( · , x) converges to tΓ( · , x) uniformly on any compact subset of
n \ {x}, where tΓ( · , x) is a fundamental matrix for tL satisfying all properties
stated in Sec. 3.5. By (3.42), we find that
gklµν := −
Bρµ (y)
tΓσνkl ( · , x) = −
Bσν (x)
lk ( · , y).
From the continuity of Γ
lk ( · , y), it follows that for x, y ∈ R
n with x 6= y, we have
gklµν = lim
Bσν (x)
lk ( · , y) = Γ
lk (x, y)
GREEN FUNCTION ESTIMATES 11
and thus by (3.41) we obtain
gklµν = lim
lk (x, y) = Γlk(x, y).
On the other hand, (3.27) yields
gklµν = lim
Bρµ (y)
tΓσνkl ( · , x) = −
Bρµ (y)
tΓkl( · , x)
and thus it follows from the continuity of tΓkl( · , x) that
gklµν = lim
Bρµ (y)
tΓkl( · , x) =
tΓkl(y, x).
We have thus shown that
Γlk(x, y) =
tΓkl(y, x), ∀k, l = 1, . . . , N, ∀x 6= y,
which is equivalent to say
(3.43) Γ(x, y) = tΓ(y, x)T , ∀x 6= y.
Therefore, we have proved the claim that Γ(x, · ) is Hölder continuous in Rn \ {x}.
So far, we have seen that there is a sequence {ρµ}
tending to 0 such that
ρµ( · , y) → Γ( · , y) in Rn \ {y}. However, by (3.42), we obtain
lk(x, y) = limν→∞
Bσν (x)
lk( · , y) = limν→∞
Bρ(y)
tΓσνkl ( · , x)
Bρ(y)
tΓkl( · , x) = −
Bρ(y)
Γlk(x, · ),
(3.44)
i.e., we have the following representation for the averaged fundamental matrix:
(3.45) Γρ(x, y) = −
Bρ(y)
Γ(x, z) dz.
Therefore, by the continuity, we obtain
(3.46) lim
ρ(x, y) = Γ(x, y), x 6= y.
3.7. Properties of fundamental matrix. We record what we obtained so far in
the following theorem:
Theorem 3.1. Assume that operators L and tL satisfy the property (H). Then,
there exists a unique fundamental matrix Γ(x, y) = (Γij(x, y))
i,j=1 (x 6= y) which is
continuous in {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ(x, · ) is locally integrable
in Rn for all x ∈ Rn and that for all f = (f1, . . . , fN)T ∈ C∞c (R
n)N , the function
u = (u1, . . . , uN )T given by
(3.47) u(x) :=
Γ(x, y)f (y) dy
belongs to Y 1,2(Rn)N and satisfies Lu = f in the sense
(3.48)
ij Dβu
f iφi, ∀φ ∈ C∞c (R
n)N .
Moreover, Γ(x, y) has the property
(3.49)
ij DβΓjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (R
n)N .
12 S. HOFMANN AND S. KIM
Furthermore, Γ(x, y) satisfies the following estimates:
‖Γ( · , y)‖Y 1,2(Rn\Br(y)) + ‖Γ(x, · )‖Y 1,2(Rn\Br(x)) ≤ Cr
2 , ∀r > 0,(3.50)
‖Γ( · , y)‖Lp(Br(y)) + ‖Γ(x, · )‖Lp(Br(x)) ≤ Cpr
2−n+ n
p , ∀p ∈ [1, n
),(3.51)
‖DΓ( · , y)‖Lp(Br(y)) + ‖DΓ(x, · )‖Lp(Br(x)) ≤ Cpr
1−n+ n
p , ∀p ∈ [1, n
),(3.52)
|{x ∈ Rn : |Γ(x, y)| > t}|+ |{y ∈ Rn : |Γ(x, y)| > t}| ≤ Ct−
n−2 ,(3.53)
|{x ∈ Rn : |DxΓ(x, y)| > t}|+ |{y ∈ R
n : |DyΓ(x, y)| > t}| ≤ Ct
n−1 ,(3.54)
|Γ(x, y)| ≤ C |x− y|
, ∀x 6= y,(3.55)
|Γ(x, y)− Γ(z, y)| ≤ C |x− z|
µ0 |x− y|
2−n−µ0 if |x− z| < |x− y| /2,(3.56)
|Γ(x, y)− Γ(x, z)| ≤ C |y − z|
µ0 |x− y|
2−n−µ0 if |y − z| < |x− y| /2,(3.57)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Proof. Let Γρ(x, y) and Γ(x, y) be constructed as above. We have already seen
that Γ is continuous in {(x, y) ∈ Rn × Rn : x 6= y} and satisfies all the properties
(3.49) – (3.57). By using the Lax-Milgram lemma as in Sec. 3.1, we find that for
all f ∈ C∞c (R
n)N , there is a unique u ∈ Y 1,2(Rn)N satisfying
ij Dβu
f ivi, ∀v ∈ Y 1,2(Rn)N .
If we set vi = Γ
ki(x, · ) above, then (3.5) together with (3.43) implies that
(3.58)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Bρ(x)
Assume that f is supported in BR(x) for some R > 0. Then, by (3.27) and (3.43)
we have
ki(x, · )f
i = lim
BR(x)
ki(x, · )f
Γki(x, · )f
By the same argument which lead to (3.14) in Section 3.2, we find that u is Hölder
continuous. Therefore, (3.47) follows by taking the limits in (3.58).
Now, it only remains to prove the uniqueness. Assume that Γ̃(x, y) is another ma-
trix such that Γ̃ is continuous on {(x, y) ∈ Rn × Rn : x 6= y} and such that Γ̃(x, · )
is locally integrable in Rn for all x ∈ Rn and that for all f ∈ C∞c (R
n)N ,
ũ(x) :=
Γ̃(x, y)f (y) dy
belongs to Y 1,2(Rn) and satisfies Lu = f in the sense of (3.48). Then by the
uniqueness in H = Y 1,2(Rn)N , we must have u = ũ. Therefore, for all x ∈ Rn we
(Γ− Γ̃)(x, ·)f = 0, ∀f ∈ C∞c (R
n)N ,
and thus we have Γ ≡ Γ̃ in {(x, y) ∈ Rn × Rn : x 6= y}. �
Theorem 3.2. Assume that the operators L and tL satisfy the property (H). If
f ∈ (L
n+2 (Rn) ∩ L
loc(R
n))N for some p > n/2, then there exists a unique u in
GREEN FUNCTION ESTIMATES 13
Y 1,2(Rn)N such that
(3.59)
ij Dβu
f ivi, ∀v ∈ Y 1,2(Rn)N .
Moreover, u is continuous and has the following representation:
(3.60) uk(x) =
Γki(x, y)f
i(y) dy, k = 1, . . . , N,
where (Γki(x, y))
k,i=1 is the fundamental matrix of L.
Proof. Since f ∈ L
n+2 (Rn)N , the same argument as appears in Sec. 3.1 implies
that there is u ∈ Y 1,2(Rn)N satisfying (3.59). If we set vi = Γ
ki(x, · ) in (3.59),
then (3.2) implies that
(3.61)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Bρ(x)
Next, note that (3.20), (3.26), and the assumption f i ∈ L
n+2 (Rn) ∩ L
loc(R
n) for
some p > n/2, imply that
ki(x, · )f
i = lim
B1(x)
ki(x, · )f
Rn\B1(x)
ki(x, · )f
B1(x)
Γki(x, · )f
Rn\B1(x)
Γki(x, · )f
Γki(x, · )f
(3.62)
Finally, by the same argument which lead to (3.14) in Sec. 3.2, we find that u is
Hölder continuous, and thus (3.60) follows from (3.61) and (3.62). �
Corollary 3.3. Suppose that f = (f1, . . . , fN )T has a bound
(3.63) |f(x)| ≤ C(1 + |x|)−(1+n/2+ǫ) ∀x ∈ Rn for some ǫ > 0.
Then, u = (u1, . . . , uN)T given by (3.60) is a unique Y 1,2(Rn)N solution of Lu = f
in Rn in the sense of (3.59).
Proof. Note that (3.63) implies f ∈ (L
n+2 (Rn) ∩ Lp(Rn))N . �
Theorem 3.4. Assume that L and tL satisfy the property (H). If f ∈ Y 1,2(Rn)N
satisfies Df ∈ L
loc(R
n)N×n for some p > n, then
(3.64) fk(x) =
DαΓki(x, · )A
ij Dβf
j , k = 1, . . . , N,
where (Γki(x, y))
k,i=1 is the fundamental matrix of L.
Proof. We denote by tΓρ the averaged fundamental matrix of tL. Recall that
columns of tΓρ belong to H. Then, by (3.5) we have
ji Dα
ik( · , x)Dβf
j = −
Bρ(x)
14 S. HOFMANN AND S. KIM
As in (3.62), the assumption Df ∈ L
loc(R
n)N for p > n, together with (3.22) and
(3.24) yields
Bρ(x)
fk = lim
B1(x)
Rn\B1(x)
ji Dα
ik( · , x)Dβf
B1(x)
Rn\B1(x)
ji Dα
tΓik( · , x)Dβf
ij DαΓki(x, · )Dβf
(3.65)
where we used (3.43) in the last step. By the Morrey’s inequality [17], f is contin-
uous and thus (3.64) follows from (3.65). �
Corollary 3.5. Assume that L, tL, L̃, and tL̃ satisfy the property (H). Denote by
Γ and Γ̃ the fundamental matrices of L and L̃, respectively. If the coefficients A
of L and Ã
ij of L̃ are Hölder continuous, then
(3.66) Γ̃lm(x, y) = Γlm(x, y) +
DαΓli(x, · )(A
ij − Ã
ij )DβΓ̃jm( · , y), x 6= y.
Proof. We denote by Γρ and Γ̃ρ (ρ < |x− y| /4) the averaged fundamental matrices
of L and L̃ respectively. Recall that columns of Γρ and Γ̃ρ belong to H. Moreover,
since we assume that the coefficients are Hölder continuous, the standard elliptic
theory, (3.38), and (3.45) implies thatDΓρ(x, · ) andDΓ̃ρ( · , y) are locally bounded.
Therefore, by setting f j = Γ̃
jm( · , y) in (3.64) we have
(3.67) Γ̃
lm(x, y) =
DαΓli(x, · )A
ij DβΓ̃
jm( · , y),
Next, set f j = Γ
lj(x, · ) and apply (3.64) with L replaced by
tL̃ to get
lm(x, y) =
t̃Γmi(y, · )
ij DβΓ
lj(x, · ).
By using (3.43) and interchanging indices (α ↔ β, i ↔ j), we obtain
(3.68) Γ
lm(x, y) =
li(x, · )Ã
ij DβΓ̃jm( · , y).
Now, set r = |x− y| /4 and split the integral (3.67) into three pieces (recall ρ < r)
Br(x)
Br(y)
Rn\(Br(x)∪Br(x))
DαΓli(x, · )A
ij DβΓ̃
jm( · , y).
Since we assume that the coefficients are Hölder continuous, it follows from the stan-
dard elliptic theory that DΓ(x, · ) and DΓ̃( · , y) are continuous (and thus bounded)
on Br(y) and Br(x) respectively. Moreover, (3.45) implies
DΓ̃ρ( · , y) → DΓ̃( · , y) uniformly on Br(x) as ρ → 0.
Therefore, as in (3.65), we may take the limit ρ → 0 in (3.67) to get
Γ̃lm(x, y) =
DαΓli(x, · )A
ij DβΓ̃jm( · , y),
GREEN FUNCTION ESTIMATES 15
Similarly, by taking the limit ρ → 0 in (3.68), we obtain
Γlm(x, y) =
DαΓli(x, · )Ã
ij DβΓ̃jm( · , y).
The proof is complete. �
Remark 3.6. We note that in terms of matrix multiplication (3.60) is written as
u(x) =
Γ(x, y)f (y) dy,
where both u,f are understood as column vectors. Also, (3.66) reads
Γ̃(x, y) = Γ(x, y) +
DαΓ(x, · )(A
αβ − Ãαβ)DβΓ( · , y).
4. Green’s matrix in general domains
4.1. Construction of Green’s matrix. In this section, we shall construct the
Green’s matrix in any open, connected set Ω ⊂ Rn, where n ≥ 3. To construct the
Green’s matrix in Ω, we need to adjust arguments in Section 3.
Henceforth, we shall denote Ωr(y) := Ω ∩Br(y) and dy := dist(y, ∂Ω). Also, as
in Section 3, we use the letter C to denote a constant depending on n, N , λ, Λ, µ0,
H0, and sometimes on an exponent p characterizing Lebesgue classes.
It is routine to check that for any given y ∈ Ω and 1 ≤ k ≤ N , the linear
functional u 7→ −
Ωρ(y)
uk is bounded on H = Y
0 (Ω)
N . Therefore, by Lax-Milgram
lemma, there exists a unique vρ = vρ;y,k ∈ H such that
(4.1)
ij Dβv
ρ Dαu
i = −
Ωρ(y)
uk, ∀u ∈ H.
Note that as in (3.3), we have
(4.2) ‖Dvρ‖L2(Ω) = ‖vρ‖H ≤ C |Ωρ(y)|
We define the “averaged Green’s matrix” Gρ( · , y) = (G
jk( · , y))
j,k=1 by
jk( · , y) = v
ρ = v
ρ;y,k.
Note that as in (3.5), we have
(4.3)
ij DβG
jk( · , y)Dαu
i = −
Ωρ(y)
uk, ∀u ∈ H.
Next, observe that as in (3.7)–(3.10), for any given f ∈ L∞c (Ω)
N , there exists a
unique u ∈ H such that
ik( · , y)f
i = −
Ωρ(y)
Moreover, as in (3.11), we have
‖Du‖L2(Ω) ≤ C ‖f‖L2n/(n+2)(Ω) .
Also, by following the argument as appears in Section 3.2, we find that if f is
supported in BR(y), then we have
‖u‖L∞(BR/4(y)) ≤ CR
2−n/p ‖f‖Lp(BR(y)) , ∀R < dy, ∀p > n/2.
16 S. HOFMANN AND S. KIM
Therefore, as in (3.16), for any f ∈ L∞c (BR(y)), R < dy, we have
BR(y)
ik( · , y)f
≤ CR2−n/p ‖f‖Lp(BR(y)) , ∀ρ < R/4, ∀p > n/2.
Therefore, as in (3.17), we see that if R < dy, then
‖Gρ( · , y)‖Lq(BR(y)) ≤ CR
2−n+n/q, ∀ρ < R/4, ∀q ∈ [1, n
Then, by following the lines in (3.18)–(3.19), we obtain
|Gρ(x, y)| ≤ C |x− y|
if |x− y| < dy/2, ∀ρ < |x− y| /3.
Next, we shall derive an estimate corresponding to (3.22). Let η ∈ C∞(Rn) be
a cut-off function such that 0 ≤ η ≤ 1, η ≡ 1 outside BR/2(y), η ≡ 0 on BR/4(y),
and |Dη| ≤ C/R, where R ≤ dy . By setting u = η
2vρ ∈ H in (4.1), we obtain
η2 |Dvρ|
≤ CR−2
BR/2(y)\BR/4(y)
≤ CR−2
BR/2(y)\BR/4(y)
|x− y|
2(2−n)
= CR−2R4−n = CR2−n, ∀ρ < R/12.
(4.4)
Therefore, we have (r = R/2)
(4.5)
Ω\Br(y)
|DGρ( · , y)|
≤ Cr2−n, ∀ρ < r/6, ∀r < dy/2.
On the other hand, (4.2) implies that if ρ ≥ r/6, then
(4.6)
Ω\Br(y)
|DGρ( · , y)|
|DGρ( · , y)|
≤ C |Ωρ(y)|
n ≤ Cr2−n.
Therefore, by combining (4.5) and (4.6), we obtain
(4.7)
Ω\Br(y)
|DGρ( · , y)|
≤ Cr2−n, ∀r < dy/2, ∀ρ > 0.
From the estimate (4.7), which corresponds to (3.22), we can derive an estimate
corresponding to (3.24) as follows. By following the lines between (3.22) and (3.23),
we obtain
(4.8) |{x ∈ Ω : |DxG
ρ(x, y)| > t}| ≤ Ct−
n−1 , ∀ρ > 0 if t > (dy/2)
Then, by following lines (3.24)–(3.25), we find (set τ = (R/2)1−n)
(4.9)
BR(y)
|DGρ( · , y)|
≤ CRp(1−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0,
Now, we will derive estimates corresponding (3.20) and (3.26). Let η be the
same as in (4.4). Note that (4.4) and (4.7) implies that for R < dy,
(4.10)
|D(ηvρ)|
η2 |Dvρ|
≤ CR2−n, ∀ρ < R/12.
Since ηvρ ∈ H = Y
0 (Ω), it follows from (4.10) and (2.13) that
(4.11)
Ω\Br(y)
2∗ ≤ Cr−n, ∀r < dy/2, ∀ρ < r/6.
GREEN FUNCTION ESTIMATES 17
On the other hand, if ρ ≥ r/6, then (4.2) implies
Ω\Br(y)
2∗ ≤ C
|Dvρ|
)2∗/2
≤ C |Ωρ|
≤ Cr−n.
(4.12)
Therefore, by combining (4.11) and (4.12), we obtain
(4.13)
Ω\Br(y)
|Gρ( · , y)| 2
≤ Cr−n, ∀r < dy/2, ∀ρ > 0.
As in Section 3.3, the above estimate (4.13) yields
(4.14) |{x ∈ Ω : |Gρ(x, y)| > t}| ≤ Ct−
n−2 , ∀ρ > 0 if t > (dy/2)
Then, as we argued in (4.9), we find (set τ = (R/2)2−n)
(4.15)
BR(y)
|Gρ( · , y)|
≤ CRp(2−n)+n, ∀R < dy, ∀ρ > 0, ∀p ∈ (0,
Now, observe that (4.9) and (4.15) in particular imply that
(4.16) ‖Gρ( · , y)‖W 1,p(Bdy (y))
≤ C(dy) for some p ∈ (1,
), uniformly in ρ.
Therefore, from (4.16) together with (4.7) and (4.13), it follows that there exist a
sequence {ρµ}
tending to 0 and functions G( · , y) and G̃( · , y) such that
ρµ( · , y) ⇀ G( · , y) in W 1,p(Bdy (y))
N×N and(4.17)
ρµ( · , y) ⇀ G̃( · , y) in Y 1,2(Ω \Bdy/2(y))
N×N as µ → ∞.(4.18)
Since G( · , y) ≡ G̃( · , y) on Bdy(y) \Bdy/2(y), we shall extend G( · , y) to entire Ω
by setting G( · , y) = G̃( · , y) on Ω \ Bdy(y) but still call it G( · , y) in the sequel.
Moreover, by applying a diagonalization process and passing to a subsequence, if
necessary, we may assume that
(4.19) Gρµ( · , y) ⇀ G( · , y) in Y 1,2(Ω \Br(y))
N×N as µ → ∞, ∀r < dy.
We claim that the following holds:
(4.20)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
To see (4.20), write φ = ηφ+(1− η)φ, where η ∈ C∞c (Bdy (y)) is a cut-off function
satisfying η ≡ 1 on Bdy/2(y). Then, (4.3), (4.17), and (4.19) yield
φk(y) = lim
Ωρµ (y)
ηφk + lim
Ωρµ (y)
(1 − η)φk
= lim
ij DβG
jk (·, y)Dα(ηφ
i) + lim
ij DβG
jk (·, y)Dα((1− η)φ
ij DβGjk( · , y)Dα(ηφ
ij DβGjk( · , y)Dα((1 − η)φ
ij DβGjk( · , y)Dαφ
i as desired.
Next, we claim thatG( · , y) = 0 on ∂Ω in the sense that for all η ∈ C∞c (Ω) satisfying
η ≡ 1 on Br(y) for some r < dy, we have
(1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
18 S. HOFMANN AND S. KIM
To see this, it is enough to show that
(4.21) (1 − η)Gρµ( · , y) ⇀ (1− η)G( · , y) in Y 1,2(Ω)N×N as µ → ∞,
for (1 − η)Gρµ( · , y) ∈ Y
0 (Ω)
N×N for all µ ≥ 1 and Y
0 (Ω) is weakly closed in
Y 1,2(Ω) by Mazur’s theorem. To show (4.21), we note that (4.19) yields
(1− η)Gkl( · , y)φ =
Gkl( · , y)(1− η)φ = lim
kl ( · , y)(1− η)φ
= lim
(1− η)G
kl ( · , y)φ, ∀φ ∈ L
n+2 (Ω),
D((1 − η)Gkl( · , y)) · ψ = −
Gkl( · , y)Dη ·ψ +
DGkl( · , y) · (1 − η)ψ
= − lim
kl ( · , y)Dη · ψ + limµ→∞
kl ( · , y) · (1− η)ψ
= lim
D((1 − η)G
kl ( · , y)) · ψ, ∀ψ ∈ L
2(Ω)N .
By using the same duality argument as in (3.29), we derive the following esti-
mates that correspond to (3.30)–(3.36):
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.22)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.23)
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < dy/2,(4.24)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dy/2)
2−n,(4.25)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dy/2)
1−n.(4.26)
Also, we obtain pointwise bound and Hölder continuity estimate for G( · , y)
corresponding to (3.38) and (3.39), respectively, as follows. Denote by vT the k-th
column of G( · , y) and set R := d̄x,y/2, where
(4.27) d̄x,y := min(dx, dy, |x− y|).
Since v is a weak solution of Lu = 0 in B3R/2(x) ⊂ Ω \ BR/2(y), it follows from
(2.10) and (4.24) that
|v(x)| ≤ CR(2−n)/2 ‖v‖L2∗(Ω\BR/2(y)) ≤ CR
which in turn implies that
(4.28) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|).
In particular, we have
(4.29) |G(x, y)| ≤ C |x− y|
if |x− y| < dx/2 or |x− y| < dy/2.
Similarly, it follows from (2.11) and (4.24) that
(4.30) [v]2Cµ0(BR(x)) ≤ CR
2−n−2µ0
B3R/2(x)
≤ CR2(2−n−µ0).
Therefore, we find that
(4.31) |G(x, y)−G(z, y)| ≤ C |x− z|
µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2,
where d̄x,y is given by (4.27).
GREEN FUNCTION ESTIMATES 19
Denote by tGσ( · , x) the averaged Green’s matrix of tL in Ω with a pole at x ∈ Ω.
Observe that we have an identity corresponding to (3.42).
(4.32) −
Ωρ(y)
tGσkl( · , x) = −
Ωσ(x)
lk( · , y).
Let tG( · , x) be a Green’s matrix of tL in Ω with a pole at x ∈ Ω that is obtained
by a sequence {σν}
ν=1 tending to 0. Then, by a similar argument as appears in
Section 3.6, we obtain
(4.33) Glk(x, y) =
tGkl(y, x), ∀k, l = 1, . . . , N, ∀x, y ∈ Ω, x 6= y,
which is equivalent to say
(4.34) G(x, y) = tG(y, x)T , ∀x, y ∈ Ω, x 6= y.
Using (4.34), we find that G(x, · ) satisfies the estimates corresponding to (4.22)–
(4.26) and (4.31). Moreover, by following the lines (3.44)–(3.45) and using (4.32)
we obtain
(4.35) Gρ(x, y) = −
Ωρ(y)
G(x, z) dz.
Therefore, by the continuity, we find
(4.36) lim
ρ(x, y) = G(x, y), ∀x, y ∈ Ω, x 6= y.
Finally, we summarize what we obtained so far in the following theorem.
Theorem 4.1. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω)
for x ∈ Ω; we set dx = ∞ if Ω = R
n. Assume that operators L and tL satisfy the
property (H). Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y))
i,j=1
(x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that
G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈
C∞c (Ω)
N , the function u = (u1, . . . , uN )T given by
(4.37) u(x) :=
G(x, y)f (y) dy
belongs to Y
0 (Ω)
N and satisfies Lu = f in the sense
(4.38)
ij Dβu
f iφi, ∀φ ∈ C∞c (Ω)
Moreover, G(x, y) has the properties that
(4.39)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy,
(4.40) (1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
Furthermore, G(x, y) satisfies the following estimates:
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.41)
‖G(x, · )‖Lp(Br(x)) ≤ Cp r
2−n+n/p, ∀r < dx, ∀p ∈ [1,
),(4.42)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < dy, ∀p ∈ [1,
),(4.43)
‖DG(x, · )‖Lp(Br(x)) ≤ Cp r
1−n+n/p, ∀r < dx, ∀p ∈ [1,
),(4.44)
20 S. HOFMANN AND S. KIM
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < dy/2,(4.45)
‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr
1−n/2, ∀r < dx/2,(4.46)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dy/2)
2−n,(4.47)
|{y ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (dx/2)
2−n,(4.48)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dy/2)
1−n.(4.49)
|{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (dx/2)
1−n,(4.50)
(4.51) |G(x, y)| ≤ Cd̄2−nx,y , where d̄x,y := min(dx, dy, |x− y|),
|G(x, y)−G(z, y)| ≤ C |x− z|
µ0 d̄2−n−µ0x,y if |x− z| < d̄x,y/2,(4.52)
|G(x, y)−G(x, z)| ≤ C |y − z|
µ0 d̄2−n−µ0x,y if |y − z| < d̄x,y/2,(4.53)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Proof. LetGρ(x, y) andG(x, y) be constructed as above. We have already seen that
G is continuous on {(x, y) ∈ Ω× Ω : x 6= y} and satisfies all the properties (4.39)
– (4.53). Also, as in the proof of Theorem 3.1, we find that for all f ∈ C∞c (Ω)
there is a unique u ∈ (Y
0 (Ω) ∩ C(Ω))
N satisfying
ij Dβu
f ivi, ∀v ∈ Y
0 (Ω)
If we set vi = G
ki(x, · ) above, then by (4.3) and (4.34), we find
(4.54)
ki(x, · )f
ji Dα
ik( · , x)Dβu
j = −
Ωρ(x)
Fix r < dx/2. By (4.17), (4.18), and (4.34), we have
ki (x, · )f
i = lim
Br(x)
ki (x, · )f
Ω\Br(x)
ki (x, · )f
B1(x)
Gki(x, · )f
Ω\B1(x)
Gki(x, · )f
Gki(x, · )f
Therefore, (4.37) follows by taking the limits in (4.54). By proceeding as in the
proof of Theorem 3.1, we also derive the uniqueness of Green’s matrix in Ω. �
4.2. Boundary regularity. Let Σ be any subset of Ω and u be aW 1,2(Ω) function.
Then we shall say u = 0 on Σ (in the sense of W 1,2(Ω)) if u is a limit in W 1,2(Ω)
of a sequence of functions in C∞c (Ω \ Σ).
We shall denote ΣR(x) := ∂Ω ∩ BR(x) for any R > 0. We shall abbreviate
ΩR = ΩR(x) and ΣR = ΣR(x) if the point x is well understood in the context.
Lemma 4.2 (Boundary Poincaré inequality). Assume that |BR \ Ω| ≥ θ |BR| for
some θ > 0. Then, for any u ∈ W 1,2(ΩR) satisfying u = 0 on ΣR, we have the
following estimate:
(4.55) ‖u‖L2(ΩR) ≤
R ‖Du‖L2(ΩR) .
GREEN FUNCTION ESTIMATES 21
Proof. Since u = 0 in ΣR, we may extend u to a W
1,2(BR) function by setting
u = 0 in S := BR \Ω. Note that Du = 0 in S. Then the lemma follows from (7.45)
in [10, p. 164]. �
Lemma 4.3 (Boundary Caccioppoli inequality). Let the operator L satisfy condi-
tions (2.2), (2.3). Suppose u is a W 1,2(ΩR)
N solutions of Lu = 0 in ΩR satisfying
u = 0 on ΣR. Then, we have
(4.56) ‖Du‖L2(Ωr) ≤
‖u‖L2(ΩR) , ∀0 < r < R,
where C = C(n,N, λ,Λ) > 0.
Proof. It is well known. �
Definition 4.4. We say that Ω satisfies the condition (S) at a point x̄ ∈ ∂Ω if
there exist θ > 0 and Ra ∈ (0,∞] such that
(4.57) |BR(x̄) \ Ω| ≥ θ |BR(x̄)| , ∀R < Ra.
We say that Ω satisfies the condition (S) uniformly on Σ ⊂ ∂Ω if there exist θ > 0
and Ra such that (4.57) holds for all x̄ ∈ Σ.
Definition 4.5. Let Ω satisfy the condition (S) at x̄ ∈ ∂Ω. We shall say that
an operator L satisfies the property (BH) if there exist µ1, H1 > 0 such that if
u ∈ W 1,2(ΩR(x̄))
N is a weak solution of the problem, Lu = 0 in ΩR(x̄) and u = 0
on ΣR(x̄), where R < Ra, then u satisfies the following estimates:
(4.58)
Ωr(x̄)
)n−2+2µ1
Ωs(x̄)
, ∀0 < r < s ≤ R.
Lemma 4.6. There exists ǫ0 = ǫ0(n, λ0,Λ0) > 0 such that if the coefficients of the
operator L in (2.1) satisfies (2.8) in Lemma 2.2, then L satisfies the property (BH)
with µ1 = µ1(n, λ0,Λ0, θ) > 0 and H1 = H1(n,N, λ0,Λ0, θ) > 0.
Proof. Throughout the proof, we shall abbreviate Ωr = Ωr(x̄) for any r > 0,
Σr = Σr(x̄), the point x̄ ∈ ∂Ω to be understood. For any s ≤ R < Ra, let
vi (i = 1, . . . , N) be a unique W 1,2(Ωs) solution of L0v
i = 0 in Ωs satisfying
vi − ui ∈ W
0 (Ωs), where L0v
i = −Dα(a
αβDβv
We claim that there exist µ2(n, λ0,Λ0, θ) > 0 and C(n, λ0,Λ0, θ) > 0 such that
the following estimate holds:
(4.59)
)n−2+2µ2
, ∀0 < r < s.
We first note that we may assume that r ≤ s/8; otherwise (4.59) becomes trivial.
Since each vi satisfies vi = 0 on Σs, it follows from Theorem 8.27 [10, pp. 203–204]
and Theorem 8.25 [10, pp. 202–203] that there is µ2 = µ2(n, λ0,Λ0, θ) > 0 and
C = C(n, λ0,Λ0, θ) > 0 such that
(4.60) osc
vi ≤ Crµ2s−µ2 sup
|vi| ≤ Crµ2s−µ2−n/2‖vi‖L2(Ωs/2).
22 S. HOFMANN AND S. KIM
In particular, the estimate (4.60) implies vi(x̄) = lim
vi(x) = 0. Then, Lemma 4.3
and Lemma 4.2 imply that for all i = 1, . . . , N (recall r < s/8)
≤ Cr−2
|vi|2 = Cr−2
|vi − vi(x̄)|2
≤ Crn−2
)n−2+2µ2
|vi|2
)n−2+2µ2
|Dvi|2,
and thus we have proved the claim.
Next, note that w := u− v belongs to W
0 (Ωs)
N and thus it satisfies
aαβDβw
(aαβδij −A
ij )Dβu
Therefore, we have
(4.61)
≤ (λ−1 ‖ǫ‖L∞)
where ǫ(x) is as defined in (2.8). By combining (4.59) and (4.61), we obtain
)n−2+2µ2
+ C0 ‖ǫ‖
, ∀0 < r < s.
Now, choose a µ1 ∈ (0, µ2). Then, from a well known iteration argument (see, e.g.,
[8, Lemma 2.1, p. 86]), it follows that there is ǫ0 such that if ‖ǫ‖L∞ < ǫ0, then
(4.58) holds. �
Theorem 4.7. Let the operator L satisfy the properties (H) and (BH). Assume
that Ω satisfies the condition (S) at x̄ ∈ ∂Ω with parameters θ,Ra. Let x ∈ Ω such
that |x− x̄| = dx ≤ R/2, where R < Ra is given. Then, any weak solution u of
Lu = 0 in ΩR(x̄) satisfying u = 0 on ΣR(x̄), we have
(4.62) |u(x)| ≤ CdµxR
1−n/2−µ ‖Du‖L2(ΩR(x̄)) , dx := dist(x, ∂Ω),
where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1).
Proof. The proof is an adaptation of a technique due to Campanato [4]. In this
proof, we shall use the notation ux,r := −
Ωr(x)
u. Also, we shall abbreviate d = dx.
Observe that
(4.63) Ωd(x) = Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄).
We may assume that R > 3d so that Ω2d(x) ⊂ ΩR(x̄); otherwise 2d ≤ R ≤ 3d and
(4.62) follows from Lemma 2.4. We estimate u(x) by
|u(x)| ≤ |u(x)− ux,2d|+ |ux,2d − ux̄,2d|+ |ux̄,2d| := I + II + III.
We shall estimate I first. For any r1 < r2 ≤ 2d, we estimate
(4.64) |ux,r1 − ux,r2 |
≤ 2 |u(z)− ux,r1|
+ 2 |u(z)− ux,r2 |
Note that since Bd(x) ⊂ Ω, we have
|Ωr(x)| ≥ Cr
n, ∀r ≤ 2d.
GREEN FUNCTION ESTIMATES 23
Therefore, by integrating (4.64) over Ωr1(x) with respect to z, we estimates
(4.65) |ux,r1 − ux,r2 |
≤ Cr−n1
|u− ux,r1 |
|u− ux,r2 |
Since u = 0 on ΣR(x̄), we may extend u to BR(x̄) as a W
1,2 function by setting
u = 0 on BR(x̄) \Ω. Therefore, by a version of Poincaré inequality (see, e.g. (7.45)
in [10, p. 164]), we have for all r ≤ 2d,
(4.66)
|u− ux,r|
|u− ux,r|
≤ Cr2
= Cr2
Therefore, by (4.65) and (4.66), we obtain
(4.67) |ux,r1 − ux,r2 |
≤ Cr−n1
Ωr1(x)
+ r22
Ωr2 (x)
Next, we claim that the following estimate holds:
(4.68)
Ωr(x)
)n−2+2µ
ΩR(x̄)
, ∀r ≤ 2d.
We first consider the case when r ≤ d. Note that in this case, we have Ωr(x) = Br(x)
and Ωd(x) = Bd(x). Since L satisfies (H), it follows from (4.63) that
(4.69)
Ωr(x)
)n−2+2µ
Ωd(x)
)n−2+2µ
Ω2d(x̄)
On the other hand, since L satisfies (BH), it follows from (4.58) that
(4.70)
Ω2d(x̄)
)n−2+2µ ∫
ΩR(x̄)
By combining (4.69) and (4.70), we obtain (4.68). Next, consider the case when
d < r. In this case, we have Ωr(x) ⊂ Ω2r(x̄), and thus it follows from (4.58)
Ωr(x)
Ω2r(x̄)
)n−2+2µ
ΩR(x̄)
We proved the claim (4.68).
Now, by using (4.68), we estimates (4.67) as follows (recall r1 < r2 ≤ 2d):
(4.71) |ux,r1 − ux,r2 |
≤ Cr−n1 (r
1 + r
2−n−2µ
ΩR(x̄)
For any r ≤ 2d, set r1 = r2
−(i+1) and r2 = r2
−i in (4.71) to get
∣ux,r2−(i+1) − ux,r2−i
≤ Cr2µ2−2µ(i+1)R2−n−2µ
ΩR(x̄)
Therefore, for 0 ≤ j < k, we obtain
∣ux,r2−k − ux,r2−j
∣ux,r2−(i+1) − ux,r2−i
≤ Crµ
2−µ(i+1)
R1−n/2−µ ‖Du‖L2(ΩR(x̄))
= C2−jµrµR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
(4.72)
24 S. HOFMANN AND S. KIM
By setting r = 2d, j = 0, and letting k → ∞ in (4.72), we obtain
(4.73) I = |u(x) − ux,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
Next, we estimate III. Since |Br(x̄) ∩Bd(x)| ≥ Cr
n for r ≤ 2d, we have
(4.74) |Ωr(x̄)| ≥ Cr
n, ∀r ≤ 2d.
Also, as in (4.66), we have for all r ≤ 2d (recall u ≡ 0 on BR(x̄) \ Ω)
(4.75)
|u− ux̄,r|
|u− ux̄,r|
≤ Cr2
= Cr2
Therefore, as in (4.67) we have for r1 < r2 ≤ 2d,
|ux̄,r1 − ux̄,r2 |
≤ Cr−n1
Ωr1(x̄)
+ r22
Ωr2 (x̄)
Then, by using the property (BH), we obtain (c.f. (4.72), (4.73))
(4.76) |û(x̄)− ux̄,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) ,
where û(x̄) := limk→∞ ux̄,2−kr. (note that (4.72) implies û(x̄) exists). It follows
from (4.74), (4.55), and (4.58) that for any r ≤ 2d, we have
|ux̄,r|
Ωr(x̄)
≤ Cr−n
Ωr(x̄)
≤ Cr2−n
Ωr(x̄)
≤ Cr2−n
)n−2+2µ
ΩR(x̄)
= Cr2µR2−n−2µ
ΩR(x̄)
and thus that û(x̄) = 0. Therefore, by (4.76) we obtain
(4.77) III = |ux̄,2d| = |û(x̄)− ux̄,2d| ≤ Cd
µR1−n/2−µ ‖Du‖L2(ΩR(x̄)) .
Finally, we estimate II.
(4.78) |ux,2d − ux̄,2d|
≤ 2 |u(z)− ux,2d|
+ 2 |u(z)− ux̄,2d|
By integrating (4.78) over Bd(x) ⊂ Ω2d(x) ∩Ω2d(x̄) with respect to z, we estimate
|ux,2d − ux̄,2d|
≤ Cd−n
Ω2d(x)
|u− ux,2d|
Ω2d(x̄)
|u− ux̄,2d|
≤ Cd2−n
Ω2d(x)
Ω2d(x̄)
≤ Cd2µR2−n−2µ
ΩR(x̄)
(4.79)
where we have used (4.66), (4.75), (4.68), and (4.58). Therefore, by combining
(4.73), (4.77), and (4.79), we obtain (4.62). �
Theorem 4.8. Let the operators L, tL satisfy the properties (H) and (BH). Assume
that Ω satisfies the condition (S) uniformly on ∂Ω with parameters θ,Ra. Denote
Rx,y := min(|x− y| , 4Ra).
GREEN FUNCTION ESTIMATES 25
Then the Green matrix G(x, y) satisfies
|G(x, y)| ≤ CdµxR
1−n/2−µ
x,y d
1−n/2
y if dx ≤ Rx,y/8,(4.80)
|G(x, y)| ≤ CdµyR
1−n/2−µ
x,y d
1−n/2
x if dy ≤ Rx,y/8,(4.81)
where C = C(n,N, λ,Λ, θ, µ0, µ1, H0, H1) > 0 and µ = min(µ0, µ1). As a conse-
quence, we have G( · , y) = 0, G(x, · ) = 0 on ∂Ω in the usual sense.
Proof. We only need to prove (4.80), for (4.81) will then follow from (4.34). Set
R = Rx,y/4, r = dy/2, and choose x̄ ∈ ∂Ω such that |x− x̄| = dx. Then, since
dy ≤ |x− y|+ dx ≤
|x− y| ,
we have
|y − x̄| ≥ |x− y| − dx ≥
|x− y| ≥ R+ r,
and thus, ΩR(x̄) ⊂ Ω\Br(y). Now, we apply Theorem 4.7 with u = G( · , y). Then,
by (4.62) and (4.24), we obtain
|G(x, y)| ≤ CdµxR
1−n/2−µ ‖DG( · , y)‖L2(Ω\Br(y)) ≤ Cd
1−n/2−µ
x,y d
1−n/2
The proof is complete. �
Remark 4.9. We note that in the scalar case, the maximum principle yields (see
[11, Theorem 1.1])
(4.82) G(x, y) ≤ C |x− y|
, ∀x 6= y ∈ Ω.
Then, by the boundary Caccioppoli inequality, we have (c.f. (4.4)–(4.7))
Ω\Br(y)
|DG( · , y)|
≤ Cr2−n, ∀r > 0.
Therefore, in the scalar case we don’t need to require that r < dy/2 (or r < dx/2)
in the proof of Theorem 4.8 and we may as well set r = |x− y| /2 to get
G(x, y) ≤ CdµxR
1−n/2−µ
x,y |x− y|
1−n/2
if dx ≤ Rx,y/8,
G(x, y) ≤ CdµyR
1−n/2−µ
x,y |x− y|
1−n/2
if dy ≤ Rx,y/8.
In particular, if G(x, y) is the Green’s function on Rn+, then we obtain
G(x, y) ≤ Cdµx |x− y|
2−n−µ
if dx ≤ |x− y| /8,
G(x, y) ≤ Cdµy |x− y|
2−n−µ
if dy ≤ |x− y| /8,
for ∂Rn+ satisfies the condition (S) with θ = 1/2 and Ra = ∞.
5. Remarks on VMO coefficients case
Definition 5.1 (Sarason [19]). For a measurable function f defined on Rn, we
shall denote fx,r = −
Br(x)
f and for 0 < δ < ∞ we define
(5.1) Mδ(f) := sup
Br(x)
∣f − fx,r
∣ ; M0(f) := lim
Mδ(f).
We shall say that f belongs to VMO if M0(f) = 0.
26 S. HOFMANN AND S. KIM
Definition 5.2. We say that the operator L satisfies the property (H)loc if there
exist µ0, H0, Rc > 0 such that all weak solutions u of Lu = 0 in BR = BR(x0) with
R ≤ Rc satisfy
(5.2)
)n−2+2µ0
, 0 < r < s ≤ R.
Similarly, we say that the transpose operator tL satisfies the property (H)loc if
corresponding estimates hold for all weak solutions u of tLu = 0 in BR with R ≤ Rc.
Lemma 5.3. Let the coefficients of the operator L in (2.1) satisfy the conditions
(2.2) and (2.3). If the coefficients belong to VMO in addition, then L satisfies the
property (H)loc.
Proof. It is well known that if the coefficients are uniformly continuous, then L
satisfies the property (H)loc; see e.g. [8, pp. 87–89]. Essentially, the same proof
carries over to the VMO coefficients case. One only needs to make a note of the
following two facts. First, a theorem of Meyers [16] implies that there is some
p = p(n,N, λ,Λ) > 2 such that if u is a weak solution of Lu = 0 in BR(x), then
Br(x)
B2r(x)
, ∀r < R/2.
Secondly, note that the John-Nirenberg theorem [13] implies that
Br(x)
∣f − fr,x
≤ C(n, q)Mδ(f), ∀r < c(n)δ, ∀q ∈ (0,∞),
where Mδ(f) is defined as in (5.1). For the details, we refer to [3, pp. 47–48]. �
In the rest of this section, we shall assume that the operators L and tL satisfy
the property (H)loc with parameters µ0, H0, Rc. We shall denote
(5.3) rx := min(dx, Rc), r̄x,y := min(d̄x,y, Rc),
where dx = dist(x, ∂Ω) and d̄x,y is as in (4.28). It is routine to check that all
estimates appearing in Section 4.1 remain valid if dx, d̄x,y are replaced by rx, r̄x,y,
respectively. Therefore, we have the following theorem:
Theorem 5.4. Let Ω be an open connected set in Rn. Denote dx := dist(x, ∂Ω) for
x ∈ Ω; we set dx = ∞ if Ω = R
n. Assume that operators L and tL satisfy the prop-
erty (H)loc. Then, there exists a unique Green’s matrix G(x, y) = (Gij(x, y))
i,j=1
(x, y ∈ Ω, x 6= y) which is continuous in {(x, y) ∈ Ω× Ω : x 6= y} and such that
G(x, · ) is locally integrable in Ω for all x ∈ Ω and that for all f = (f1, . . . , fN)T ∈
C∞c (Ω)
N , the function u = (u1, . . . , uN )T given by
(5.4) u(x) :=
G(x, y)f (y) dy
belongs to Y
0 (Ω)
N and satisfies Lu = f in the sense
(5.5)
ij Dβu
f iφi, ∀φ ∈ C∞c (Ω)
Moreover, G(x, y) has the properties that
(5.6)
ij DβGjk( · , y)Dαφ
i = φk(y), ∀φ ∈ C∞c (Ω)
GREEN FUNCTION ESTIMATES 27
and that for all η ∈ C∞c (Ω) satisfying η ≡ 1 on Br(y) for some r < dy,
(5.7) (1− η)G( · , y) ∈ Y
0 (Ω)
N×N .
Furthermore, G(x, y) satisfies the following estimates: For rx, ry, r̄x,y as in (5.3),
‖G( · , y)‖Lp(Br(y)) ≤ Cp r
2−n+n/p, ∀r < ry, ∀p ∈ [1,
),(5.8)
‖G(x, · )‖Lp(Br(x)) ≤ Cp r
2−n+n/p, ∀r < rx, ∀p ∈ [1,
),(5.9)
‖DG( · , y)‖Lp(Br(y)) ≤ Cp r
1−n+n/p, ∀r < ry , ∀p ∈ [1,
),(5.10)
‖DG(x, · )‖Lp(Br(x)) ≤ Cp r
1−n+n/p, ∀r < rx, ∀p ∈ [1,
),(5.11)
‖G( · , y)‖Y 1,2(Ω\Br(y)) ≤ Cr
1−n/2, ∀r < ry/2,(5.12)
‖G(x, · )‖Y 1,2(Ω\Br(x)) ≤ Cr
1−n/2, ∀r < rx/2,(5.13)
|{x ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (ry/2)
2−n,(5.14)
|{y ∈ Ω : |G(x, y)| > t}| ≤ Ct−
n−2 , ∀t > (rx/2)
2−n,(5.15)
|{x ∈ Ω : |DxG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (ry/2)
1−n,(5.16)
|{y ∈ Ω : |DyG(x, y)| > t}| ≤ Ct
n−1 , ∀t > (rx/2)
1−n,(5.17)
(5.18) |G(x, y)| ≤ Cr̄2−nx,y , ∀x, y ∈ Ω,
|G(x, y)−G(z, y)| ≤ C |x− z|
µ0 r̄2−n−µ0x,y if |x− z| < r̄x,y/2,(5.19)
|G(x, y)−G(x, z)| ≤ C |y − z|
µ0 r̄2−n−µ0x,y if |y − z| < r̄x,y/2,(5.20)
where C = C(n,N, λ,Λ, µ0, H0) > 0 and Cp = Cp(n,N, λ,Λ, µ0, H0, p) > 0.
Remark 5.5. Dolzmann-Müller [6] derived a global estimate
(5.21) |G(x, y)| ≤ C |x− y|
∀x, y ∈ Ω, x 6= y,
assuming that Ω is a bounded C1 domain. We have not attempted to derive the
corresponding estimate here. However, we would like to point out that the constant
C in their estimate depends on the domain (e.g., the diameter of the domain and
also some characteristics of ∂Ω) while our interior estimate (5.18) does not.
References
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Manuscripta Math. 88 (1995), no. 2, 261–273.
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coefficients. Z. Anal. Anwendungen 5 (1986), no. 6, 507–531.
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(S. Hofmann) Mathematics Department, University of Missouri, Columbia, Missouri
65211, United States of America
E-mail address: hofmann@math.missouri.edu
(S. Kim) Centre for Mathematics and its Applications, The Australian National
University, ACT 0200, Australia
E-mail address: seick.kim@maths.anu.edu.au
	1. Introduction
	2. Preliminaries
	2.1. Strongly elliptic systems
	2.2. Function spaces Y1,2() and Y1,20()
	3. Fundamental matrix in Rn
	3.1. Averaged fundamental matrix
	3.2. L estimates for averaged fundamental matrix
	3.3. Uniform weak-Lnn-2 estimates for bold0mu mumu (,y)
	3.4. Uniform weak-Lnn-1 estimates for Dbold0mu mumu (y)
	3.5. Construction of the fundamental matrix
	3.6. Continuity of the fundamental matrix
	3.7. Properties of fundamental matrix
	4. Green's matrix in general domains
	4.1. Construction of Green's matrix
	4.2. Boundary regularity
	5. Remarks on VMO coefficients case
	References
