# Zeros for the Gradients of WeaklyA-Harmonic Tensors

Authors: Yuxia Tong; Jiantao Gu; Shenzhou Zheng
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

The Caccioppoli inequality of weaklyA-harmonic tensors has been proved, which can be used to consider the weak reverse Hölder inequality, regularity property, and zeros of weakly A-harmonic tensors.


---

## Body

## 1. Introduction

In this paper, we consider theA-harmonic equation for differential forms
(1)
d
*
A
(
x
,
d
u
)
=
0
,
where A
:
Ω
×
∧
l
(
R
n
)
→
∧
l
+
1
(
R
n
) satisfies the conditions
(2)
|
A
(
x
,
ξ
)
|
≤
β
|
ξ
|
p
-
1
,
〈
A
(
x
,
ξ
)
,
ξ
〉
≥
α
|
ξ
|
p
for almost every x
∈
Ω and all ξ
∈
∧
l
(
R
n
). Here, α
,
β
>
0 are constants and 1
<
p
<
∞ is a fixed exponent associated with (1). u
∈
W
loc
⁡
1
,
p
(
Ω
,
∧
l
-
1
) is an A-harmonic tensor in Ω if u satisfies (1) in Ω.There has been remarkable work [1–10] in the study of (1). When u is a 0-form, that is, u is a function, (1) is equivalent to
(3)
div
⁡
A
(
x
,
∇
u
)
=
0
.
Lots of results have been obtained in recent years about different versions of the A-harmonic equation; see [11–15].In 1995, Stroffolini [16] first introduced weakly A-harmonic tensors and gave the higher integrability result of weakly A-harmonic tensors. The wordweak means that the integrable exponent r of u is smaller than the natural exponent p. In 2010, Gao and Wang [17] gave an alternative proof of the higher integrability result of weakly A-harmonic tensors by introducing the definition of weak W
T
2-class of differential forms.Definition 1 (see [16, 17]).
A very weak solution to (1) (also called weakly A-harmonic tensor) is an element u of the Sobolev space W
loc
⁡
1
,
r
(
Ω
,
∧
l
-
1
) with max
⁡
{
1
,
p
-
1
}
≤
r
<
p such that
(4)
∫
Ω
‍
〈
A
(
x
,
d
u
)
,
d
φ
〉
d
x
=
0
for all φ
∈
W
1
,
(
r
/
(
r
-
p
+
1
)
)
(
Ω
,
∧
l
-
1
) with compact support.
Under some conditions, the present paper proves that almost every zero for the gradients of weaklyA-harmonic tensor u has infinite order. To do this, we need to give the Caccioppoli inequality and the weak reverse Hölder inequality of weakly A-harmonic tensors.We keep using the traditional notation.LetΩ be a connected open subset of R
n, let e
1
,
e
2
,
…
,
e
n be the standard unit basis of R
n, and let ∧
l
=
∧
l
(
R
n
) be the linear space of l-covectors, spanned by the exterior products e
I
=
e
i
1
∧
e
i
2
∧
⋯
∧
e
i
l, corresponding to all ordered l-tuples I
=
(
i
1
,
i
2
,
…
,
i
l
), 1
≤
i
1
<
i
2
<
⋯
<
i
l
≤
n, l
=
0,1
,
…
,
n. Let R
=
R
1. The Grassman algebra ∧
=
⊕
∧
l is a graded algebra with respect to the exterior products. For α
=
∑
‍
α
I
e
I
∈
∧ and β
=
∑
‍
β
I
e
I
∈
∧, the inner product in ∧ is given by 〈
α
,
β
〉
=
∑
‍
α
I
β
I with summation over all l-tuples I
=
(
i
1
,
i
2
,
…
,
i
l
) and all integers l
=
0,1
,
…
,
n. The Hodge star operator ⋆
:
∧
→
∧ is denoted by the rules ⋆
1
=
e
1
∧
e
2
∧
⋯
∧
e
n and α
∧
⋆
β
=
β
∧
⋆
α
=
〈
α
,
β
〉
(
⋆
1
) for all α
,
β
∈
∧. The norm of α
∈
∧ is given by the formula |
α
|
2
=
〈
α
,
α
〉
=
⋆
(
α
∧
⋆
α
)
∈
∧
0
=
R. The Hodge star is an isometric isomorphism on ∧ with ⋆
:
∧
l
→
∧
n
-
l and ⋆
⋆
(
-
1
)
l
(
n
-
l
)
:
∧
l
→
∧
l. Balls are denoted by B and ρ
B is the ball with the same center as B and with diam
⁡
(
ρ
B
)
=
ρ
diam
⁡
(
B
). We do not distinguish balls from cubes throughout this paper. The n-dimensional Lebesgue measure of a set E
⊆
R
n is denoted by |
E
|.Differential forms are important generalizations of real functions and distributions; note that a 0-form is the usual function inR
n. A differential l-form ω on Ω is a Schwartz distribution on ω with values in ∧
l
(
R
n
). We use D
′
(
Ω
,
∧
l
) to denote the space of all differential l-forms ω
(
x
)
=
∑
I
‍
ω
I
(
x
)
d
x
I
=
∑
‍
ω
i
1
i
2
,
…
,
i
l
(
x
)
d
x
i
1
∧
d
x
i
2
∧
⋯
∧
d
x
i
l. We write L
p
(
Ω
,
∧
l
) for the l-forms with ω
I
∈
L
p
(
Ω
,
R
) for all ordered l-tuples I. Thus, L
p
(
Ω
,
∧
l
) is a Banach space with norm
(5)
∥
ω
∥
p
,
Ω
=
(
∫
Ω
‍
|
ω
(
x
)
|
p
d
x
)
1
/
p
=
(
∫
Ω
‍
(
∑
‍
|
ω
I
(
x
)
|
2
)
p
/
2
d
x
)
1
/
p
.
For ω
∈
D
′
(
Ω
,
∧
l
), the vector-valued differential form ∇
ω
=
(
∂
ω
/
∂
x
1
,
…
,
∂
ω
/
∂
x
n
) consists of differential forms ∂
ω
/
∂
x
i
∈
D
′
(
Ω
,
∧
l
) where the partial differentiations are applied to the coefficients of ω. As usual, W
1
,
p
(
Ω
,
∧
l
) is used to denote the Sobolev space of l-forms, which equals L
p
(
Ω
,
∧
l
)
⋂
‍
L
1
p
(
Ω
,
∧
l
) with norm
(6)
∥
ω
∥
W
1
,
p
(
Ω
,
∧
l
)
=
∥
ω
∥
W
1
,
p
(
Ω
,
∧
l
)
=
diam
⁡
(
Ω
)
-
1
∥
ω
∥
p
,
Ω
+
∥
∇
ω
∥
p
,
Ω
.
The notations W
loc
⁡
1
,
p
(
Ω
,
R
) and W
loc
⁡
1
,
p
(
Ω
,
∧
l
) are self-explanatory. We denote the exterior derivative by d
:
D
′
(
Ω
,
∧
l
)
→
D
′
(
Ω
,
∧
l
+
1
) for l
=
0,1
,
…
,
n. Its formal adjoint operator d
⋆
:
D
′
(
Ω
,
∧
l
+
1
)
→
D
′
(
Ω
,
∧
l
) is given by d
⋆
=
(
-
1
)
n
l
+
1
⋆
d
⋆ on D
′
(
Ω
,
∧
l
+
1
),  l
=
0,1
,
…
,
n. A differential l-form u
∈
D
′
(
Ω
,
∧
l
) is called a closed form if d
u
=
0 in Ω. It is called exact if there exists a differential form α
∈
D
′
(
Ω
,
∧
l
-
1
) such that u
=
d
α. Poincaré Lemma implies that exact forms are closed.From [1, 18], if D
⊂
R
n is a bounded, convex domain, to each y
∈
D there corresponds a linear operator K
y
:
C
∞
(
D
,
∧
l
)
→
C
∞
(
D
,
∧
l
-
1
) defined by
(7)
(
K
y
ω
)
(
x
;
ξ
1
,
…
,
ξ
l
-
1
)
=
∫
0
1
‍
t
l
-
1
ω
(
t
x
+
y
-
t
y
;
x
-
y
,
ξ
1
,
…
,
ξ
l
-
1
)
d
t
and a decomposition ω
=
d
(
K
y
ω
)
+
K
y
(
d
ω
). A homotopy operator T
:
C
∞
(
D
,
∧
l
)
→
C
∞
(
D
,
∧
l
-
1
) is defined by averaging K
y over all points y in D; that is,
(8)
T
ω
=
∫
D
‍
φ
(
y
)
K
y
ω
d
y
,
where φ
∈
C
0
∞
(
D
) is normalized by ∫
D
‍
φ
(
y
)
d
y
=
1. Then, there is also a decomposition
(9)
ω
=
d
(
T
ω
)
+
T
(
d
ω
)
.
The l-form ω
D
∈
D
′
(
D
,
∧
l
) is defined by
(10)
ω
D
=
{
|
D
|
-
1
∫
D
‍
ω
(
y
)
d
y
if
l
=
0
d
(
T
ω
)
if
l
=
1,2
,
…
,
n
for all ω
∈
L
p
(
D
,
∧
l
). Clearly, ω
D is a closed form and for l
>
0, ω
D is an exact form.


## 2. The Caccioppoli Inequality of WeaklyA-Harmonic Tensors

We need the following elementary inequality.Lemma 2 (see [19]).
SupposeX and Y are vectors of an inner product space. Then
(11)
|
|
X
|
-
ɛ
X
-
|
Y
|
-
ɛ
Y
|
≤
1
+
ɛ
1
-
ɛ
2
ɛ
|
X
-
Y
|
1
-
ɛ
for 0
≤
ɛ
<
1, and
(12)
|
|
X
|
ɛ
X
-
|
Y
|
ɛ
Y
|
≤
(
1
+
ɛ
)
(
|
Y
|
+
|
X
-
Y
|
)
ɛ
|
X
-
Y
|
for ɛ
≥
0.Next is the caccioppoli inequality of weaklyA-harmonic tensors.Theorem 3.
Letu
∈
D
′
(
Ω
,
∧
l
-
1
) be a weakly A-harmonic tensor in a domain Ω
∈
R
n and d
u
∈
L
r
(
D
,
∧
l
), l
=
1
,
…
,
n. Then, there exists a constant C, independent of u, such that
(13)
∫
B
ρ
‍
|
d
u
|
r
d
x
≤
C
(
n
,
p
,
β
α
)
∫
B
R
‍
|
u
-
c
R
-
ρ
|
r
d
x
for all balls B
⊂
Ω and all closed forms c, where 0
<
ρ
<
R.Proof .
Letu
∈
W
loc
⁡
1
,
r
(
Ω
,
∧
l
-
1
) be a very weak solution of (1). Fix R
0
:
R
0
≤
d
=
dist
⁡
(
x
0
,
∂
Ω
) for all x
0
∈
Ω. Let B
R
=
B
R
(
x
0
)
⊂
⊂
Ω and 0
<
R
/
2
≤
τ
<
t
≤
R be arbitrarily fixed cube. Fix a cutoff function η
(
x
)
∈
C
0
∞
(
B
R
) such that supp
⁡
η
⊂
B
t, 0
≤
η
≤
1, |
∇
η
|
≤
C
(
n
)
/
(
t
-
τ
), and η
≡
1 on B
τ. Consider the exact form of η
(
u
-
c
), where c
∈
D
′
(
Ω
,
∧
l
-
1
) with d
c
=
0. With the aid of the Hodge decomposition [18],
(14)
|
d
(
η
(
u
-
c
)
)
|
r
-
p
d
(
η
(
u
-
c
)
)
=
d
φ
+
h
,
where d
φ, h
∈
L
r
/
(
r
-
p
+
1
)
(
B
t
,
∧
l
), and
(15)
∥
h
∥
r
/
(
r
-
p
+
1
)
≤
C
(
n
)
(
p
-
r
)
∥
d
(
η
(
u
-
c
)
)
∥
r
r
-
p
+
1
.
Then we have
(16)
∥
d
φ
∥
r
/
(
r
-
p
+
1
)
≤
∥
d
(
η
(
u
-
c
)
)
|
r
-
p
d
(
η
(
u
-
c
)
)
∥
r
/
(
r
-
p
+
1
)
+
∥
h
∥
r
/
(
r
-
p
+
1
)
≤
∥
d
(
η
(
u
-
c
)
)
|
∥
r
r
-
p
+
1
+
C
(
n
)
|
p
-
r
|
∥
d
(
η
(
u
-
c
)
)
∥
r
r
-
p
+
1
≤
C
(
n
)
∥
d
(
η
(
u
-
c
)
)
∥
r
r
-
p
+
1
.
We can use φ
∈
W
1
,
r
/
(
r
-
p
+
1
)
(
Ω
,
∧
l
-
1
) as a test function for (4). Then, by Definition 1,
(17)
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
|
d
(
η
(
u
-
c
)
)
|
r
-
p
d
(
η
(
u
-
c
)
)
-
h
〉
d
x
=
0
.
Let
(18)
E
=
|
d
(
η
(
u
-
c
)
)
|
r
-
p
d
(
η
(
u
-
c
)
)
-
|
η
d
(
u
-
c
)
|
r
-
p
η
d
(
u
-
c
)
;
using Lemma 2 yields
(19)
|
E
|
≤
2
p
-
r
p
-
r
+
1
r
-
p
+
1
|
(
u
-
c
)
d
η
|
r
-
p
+
1
.
Then (17) becomes
(20)
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
|
η
d
(
u
-
c
)
|
r
-
p
η
d
(
u
-
c
)
〉
d
x
≤
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
h
〉
d
x
-
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
E
〉
d
x
.
Noticing that c satisfies d
c
=
0, then by the condition (2) we get
(21)
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
|
η
d
(
u
-
c
)
|
r
-
p
η
d
(
u
-
c
)
〉
d
x
=
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
|
η
d
u
|
r
-
p
η
d
u
〉
d
x
≥
α
∫
B
τ
‍
|
d
u
|
r
d
x
.
Combining the above inequality with (20), we get
(22)
α
∫
B
τ
‍
|
d
u
|
r
d
x
≤
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
h
〉
d
x
-
∫
B
t
‍
〈
A
(
x
,
d
u
)
,
E
〉
d
x
=
I
1
+
I
2
.
In the following we will estimate the right side of (22). By (2), the Hölder inequality, and (15),
(23)
|
I
1
|
≤
∫
B
t
‍
|
A
(
x
,
d
u
)
|
|
h
|
d
x
≤
β
∫
B
t
‍
|
d
u
|
p
-
1
|
h
|
d
x
≤
β
(
∫
B
t
‍
|
d
u
|
r
)
(
p
-
1
)
/
r
(
∫
B
t
‍
|
h
|
r
/
(
r
-
p
+
1
)
d
x
)
(
r
-
p
+
1
)
/
r
≤
β
C
(
n
)
(
p
-
r
)
(
∫
B
t
‍
|
d
u
|
r
)
(
p
-
1
)
/
r
×
(
∫
B
t
‍
|
d
(
η
p
(
u
-
c
)
)
|
r
d
x
)
(
r
-
p
+
1
)
/
r
.
For
(24)
(
∫
B
t
‍
|
d
(
η
p
(
u
-
c
)
)
|
r
d
x
)
(
r
-
p
+
1
)
/
r
=
(
∫
B
t
‍
|
η
p
d
u
+
p
η
p
-
1
(
u
-
c
)
d
η
|
r
d
x
)
(
r
-
p
+
1
)
/
r
≤
C
(
p
,
r
)
(
∫
B
t
‍
|
η
d
u
|
r
d
x
)
(
r
-
p
+
1
)
/
r
+
C
(
p
,
r
)
(
∫
B
t
‍
|
(
u
-
c
)
d
η
|
r
d
x
)
(
r
-
p
+
1
)
/
r
≤
C
(
p
,
r
)
(
∫
B
t
‍
|
d
u
|
r
d
x
)
(
r
-
p
+
1
)
/
r
+
C
(
n
,
p
,
r
)
(
∫
B
t
‍
|
u
-
c
t
-
τ
|
r
d
x
)
(
r
-
p
+
1
)
/
r
,
(23) and (24) with Young’s inequality yield
(25)
|
I
1
|
≤
β
C
(
n
,
p
,
r
)
(
p
-
r
)
×
∫
B
t
‍
|
d
u
|
r
d
x
+
β
C
(
n
,
p
,
r
)
(
p
-
r
)
ɛ
∫
B
t
‍
|
d
u
|
r
d
x
+
β
C
(
ɛ
,
n
,
p
,
r
)
(
p
-
r
)
∫
B
t
‍
|
u
-
c
t
-
τ
|
r
d
x
.
Next we estimate I
2. By (2), the Hölder inequality, (19), and Young’s inequality,
(26)
|
I
2
|
≤
∫
B
t
‍
|
A
(
x
,
d
u
)
|
|
E
|
d
x
≤
2
p
-
r
p
-
r
+
1
r
-
p
+
1
β
∫
B
t
‍
|
d
u
|
p
-
1
|
(
u
-
c
)
d
η
|
r
-
p
+
1
d
x
≤
β
C
(
n
,
p
,
r
)
(
∫
B
t
‍
|
d
u
|
r
d
x
)
(
p
-
1
)
/
r
×
(
∫
B
t
‍
|
(
u
-
c
)
d
η
|
r
d
x
)
(
r
-
p
+
1
)
/
r
≤
ɛ
β
C
(
n
,
p
,
r
)
∫
B
t
‍
|
d
u
|
r
d
x
+
β
C
(
n
,
p
,
r
,
ɛ
)
×
∫
B
t
‍
|
u
-
c
t
-
τ
|
r
d
x
.
Combining (22), (25), and (26), we get
(27)
α
∫
B
τ
‍
|
d
u
|
r
d
x
≤
β
C
(
n
,
p
,
r
)
(
(
p
-
r
)
(
1
+
ɛ
)
+
ɛ
)
∫
B
t
‍
|
d
u
|
r
d
x
+
β
C
(
ɛ
,
n
,
p
,
r
)
(
(
p
-
r
)
+
1
)
∫
B
t
‍
|
u
-
c
t
-
τ
|
r
d
x
.
Let p
-
r and ɛ small enough to let
(28)
β
C
(
n
,
p
,
r
)
(
(
p
-
r
)
(
1
+
ɛ
)
+
ɛ
)
α
=
θ
<
1
;
then we have
(29)
∫
B
τ
‍
|
d
u
|
r
d
x
≤
θ
∫
B
t
‍
|
d
u
|
r
d
x
+
C
(
n
,
p
,
r
,
β
α
)
∫
B
t
‍
|
u
-
c
t
-
τ
|
r
d
x
.
Next we will refine the inequality (29). Let
(30)
0
<
ρ
,
t
<
R
,
f
(
t
)
=
∫
B
t
‍
|
d
u
|
r
d
x
,
A
=
∫
B
R
‍
|
u
-
c
R
-
ρ
|
r
d
x
.
Choosing ɛ
∈
(
0,1
) satisfied ε
r
>
θ. Let
(31)
t
0
=
ρ
,
t
i
+
1
=
t
i
+
(
1
-
ɛ
)
ε
i
(
R
-
ρ
)
,
i
=
0,1
,
2
,
…
then when k
→
∞, t
k
→
R. We deduce from (29) that
(32)
f
(
ρ
)
=
f
(
t
0
)
≤
θ
f
(
t
1
)
+
C
∫
B
t
1
‍
|
u
-
c
(
1
-
ɛ
)
ɛ
0
(
R
-
ρ
)
|
r
d
x
≤
θ
f
(
t
1
)
+
C
A
(
1
-
ɛ
)
r
≤
θ
k
f
(
t
k
)
+
C
A
(
1
-
ɛ
)
r
∑
i
=
0
k
-
1
‍
(
θ
ɛ
-
r
)
i
.
Let k
→
∞ yield
(33)
∫
B
ρ
‍
|
d
u
|
r
d
x
≤
C
(
n
,
p
,
r
,
β
α
)
∫
B
R
‍
|
u
-
c
R
-
ρ
|
r
d
x
.
Finally, in our case, r is sufficiently close to p; we can estimate C
(
n
,
p
,
r
,
β
/
α
) independently of r.Especially, letρ
=
R
/
2, c
=
u
B
R, and then (13) becomes
(34)
∫
B
R
/
2
‍
|
d
u
|
r
d
x
≤
C
(
n
,
p
,
β
α
)
∫
B
R
‍
|
u
-
u
B
R
R
|
r
d
x
,
or
(35)
(
⨍
B
R
/
2
|
d
u
|
r
d
x
)
1
/
r
≤
C
(
n
,
p
,
β
/
α
)
R
(
⨍
B
R
|
u
-
u
B
R
|
r
d
x
)
1
/
r
.


## 3. Zeros for the Gradients of WeaklyA-Harmonic Tensors

We need the following Poincaré inequality.Lemma 4 (see [16]).
LetD be a cube or a ball, and ω
∈
L
s
(
D
,
∧
l
) with d
ω
∈
L
s
(
D
,
∧
l
+
1
). Then
(36)
1
diam
⁡
D
(
⨍
D
|
ω
-
ω
D
|
s
)
1
/
s
≤
C
(
n
,
s
)
(
⨍
D
|
d
ω
|
n
s
/
(
n
+
s
-
1
)
)
(
n
+
s
-
1
)
/
n
s
.
Here we denote by ⨍
D the integral mean over D.Using the Caccioppoli inequality (13) and Lemma 4, we can get the weak-reverse Hölder inequality of weakly A-harmonic tensors.Theorem 5.
Letu
∈
D
′
(
Ω
,
∧
l
-
1
) be a weakly A-harmonic tensor in a domain Ω
∈
R
n, and d
u
∈
L
r
(
D
,
∧
l
), l
=
1
,
…
,
n. Then there exists a constant C, independent of u and R, such that
(37)
(
⨍
B
R
/
2
|
d
u
|
r
d
x
)
1
/
r
≤
C
(
n
,
p
,
r
,
β
α
)
(
⨍
B
R
|
d
u
|
n
r
/
(
n
+
r
-
1
)
d
x
)
(
n
+
r
-
1
)
/
n
r
for all balls B
⊂
Ω.Proof.
By Lemma4,
(38)
(
⨍
B
R
|
u
-
u
c
|
r
d
x
)
1
/
r
≤
C
(
n
,
r
)
R
(
⨍
B
R
|
d
u
|
n
r
/
(
n
+
r
-
1
)
d
x
)
(
n
+
r
-
1
)
/
n
r
.
Then, by (35), we get
(39)
(
⨍
B
R
/
2
|
d
u
|
r
d
x
)
1
/
r
≤
C
(
n
,
p
,
r
,
β
α
)
(
⨍
B
R
|
d
u
|
n
r
/
(
n
+
r
-
1
)
d
x
)
(
n
+
r
-
1
)
/
n
r
.Next we consider the main results of this paper.Definition 6 (see [20]).
A pointx
0
∈
Ω is said to be an essential zero of a function h
∈
L
loc
⁡
1
(
Ω
) if
(40)
lim
⁡
R
→
0
1
R
n
∫
Q
(
x
0
,
R
)
‍
|
h
(
x
)
|
d
x
=
0
,
where Q
(
x
0
,
R
) denotes the cube centered at x
0 of side length 2
R. The order of the essential zero is defined to be
(41)
N
(
x
0
)
=
sup
⁡
{
α
:
lim
⁡
R
→
0
1
R
n
+
α
∫
Q
(
x
0
,
R
)
‍
|
h
(
x
)
|
d
x
=
0
}
.Lemma 7 (see [20]).
Leth
∈
L
loc
⁡
1
(
Ω
) satisfy the weak-reverse Hölder inequality
(42)
1
|
Q
|
∫
Q
‍
h
p
d
x
≤
A
p
(
1
|
2
Q
|
∫
2
Q
‍
h
d
x
)
p
for all cubes Q
⊂
2
Q
⊂
Ω and some 1
<
p
<
∞, with a constant A
P independent of the cube. Then almost every zero of h has infinite order.Theorem 8.
There exist exponentsr
0
=
r
0
(
n
,
p
,
β
/
α
)
∈
(
1,2
); if r
>
r
0, then for the weakly A-harmonic tensor u
∈
W
loc
⁡
1
,
r
(
Ω
,
∧
l
-
1
) almost every zero of d
u has infinite order.Proof.
By the weak-reverse Hölder inequality (37) and Lemma 7, we get the desired result.



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*Source: 231248-2014-05-12.xml*