Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 610530, 10 pages
doi:10.1155/2009/610530
Research Article
An Improved Hardy-Rellich Inequality with
Optimal Constant
Ying-Xiong Xiao1 and Qiao-Hua Yang2
1
2
School of Mathematics and Statistics, Xiaogan University, Xiaogan, Hubei 432000, China
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Correspondence should be addressed to Ying-Xiong Xiao, xyx21cn@163.com
Received 25 May 2009; Accepted 11 September 2009
Recommended by Siegfried Carl
We show that a Hardy-Rellich inequality with optimal constants on a bounded domain can be
refined by adding remainder terms. The procedure is based on decomposition into spherical
harmonics.
Copyright q 2009 Y.-X. Xiao and Q.-H. Yang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Hardy inequality in RN reads, for all u ∈ C0∞ RN and N ≥ 3,
N − 22
|∇u| dx ≥
4
RN
u2
2
RN
|x|2
1.1
dx,
and N − 22 /4 is the best constant in 1.1 and is never achieved. A similar inequality with
the same best constant holds if RN is replaced by an arbitrary domain Ω ⊂ RN and Ω contains
the origin. Moreover, Brezis and Vázquez 1 have improved it by establishing that for u ∈
C0∞ Ω,
Ω
|∇u|2 dx ≥
N − 22
4
u2
Ω |x|
dx Λ−Δ, 2
2
ωN
|Ω|
2/N Ω
u2 dx,
1.2
2
Journal of Inequalities and Applications
where ωN and |Ω| denote the volume of the unit ball B1 and Ω, respectively, and Λ−Δ, 2 is
the first eigenvalue of the Dirichlet Laplacian of the unit disc in R2 . In case Ω is a ball centered
at zero, the constant Λ−Δ, 2 in 1.2 is sharp.
Similar improved inequalities have been recently proved if instead of 1.1 one
considers the corresponding Lp Hardy inequalities. In all these cases a correction term is
added on the right-hand side see, e.g., 2–4.
On the other hand, the classical Rellich inequality states that, for N ≥ 5,
RN
|Δu|2 dx ≥
NN − 4
4
2 u2
RN
|x|
4
u ∈ C0∞ RN ,
dx,
1.3
and NN − 4/42 is the best constant in 1.3 and is never achieved see 5. And, more
recently, Tertikas and Zographopoulos 6 obtained a stronger version of Rellich’s inequality.
That is, for all u ∈ C0∞ RN ,
RN
|Δu|2 dx ≥
N2
4
|∇u|2
RN
|x|2
dx,
1.4
N ≥ 5.
Both inequalities are valid when RN is replaced by a bounded domain Ω ⊂ RN containing
the origin and the corresponding constants are known to be optimal. Recently, Gazzola et al.
4 have improved 1.3 by establishing that for Ω ⊂ BR 0 and u ∈ C0∞ Ω,
Ω
|Δu|2 dx ≥
NN − 4
4
2 u2
Ω |x|
dx 4
NN − 4
Λ−Δ, 2R−2
2
2
−4
Λ −Δ , 4 R
u2 dx,
u2
Ω |x|
2
dx
1.5
Ω
where
Λ −Δ2 , 4 inf
4
B1
4
u∈W 2,2 B1 \{0}
Δu2 dx
4
B1
u2 dx
1.6
,
4
and B1 is the unit ball in R4 . Our main concern in this note is to improve 1.4. In fact we
have the following theorem.
Theorem 1.1. There holds, for N ≥ 5 and u ∈ C0∞ Ω,
Ω
|Δu|2 dx ≥
N2
4
|∇u|2
Ω
|x|2
dx Λ−Δ, 2
ωN
|Ω|
2/N Inequality 1.7 is optimal in case Ω is a ball centered at zero.
Combining Theorem 1.1 with 1.2, we have the following.
Ω
|∇u|2 dx.
1.7
Journal of Inequalities and Applications
3
Corollary 1.2. There holds, for N ≥ 5 and u ∈ C0∞ Ω,
ωN 2/N
u2
N − 22
Λ−Δ,
2
dx
dx
2
2
4
|Ω|
Ω |x|
Ω |x|
4/N 2 ωN
u2 dx.
Λ−Δ, 2
|Ω|
Ω
N2
|Δu| dx ≥
4
Ω
2
|∇u|2
1.8
Next we consider analogous inequality 1.5. The main result is the following theorem.
Theorem 1.3. Let N ≥ 8 and let Ω ⊂ RN be such that Ω ⊂ BR 0. Then for every u ∈ C0∞ Ω one
has
Ω
|Δu|2 dx ≥
|∇u|2
NN − 8
Λ−Δ, 2R−2
2
4
|x|
Ω
2
−4
Λ −Δ , 4 R
u2 dx.
N2
4
dx u2
Ω |x|
2
dx
1.9
Ω
Remark 1.4. Since
|∇u|2
Ω
|x|
2
dx ≥
N − 42
4
u2
Ω |x|
dx Λ−Δ, 2
4
ωN
|Ω|
2/N u2
2
Ω |x|
dx,
N ≥ 5,
1.10
inequality 1.5 is implied by 1.9 in case of N ≥ 8.
2. The Proofs
To prove the main results, we first need the following preliminary result.
Lemma 2.1. Let N ≥ 5 and u ∈ C0∞ RN . Set r |x|. If ux is a radial function, that is, ux ur, then
N2
|Δu| dx −
4
RN
|∇u|2
2
RN
N − 22
dx
dx
−
|∇u
|
r
4
|x|2
RN
u2r
2
RN
|x|2
dx.
2.1
Proof. Observe that if ux ur, then
|∇u| |ur |,
Δu d2 u N − 1 du
·
.
r
dr
dr 2
2.2
4
Journal of Inequalities and Applications
Therefore, we have
RN
2
urr N − 1 ur dx
r
RN
u2r
urr ur
2
2
dx
urr dx N − 1
dx 2N − 1
2
r
RN
RN r
RN
u2r
1 d u2r
·
dx.
u2rr dx N − 12
dx
−
1
N
2
dr
RN
RN r
RN r
|Δu|2 dx 2.3
Though integration by parts, when n ≥ 3,
∞
d u2r
u2r
1 d u2r
N−2
·
dx dr −N − 2
dσ r
·
dx,
2
dr
dr
0
RN r
SN−1
RN r
2.4
and hence
RN
|Δu|2 dx −
N2
4
|∇u|2
|x|
RN
2
dx RN
u2rr dx −
N − 22
4
u2r
dx
2
RN r
N − 22
|∇ur | dx −
4
RN
u2r
2
RN
|x|2
2.5
dx.
By Lemma 2.1 and inequality 1.2, we have, when restricted to radial functions,
Ω
|Δu|2 dx −
N2
4
|∇u|2
Ω
|x|2
dx ≥ Λ−Δ, 2
ωN
|Ω|
2/N Ω
|∇u|2 dx.
2.6
Our next step is to prove the following. If ux is not a radial function, inequality 2.6 also
holds.
Let u ∈ C0∞ Ω. If we extend u as zero outside Ω, we may consider u ∈ C0∞ RN .
Decomposing u into spherical harmonics we get
u
∞
k
0
uk :
∞
fk rφk σ,
2.7
k
0
where φk σ are the orthonormal eigenfunctions of the Laplace-Beltrami operator with
responding eigenvalues
ck kN k − 2,
k ≥ 0.
2.8
Journal of Inequalities and Applications
5
k
k−1
The functions fk r belong to C0∞ Ω, satisfying
fk r Or and fk r Or as r → 0.
In particular, φ0 σ 1 and u0 r 1/|∂Br | ∂Br u dσ, for any r > 0. Then, for any k ∈ N, we
have
ck
Δuk Δfk r − 2 fk r φk σ.
r
2.9
So
2
ck
Δfk r − 2 fk r dx,
r
RN
RN
∇fk r2 ck f 2 r dx.
|∇uk |2 dx r2 k
RN
RN
|Δuk |2 dx 2.10
In addition,
RN
|Δu|2 dx k
0
2
RN
∞ |∇u| dx RN
∞ k
0
|Δuk |2 dx k
0
2
RN
∞ |∇uk | dx RN
∞ k
0
Δfk r −
RN
ck
fk r
r2
2
dx,
∇fk r2 ck f 2 r dx.
r2 k
2.11
Using equality 2.10, we have that see, e.g., 6, page 452
|Δuk |2 dx RN
RN
2
fk dx N − 1 2ck ck ck 2N − 4
|∇uk |2
RN
|x|2
∇fk r2
dx r2
RN
RN
r −2 fk dx
2
RN
r −4 fk2 dx,
dx ck
fk2 r
RN
r4
2.12
dx.
Therefore, we have that, by 2.12,
RN
|Δuk |2 dx −
N2
4
|∇uk |2
RN
|x|2
dx
2
fk
N − 22
−
dx
2
4
RN
RN r
⎡
2 ⎤
2
2
fk
fk
N − 8N 32
ck ⎣2
dx ck −
dx⎦.
2
4
4
RN r
RN r
2
fk dx
2.13
6
Journal of Inequalities and Applications
Lemma 2.2. There holds, for N ≥ 4 and k ≥ 1,
2
2
2
fk
fk
fk
N 2 − 8N 32
ωN 2/N
2
dx ck −
dx ≥ 2Λ−Δ, 2
dx.
2
2
4
4
|Ω|
Ω r
Ω r
Ω r
2.14
Proof. Set gk fk /r. Then gk satisfies gk r Or k−1 and gk r Or k−2 as r → 0.
Moreover, since fk r belong to C0∞ Ω, we have that
Ω
2
gk dx
2
fk
r2
2
fk
Ω
r2
2
fk
Ω
Ω
r2
2
fk fk
fk
dx − 2
dx dx
3
4
r
Ω
Ωr
dx N − 3
fk2
Ω
dx N − 3
r4
gk2
Ω
r2
2.15
dx
dx.
Here we use the fact when N ≥ 4 and k ≥ 1,
2
∞
d fk2
fk fk
fk
N−4
2
dr
−N
−
4
dx
dσ
r
·
dx.
3
4
dr
0
Ω r
SN−1
Ωr
2.16
Using inequalities 1.2 and 2.15, we have that, for N ≥ 4 and k ≥ 1,
2
fk 2
fk
N 2 − 8N 32
dx ck −
dx
2
2
4
4
Ω r
Ω r
2
gk
2
N2 8
2
dx
gk dx ck −
2
4
Ω
Ωr
N − 22
≥
2
gk2
ωN
dx 2Λ−Δ, 2
2
|Ω|
r
Ω
2/N Ω
gk2 dx
N2 8
ck −
4
ωN 2/N
dx 2Λ−Δ, 2
gk2 dx
2
|Ω|
Ωr
Ω
2
gk
ωN 2/N
N 2 − 8N 4c1
dx 2Λ−Δ, 2
gk2 dx
≥
2
4
|Ω|
Ωr
Ω
N 2 − 8N 4ck
4
gk2
gk2
Ω
r2
dx
Journal of Inequalities and Applications
7
2
gk
ωN 2/N
N 2 − 4N − 4
dx 2Λ−Δ, 2
gk2 dx
2
4
|Ω|
Ωr
Ω
ωN 2/N
≥ 2Λ−Δ, 2
gk2 dx
|Ω|
Ω
2/N 2
fk
ωN
2Λ−Δ, 2
dx.
2
|Ω|
Ω r
2.17
An immediate consequence of the inequalities 2.13 and Lemma 2.2 is the following
result. For k ≥ 1,
N2
|Δuk | dx −
4
RN
|∇uk |2
2
≥
fk dx −
2
RN
|x|2
RN
N − 2
4
dx
2
fk 2
RN
r2
dx 2ck Λ−Δ, 2
ωN
|Ω|
2/N 2
fk dx.
2
Ω r
2.18
Using inequalities 2.18 and Lemma 2.1, we have that, since fk r ∈ C0∞ Ω, for k ≥ 1,
N2
|Δuk | dx −
4
RN
|∇uk |2
2
|x|2
2/N RN
dx
fk 2
ωN
ωN 2/N
≥ Λ−Δ, 2
dx
fk dx 2ck Λ−Δ, 2
2
|Ω|
|Ω|
RN
Ω r
fk 2
ωN 2/N
≥ Λ−Δ, 2
dx
fk dx ck
2
|Ω|
RN
Ω r
ωN
Λ−Δ, 2
|Ω|
2/N RN
2.19
|∇uk |2 dx.
Inequality 2.19 implies that, if ux is not a radial function, then
Ω
|Δu|2 dx −
N2
4
|∇u|2
Ω
|x|2
dx ≥ Λ−Δ, 2
ωN
|Ω|
2/N Ω
|∇u|2 dx.
2.20
Proof of Theorem 1.1. Using inequality 2.6 and 2.20, we have that, for N ≥ 5 and u ∈
C0∞ Ω,
Ω
|Δu|2 dx ≥
N2
4
|∇u|2
Ω
|x|2
dx Λ−Δ, 2
ωN
|Ω|
2/N Ω
|∇u|2 dx.
2.21
8
Journal of Inequalities and Applications
In case Ω is a ball centered at zero, a simple scaling allows to consider the case Ω B1 . Set
H
inf
u∈C0∞ B1 \{0}
2 2
2
2
dx
dx
−
N
/4
/|x|
|Δu|
|∇u|
B1
B1
.
2
dx
B1 |∇u|
2.22
Using Lemma 2.1 and inequality 1.2, we have that H ≤ Hradial Λ−Δ, 2. On the other
hand, we have, by inequality 2.21, H ≥ Λ−Δ, 2. Thus H Λ−Δ, 2. The proof is complete.
Proof of Theorem 1.3. A scaling argument shows that we may assume R 1 and Ω B1 B.
Step 1. Assume u is radial, r |x| and vr |x|N−4/2 ur, then see 6, Lemma 2.3
N2
|Δu| dx −
4
B
|∇u|2
2
B
|x|2
dx |Δv|2
B |x|
dx N−4
NN − 8
− NN − 4
4
vr2
N−2
B |x|
dx,
2.23
and see 6, 6.4
|Δv|2
dx N−4
B |x|
2
vrr
B |x|
dx N − 1N − 3
N−4
vr2
N−2
B |x|
2.24
dx.
Therefore
|Δu|2 dx −
B
N2
4
|∇u|2
B
|x|2
dx vr2
vr2
NN − 8
dx
3
dx
dx.
N−4
N−2
N−2
4
B |x|
B |x|
B |x|
2
vrr
2.25
Since v is radial,
dx
3
N−4
2
vrr
B |x|
vr2
B |x|
dx ≥ Λ−Δ, 2
N−2
vr2
B |x|
dx N−2
ΣN
Σ4
ΣN
Σ4
B4
B4
v2
B |x|
N−2
2
vrr
dx 3
dx;
ΣN
Σ4
vr2
B4
|Δrad,4 v|2 dx
ΣN 2
≥
Λ −Δ , 4
v2 dx
Σ4
B4
v2
Λ −Δ2 , 4
dx,
N−4
B |x|
|x|2
dx
2.26
Journal of Inequalities and Applications
9
where Σk denote the surface area of the unit sphere in Rk , B4 is the unit ball in R4 , and
∂2
3 ∂
2
r ∂r
∂r
Δrad,4 2.27
is the radial Laplacian in R4 .
Therefore, for N ≥ 8,
N 2 |∇u|2
dx
|Δu| dx −
4 B |x|2
B
v2
v2
NN − 8 2
Λ
−Δ
dx
,
4
dx
≥ Λ−Δ, 2
N−2
N−4
4
B |x|
B |x|
u2
NN − 8 2
Λ
−Δ
Λ−Δ, 2
dx
,
4
u2 dx.
2
4
B |x|
B
2
2.28
Step 2. For u ∈ C0∞ B, set
u
∞
∞
fk rφk σ.
uk :
k
0
2.29
k
0
We get, by 2.18,
N2
|Δuk | dx −
4
B
|∇uk |2
2
B
|x|2
dx ≥
B
2
fk dx
N − 22
−
4
2
N2
Δfk dx −
4
B
2
fk
B
r2
2
∇fk B
|x|2
dx
2.30
dx.
In getting the last equality, we used Lemma 2.1.
Using inequality 1.9 for radial functions from step 1,
N2
|Δuk | dx −
4
B
|∇uk |2
2
≥ Λ−Δ, 2
dx
fk2
B |x|
Λ−Δ, 2
|x|2
B
dx 2
u2k
B |x|
dx 2
NN − 8 Λ −Δ2 , 4
fk2 dx
4
B
NN − 8 Λ −Δ2 , 4
u2k dx,
4
B
2.31
10
Journal of Inequalities and Applications
one obtains, by 2.11,
N2
|Δu| dx −
4
B
NN − 8 2
Λ −Δ , 4
dx ≥ Λ−Δ, 2
dx u2 dx
2
4
|x|2
B |x|
B
|∇u|2
2
B
u2
2.32
which demonstrates inequality 1.9.
Acknowledgment
This work was supported by National Science Foundation of China under Grant no.
10571044.
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Zeitschrift, vol. 227, no. 3, pp. 511–523, 1998.
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