Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 874631, 12 pages
doi:10.1155/2009/874631
Research Article
Symmetrization of Functions and the Best Constant
of 1-DIM Lp Sobolev Inequality
Kohtaro Watanabe,1 Yoshinori Kametaka,2 Atsushi Nagai,3
Hiroyuki Yamagishi,4 and Kazuo Takemura3
1
Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu,
Yokosuka 239-8686, Japan
2
Graduate School of Mathematical Sciences, Faculty of Engineering Science, Osaka University,
1–3 Matikaneyamatyo, Toyonaka 560-8531, Japan
3
Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, 2-11-1 Shinei,
Narashino 275-8576, Japan
4
Department of Monozukuri Engineering, Tokyo Metropolitan College of Industrial Technology,
1-10-40 Higashi-ooi, Shinagawa, Tokyo 140-0011, Japan
Correspondence should be addressed to Kohtaro Watanabe, wata@nda.ac.jp
Received 25 June 2009; Accepted 16 October 2009
Recommended by Kunquan Lan
m,p
The best constants Cm, p of Sobolev embedding of W0 −s, s into L∞ −s, s for m 1, 2, 3 and
1 < p are obtained. A lemma concerning the symmetrization of functions plays an important role
in the proof.
Copyright q 2009 Kohtaro Watanabe et al. 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
m,p
Let W0 −s, s be a Sobolev space which consists of the functions whose derivatives up to
m − 1 vanish at x ±s, that is,
m,p
W0 −s, s : u | ui ∈ Lp −s, s i 0, . . . , m, uj ±s 0 j 0, . . . m − 1 ,
1.1
where ui denotes ith derivative of u in a distributional sense. The purpose of this paper is to
investigate the best constant Cm, p of Lp Sobolev inequality
sup |ux|
−s≤x≤s
m,p
≤ C
s 1/p
m p
,
u x
dx
−s
where u ∈ W0 −s, s and 1 < p. The result is as follows.
1.2
2
Journal of Inequalities and Applications
Theorem 1.1. The best constant of inequality 1.2 is
C 1, p 2−q−1/q s1/q ,
C 2, p C 3, p 21−2q/q
α
1.3
sq1/q
1/q ,
22q−1/q q 1
xq α − xq dx 0
s
1.4
1/q
xq x − αq dx
1.5
,
α
where q satisfies 1/p 1/q 1 and α in 1.5 is the unique solution of the equation
−
s
xq x − αq−1 dx α
α
xq α − xq−1 dx 0,
1.6
0
satisfying 0 < α < s.
For special values of p, the solution of 1.6 can be written explicitly, and C3, p is
expressed as follows.
Corollary 1.2. One has
s5/2
,
27/2 · 51/2
√ −1/3
√ 1/3 1/4
4
s9/4
C 3,
2−1/3 51/3 13 3 21
.
11/4 1/2 1/2 −1 − 21/3 52/3 13 3 21
3
2
·3 ·7
1.7
C3, 2 The best constants C1, p and C2, p were recently obtained by Oshime 1. This
paper gives an alternative proof which simplifies the derivation process of C1, p and C2, p
and computes further constant C3, p. To compute these constants, the following lemma with
respect to the symmetrization of functions plays an important role.
Lemma 1.3. Let m be an integer satisfying 1 ≤ m ≤ 3 and let us define the functional S as follows:
sup−s≤x≤s |ux|
Su : p 1/p
s m
u x
dx
−s
m,p
u ∈ W0 −s, s, u /0 .
1.8
m,p
Then, for an arbitrary u ∈ W0 −s, s, there exists an element u∗ which belongs to the following
m,p
m,p
sub-space W∗ of W0 −s, s:
m,p
W∗
:
u∈
m,p
W0 −s, s
| max |ux| u0, ux u|x|
−a≤x≤a
−s ≤ x ≤ s
1.9
Journal of Inequalities and Applications
3
such that
Su ≤ Su∗ .
1.10
Remark 1.4. The proof of this lemma see Section 3 does not apply to the case m ≥ 4, since
the relation
i y 0
u
i y − 0 u
1.11
may fail to hold for i ≥ 4 see 3.4–3.6. Hence the problem to obtain Cm, p for m ≥ 4
essentially still remains.
The existence of the maximizer of S can be seen in the proof of Theorem 1.1, where
we construct it concretely, but here we would like to see this briefly though the proof of the
following lemma.
Lemma 1.5. Assume that the assertion of Lemma 1.3 holds, then the maximizer of S exists.
Proof. Let R be sufficiently large, and let W and W be as
m,p
W : u ∈ W0 −s, s | u0 1 ,
W :
m,p
u ∈ W0 −s, s | um Lp −s,s
≤R .
1.12
From Lemma 1.3, we see that if the maximizer exists, it is the element of W : W ∩ W . So,
we can restrict the definition domain of S to W. Since W is convex and strongly closed by
m,p
Sobolev inequality in W0 −s, s, it is weakly closed. In addition, W is weakly compact,
m,p
so W is also weakly compact. Moreover, · is weakly lower-semicontinuous in W0 −s, s,
and hence 1/S attains its minimum in W. This proves the lemma.
Finally, we introduce some studies related to the present paper. When p 2 Hilbertian
Sobolev space case, the best constants for the embeddings of W m,2 a, b into L∞ a, b for
various conditions were treated in Richardson 2, Kalyabin 3, and 4–8; see also references
of these literatures. On the other hand, for the case p /
2, few literature seems to be available.
In 9, Kametaka, Oshime, Watanabe, Yamagishi, Nagai, and Takemura obtained the best
m,p
constant of 1.2 when u belongs to a subspace WP of W m,p 0, 1 which consists of periodic
functions
m,p
WP
:
u∈W
m,p
i
i
0, 1 | u 0 u 1 0 ≤ i ≤ m − 1,
1
uxdx 0 ,
1.13
0
as
C m, p ⎧
⎨
bm ·
Lp/p−1 0,1
m 2n − 1, n 1, 2, 3, . . .,
⎩
b α; ·
p/p−1
m
L
0,1
m 2n, n 1, 2, 3, . . .,
1.14
4
Journal of Inequalities and Applications
where bm · is a Bernoulli polynomial, bm α; · bm · − bm α, and α is an unique solution of
the equation
α
−1
m−1
bm α; xdx
1/p−1
0
−
1/2
−1m bm α; xdx
1/p−1
0
1.15
α
in the interval 0 < α < 1/2. Moreover, in 1, Oshime obtained the best constant C1, p and
C2, p. Other topics on this subject, especially the best constant of Sobolev inequalities on
Riemannian manifolds, are seen in Hebey 10.
2. Proof of Theorem 1.1
First, we prepare the following lemma.
m,p
Lemma 2.1. Let u ∈ W0 −s, s and H be a function satisfying
⎧
1
⎪
⎪
⎨ ,
2
Hx :
⎪
⎪
⎩− 1 ,
2
−s ≤ x < 0,
0 ≤ x ≤ s,
m 1
⎧ 1
s
⎪
⎪
,
⎨− x 2
2
⎪
⎪
⎩ 1 x− s ,
2
2
−s ≤ x < 0,
m 2
⎧
1
⎪
⎪
⎨ xx α,
4
⎪
⎪
⎩− 1 xx − α,
4
−s ≤ x < 0,
0 ≤ x ≤ s,
2.1
0 ≤ x ≤ s,
m 3
then, it holds that
u0 s
−s
um xHxdx,
where α is an arbitrary constant (later, in Lemma 2.3, one fixes the value of α to satisfy 1.6).
Proof. By integration by parts, we obtain the result.
2.2
Journal of Inequalities and Applications
5
From Lemma 1.3, to obtain the best constant of 1.2, we can restrict the definition
m,p
m,p
domain of the functional S to the nonzero element of W∗ . Now, let u ∈ W∗ , then from
Lemma 2.1 and Hölder’s inequality, we have for m 1, 2, 3,
sup |ux| u0 ≤ H
Lq −s,s um Lp −s,s
−s≤x≤s
.
2.3
So, we have
C m, p ≤ H
Lq −s,s ,
2.4
m,p
and the equality holds in 2.4 if and only if there exists u ∈ W∗
satisfying
um x sgn Hx |Hx|q/p sgn Hx |Hx|q−1 .
2.5
To confirm the existence of such u, we use the following lemmas.
Lemma 2.2. Let f ∈ C−s, s satisfy
s
−s
xi fxdx 0
i 0, 1, . . . , m − 1,
2.6
then the solution u of
um x fx
2.7
m,p
exists in W0 −s, s.
Proof . Let us define u as
x
ux :
x − tm−1
ftdt.
−s m − 1!
2.8
Clearly u is Cm −s, s, and
⎧
x
⎪
x − tm−1−i
⎪
⎨
ftdt 0 ≤ i ≤ m − 1,
ui x −s m − 1 − i!
⎪
⎪
⎩fx
i m.
Moreover, from the assumption, it holds that ui ±s 0 0 ≤ i ≤ m − 1.
Lemma 2.3. The solution α of 1.6 uniquely exists.
2.9
6
Journal of Inequalities and Applications
Proof. Let
fα : −
s
x x − α
q
q−1
dx α
α
xq α − xq−1 dx.
2.10
0
Since
s
f α q − 1
f0 −
fs s
s
x x − α
q
q−2
α
dx q − 1
α
xq α − xq−2 dx > 0,
0
2.11
x2q−1 dx < 0,
0
xq s − xq−1 dx > 0
0
the assertion is proved.
Using Lemma 2.2 and 5, we obtain the following lemma.
m,p
Lemma 2.4. Let α be a solution of 1.6 (when m 3), then the solution of 2.5 belongs to W∗
for m 1, 2, 3.
Proof. First, we prove the case m 2 and 3. For simplicity, let us put Hx
q−1
sgn Hx|Hx| . Note that in these cases H is a continuous function on −s, s.
is an even function, so integration of xHx
1 In the case m 2, H
over the interval
−s, s vanishes. In addition,
s
1 q−1
s/2 2
x
s q−1
dx −
2
s/2
0
1 2q−1
−x −
s q−1
dx 0
2
2.12
holds, so the integration of Hx
over the interval −s, s also vanishes. Hence, by Lemma 2.2,
m,p
the solution u of 2.5 belongs to W0 −s, s. Properties max−s≤x≤s |ux| u0 and ux is an even function and 2.12. So, we have proven u ∈ W∗m,p .
u−x follow from the fact that H
is an odd function, so integrations of Hx
2 In the case m 3, H
and x2 Hx
over the interval −s, s vanish. Moreover, from 1.6, the integration of xHx
over the
interval −s, s also vanishes. Hence, again by Lemma 2.2, we have the solution of 2.5 which
m,p
belongs to W0 −s, s. The remaining part is the same as case i.
3 In the case m 1, let us define u as follows:
⎧
1
⎪
⎪
−s ≤ x < 0,
⎨ q−1 x s
2
ux ⎪
⎪ 1
⎩
− q−1 x − s 0 ≤ x ≤ s.
2
2.13
Journal of Inequalities and Applications
7
Clearly it holds that u satisfies 2.5 a.e., u±s 0, max−s≤x≤s |ux| u0 and ux u−x.
To see u ∈ W 1,p −s, s, let ϕ be an arbitrary element of C0∞ −s, s. Since
s
−s
u xϕxdx
0
−s
−
−
s
0
1
2q−1
1
2q−1
x sϕ xdx s
0
1
2q−1
x − sϕ xdx
0
−s
1
2q−1
ϕxdx
2.14
ϕxdx,
we have, in a distributional sense,
⎧
1
⎪
⎪
⎨ q−1
2
u x ⎪
⎪
⎩− 1
2q−1
−s ≤ x < 0,
2.15
0 ≤ x ≤ s.
Therefore, u ∈ W 1,p −s, s. This proves the case m 1.
Proof of Theorem 1.1. From Lemma 1.3, and the argument of this section, especially
Lemma 2.4, Cm, p H
Lq −s,s . This proves Theorem 1.1.
Proof of Corollary 1.2 . In this case, 1.6 becomes
35s3 − 120αs2 140α2 s − 56α3 0.
2.16
So, we can explicitly solve this equation with respect to α. Substituting to 1.5, we obtain the
result.
3. Proof of Lemma 1.3
Now, all we have to do is to prove Lemma 1.3.
Proof of Lemma 1.3. To avoid the complexity of notation, in the follwings, we fix s 1. Let u
m,p
be an arbitrary element of W0 −1, 1 1 ≤ m ≤ 3, and let
max |ux| u y .
−1≤x≤1
3.1
Here, we can assume y ≥ 0, since if it does not, ux can be replaced by u−x.
i There is the case
1 y p
m p
u x
dx ≥ um x
dx.
−1
y
3.2
8
Journal of Inequalities and Applications
Let us define u
as
⎧
⎪
0
−1 ≤ x < −1 2y ,
⎪
⎪
⎨ u
x : u 2y − x
−1 2y ≤ x < y ,
⎪
⎪
⎪
⎩
ux
y≤x≤1 .
3.3
We have
u
y−0 u
y0 u y ,
y 0 0,
u
y − 0 u
u
y − 0 u
y 0 u y ,
3.4
3.5
3.6
when m 3, 3.4 and 3.5 when m 2, and 3.4 when m 1. Further, let us define
u∗ x :
⎧ ⎨u
xy
⎩0
−1 y ≤ x ≤ 1 − y ,
1 − y < |x| ≤ 1 .
3.7
i
m,p
Then u∗ ∈ W∗ , since max−1≤x≤1 |u∗ x| u∗ 0, u∗ ±1 00 ≤ i ≤ m − 1. Moreover, from
m
3.2, we have u∗ Lp −1,1 ≤ um Lp −1,1 . In addition, clearly u∗ 0 max−1≤x≤1 |u∗ x| uy max−1≤x≤1 |ux|. So, in the case i, we have proven the lemma.
ii There is the case
1 y p
m p
u x
dx < um x
dx.
−1
3.8
y
Let t be an element satisfying 0 ≤ t ≤ y, and let
x −1 t1
x 1 −1 ax 1.
y1
3.9
Further, let Ux be
x 1
−1
U x : u
a
−1 ≤ x ≤ t .
3.10
So,
∂m
x U
−m m
−m m x 1
−1 ,
x a ∂x ux|xx 1/a−1 a u
a
3.11
Journal of Inequalities and Applications
9
and hence we obtain
t
−1
m p ∂ U x dx a−mp
x
t p
m x 1
u
− 1 dx .
a
−1
3.12
By putting x −1 ax 1 the right-hand side of 3.12 becomes
a
−mp1
y m p
u x
dx.
3.13
−1
Similarly, let us put
x 1 1−t
x − 1 1 bx − 1
1−y
3.14
and define Ux as
x −1
1
U x : u
b
t ≤ x ≤ 1 .
3.15
So,
−m m
−m m x − 1
∂m
1
,
U
x
∂
ux|
b
u
b
x
xx −1/b1
x
b
3.16
and hence we obtain
1
t
m p ∂ U x dx b−mp
x
a
p
m x − 1
u
1 dx .
b
3.17
t
By putting x 1 bx − 1 the right-hand side of 3.17 becomes
b−mp1
1 m p
u x
dx.
3.18
y
Let us put
A :
y m p
u x
dx,
−1
B :
1 m p
u x
dx
3.19
y
and define
ft :
t1
y1
−mp1
A
1−t
1−y
−mp1
B.
3.20
10
Journal of Inequalities and Applications
Note that
p
f y A B um p
L −1,1
.
3.21
The derivative of f is
t 1 −mp
1 − t −mp
1
1
f t mp − 1 −
A
B .
y1 y1
1−y 1−y
3.22
a The case
1≤
1−y
1y
mp−1
B
.
A
3.23
In this case, we have
f t ≥ 0
mp−1
mp−1
←→ y 1
t 1−mp A ≤ 1 − y
1 − t−mp B
←→
1−t
1t
mp
≤
1−y
1y
mp−1
3.24
B
.
A
Since
max
0≤t≤y
1−t
1t
mp
1,
3.25
from the assumption 3.23, f is monotone increasing. So, we have
min ft f0 0≤t≤y
0 1
p
m p
U x
dx Um x
dx.
−1
3.26
0
But, from 3.23, it holds that
1
0 −mp1
−mp1
p
1
1
m p
A≤
B Um x
dx.
U x
dx y
1
1
−
y
−1
0
3.27
Journal of Inequalities and Applications
11
So, if we put
u∗ x :
⎧
⎨Ux
−1 ≤ x ≤ 0,
3.28
⎩U−x 0 ≤ x ≤ 1,
m p
m,p
p
as case i, we have u∗ ∈ W∗ and u∗ Lp −1,1 ≤ f0 ≤ fy um Lp −1,1 . In addition,
u∗ 0 max−1≤x≤1 |u∗ x| uy max−1≤x≤1 |ux|. So, we have proven the case ii-a.
b The case
1−y
1y
mp−1
B
< 1.
A
3.29
In this case, we have
f t mp mp − 1
1
1
t 1 −mp−1
1 − t −mp−1
A
B > 0.
y1 y1
1−y 1−y
3.30
Moreover
mp−1
mp−1 f 0 mp − 1 − 1 y
A 1−y
B < 0,
3.31
1
1
A
B > 0,
f y mp − 1 −
y1
1−y
since we have 3.29 and the assumption 3.8 A < B, respectively. Therefore, there exists
t0 0 < t0 < y such that f t0 0. Let us define the constant M as
M :
1−y
1y
mp−1/mp B
A
1/mp
3.32
,
then t0 1 − M/1 M. Now we have
1 t0 m p
m p
dx
<
x
U
U x
dx,
3.33
−1
t0
since
t0 1 m p
m p
U x
dx <
U x
dx
−1
t0
←→
1 − t0
1−y
−mp1
B<
t0 1
y1
−mp1
A ←→
←→ Mmp < Mmp−1 ←→ M < 1 ←→ 3.29.
1−y
1y
mp−1
B
<
A
1 − t0
1 t0
mp−1
3.34
12
Journal of Inequalities and Applications
Let us define u
as
⎧
⎪
0
−1 ≤ x ≤ 2t0 − 1,
⎪
⎪
⎨
u
x : U2t0 − x 2t0 − 1 ≤ x ≤ t0 ,
⎪
⎪
⎪
⎩
Ux
t0 ≤ x ≤ 1,
⎧
⎨u
x t0 −1 t0 ≤ x ≤ 1 − t0 ,
u∗ :
⎩0
1 − t0 < |x| ≤ 1.
m,p
Then, again as case i, we have u∗ ∈ W∗
p
Um Lp −1,1 ft0 <
max−1≤x≤1 |ux|. So, we
p
um Lp −1,1 .
m p
3.35
p
and by 3.33, u∗ Lp −1,1 um Lp −1,1 ≤
fy In addition, u∗ 0 max−1≤x≤1 |u∗ x| uy have proven the case ii-b. This completes the proof.
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