Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 875057, 17 pages
doi:10.1155/2011/875057
Research Article
The Best Constant of Sobolev
Inequality Corresponding to
Clamped Boundary Value Problem
Kohtaro Watanabe,
1
Yoshinori Kametaka,
2
Hiroyuki Yamagishi,
3
Atsushi Nagai,
4
and Kazuo Takemura
4
1
Department of Computer Science, National Defense Academy, 1-10-20 Hashirimizu, Yokosuka
239-8686, Japan
2
Division of Mathematical Sciences, Graduate School of Engineering Science, Osaka University,
1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan
3
Tokyo Metropolitan College of Industrial Technology, 1-10-40 Higashi-ooi, Shinagawa,
Tokyo 140-0011, Japan
4
Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University,
2-11-1 Shinei, Narashino 275-8576, Japan
Correspondence should be addressed to Kohtaro Watanabe, [email protected]
M
u | u
M
∈ L
2
−s, s
,u
i
±s
0
0 ≤ i ≤ M − 1
,
u, v
M
s
−s
follows:
S
u
sup
|y|≤s
u
y
2
u
2
M
.
1.2
To obtain the supremum of S i.e., the best constant of Sobolev inequality,weconsiderthe
following clamped boundary value problem:
−1
, we have the following
proposition. The result is expressed by the monomial K
j
x:
K
j
x
K
j
M; x
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x
2M−1−j
2M − 1 − j
!
0 ≤ j ≤ 2M − 1
dy
−s<x<s
, 1.4
where Green’s function Gx, yGM; x, y−s<x,y<s is given by
G
x, y
−1
M
2
K
0
x − y
D
−1
K
ij
2s
K
i
s y
K
j
s − x
0
1.5
−1
M
D
. 1.6
D is the determinant of M × M matrix K
ij
2s0 ≤ i, j ≤ M − 1, x ∧ y minx, y,and
x ∨ y maxx, y.
Boundary Va lue Problems 3
With the aid of Proposition 1.1, we obtain the following theorem. The p roof of
Proposition 1.1 isshowninAppendicesA and B.
Theorem 1.2. i The supremum CM; −s, s (abbreviated as CM if there is no confusion) of the
Sobolev functional S is given by
C
M; −s, s
sup
u∈H, u
/
≡ 0
S
u
max
|y|≤s
G
y, y
G
3
24
,C
3, −s, s
s
5
640
,C
4, −s, s
s
7
32256
, 1.8
ii CM; −s, s is attained by u Gx, 0,thatis,SGx, 0 CM; −s, s.
Clearly, Theorem 1.2i, ii is rewritten equivalently as follows.
Corollary 1.3. Let u ∈ H, then the best constant of Sobolev inequality (corresponding to the
embedding of H into L
∞
−s, s)
sup
|y|≤s
p
x
u 0
−s ≤ x ≤ s
, 1.10
where px ∈ L
1
−s, s ∩ C−s, s. If the above equation has two points s
1
and s
2
in −s, s
satisfying us
1
0 us
2
, then these points are said to be conjugate. It is wellknown that if
there exists a pair of conjugate points in −s, s, then the classical Lyapunov inequality
s
−s
p
x
and s
2
conjugate if there exists a nontrivial
C
2M
−s, s ∩ C
M−1
−s, s solution of 1.12 satisfying
u
i
s
1
0 u
i
s
2
i 0, ,M− 1
.
1.13
We point out that the constant which appears in the generalized Lyapunov inequality by
Levin 3 and Das and Vatsala 4 is the reverse of the Sobolev best embedding constant.
Corollary 1.4. Ifthereexistsapairofconjugatepointson−s, s with respect to 1.12,then
s
−s
M
x
2
dx
s
2
s
1
p
x
u
x
2
dx ≤
sup
s
1
≤x≤s
2
|
u
M
x
2
dx
s
2
s
1
p
x
dx.
1.15
In the second inequality, the equality holds for the function which attains the Sobolev best
constant, so especially it is not a constant function. Thus, for this function, the first inequality
is strict, and hence we obtain
1
C
M; s
1
,s
2
s
2
−s
1
p
x
dx ≤
s
−s
p
x
dx, 1.17
we obtain the result.
Here, at the end of this section, we would like to mention some remarks about
1.12. The generalized Lyapunov inequality of the form 1.14 was firstly obtained by
Levin 3 without proof; see Section 4 of Reid 6. Later, Das and Vatsala 4 obtained
thesameinequality1.14 by constructing Green’s function for BVP
M
. The expression
of the Green’s function of Proposition 1.1 is different from that of 4. The expression of
Boundary Va lue Problems 5
, 2.1
2
∂
i
x
G
x, y
x±s
0
0 ≤ i ≤ M − 1, −s<y<s
, 2.2
3
∂
i
x
G
x, y
−s<x<s
,
2.3
4
∂
i
x
G
x, y
xy0
− ∂
i
x
G
x, y
xy−0
⎧
⎪
−1
M
2
sgn
x − y
k
K
k
x − y
D
−1
2s
K
i
s y
−1
k
K
kj
s − x
0
.
2.5
6 Boundary Value Problems
For k 2M,notingthefactK
j
x0 2M ≤ j,wehave1.Next,for0≤ k ≤ M − 1and
−s<y<s,wehavefrom2.5
K
ij
2s
K
i
s − y
K
kj
0
0
Since K
k
0, ,K
kM−1
0 0, ,0,wehave
−1
Mk
2 ∂
k
x
G
x, y
x−s
K
k
s y
D
−1
K
ij
2s
K
i
s y
0 ··· 0 −K
k
s y
0.
2.7
Note that subtracting the kth row from Mth row, the second equality holds. Equation
∂
k
M
2
1 −
−1
k
K
k
0
⎧
⎪
⎨
⎪
⎩
0
0 ≤ k ≤ 2M − 2
,
−1
M
·,y
M
s
−s
u
M
x
∂
M
x
G
x, y
dx
−s ≤ y ≤ s
. 2.9
Proof. For functions u ux and v vxGx, y with y arbitrarily fixed in −s ≤ y ≤ s,we
have
u
M
v
M
−s
u
M
x
v
M
x
dx −
s
−s
u
x
−1
M
v
2M
x
dx
⎡
xs
xy0
M−1
j0
−1
M−1−j
u
j
s
v
2M−1−j
s
− u
j
−s
v
2M−1−j
Using 1, 2,and4 in Lemma 2.1,wehave2.9.
3. Sobolev Inequality
In this section, we give a proof of Theorem 1.2 and Corollary 1.3.
Proof of Theorem 1.2 and Corollary 1.3. Applying Schwarz inequality to 2.9,wehave
u
y
2
≤
s
−s
∂
M
x
G
x, y
x
2
dx. 3.1
Note that the last equality holds from 2.9; that is, substituting 2.9, u·G·,y.Letus
assume that
C
M; −s, s
C
M
max
|y|≤s
G
y, y
G
0, 0
, 3.2
holds this will be proved in the next section. From definition of CM,wehave
sup
|y|≤s
|G
y, 0
|
2
≤ C
M
s
−s
∂
M
x
G
x, 0
G
y, 0
2
≤ C
M
s
−s
∂
M
x
G
x, 0
2
M
x
G
x, 0
2
dx,
3.6
which completes the proof of Theorem 1.2 and Corollary 1.3.
Thus, all we have to do is to prove 3.2.
4. Diagonal Value of Green’s Function
In this section, we consider the diagonal value of Green’s function, that is, Gx, x.From
Proposition 1.1,wehaveforM 1, 2, 3
G
1; x, x
s
2
− x
2
2s
,G
M;1xK
0
M;1−x.
Precisely, we have the following proposition.
Proposition 4.1. Consider
G
x, x
−1
M
D
−1
K
ij
2s
K
i
s − x
K
0
s − x
2
M − 1
M − 1
1
K
0
2s
s
2
− x
2
2M−1
{
2M − 1
K
ij
2s
K
i
s
K
j
s
0
s
2M−1
2
Lemma 4.2. Let uxc
1
Gx, x,where
c
−1
1
−1
M
2
2M − 1
2M − 1
D
−1
− u
22M−1
1
−s<x<s
, 4.5
u
i
±s
0
0 ≤ i ≤ 2M − 2
, 4.6
u
2M−1
s
−
2
M − 1
M − 1
0
s
xK
0
s − x satisfy BVP2M − 1in case of fx1−s<x<s.Sowehave
c
1
G
x, x
c
2
K
0
s x
K
0
s − x
−s<x<s
, 4.8
2
M − 1
M
D
−1
v
x
,v
x
K
ij
2s
K
i
s − x
K
j
x
,w
k,l
x
K
ij
2s
K
li
s − x
K
k−lj
s x
0
x
2M−2
l0
−1
l
2
2M − 1
l
w
22M−1,l
x
−
2
2M − 1
2M − 1
s x
K
2M2M−2−lj
s x
0
0 ≤ l ≤ 2M − 2
. 4.13
The third term also vanishes because
K
li
s − x
0
2M ≤ l ≤ 2
2M − 1
. 4.14
Thus, we have
v
22M−1
2s
K
2M−1i
s − x
K
2M−1j
s x
0
.
4.15
Hence, we have
− u
22M−1
x
− c
1
−1
M
D
−1
v
22M−1
x
1, 4.16
by which we obtain 4.5.Next,for0≤ k ≤ M − 1, we have
v
k
s
2s
K
li
0
K
k−lj
2s
0
. 4.17
Boundary Va lue Problems 11
Since 0 ≤ l i ≤ 2M − 2, we have w
k,l
s0. Thus, we have v
k
s0 0 ≤ k ≤ M − 1.For
M ≤ k ≤ 2M − 2, we have
v
k
s
w
k,l
s
. 4.18
The first term vanishes because K
li
00 0 ≤ l ≤ M − 1. Next, we show that the second
term also vanishes. Let
w
k,l
s
1
K
2M−lj
2s
0
.
.
.
.
.
.
K
M−1j
2s
0
K
k−lj
2s
0
−s0 0 ≤ k ≤ 2M − 2.Hence,wehave
4.6. Finally, we will show 4.7.Fork 2M − 1, noting K
li
00 0 ≤ l ≤ M − 1,wehave
v
2M−1
s
2M−1
lM
−1
l
2M − 1
l
w
2M−1,l
s
, 4.20
where
w
2M−1,l
K
j
2s
0
.
.
2s
0
K
2M−1−lj
2s
0
.
.
.
.
.
K
2M−2−lj
2s
0
K
2M−1−lj
2s
1
K
2M−lj
2s
0
.
.
.
.
.
.
K
Thus, we obtain w
2M−1,l
s−D M ≤ l ≤ 2M − 1.Hencewehave
v
2M−1
1
2M−1
lM
−1
l
2M − 1
l
w
2M−1,l
s
− D
2M−1
lM
2
M − 1
M − 1
,
4.22
that is,
u
2M−1
s
c
1
−1
M
D
−1
v
2M−1
s
−
2M − 1
!
2
s
2
− x
2
2M−1
.
4.24
Differentiating ux k times, we have
u
k
x
c
2
k
l0
−1
l
c
2
2
2M − 1
2M − 1
1. 4.26
Thus, we have 4.5.If0≤ k ≤ 2M − 2, then we have
u
k
s
c
2
k
l0
−1
l
k
l
K
−1
l
2M − 1
l
K
2M−1−l
2s
K
l
0
− c
2
K
0
2s
. 4.28
This proves Lemma 4.3.
Boundary Va lue Problems 13
Appendices
A. Deduction of 1.5
N
⎛
⎜
⎜
⎜
⎝
01
0
.
.
.
.
.
.
1
0
⎞
⎟
⎟
⎟
⎠
2M × 2M nilpotent matrix
,
A.1
BVP
M
x
K
ij
x
, K
0
⎛
⎝
1
···
1
⎞
⎠
K
0
−1
, A.3
then i, j satisfy 0 ≤ i, j ≤ 2M − 1. Ex satisfies the initial value problem E
y
dy,
u
x
E
x − s
u
s
−
s
x
E
x − y
e
−1
M
f
M
K
i
x − y
f
y
dy,
u
i
x
2M−1
j0
K
ij
x − s
u
2M−1−j
s
j0
K
ij
x s
u
2M−1−j
−s
x
−s
−1
M
K
i
x − y
f
y
dy,
u
f
y
dy.
A.6
In particular, if i 0, then we have
u
0
x
M−1
j0
K
j
x s
u
2M−1−j
−s
x
2M−1−j
s
−
s
x
−1
M
K
0
x − y
f
y
dy.
A.7
On the other hand, using the boundary conditions A.2 again, we have
0 u
i
s
dy,
0 u
i
−s
M−1
j0
K
ij
−2s
u
2M−1−j
s
−
s
−s
−1
M
K
i
K
ij
−1
2s
K
i
s − y
f
y
dy,
u
2M−1−i
s
s
−s
−1
s
−s
−1
M
K
j
x s
K
ij
−1
2s
K
i
s − y
f
y
s
−s
−1
M
K
j
x − s
K
ij
−1
−2s
K
i
−s − y
f
where Green’s function Gx, y is given by
G
x, y
−1
M
2
K
0
x − y
−
K
j
x s
i
−s − y
.
A.11
Using properties K
i
−x−1
i1
K
i
x,wehave
K
j
x − s
−
K
j
s − x
−1
i
δ
ij
K
ij
2s
−1
i
δ
ij
,
K
i
−s − y
where δ
ij
is Kronecker’s delta defined by δ
ij
1 i j, 0 i
/
j. Inserting these three
relations into A.11,wehave
G
x, y
−1
M
2
K
0
x − y
−
ij
−1
2s
K
i
s y
.
A.13
Applying the relation
t
a
A
−1
b −
A
b
t
a
K
i
s − y
K
j
s x
0
−
K
ij
2s
K
i
−s<x<s
,
u
i
−s
−1
i1
K
i
s y
,u
i
s
K
i
s − y
0 ≤ i ≤ M − 1
0
−
K
ij
2s
K
i
s y
K
j
s − x
0
K
kj
s x
0
−
−1
k
K
ij
2s
K
i
s y
K
ij
2s
K
i
s − y
K
kj
0
0
−
−1
−1
k
D
−1
K
ij
2s
K
i
s y
0 ··· 0 −K
k
s y
−s
−1
i1
K
i
s y
,v
i
s
K
i
s − y
0 ≤ i ≤ M − 1
.
B.6
which is the same equation as B.2.Hence,wehavevxux.
References
1 C W. H a, “Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type,” Proceedings
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