Hindawi Publishing Corporation
Boundary Value Problems
Volume 2010, Article ID 690342, 21 pages
doi:10.1155/2010/690342
Research Article
Extremal Values of Half-Eigenvalues for
p-Laplacian with Weights in L
1
Balls
Ping Yan
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Correspondence should be addressed to Ping Yan,
Received 24 May 2010; Accepted 21 October 2010
Academic Editor: V. Shakhmurov
Copyright q 2010 Ping Yan. 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.
For one-dimensional p-Laplacian with weights in L
γ
: L
γ
0, 1, R1 ≤ γ ≤∞ balls, we are
interested in the extremal values of the mth positive half-eigenvalues associated with Dirichlet,
Neumann, and generalized periodic boundary conditions, respectively. It will be shown that the
extremal value problems for half-eigenvalues are equivalent to those for eigenvalues, and all these
extremal values are given by some best Sobolev constants.
1. Introduction
Occasionally, we need to solve extremal value problems for eigenvalues. A classical example
studied by Krein 1 is the infimum and the supremum of the mth Dirichlet eigenvalues of
Hill’s operator with positive weight
inf
1
0
w
t
dt r
. 1.2
In this paper, we always use superscripts D, N, P,andG to indicate Dirichlet, Neumann,
periodic and generalized periodic boundary value conditions, respectively. Similar extremal
value problems for p-Laplacian were studied by Yan and Zhang 2. For Hill’s operator with
weight, Lou and Yanagida 3 studied the minimization problem of the positive principal
2 Boundary Value Problems
Neumann eigenvalues, which plays a crucial role in population dynamics. G iven constants
κ ∈ 0, ∞ and α ∈ 0, 1, denote
S
κ,α
:
ω ∈L
∞
: −1 ≤ ω ≤ κ, ω
0,
1
0
ω
topology is well understood, and so is the Fr
´
echet
differentiable dependence. Many of these results are summarized in 4. It is remarkable that
this step cannot be answered immediately by such a continuity results, because the space of
weights is infinite-dimensional. The second step is to find the minimizers/maximizers. This
step is tricky and it depends on the problem studied. For L
1
weights the solution is suggested
by the Pontrjagin’s Maximum Principle 5, Sections 48.6–48.8.
For Sturm-Liouville operators and Hill’s operators Zhang 6 proved that the eigenval-
ues are continuous in potentials in the sense of weak topology w
γ
. Such a stronger continuity
result has been generalized to eigenvalues and half-eigenvalues on potentials/weights for
scalar p-Laplacian associated with different types of boundary conditions see 7–10.
As an elementary application of such a stronger continuity, the proof of the first step,
that is, the existence of minimizers or maximizers, of the extremal value problems as in 1–3
was quite simplified in 9, 10.
Based on the continuity of eigenvalues in weak topology and the Fr
´
echet differentia-
bility, some deeper results have also been obtained by Zhang and his coauthors in 10 –12 by
using variational method, singular integrals and limiting approach.
The extremal values of eigenvalues for Sturm-Liouville operators with potentials in L
1
balls were studied in 11, 12. For γ ∈ 1, ∞, r ≥ 0andm ∈ Z
: {0, 1, 2, }, denote
L
: sup
λ
F
m
q
: q ∈L
γ
,
q
γ
≤ r
,
1.5
where the superscript F denotes N or P if m 0andD or N if m>0. By the limiting approach
γ ↓ 1, the most important extremal values in L
1
balls are proved to be finite real numbers, and
they can be evaluated explicitly by using some elementary functions Z
0
r, Z
1
γ
,w≥ 0,
w
γ
≤ r
inf
μ
D
m
w
: w ∈L
γ
,w≥ 0,
w
γ
r
m
p
·
K
p
α
, ∀α ∈
1, ∞
.
1.7
Moreover, the infimum can be attained by some weight if only γ ∈ 1, ∞. By letting the
radius r ↓ 0
one sees that the supremum
sup
μ
D
m
w
: w ∈L
γ
,w≥ 0,
w
γ
≤ r
∞, 1.8
D
m,γ
r
E
N
m,γ
r
E
G
m,γ
r
m
p
·
K
pγ
∗
,p
r
1.9
for any γ ∈ 1, ∞, m ∈ N and r>0. It will also be proved that
E
F
m,γ
. It is possible that for some weights in L
γ
balls the mth
positive half-eigenvalue does not exist; see Remark 2.3. So it is impossible to utilize directly
the continuous dependence of half-eigenvalues in weights in weak topology or the Fr
´
echet
differential dependence, as done in 10–12. Some more fundamental continuous results
in weak topology and differentiable results in Lemma 2.1 will be used instead. We will
first show two facts. One is the monotonicity of the half-eigenvalues on the weights a, b.
The other is the infimum H
F
m,γ
r can be attained by some weights for any γ ∈ 1, ∞.As
consequence of these two facts, for each minimizer a
γ
,b
γ
, one sees that a
γ
and b
γ
do not
overlap if γ ∈ 1, ∞. Moreover the extremal problem for half-eigenvalues is reduced to that
for eigenvalues. Roughly speaking, for any γ ∈ 1, ∞ and r>0 we have
H
F
m,γ
E
F
0,γ
r
0, ∀F ∈
{
N, G
}
.
1.12
Based on some topological fact on L
γ
balls, the extremal values in L
1
balls can be obtained by
the limiting approach γ ↓ 1. Consequently 1.11 and 1.12 also hold for γ 1.
2. Preliminary Results and Extremal Value Problems
Denote by φ
p
· the scalar p-Laplacian and let x
±
·max{±x·, 0}. Let us consider the
positive half-eigenvalues of
φ
p
x
0
x
1
0, D
x
0
x
1
0, N
x
0
± x
1
x
,
x
0
,y
0
1, 0
.
2.2
The functions cos
p
θ and sin
p
θ are the so-called p-cosine and p-sine. They share several
remarkable relations as ordinary trigonometric functions, for instance
Boundary Value Problems 5
i both cos
p
θ and sin
p
θ are 2π
θ|
p
p − 1|sin
p
θ|
p
∗
≡ 1.
By setting φ
p
x
−y and introducing the Pr
¨
ufer transformation x r
2/p
cos
p
θ, y
r
2/p
∗
sin
p
θ, the scalar equation
φ
p
x
A
t, θ; a, b
:
⎧
⎨
⎩
a
t
cos
p
θ
p
p − 1
sin
p
θ
p
∗
if cos
p
θ<0,
2.5
log r
G
t, θ; a, b
:
⎧
⎪
⎨
⎪
⎩
p
2
a
t
− 1
φ
p
θ
φ
p
∗
sin
p
θ
if cos
p
θ<0.
2.6
For any ϑ
0
∈ R, denote by θt; ϑ
0
,a,b,rt; ϑ
0
,a,b, t ∈ 0, 1, the unique solution of 2.5
2.6 satisfying θ0; ϑ
0
,a,bϑ
0
and r0; ϑ
0
,a,b1. Let
Θ
m
a, b the set of nonnegative half-eigenvalues of 2.1
2.2 for which the corresponding half-eigenfunctions have precisely m zeroes in the interval
0, 1. Define
Θ
a, b
: max
ϑ
0
∈0,2π
p
{
Θ
ϑ
0
,a,b
− ϑ
0
}
max
ϑ
0
∈R
{
Θ
min
ϑ
0
∈R
{
Θ
ϑ
0
,a,b
− ϑ
0
}
, 2.9
λ
L
m
λ
L
m
a, b
: min
λ>0 | Θ
λa, λb
L/R
m
a, b
,
λ
L/R
m
a, b
∈ Σ
m
a, b
2.12
if only these numbers exist.
Lemma 2.1 see 7, 8. Denote by w
γ
the weak topology in L
γ
.Then
iΘϑ, a, b is jointly continuous in ϑ, a, b ∈ R × L
γ
,w
γ
2
. The derivatives of
Θϑ, a, b at ϑ,ata ∈L
γ
and at b ∈L
γ
(in the Fr
´
echet sense), denoted, respectively,
by ∂
ϑ
Θ, ∂
a
Θ, and ∂
b
Θ,are
∂
ϑ
Θ
ϑ, a, b
1
R
2
ϑ, a, b
X
p
−
∈C
0
⊂
L
γ
,
·
γ
∗
,
2.14
where C
0
: C0, 1, R and
X X
t
X
t; ϑ, a, b
:
2
∈L
1
, write a 0ifa ≥ 0and
1
0
atdt > 0. Write a
1
,b
1
≥ a
2
,b
2
if a
1
≥ a
2
and b
1
≥ b
2
.Writea
1
,b
1
a
2
,
a
,b
0, 0
}
.
2.16
Theorem 2.2. Suppose a, b ∈W
1
. There hold the following results.
i All positive Dirichlet half-eigenvalues of 2.1 consist of two sequences {λ
D
m
a, b}
m∈N
and
{λ
D
m
b, a}
m∈N
,whereλ
< ···<λ
D
m
a, b
< ···
−→ ∞
.
2.17
Boundary Value Problems 7
ii All nonnegative Neumann half-eigenvalues of 2.1 consist of two sequences
{λ
N
m
a, b}
m∈Z
and {λ
N
m
b, a}
m∈Z
,whereλ
N
m
a, b is determined by
< ···<λ
N
m
a, b
< ···
−→ ∞
.
2.18
Moreover,
λ
N
0
a, b
> 0 ⇐⇒ a
0,
1
0
a
t
< 0.
a, b
,
λ
L/R
m
a, b
⊂ Σ
m
a, b
⊂
λ
L
m
a, b
,
λ
R
m
a, b
,
0
λ
L
0
≤ λ
R
0
< λ
L
1
≤ λ
R
1
< ···< λ
L
m
≤ λ
R
m
< ···
−→ ∞
,
0 ≤
R
0
a, b
> 0 ⇐⇒
1
0
a
t
dt < 0 or
1
0
b
t
dt < 0.
2.23
Proof. Compared with results in 8, we need only prove
Σ
2m1
a, b
that only finite of these positive half-eigenvalues exists. We refer this to Remark 2.4 in 8.
In other cases, for example if a<0andb<0, there exist no positive half-eigenvalues. Since we
are going to study the infimum of positive half-eigenvalues, if one of these half-eigenvalues,
say λ
D
m
a, b, does not exist, we define λ
D
m
a, b∞ for simplicity.
Theorem 2.4. Suppose a, b ∈L
1
. There hold the following results.
i If λ
L
m
a, b < ∞ for some m ∈ N,then
λ ≥ λ
L
m
a, b
, ∀λ ∈ Σ
m
a, b
;
p
for some μ>0andm ∈ N, then there exists δ>0 such that
Θ
λa, λb
>mπ
p
, ∀λ ∈
μ, μ δ
. 2.27
2 If
Θμa, μbmπ
p
for some μ>0andm ∈ Z
, then there exists δ ∈ 0,μ such that
Θ
λa, λb
<mπ
p
, ∀λ ∈
μ − δ, μ
.
⎪
⎨
⎪
⎩
≤ mπ
p
if 0 ≤ λ<λ
R
m
,
>mπ
p
if λ>λ
R
m
∀m ∈ Z
2.30
if
λ
R
m
a, b < ∞.
Boundary Value Problems 9
Step 3. Suppose λ
L
m
a, b < ∞ for some m ∈ N. For any λ ∈ Σ
m
.
2.32
It follows from 2.29 that λ ≥ λ
L
m
a, b, which completes the proof of i.Resultsii can be
proved analogously by using 2.30.
In the product space L
γ
×L
γ
,1≤ γ ≤∞, one can define the norm |·|
γ
as
|
a, b
|
γ
:
1
0
|
a
t
a, b
|
∞
: lim
γ →∞
|
a, b
|
γ
max
{
a
∞
,
b
∞
}
, ∀
a, b
∈L
∞
B
γ
δ
a, b
:
a
1
,b
1
∈L
γ
×L
γ
:
|
a
1
− a, b
1
− b
|
γ
γ
r
:
a, b
∈
S
γ
r
: a ≥ 0,b≥ 0
,
B
γ
r
:
a ∈L
γ
:
r
: inf
λ
D
m
a, b
:
a, b
∈
B
γ
r
, ∀m ∈ N, 2.35
H
N
m,γ
r
λ ∈ Σ
m
a, b
:
a, b
∈
B
γ
r
, ∀m ∈ N, 2.37
H
N
0,γ
r
: inf
λ
N
m
> 0:
a, b
∈
B
γ
r
. 2.39
Remark 2.5. i It follows from Theorem 2.2 that all the extremal values defined by 2.35–
2.39 are fi nite.
ii Although there may exist nonvariational half-eigenvalues in Σ
m
a, bcf. 13,
Theorem 2.4 shows that
λ
L
m
a, b
inf Σ
m
r
, ∀m ∈ N. 2.41
Notice that if a b, then the half-eigenvalue problem of 2.1 is equivalent to the
eigenvalue problem of
φ
p
x
λa
t
φ
p
x
0, a.e.t∈
0, 1
.
2.42
If a
: λ
D
m
a, a
,λ
N
m
a
: λ
N
m
a, a
,
λ
L
m
a
: λ
L
m
r
: inf
λ
D
m
a
: a ∈ B
γ
r
, ∀m ∈ N, 2.44
E
N
m,γ
r
: inf
λ
N
m
a
λ
L
m
a
: a ∈ B
γ
r
, ∀m ∈ N,
2.46
E
N
0,γ
r
: inf
λ
N
m
a
> 0:a ∈ B
Theorem 3.1. For any γ ∈ 1, ∞, m ∈ N and r>0, one has
E
D
m,γ
r
m
p
·
K
pγ
∗
,p
r
.
3.1
If γ ∈ 1, ∞,thenE
D
m,γ
r can be attained by some weight, called a minimizer, and each minimizer is
contained in S
γ
r.Ifγ 1,thenE
D
m,γ
r cannot be attained by any weight in B
m,γ
r
inf
λ
D
m
w
: w ∈L
γ
,w≥ 0,
w
γ
≤ r
. 3.3
Now the theorem can be completed by the proof of 10, Theorem 5.6;seealso1.6.
Lemma 3.2. Given a ∈L
γ
, define a
s
t : as t for any s, t ∈ R.Then
λ
L
a
max
s∈R
λ
D
m
a
s
max
s∈R
λ
N
m
a
s
, ∀m ∈ N,
λ
R
0
γ
r for any s ∈ R. One can obtain the
following theorem immediately from Theorem 3.1 and Lemma 3.2.
Theorem 3.4. There holds 1.9 for any γ ∈ 1, ∞, m ∈ N and r>0.Ifγ ∈ 1, ∞, any extremal
value involved in 1.9 can be attained by some weight, and each minimizer is contained in S
γ
r.If
γ 1, none of these extremal values can be attained by any weight in B
γ
r.
However, we cannot characterize E
N
0,γ
and E
G
0,γ
by using Theorem 3.1 and Lemma 3.2,
because λ
D
0
a does not exist for any weight a ∈L
γ
.
Theorem 3.5. There holds 1.10 for any γ ∈ 1, ∞ and r>0.
Proof. Choose a sequence of weights
a
k
t
,
k>2. 3.5
Then a
k
∈ B
γ
r, a
k
0and
1
0
a
k
tdt < 0. It follows from Theorem 2.2ii that ν
k
:
λ
N
0
a
k
λ
N
0
a
k
,a
0
t
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
rt∈
0,
1
2
,
−rt∈
1
2
, 1
,
a.e.t∈
0
a
> 0 ⇐⇒ λ
N
0
a
⇐⇒
1
0
a
t
dt < 0.
3.9
Combining Lemma 3.2 and the definitions in 2.47 and 2.48, one has E
G
0,γ
rE
N
0,γ
r0,
completing the proof of the theorem.
Boundary Value Problems 13
4. Infimum of Half-Eigenvalues with Weights in L
,then
i λ
D
m
a
0
,b
0
< ≤λ
D
m
a
1
,b
1
for any m ∈ N,
ii λ
N
m
a
0
,b
0
< ≤λ
N
m
a
1
,b
1
N
m
a, b for
arbitrary large m ∈ N. Employing the boundary value conditions and Fr
´
echet differentiability
of Θϑ, a, b in weights Lemma 2.1iii, one can prove the following lemma.
Lemma 4.2. Given a
i
,b
i
∈L
γ
, i 0, 1, γ ∈ 1, ∞. Suppose a
0
,b
0
≥a
1
,b
1
,then
i if λ
D
m
a
1
,b
1
< ∞ for some m ∈ N,thenλ
m
a
1
,b
1
;
iii if a
1
0 and
1
0
a
0
tdt < 0,then0 <λ
N
0
a
0
,b
0
< ≤λ
N
0
a
1
,b
1
0
≥
a
1
,b
1
⇒ Θ
ϑ, a
0
,b
0
<
≤
Θ
ϑ, a
1
,b
1
, ∀ϑ ∈ R. 4.1
eigenvalues.
Lemma 4.3. Given a, b, a
i
,b
i
∈L
γ
, i 0, 1, γ ∈ 1, ∞. There hold the following results.
i If λ
L
m
a, b < ∞ for some m ∈ N,thenλ
L
m
a
,b
≤ λ
L
m
a, b.
ii If
λ
R
m
a, b < ∞ for some m ∈ N,thenλ
R
m
a
<
≤λ
L
m
a
1
,b
1
.
iv If a
0
,b
0
≥a
1
,b
1
≥ 0, 0 and λ
R
m
a
1
,b
1
< ∞ for some m ∈ N,thenλ
R
m
a
0
,b
Notice that Θϑ, a, b−ϑ is 2π
p
-periodic in ϑ ∈ R. Combining the definition of Θa, b in 2.8,
one has
Θ
λa
,λb
≥ Θ
λa, λb
, ∀λ ≥ 0. 4.4
By Lemma 2.1ii, Θ
0 · a, 0 · a ∈ 0,π
p
and Θλa, λb is continuous in λ ∈ R. As functions
of λ ∈ 0, ∞, the smooth curve Θ
λa
,λb
lies above Θλa, λb. By the definition of λ
L
m
a, b
in 2.10,ifλ
m
a, b
τ
,λ
N
m
τa,τb
λ
N
m
a, b
τ
,λ
L
m
τa,τb
λ
L
m
0, ∞
, ∀γ ∈
1, ∞
, ∀m ∈ N,
4.6
where F denotes D, N or G.
Theorem 4.4. Given γ ∈ 1, ∞, r>0, m ∈ N and F ∈{D, N, G}.ThenH
F
m,γ
r > 0 and it can be
attained by some weights. Moreover, any minimizer a
F
,b
F
∈
S
γ
r.
Proof. We only prove for the case F G, other cases can be proved analogously. There exists
a sequence of weights a
n
,b
n
∈
∈ 0, 2π
p
, n ∈ N, such that
Θ
ϑ
n
,ν
n
a
n
,ν
n
b
n
− ϑ
n
mπ
p
,
Θ
ϑ, ν
n
a
n
,ν
n
b
→ ϑ
0
and
a
n
,b
n
−→
a
0
,b
0
∈
B
γ
r
, in
L
γ
,w
γ
b
0
− ϑ ≤ mπ
p
, ∀ϑ ∈
0, 2π
p
.
4.10
Thus Θ
ν
0
a
0
,ν
0
b
0
mπ
p
. It follows from 2.10 and 2.13 that r : |a
0
,b
0
|
γ
> 0and
≥ inf
λ
L
m
a, b
:
a, b
∈
B
γ
r
ν
0
H
G
m,γ
r
. 4.12
m,γ
r can be obtained if only γ ∈ 1, ∞.
In the following we will study the property of the minimizers.
Theorem 4.5. Given γ ∈ 1, ∞, r>0, m ∈ N, and F ∈{D, N, G},ifa, b is the minimizer of
H
F
m
r,thena, b ∈
S
γ
r. Moreover, a and b do not overlap, that is,
a
t
0 a.e.t∈ J
b
:
{
t | b
t
> 0
}
,
b
⎨
⎪
⎩
|
a
t
|
2
, if t ∈ J
0
,
a
t
, otherwise,
b
1
t
⎧
⎨
⎩
b
t
<λ
L
m
a, b
H
F
m,γ
r
,
4.16
which is in contradiction to the definition of H
F
m,γ
r.Thusa is nonnegative. Analogously b
is also nonnegative. Then it follows from Theorem 4.4 that a, b ∈
S
γ
r.
Step 2 Nonoverlap.Ifa and b overlap, then a, b 0, 0, that is, there exists J
0
⊂ 0, 1 with
positive measure such that
a
for some
J
0
with positive measure. Let
a
1
t
a
t
,b
1
t
⎧
⎨
⎩
0ift ∈
J
0
,
b
νb
1
t
φ
p
X
−
0. 4.20
Boundary Value Problems 17
Therefore λ
L
m
a
1
,b
1
≤ ν H
G
m,γ
r. It follows that
H
G
m,γ
a
:
{
a
t
> 0
}
, 4.22
X
t
< 0 a.e.t∈ J
b
:
{
b
t
> 0
}
. 4.23
Proof. If 4.23 does not hold. Then there exist
J
0
⊂ J
,b
0
is the minimizer of ν : H
F
m
r
and X is the corresponding half-eigenfunction. Let w
0
a
0
b
0
.ByTheorem 4.5, a
0
,b
0
∈
S
γ
r and a
0
and b
0
do not overlap, thus
w
0
γ
0.
4.26
Hence
H
F
m,γ
r
ν ≥ E
F
m,γ
r
.
4.27
On the other hand, for any w ∈ B
γ
r and λ ∈ R, one has |w
,w
−
|
γ
w
γ
and
φ
p
x
λw
−
t
φ
p
x
−
0.
4.28
18 Boundary Value Problems
Take the notations λ
G
m
a, b : λ
L
m
a, b and λ
G
m
: w ∈ B
γ
r
inf
λ
F
m
w
,w
−
: w ∈ B
γ
r
≥ inf
λ
F
m
0
a, b does not change its sign.
4.4. The Infimum in L
1
Balls
We cannot handle extremal problem in L
1
balls in the same way as done for L
γ
γ>1 case,
because L
1
balls are not sequentially compact even in the sense of weak topology.
Lemma 4.8. Given γ ∈ 1, ∞, r>0, and m ∈ N, there hold the following properties.
i If λ
L
m
a
0
,b
0
< ∞, then there exists δ>0 such that
λ
L
m
a, b
< ∞, ∀
a, b
∈
B
γ
δ
a
0
,b
0
.
4.31
Proof. i Suppose λ
L
m
a
0
,b
0
< ∞.ByTheorem 2.4i there exist ε>0andν>λ
L
m
a
0
,b
0
By Lemma 2.1i, that is, the continuous dependence of Θϑ, a, b in the weights a, b, there
exists δ>0 such that
Θ
ϑ
0
,νa,νb
− ϑ
0
>mπ
p
ε, ∀
a, b
∈
B
γ
δ
a
0
,b
0
.
4.34
Boundary Value Problems 19
<ν<∞, ∀
a, b
∈
B
γ
δ
a
0
,b
0
,
4.36
completing the proof of i.
ii Suppose μ : λ
D/N
m
a
0
,b
0
< ∞.LetX be the half-eigenfunction corresponding to
μ. Then X satisfies Dirichlet or Neumann boundary value conditions and
φ
0
a
0
X
p
b
0
X
p
−
dt
1
μ
1
0
X
p
dt > 0.
4.38
Let ϑ
D
X
p
−
dt > 0.
4.39
Notice that Θϑ
D/N
,μa
0
,μb
0
ϑ
D/N
mπ
p
. Then there exist ε>0andν>μ λ
D/N
m
a
0
,b
0
such that
Θ
ϑ
D/N
,νa
and hence λ
D/N
m
a, b < ∞, completing the proof of ii.
As a function of α,Kα, p is continuous in α ∈ 1, ∞. Explicit formula of Kα, p can
be found in 17, Theorem 4.1. For instance, Kp, pπ
p
p
and K∞,p2
p
.
Theorem 4.9. For any r>0, 1.11 holds for γ 1, that is,
H
F
m,1
r
E
F
m,1
r
2m
p
r
2m
p
r
.
4.43
Any a
0
,b
0
∈
B
1
r can be approximated by elements in
B
γ
r, γ>1, in the sense that there
exists a
γ
,b
γ
∈
B
γ
a
0
t
|
1/γ
· sign
a
0
t
,b
γ
r
1/γ
∗
|
b
0
t
|
1/γ
· sign
b
0
exists for any a, b ∈ B
γ
δ
a
0
,b
0
. We can assume that λ
F
m
a
γ
,b
γ
exists for any γ ∈ 1, ∞
due to 4.44. Furthermore, by Lemma 2.1iii, one can prove that λ
F
m
a, b is continuously
differentiable in a, b ∈ B
γ
δ
a
0
,b
0
in |·|
1
topology. In particular, λ
F
γ
≥ lim
γ↓1
H
F
m,γ
r
,
4.47
and therefore,
H
F
m,1
r
inf
λ
F
m
a
0
,b
0
On the other hand, we prove
H
F
m,1
r
≤
2m
p
r
.
4.49
Notice that
B
γ
2
1/γ−1
r ⊂
B
1
r for all γ>1andallr>0, because
|
a, b
b
γ
γ
1/γ
4.50
Boundary Value Problems 21
for any a, b ∈L
γ
×L
γ
.Thusweobtain
H
F
m,1
r
≤ H
F
m,γ
2
1−1/γ
r
m
6 M. Zhang, “Continuity in weak topology: higher order linear systems of ODE,” Science in China A,
vol. 51, no. 6, pp. 1036–1058, 2008.
7 W. Li and P. Yan, “Continuity and continuous differentiability of half-eigenvalues in potentials,” to
appear in Communication in Contemporary Mathematics.
8 W. Li and P. Yan, “Various half-eigenvalues of scalar p-Laplacian with indefinite integrable weights,”
Abstract and Applied Analysis, vol. 2009, Article ID 109757, 27 pages, 2009.
9 G. Meng, P. Yan, and M. Zhang, “Spectrum of one-dimensional p-Laplacian with an indefinite
integrable weight,” Mediterranean Journal of Mathematics, vol. 7, no. 2, pp. 225–248, 2010.
10 P. Yan and M. Zhang, “Continuity in weak topology and extremal problems of eigenvalues of the
p-Laplacian,” to appear in Transactions of the American Mathematical Society.
11 M. Zhang, “Extremal values of smallest eigenvalues of Hill’s operators with potentials in L
1
balls,”
Journal of Differential Equations, vol. 246, no. 11, pp. 4188–4220, 2009.
12 Q. Wei, G. Meng, and M. Zhang, “Extremal values of eigenvalues of Sturm-Liouville operators with
potentials in L
1
balls,” Journal of Differential Equations, vol. 247, no. 2, pp. 364–400, 2009.
13 P. A. Binding and B. P. Rynne, “Variational and non-variational eigenvalues of the p-Laplacian,”
Journal of Differential Equations, vol. 244, no. 1, pp. 24–39, 2008.
14 M. Zhang, “The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with
periodic potentials,” Journal of the London Mathematical Society. Second Series, vol. 64, no. 1, pp. 125–143,
2001.
15 V. I. Arnold, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics,
Springer, New York, NY, USA, 2nd edition, 1989.
16 H. Broer and M. Levi, “Geometrical aspects of stability theory for Hill’s equations,” Archive for Rational
Mechanics and Analysis, vol. 131, no. 3, pp. 225–240, 1995.
17 M. Zhang, “Nonuniform nonresonance of semilinear differential equations,” Journal of Differential
Equations, vol. 166, no. 1, pp. 33–50, 2000.
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