Theoretical Mathematics & Applications, vol.3, no.2, 2013, 39-48
ISSN: 1792-9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Universal inequalities for eigenvalues
of elliptic operators in divergence
form on domains in complete
noncompact Riemannian manifolds
Yanli Li1 and Feng Du2
Abstract
In this paper, we study the eigenvalue problem of elliptic operators in
divergence form, and obtain some universal inequalities for eigenvalues
of elliptic operators in divergence form on domains in complete simple
connected noncompact Riemannian manifolds admitting special functions which include hyperbolic space. Especially, by using our universal
inequalities, we can get the different universal inequalities including
Yang inequality.
Mathematics Subject Classification: 35P15, 53C42, 58G25
Keywords: Elliptic operators in divergence form, eigenvalues, universal inequalities, complete noncompact Riemannian manifolds, hyperbolic space
1
School of Electronic and Information Science, Jingchu University of Technology, Hubei
Jingmen 448000, P.R. China.
2
School of Mathematics and Physics Science, Jingchu University of Technology, Hubei
Jingmen 448000, P.R. China.
Article Info: Received : April 20, 2013. Revised : May 30, 2013
Published online : June 25, 2013
40
1
Universal inequalities for eigenvalues
Introduction
Let Ω be a bounded domain in an n-dimensional complete Riemannian
manifold M . Let ∆ be the Laplacian operator acting on functions on M and
consider the following eigenvalues problem for the Laplacian operator

 ∆u = −λu,
in Ω,
(1.1)

u = 0,
on ∂Ω,
it is known that this eigenvalue problem has a discrete spectrum,
0 < λ 1 < λ 2 ≤ · · · ≤ λk ≤ · · · ,
where each eigenvalue is repeated with its multiplicity. When M = Rn , ∆ =
Pn ∂ 2
i=1 ∂x2 , Payne-Pólya-Weinberger [11] in 1956 proved
i
k
4 X
λk+1 − λk ≤
λi .
kn i=1
(1.2)
In 1980, Hile-Protter [9] strengthened (1.1), and proved
k
λi
kn X
≤
.
4
λ
−
λ
k+1
i
i=1
(1.3)
In 1991, Yang [13] gave the following much stronger inequality
k
k
X
4X
(λk+1 − λi )2 ≤
(λk+1 − λi )λi .
n
i=1
i=1
From inequality (1.3), we can get a weaker but explicit form
!
µ
¶ Ã k
4 1 X
λk+1 ≤ 1 +
λi .
n k i=1
(1.4)
(1.5)
These inequalities are called universal inequalities because they do not involve
domain dependence.
In the following, we introduce an elliptic operator in divergence form in
Riemannian manifold. Let (M, h, i) be an n-dimensional compact Riemannian
manifold with boundary ∂M (possibly empty). Let A : M → End(T M ) be a
41
Yanli Li and Feng Du
smooth symmetric and positive definite section of the bundle of all endomorphisms of T M . Let V be a nonnegative continuous function on M . Denote by
∆ and ∇ the Laplacian and the gradient operator of M respectively. Define
Lu = −div(A∇u) + V u,
(1.6)
where for a vector field X on M , divX denotes the divergence of X. The
operator L defined in (1.5) is an elliptic operator in divergence form. It is easy
to see that the Laplacian operator and Schrödinger operator are it’s special
cases. In 2010, do Carmo-Wang-Xia [6] considered the eigenvalue problem of
the elliptic operator in divergence form with weight such that

 Lu = λρu,
in Ω,
(1.7)
 u = 0,
on ∂Ω,
where ρ is a weight function which is positive and continuous on M . They got
a Yang type inequality
µ µ
¶
¶
k
4ξ22 ρ22 X
1
V0
n2 H02
(λk+1 − λi ) ≤
(λk+1 − λi )
λi −
+
,
2
nρ
ξ
ρ
4ρ
1
2
1
1
i=1
i=1
k
X
2
(1.8)
where ξ1 I ≤ A and tr(A) ≤ nξ2 throughout M , ρ1 ≤ ρ(x) ≤ ρ2 , ∀x ∈ M ,
I is the identity map, ξ1 , ξ2 , ρ1 , ρ2 are positive constants, H0 = maxM (|H|),
V0 = minM (V ), and |H| is the mean curvature of M immersed into some
Euclidean space RN . Recently, Du-Wu-Li[7] considered the problem (1.7) on
complete Riemannian manifolds, they proved the inequality
k
X
i=1
Ã
! 12
k
2ρ2 nξ2 X
f (λi ) ≤
g(λi )
nρ1 ξ1 i=1
à k
¶! 21
µ
X
(f (λi ))2
S
,
×
λi +
(λ
−
λ
)g(λ
)
ρ
k+1
i
i
1
i=1
³
n2 ξ1
|H|2
4
(1.9)
´
where (f, g) ∈ =λk+1 ,S = supM
− V , and H is the mean curvature
vector field.
In this paper, we will consider the eigenvalue problem (1.7) on bounded domains in the complete simply connected noncompact Riemannian manifolds.
Then we obtain.
42
Universal inequalities for eigenvalues
Theorem 1.1. Let (M, h, i) be an n(n ≥ 3)-dimensional complete noncompact
simply connected Riemannian manifold with Sectional curvature Sec satisfying
−K 2 ≤ Sec ≤ −k 2 . For a bounded domain Ω, Let A : Ω → End(T Ω) be a
smooth symmetric and positive definite section of the bundle of all endomorphisms of T Ω, and assume that ξ1 I ≤ A ≤ ξ2 I throughout Ω, ∀x ∈ Ω, ρ1 ≤
ρ(x) ≤ ρ2 , here ξ1 , ξ2 , ρ1 , ρ2 are positive constants. Let λi be the ith eigenvalue
of the eigenvalue problem (1.7), then for any (f, g) ∈ =λk+1 , we have
k
X
i=1
k
k
o 12 n X
n 1 X
(f (λi ))2
g(λi ) ×
f (λi ) ≤
ρ2 ξ2 i=1
(λk+1 − λi )g(λi )
i=1
µ
¶ o1
1
2
2
2
2
(1.10)
(ρ2 λi − V0 ) − (n − 1)k + 2(n − 1)K
ξ1
where V0 = minx∈Ω V (x).
From Theorem 1.1, we can get the following.
Corollary 1.2. Under the assumption of Theorem 1.1, if M is a hyperbolic
space H n (−1), we have
k
X
(
f (λi ) ≤
i=1
×
) 21
k
1 X
g(λi )
ρ2 ξ2 i=1
( k
X
(f (λi ))2
i=1
(λk+1 − λi )g(λi )
µ
¶) 12
1
(ρ2 λi − V0 ) − (n − 1)2 (1.11)
,
ξ1
where V0 = minx∈Ω V (x).
Remark. Taking (f (λi ), g(λi )) = ((λk+1 − λi )2 , (λk+1 − λi )2 ) in (1.11), we can
get a Yang inequality that
k
X
i=1
2
(λk+1 − λi )
µ
¶
k
1 X
1
2
≤
(λk+1 − λi )
(ρ2 λi − V0 ) − (n − 1)
ρ2 ξ2 i=1
ξ1
So, by choosing different (f (λi ), g(λi )), we can get different universal inequalities.
43
Yanli Li and Feng Du
2
Preliminaries
In this section, firstly, we shall introduce a family of couples of functions
[8].
Definition 2.1. Let λ > 0, a couple (f, g) of functions defined on [0, λ] belongs
to =λ as that
(i) f and g are positive,
(ii) f and g satisfy the following condition, for any x, y ∈ [0, λ], such that
x 6= y,
µ
¶2 µ
¶µ
¶
g(x) − g(y)
f (x) − f (y)
(f (x))2
(f (y))2
+
+
≤ 0.
x−y
g(x)(λ − x) g(y)(λ − y)
x−y
(2.1)
We can easily find that g(x) is a nonincreasing function.
In the following, we will introduce a lemma which is obtained by Du-WuLi[7].
Lemma 2.1. Let (M, h, i) be an n-dimensional compact Riemannian manifold with boundary ∂M (possibly empty). Let λi be the ith eigenvalue of the
eigenvalue problem of elliptic operators in divergence form with weight ρ such
that
Lu = λρu in M, u = 0 on ∂M,
and ui be the orthonormal eigenfunction corresponding to λi , that is,
Lui = λi ρui in M,
and ui = 0 on ∂M,
Z
ρui uj = δij , ∀ i, j = 1, 2, · · ·
M
2
Then for any h ∈ C (M ) and (f, g) ∈ =λk+1 , we have
k
X
i=1
f (λi )kui ∇hk
2
≤ δ
k
X
i=1
Z
g(λi )
M
u2i h∇h, A∇hi
+
k
X
i=1
°
¶°2
µ
°
° 1
1
° ,
°√
u
∆h
h∇u
,
∇hi
+
i
i
°
° ρ
2
(f (λi ))2
δ(λk+1 − λi )g(λi )
(2.2)
44
Universal inequalities for eigenvalues
where δ is any positive constant and kf k2 =
R
M
f 2.
From this Lemma, if we can find a “nice” function on bounded domains
in complete Riemannian manifolds and take it in (2.2) and (2.3), we can get
universal inequalities in complete Riemannian manifolds. In the following, we
will introduce a function on a bounded domain in complete noncompact simple
connected Riemannian manifolds(for the more details about this function, we
refer [5] ).
Let (M, h, i) be an n-dimensional complete noncompact Riemannian manifold, and Ω be a bounded connected domain in M . Sec(M ) is the section
curvature satisfying −K 2 ≤ Sec(M ) ≤ −k 2 , where k, K is constants and
0 ≤ k ≤ K. When p is not in Ω, define the distance function r(x) = d(x, p),
then
(n − 1)k
coshkr
coshKr
≤ ∆r ≤ (n − 1)K
,
sinhkr
sinhKr
(2.3)
because of ∂r ∆r = −|Hess r|2 − Ric(∂r , ∂r ), so
−∂r ∆r ≤ (n − 1)K 2
coshKr
− (n − 1)k 2 ,
sinhKr
(2.4)
where Hess, Ric are the Hessian operator and Ricci curvature operator, respectively.
3
Proofs of the main results
In this section, we will give the proofs of Theorem 1.1 and Corollary 1.2.
Proof of Theorem 1.1. Taking h = r in the (2.2), we can get
k
X
i=1
2
f (λi )kui k
≤ δ
k
X
i=1
Z
g(λi )
M
u2i h∇r, A∇ri
+
k
X
i=1
°
¶°2
µ
°
° 1
1
° ,
°√
u
∆r
h∇u
,
∇ri
+
i
i
°
° ρ
2
(f (λi ))2
δ(λk+1 − λi )g(λi )
(3.1)
45
Yanli Li and Feng Du
Because of ρ1 ≤ ρ(x) ≤ ρ2 , ξ1 I ≤ A ≤ ξ2 I, and (2.1), we have
°
°
° 1 °2
° √ ui ° ≥ 1
° ρ °
ρ2
ξ1 ≤ h∇r, A∇ri ≤ ξ2 .
(3.2)
(3.3)
Take (3.2) and (3.3) into (3.1), we have
k
k
k
X
1 X
ξ2 X
(f (λi ))2
f (λi ) ≤
δ
g(λi ) +
ρ2 i=1
ρ1 i=1
4δ(λk+1 − λi )g(λi )
i=1
°
°2
°
° 1
° √ (2h∇ui , ∇ri + ui ∆r)°
° ρ
°
It is known that
Z
Z
ui (−div(A∇ui ) + V ui )
ui L(ui ) =
λi =
ZM
M
=
ZM
¡
1
|∇ui | ≤
ξ1
M
2
¢
¢
ξ1 |∇ui |2 + V u2i ,
µ
¶
M
Z
hAui , ui i + V u2i
¡
≥
which yields
(3.4)
V0
λi −
ρ2
.
From (2.4) and (2.5), we can get
°2
°
°
° 1
° √ (2h∇ui , ∇ri + ui ∆r)°
°
° ρ
Z
1
=
(2h∇r, ∇ui i + ui ∆r)2
Ω ρ
µ Z
¶
Z
Z
1
2
2
2
2
≤
4 h∇r, ∇ui i − ui (∆r) − 2 ui h∇r, ∇∆ri
ρ1
Ω
Ω
¶
µZ Ω
Z
Z
1
2
2
2
2
|∇ui | − ui (∆r) − 2 ui ∂r (∆r)
≤
ρ1
Ω
Ω
Ω
µ
¶
2 2 Z
1
V0
(n − 1) k
cosh2 kr
≤
λi −
−
u2i
ρ1 ξ1
ρ2
ρ1
sinh2 kr
Ω
Z
2
2(n − 1)k 2
2(n − 1)K 2
2 cosh Kr
ui
−
+
ρ
ρ1 ρ2
sinh2 Kr
µ 1
¶Ω
2
2
1
V0
(n − 1)k
2(n − 1)K 2
≤
λi −
−
+
ρ1 ξ1
ρ2
ρ1 ρ2
ρ1 ρ2
Z
2 Z
2
(n − 1)
2(n − 1)
k
K2
2
2
−
+
ui
u
i
ρ1
ρ1
sinh2 kr
sinh2 Kr
Ω
Ω
(3.5)
(3.6)
46
Universal inequalities for eigenvalues
Since K ≥ k ≥ 0, r > 0, we obtain
K
k
≤
,
sinhKr
sinhkr
(3.7)
Since n ≥ 3, we get
(n − 1)2
k2
K2
k2
−
2(n
−
1)
≥
(n
−
1)(n
−
3)
≥ 0,
sinh2 kr
sinh2 Kr
sinh2 kr
so, by (3.7)-(3.9), we can get
°
°2
°1
°
° (ui ∆r + 2h∇r, ∇ui i)°
°ρ
°
µ
¶
1
V0
(n2 − 1)k 2 2(n − 1)K 2
≤
λi −
−
+
ρ1 ξ1
ρ2
ρ1 ρ2
ρ1 ρ2
(3.8)
(3.9)
Taking (3.10) into (3.4), we obtain
k
k
k
X
(f (λi ))2
1 X
ξ2 X
f (λi ) ≤
δ
g(λi ) +
ρ2 i=1
ρ1 i=1
4δ(λk+1 − λi )g(λi )
i=1
µ
µ
¶
¶
1
V0
(n2 − 1)k 2 2(n − 1)K 2
×
λi −
−
+
(3.10)
,
ρ1 ξ1
ρ2
ρ1 ρ2
ρ1 ρ2
above inequality implies that
k
X
i=1
k
k
X
(f (λi ))2
ρ2 ξ2 X
f (λi ) ≤
δ
g(λi ) +
ρ1 i=1
4δ(λk+1 − λi )g(λi )
i=1
µ
¶
1 1
2
2
2
×
(ρ2 λi − V0 ) − (n − 1)k + 2(n − 1)K
(3.11)
ρ1 ξ1

1
Pk
µ
¶ 2
(f (λi ))2
 1
1
i=1 (λk+1 −λi )g(λi )
δ=
(ρ2 λi − V0 ) − (n2 − 1)k 2 + 2(n − 1)K 2
Pk
 4ρ2 ξ2

ξ
1
i=1 g(λi )
(3.11) becomes
k
X
i=1
k
k
n 1 X
o 12 n X
(f (λi ))2
f (λi ) ≤
g(λi ) ×
ρ2 ξ2 i=1
(λk+1 − λi )g(λi )
i=1
µ
¶ o1
1
2
2
2
2
(ρ2 λi − V0 ) − (n − 1)k + 2(n − 1)K
(3.12)
ξ1
This completes the proof of Theorem 1.1.
Yanli Li and Feng Du
47
Proof of Corollary 1.2. Taking K = k = 1 in (3.12), we immediately obtain
(1.11).
ACKNOWLEDGEMENTS. The research work is supported by Key Laboratory of Applied Mathematics of Hubei Province and The research project
of Jingchu University of Technology.
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Universal inequalities for eigenvalues
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