DOI: 10.2478/auom-2014-0038
An. Şt. Univ. Ovidius Constanţa
Vol. 22(2),2014, 129–139
A new proof of the bound for the first Dirichlet
eigenvalue of the Laplacian operator
Chang-Jun Li and Xiang Gao*
Abstract
In this paper, we present a new proof of the upper and lower bound
estimates for the first Dirichlet eigenvalue λD
1 (B (p, r)) of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s
gradient estimate for the heat equation.
1
Introduction and Main Results
Let (M n , g) be an n-dim C ∞ complete Riemannian manifold, and ∆ denote
the Laplacian operator. For the compact manifold, it is well known that the
eigenvalue problem −∆ϕ = λϕ has discrete eigenvalues, which are listed as
0 = λ0 (M n ) < λ1 (M n ) ≤ λ2 (M n ) ≤ · · · .
Moreover we call λi (M n ) the i th eigenvalue and call a function ϕi satisfying ∆ϕi = −λi ϕi the i th eigenfunction.
Recall that the first eigenvalue λ1 (M n ) for the closed Riemannian manifold M n is defined by
R
2
n |∇φ| dµ
,
(1)
λ1 (M n ) = inf MR
φ∈Ω
φ2 dµ
Mn
Key Words: Laplacian operator, Li-Yau gradient estimate, the first Dirichlet eigenvalue.
2010 Mathematics Subject Classification: Primary 53C25; Secondary 53C44.
Received: August, 2012.
Revised: November, 2012.
Accepted: November, 2013.
*The corresponding author: gaoxiangshuli@126.com
This work is supported by the National Natural Science Foundation of China (NSFC)
11301493, 11101267 and Fundamental Research Funds for the Central Universities.
129
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
where Ω is the completing Hilbert space of
Z
∞
n Ω0 = φ ∈ C (M ) 130
φdµ = 0
Mn
under the norm
2
Z
kφk1 =
Mn
φ2 dµ +
Z
2
|∇φ| dµ.
Mn
We denote the geodesic ball with center p and radius r by B (p, r) in
the n-dim manifold M n . Then the first Dirichlet eigenvalue λD
1 (B (p, r)) of
Laplacian operator can be denoted as
R
2
|∇φ| dµ
B(p,r)
D
R
,
(2)
λ1 (B (p, r)) =
inf
φ2 dµ
φ∈H02 (B(p,r))
B(p,r)
where H02 (B (p, r)) is the completing Hilbert space of C0∞ (B (p, r)) under the
norm
Z
Z
2
2
2
kφk1 =
φ dµ +
|∇φ| dµ.
B(p,r)
B(p,r)
The upper and lower bound for the first eigenvalue of Laplacian operator
are very useful in geometry analysis and PDE. In [1], S. Y. Cheng used the
approach of Jacobi fields to obtain an upper bound for the first Dirichlet eigenvalue λD
1 (B (p, r)) of Laplacian operator, which are the following comparison
theorems.
Theorem 1.1 (Cheng). Let (M n , g) be a complete Riemannian manifold satisfying Rc ≥ (n − 1) K, then
D
λD
1 (B (p, r)) ≤ λ1 (BK (p, r)) ,
n
, and equality holds
where BK (p, r) is the geodesic ball in the space form MK
iff B (p, r) is isometric to BK (p, r).
Corollary 1.2 (Cheng). Let (M n , g) be a compact Riemannian manifold satisfying Rc ≥ 0, then
dM n
Cn
λ1 (M n ) ≤ λD
B
p,
≤ 2 ,
K
1
2
dM n
where Cn = 2n (n + 4) and dM n is the diameter of M n .
For the lower bound of the first eigenvalue λ1 (M n ) for closed Riemannian
manifold M n , in 1980, P. Li and S. T. Yau [3] used the approach of gradient
estimates to derive a lower bound as follows:
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
131
Theorem 1.3 (Li-Yau). Let (M n , g) be a closed Riemannian manifold satisfying Rc ≥ 0, then
π2
,
λ1 (M n ) ≥
2dM n
where dM n denotes the diameter of M n .
Moreover, their result was improved to a sharp lower bound by J. Q. Zhong
and H. C. Yang [5] as follows. Their proofs both rely on the gradient estimate
and the maximum principle.
Theorem 1.4 (Zhong-Yang). Let (M n , g) be a closed Riemannian manifold
satisfying Rc ≥ 0, then
π2
.
λ1 (M n ) ≥
dM n
In this paper, we will present a new and simpler proof of upper and lower
bound estimates for the first Dirichlet eigenvalue λD
1 (B (p, r)) of the Laplacian
operator for the manifold with Ricci curvature Rc ≥ −K, where K ≥ 0. Our
approach is applying the following gradient estimate which leads to the famous
Li-Yau Harnack estimate for the heat equation [4].
Theorem 1.5 (Li-Yau). Let (M n , g) be a complete Riemannian manifold with
Ricci curvature Rc (g) ≥ −K, and let B (p, 2r) denote the geodesic ball with
center p and radius 2r in M n . If u : B (p, 2r) × [0, ∞) −→ R is a positive
solution to the heat equation
∂
u (x, t) = ∆u (x, t) ,
∂t
then in B (p, r) we have
!
2
√
|∇u|
nα2 K
ut
C (n) α2
nα2
α2
sup
Kr
+
−
α
≤
+
+
, (3)
2
2
2
u
u
r
α −1
2 (α − 1)
2t
B(p,r)
where α > 1 and C (n) is a constant depended on n.
Remark 1. To use Theorem 1.5, we will deal with the solution to the heat
D
equation with the form of u(x)e−λ1 (B(p,r))t , where u(x) is the eigenfunction
corresponding to the first Dirichlet eigenvalue λD
1 (B (p, r)) of the Laplacian
operator, and then apply the Li-Yau’s gradient estimate to the solution above
and prove an integral estimate as follows.
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
132
Theorem 1.6. Let (M n , g) be a complete Riemannian manifold with Ricci
curvature Rc (g) ≥ −K, then for any geodesic ball B (p, r) in M n and arbitrary ε > 0, we have
Z t2
ε
|∇ log u|2 dt
t2 − t1 t1
!
!
√
K
4C (n) α
α2
nαK
nα
D
≥λ1 (B (p, r)) −
+
r +
−
2
2
r
α −1
2
2 (α − 1)
2t1
2
1
1 d (x1 , x2 )
−
α+
4
ε (t2 − t1 )2
for arbitrary constant speed minimal geodesic γ(t) joining points x1 and x2 such
that x1 , x2 ∈ B p, 2r and x1 = γ(t1 ), x2 = γ(t2 ).
By using Theorem 1.6, we can derive the following upper and lower bound
estimates for the first Dirichlet eigenvalue λD
1 (B (p, r)) of Laplacian operator.
Theorem 1.7. Let (M n , g) be a complete Riemannian manifold with Ricci
curvature Rc (g) ≥ −K, then for any geodesic ball B (p, r) in M n , we have
C4
C2
C2
C1
+
+
+ C3 ≤ λD
+ C3 ,
(4)
1 (B (p, r)) ≤ 2
d2B p, r
dB (p, r )
dB p, r
dB (p, r )
( 2)
( 2)
2
2
where dB (p, r ) denotes the diameter of B p, 2r and Ci (1 ≤ i ≤ 4) are some
2
constants only depending on the choice of p and r, which can be seen in the
proof.
Remark 2. Recall that for the closed Riemannian manifold, the eigenfunction u(x) corresponding to the first eigenvalue λ1 (M n ) of the Laplacian operator satisfies
Z
u(x)dµ = 0,
Mn
which implies that u(x) can not always be positive. But the Li-Yau’s gradient
estimate (3) only satisfies for the positive solution of the heat equation, thus
in this paper, we deal with the upper and lower bound estimates for the first
Dirichlet eigenvalue which can always be positive.
The paper is organized as follows: In section 2, we prove Theorem 1.6
by using the Li-Yau’s gradient estimate for heat equations. Based on this, in
section 3, we prove Theorem 1.7.
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
2
133
Proof of Theorem 1.6
Proof of Theorem 1.6. To prove Theorem 1.6, we consider the function in the
form of
D
ũ(x, t) = u(x)e−λ1 (B(p,r))t ,
where u(x) is the eigenfunction corresponding to the first Dirichlet eigenvalue
λD
1 (B (p, r)) of the Laplacian operator. It is clear that ũ(x, t) is actually a
solution to the heat equation, in fact
∂
−λD
1 (B(p,r))t
ũ(x, t) = −λD
1 (B (p, r)) u(x)e
∂t
D
= ∆(u(x)e−λ1 (B(p,r))t )
= ∆(ũ(x, t)).
On the other hand, let ϕ be the eigenfunction corresponding
to the
first Dirich p 2
2ϕ·∇ϕ
√ let eigenvalue λD
1 (B (p, r)), since |∇ |ϕ|| = ∇ ϕ = 2 ϕ2 ≤ |∇ϕ|, we
have
R
2
|∇φ| dµ
B(p,r)
D
R
λ1 (B (p, r)) =
inf
φ2 dµ
φ∈H02 (B(p,r))
B(p,r)
R
2
|∇ |ϕ|| dµ
B(p,r)
≤ R
2
|ϕ| dµ
B(p,r)
R
2
|∇ϕ| dµ
B(p,r)
R
≤
ϕ2 dµ
B(p,r)
= λD
1 (B (p, r)) ,
which implies that |ϕ| is also an eigenfunction corresponding to the first Dirichlet eigenvalue λD
1 (B (p, r)). Thus without loss of generality, we would suppose
that the eigenfunction corresponding to the first Dirichlet eigenvalue of Laplacian operator is nonnegative. Then by the strong maximum principle for the
Laplacian operator, we can assume that the eigenfunction is positive. Thus
D
ũ(x, t) = u(x)e−λ1 (B(p,r))t is actually a positive solution to the heat equation.
Then by using the Li-Yau’s estimate (3) to ũ(x, t) we have
!
√
2
4C (n) α2
nα2 K
nα2
|∇e
u|
u
et
α2
K
−α ≤
+
r +
+
2
2
2
u
e
u
e
r
α −1
2
2 (α − 1)
2t
satisfies in B p, 2r . Then for a constant speed
minimal geodesic γ(t) joining
r
points x1 and x2 such that x1 , x2 ∈ B p, 2 and x1 = γ(t1 ), x2 = γ(t2 ), we
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
134
have
ũ(x2 , t2 )
log
=
ũ(x1 , t1 )
Z
t2
d
log ũ (γ (t) , t)dt
dt
t1
Z t2 ∇e
u dγ
u
et
+
,
dt
=
u
e
u
e dt
t1
2 !
Z t2
2
α dγ u
et
1 |∇e
u|
− ≥
−
(t) dt
u
e
α u
e2
4 dt
t1
!
2
Z
Z 2
1 t2 |∇e
u|
u
et
α t2 dγ =−
−
α
dt
−
(t)
dt
α t1
u
e2
u
e
4 t1 dt
!
!
√
Z t2
4C (n) α
K
nα
α2
nαK
≥−
+
r +
+
dt
r2
α2 − 1
2
2 (α − 1)
2t
t1
2
Z α t2 dγ −
(t) dt
4 t1 dt
The second inequality we use the Li-Yau’s gradient estimate, and it follows
from γ(t) being a constant speed minimal geodesic that
dγ d(x1 , x2 )
dt (t) ≡ t2 − t1 .
Thus we have
D
log u (x2 ) − λD
1 (B (p, r)) t2 − log u (x1 ) + λ1 (B (p, r)) t1
!
!
√
α2
K
nαK
4C (n) α
+
r +
(t2 − t1 )
≥−
r2
α2 − 1
2
2 (α − 1)
−
nα
α d(x1 , x2 )2
(log t2 − log t1 ) −
2
4 t2 − t1
Then for the path γ(t) which is a constant speed minimal geodesic jointing
points x1 , x2 such that γ(t1 ) = x1 and γ(t2 ) = x2 we have
log u (x2 ) − log u (x1 ) = log u (γ(t2 )) − log u (γ(t1 ))
Z t2 dγ
dt
=
∇ log u,
dt
t1
Z t2
Z t2
1 dγ 2
≤
ε|∇ log u|2 dt +
| | dt
t1
t1 4ε dt
Z t2
d(x1 , x2 )2
.
=
ε|∇ log u|2 dt +
4ε(t2 − t1 )
t1
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
Note that
log t2 − log t1 =
1
1
(t2 − t1 ) ≤ (t2 − t1 ),
ξ
t1
135
(5)
where ξ ∈ (t1 , t2 ). Then by using these two inequalities we have
Z
t2
d(x1 , x2 )2
− λD
1 (B (p, r)) (t2 − t1 )
4ε(t2 − t1 )
!
!
√
α2
K
nαK
nα
+
r +
(t2 − t1 ) −
(t2 − t1 )
α2 − 1
2
2 (α − 1)
2t1
ε|∇ log u|2 dt +
t1
4C (n) α
r2
≥−
−
α d(x1 , x2 )2
4 t2 − t1
which implies that
Z t2
ε
|∇ log u|2 dt
t2 − t1 t1
≥λD
1
−
(B (p, r)) −
1
4
α+
1
ε
4C (n) α
r2
!
!
√
K
α2
nαK
nα
+
r +
−
2
α −1
2
2 (α − 1)
2t1
2
d (x1 , x2 )
2
(t2 − t1 )
for arbitrary ε > 0.
3
Proof of Theorem 1.7
In this section, we prove Theorem 1.7 by using Theorem 1.6.
Proof. For the geodesic ball B p, 2r , by using the strong maximum principle
for the Laplacian operator, we have


!
u2 ≥ CB (p, r ) 
2
sup
x∈B (p, r2 )
u − u u −
inf
u ,
(6)
x∈B (p, r2 )
where CB (p, r ) is a positive constant only depending on the choice of p and 2r .
2
Then for arbitrary x, y ∈ B p, 2r , we can select the appropriate s and t
such that
u (x) = u (γ (s)) = inf u
x∈B (p, r2 )
136
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
and
u (y) = u (γ (t)) =
sup
u.
x∈B (p, r2 )
Then it follows from (6) that
log u (y) − log u (x)
Z t
dγ
∇ log u,
=
dt
dt
s
Z u(y)
du
≤
u(x) |u|
Z sup
u
x∈B (p, r )
2
v
≤

u
inf
u u
r
x∈B (p, )
2
u
tCB (p, r ) 
2
=q
π
du

sup
u − u u −
x∈B (p, r2 )
!
inf
u
x∈B (p, r2 )
.
CB (p, r )
2
Then substituting this and (5) into
D
log u (x2 ) − λD
1 (B (p, r)) t2 − log u (x1 ) + λ1 (B (p, r)) t1
!
!
√
nαK
4C (n) α
α2
K
≥−
+
r +
(t2 − t1 )
r2
α2 − 1
2
2 (α − 1)
−
nα
α d(x1 , x2 )2
(log t2 − log t1 ) −
,
2
4 t2 − t1
we have
nα
(t2 − t1 )
2t1
CB (p, r )
2

!
√
2
α
K
nαK 
+
d
+
(t2 − t1 )
r
α2 − 1
2 B (p, 2 )
2 (α − 1)
λD
1 (B (p, r)) (t2 − t1 ) − q

≤
4C (n) α
d2B p, r
( 2)
π
−
d2
α B (p, r2 )
,
+
4 t2 − t1
where dB (p, r ) denotes the diameter of B p, 2r . This implies that
2
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
λD
1 (B (p, r))


!
√
2
K
nαK
4C
(n)
α
α
 + nα
+
dB (p, r ) +
≤ 2
2
2
dB p, r
α −1
2
2 (α − 1)
2t1
( 2)
d2
α B (p, r2 )
+q
+
4 (t2 − t1 )2
CB (p, r ) (t2 − t1 )
2
π
for any t1 < t2 , and
λD
1 (B (p, r))


!
√
2
4C
(n)
α
K
α
nαK
 + nα
≥ 2
+
dB (p, r ) +
2
2
dB p, r
α −1
2
2 (α − 1)
2t1
( 2)
d2
α B (p, r2 )
+
−q
4 (t1 − t2 )2
CB (p, r ) (t1 − t2 )
2
π
for any t1 > t2 .
Furthermore, by selecting the parameters
t1 = d2B (p, r )
2
and


t2 = 1 + q
1
CB (p, r )
 d2
B (p, r2 )
,
2
and substituting them into above, we have
λD
1 (B (p, r))


!
√
2
4C
(n)
α
α
K
nαK
nα
+
≤ 2
+
dB (p, r ) +
2
2
2
dB p, r
α −1
2
2 (α − 1)
2dB p, r
( 2)
( 2)
αC
B (p, r2 )
π
+ 2
+
2
dB p, r
4dB p, r
( 2)
( 2)
C1
C2
= 2
+
+ C3 .
dB p, r
dB (p, r )
( 2)
2
137
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
138
For the same reason, we can substitute the parameters


1
 d2 r
t1 =  1 + q
B (p, 2 )
CB (p, r )
2
and
t2 = d2B (p, r )
2
into the same inequality to obtain that
λD
1 (B (p, r))


!
√
2
4C
(n)
α
α
K
nαK

≥ 2
+
+
d
r
dB p, r
α2 − 1
2 B (p, 2 )
2 (α − 1)
( 2)
q
nα CB (p, r )
αCB (p, r )
π
2
2
−
+
+ 2
2
q
d
4d
B (p, r2 )
B (p, r2 )
2 1 + CB (p, r ) d2B p, r
( 2)
2
=
C4
C2
+ C3 .
+
d2B p, r
dB (p, r )
( 2)
2
References
[1] S. Y. Cheng, Eigenvalue comparison and its geometric application, Math.
Z. 143 (1975), 289-297.
[2] B. Chow, P. Lu, L. Ni, Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics, 3, Science Press and Graduate Studies in Mathematics, 77, American Mathematical Society (co-publication), 2006.
[3] P. Li and S. T. Yau, Eigenvalues of a compact Riemannian manifold, AMS
Proc. Symp. Pure Math. 36 (1980), 205-239.
[4] R. Schoen and S. T. Yau, Lectures on differential geometry. Conference
Proceedings and Lecture Notes in Geometry and Topology, I. International Press, Cambridge, MA, 1994.
[5] J. Q. Zhong and H. C. Yang, On the estimate of first eigenvalue of a
compact Riemannian manifold, Sci. Sinica Ser. A 27 (1984), 1265-1273.
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
Chang-Jun Li,
School of Mathematical Sciences,
Ocean University of China,
Lane 238, Songling Road, Laoshan District, Qingdao City,
Shandong Province, 266100, People’s Republic of China.
Email: licj@ouc.edu.cn
Xiang Gao,
School of Mathematical Sciences,
Ocean University of China,
Lane 238, Songling Road, Laoshan District, Qingdao City,
Shandong Province, 266100, People’s Republic of China.
Email: gaoxiangshuli@126.com
139
A NEW PROOF OF THE BOUND FOR THE FIRST DIRICHLET EIGENVALUE
140