Lipschitz property of harmonic function on graphs

J. Math. Anal. Appl. 366 (2010) 673–678

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Lipschitz property of harmonic function on graphs ✩ Yong Lin a , Lifeng Xi b,∗ a b

Department of Mathematics, Information School, Renmin University of China, Beijing, 100872, PR China Institute of Mathematics, Zhejiang Wanli University, Ningbo, Zhejiang, 315100, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 23 July 2009 Available online 30 December 2009 Submitted by P. Koskela

This paper proves the local Lipschitz property for harmonic (or positive subharmonic) functions on graphs. An example is also obtained to show that the global Lipschitz property of harmonic function on graphs does not hold. © 2009 Elsevier Inc. All rights reserved.

Keywords: Lipschitz property Harmonic function Laplace operator Graph

1. Introduction Suppose G is an infinite graph with vertices set V and edges set E. For two distinct vertices x and y, we write x ∼ y and say that x and y are neighbors, if there is an edge between x and y. For x ∈ V , let d x denote the degree of x, i.e., the number of neighbors of x. A path σ , joining vertices x and y, is a sequence x0 = x, x1 , . . . , xn = y of vertices such that xi ∼ xi +1 for any i, where l(σ ) = n is the length of σ . The graph G is said to be connected, if every pair of vertices can be joined by a path. In this paper we always assume that G is connected and has uniformly bounded vertices degree d = supx∈ V dx < ∞. In this case, we introduce a distance ρ : V × V → {0} ∪ N by setting ρ (x, x) = 0 and for x = y, ρ (x, y ) = inf l(σ ), where infimum is taken over all paths σ joining x and y. For vertex q ∈ V and real number R ∈ N, we denote B R (q) = {x ∈ G: ρ (x, q)  R }. Let V R = {u | u : V → R}. The Laplace operator  of graph G is defined by

u (x) =

1  d x y ∼x



u ( y ) − u (x)

for any u ∈ V R .

We say the  function is harmonic (resp. subharmonic) if u = 0 (u  0). The square of absolute value of gradient of u is ∇ u 2 (x) = y∼x (u ( y ) − u (x))2 . Please refer to [2,3,8] for further details. The aim of this paper is to study the Lipschitz property of harmonic functions on graphs. In [5], Koskela, Rajala and Shanmugalingam establish the local Lipschitz property of Cheeger-harmonic functions in certain metric spaces that endowed with a doubling measure supporting a (1, 2)-Poincaré inequality and in addition supporting a corresponding Sobolev–Poincaré type inequality for the modification of the measure obtained via heat kernel. For Alexandrov spaces, Petrunin [6] also proved that the harmonic functions are locally Lipschitz. For Riemannian manifolds whose Ricci curvature is bounded from below, the local Lipschitz property follows from the Cheng–Yau gradient estimate [1]. ✩ This work is partially supported by National Natural Science Foundation of China (Grant Nos. 10671180 and 10871202), Natural Science Foundation of Ningbo (Grant No. 2009A610077) and Morningside Center of Mathematics in Beijing. Corresponding author. E-mail addresses: [email protected] (Y. Lin), [email protected] (L. Xi).

*

0022-247X/$ – see front matter doi:10.1016/j.jmaa.2009.12.037

©

2009 Elsevier Inc. All rights reserved.

674

Y. Lin, L. Xi / J. Math. Anal. Appl. 366 (2010) 673–678

On graphs, Lin and Yau [4] proved that the Ricci curvature in the sense of Bakry and Emery is uniformly bounded below by −1/2. Therefore it is expected that the local Lipschitz property holds on graph. In this paper, we prove that Theorem 1. Suppose u is a harmonic or positive subharmonic function on G, then for any λ ∈ N, q ∈ V and any a, b ∈ B λ (q),

√    u (a) − u (b)  2 d 1 3 λ2

1/2



u 2 ( z)

· ρ (a, b),

(1.1)

z∈ B 6λ (q)

where d = supx∈ V dx < ∞. Remark 1. If u is harmonic or positive subharmonic with



z∈ V

u 2 ( z) < ∞, then u is constant by (1.1).

The paper is organized as follows. Theorem 1 is proved in Section 2. In Section 3, an example is obtained to illustrate that the global Lipschitz property does not holds for harmonic function on graphs. 2. Local Lipschitz property of harmonic functions In this section, we will prove Theorem 1. The idea of our proof comes from [7]. Firstly, we introduce the notion of cut-off function. Definition 1. Let q ∈ V and R , r be integers with R > r + 1. A cut-off function ϕ associated to the region C R ,r (q) = {x ∈ V : r  ρ (x, q)  R } is a function ϕ : V → [0, 1] satisfying the following requirements: (1) ϕ (x) = 1 when ρ (x, q)  r and ϕ (x) = 0 when ρ (x, q)  R; c with constant c > 0. (2) For every vertex x ∈ V , ∇ ϕ (x)  R − r Let

ϕ be a cut-off function associated to C R ,r (q). Then for any g ∈ V R ,    



2

2 g 2 (x) + g 2 ( y ) ϕ ( y ) − ϕ (x)  2 g 2 (x) ϕ ( y ) − ϕ (x) [7], B R (q) y ∼x



dx ϕ 2 (x) g 2 (x) = −

B R (q)

 

1 2

(2.1)

y ∼x

B R (q)





ϕ 2 ( y ) − ϕ 2 (x) g 2 ( y ) − g 2 (x) .

(2.2)

x∈ B R (q) y ∼x

ϕ ( y ) = 0 for any y with ρ (q, y )  R, therefore   ϕ 2 ( y ) − ϕ 2 (x) g 2 ( y ) − g 2 (x)

To obtain (2.2), we notice that

−

1   2

x∈ B R (q) y ∼x

1

=

2







g 2 ( y ) − g 2 (x) −

y ∼x

x∈ B R (q)



=

ϕ 2 (x) 2

ϕ (x)



2

ϕ 2( y)

y ∈ B R (q)





g 2 ( y ) − g 2 (x)

x∼ y

  g 2 ( y ) − g 2 (x) = dx ϕ 2 (x) g 2 (x).

y ∼x

x∈ B R (q)



1

B R (q)

Suppose u (x) is a harmonic or positive subharmonic function on G, then

∇ u 2 (x)  dx · u 2 (x).

(2.3)

To obtain (2.3), we notice that if u (x) is harmonic or positive subharmonic, then



u (x)



u ( y )  dx · u 2 (x),

y ∼x

therefore, we have

∇ u 2 (x) =



2

u ( y ) − u (x)

y ∼x

=



u 2 ( y ) − 2u (x)

y ∼x



 y ∼x

 y ∼x

2

2

u( y) +



u 2 (x)

y ∼x

u ( y ) − 2dx · u (x) + dx · u 2 (x) =

 y ∼x



u 2 ( y ) − u 2 (x) = dx · u 2 (x).

Y. Lin, L. Xi / J. Math. Anal. Appl. 366 (2010) 673–678

From (2.2) and (2.3), we have



ϕ 2 (x)



2

u ( y ) − u (x)



y ∼x

B R (q)



675

dx ϕ 2 (x)u 2 (x)

B R (q)

=−

1   2





ϕ 2 ( y ) − ϕ 2 (x) u 2 ( y ) − u 2 (x) .

(2.4)

B R (q) y ∼x

We square (2.4) and apply Cauchy–Schwartz inequality to get the following:



ϕ 2 (x)



=  

    1 4



x∈C R ,r (q) y ∼x

4



     2 ϕ ( y ) + ϕ (x) u ( y ) − u (x) ϕ ( y ) − ϕ (x) u ( y ) + u (x)

x∈C R ,r (q) y ∼x

    1 4

2

ϕ 2 ( y ) − ϕ 2 (x) u 2 ( y ) − u 2 (x)

    1

2 

ϕ ( y ) + ϕ (x)

2

u ( y ) − u (x)

1/2    2  2 1/2 2 · ϕ ( y ) − ϕ (x) u ( y ) + u (x)

y ∼x

x∈C R ,r (q)

    x∈C R ,r (q) y ∼x



2

y ∼x

B R (q)



2

u ( y ) − u (x)

   

y ∼x

    2  2  2 2 2 2 ϕ ( y ) + ϕ (x) u ( y ) − u (x) u ( y ) + u (x) ϕ ( y ) − ϕ (x) · x∈C R ,r (q) y ∼x

    2  2  2 2 2 2 · . ϕ ( y ) + ϕ (x) u ( y ) − u (x) u ( y ) + u (x) ϕ ( y ) − ϕ (x)

x∈C R ,r (q) y ∼x

x∈ B R (q) y ∼x

It follows from (2.1) and (2) of Definition 1 that



ϕ 2 (x)



2

y ∼x

B R (q)





 2

u 2 (x)

x∈ B R (q)



2

u ( y ) − u (x)

2c

 

2

( R − r )2



2

ϕ ( y ) − ϕ (x)

    2  2 ϕ ( y ) + ϕ 2 (x) u ( y ) − u (x) ·

y ∼x

x∈C R ,r (q) y ∼x

    2  2 u 2 (x) · ϕ ( y ) + ϕ 2 (x) u ( y ) − u (x) , x∈C R ,r (q) y ∼x

x∈ B R (q)

where

 

 2 ϕ ( y ) + ϕ (x) u ( y ) − u (x) = 2

2

x∈C R ,r (q) y ∼x

  x∈C R ,r (q)

 y ∼x, y ∈ B R (q)

In fact, for x ∈ B R (q) and y ∈ / B R (q) with y ∼ x, we have

ρ (x, q) = R , 

ρ ( y , q) = R + 1 and ϕ (x) = ϕ ( y ) = 0,

ϕ 2 ( y ) + ϕ 2 (x))(u ( y ) − u (x))2 = 0, i.e.,    2 ϕ 2 ( y ) + ϕ 2 (x) u ( y ) − u (x)

which implies

y ∼x, y ∈ / B R (q) (

x∈C R ,r (q) y ∼x

 

=

x∈C R ,r (q)

 

(x, y )∈ B R (q)× B R (q) x∼ y

2



x∈ B R (q)

 2 ϕ ( y ) + ϕ (x) u ( y ) − u (x) 2

2

y ∼x, y ∈ B R (q)







ϕ 2 (x)



2

ϕ 2 ( y ) + ϕ 2 (x) u ( y ) − u (x) 



y ∼x, y ∈ B R (q)

2

u ( y ) − u (x)



+

 y ∼x, y ∈ / B R (q)

 .

676

Y. Lin, L. Xi / J. Math. Anal. Appl. 366 (2010) 673–678



2

ϕ 2 (x)



2

u ( y ) − u (x) .

y ∼x

x∈ B R (q)

Therefore,

 

ϕ 2 (x)

2

 

 

4c 2

( R − r )2

2

u ( y ) − u (x)

y ∼x

x∈ B R (q)





u 2 (x) ·

x∈ B R (q)

ϕ 2 (x)



2

u ( y ) − u (x)

.

y ∼x

x∈ B R (q)

We have



ϕ 2 (x)



2

u ( y ) − u (x)

( R − r )2

y ∼x

x∈ B R (q)

Let R = 2r. Since



4c 2



u 2 (x).

(2.5)

x∈ B R (q)

ϕ (x) = 1 when ρ (x, y )  r, (2.5) implies

 

2

u ( y ) − u (x)



x∈ B r (q) y ∼x



4c 2 r2

u 2 ( z).

(2.6)

z∈ B 2r (q)

It means for any w ∈ B r (q),

∇ u 2 ( w ) 



4c 2 r2

u 2 ( z).

(2.7)

z∈ B 2r (q)

Let R = 2r = 6λ. Suppose vertices a, b ∈ B λ (q), then ρ (a, b)  2λ. As a result, there are vertices w 0 = a, w 1 , . . . , w ρ (x, y ) = b in B 3λ (q) = B r (q) such that w i ∼ w i +1 for all i. It follows from the gradient estimate (2.7) that

  u (a) − u (b)  2c



1

(3λ)2



1/2 u ( z) · ρ (a, b), 2

(2.8)

z∈ B 6λ (q)

for any a, b ∈ B λ (q). Here, we let

ϕ (x) =

⎧ ⎨ 1, ⎩

R −ρ (x,q) , R −r

0,

Then

if ρ (x, q)  r ; if r  ρ (x, q)  R ; if ρ (x, q)  R .

√ ∇ ϕ (x) 

d

R −r

(2.9)

,

where d = supx∈ V dx < ∞. That means c =

√    u (a) − u (b)  2 d 1 3 λ2



√

d in this case, which implies

1/2

u 2 ( z)

· ρ (a, b),

(2.10)

z∈ B 6λ (q)

for any a, b ∈ B λ (q). This completes the proof of Theorem 1. 3. Non-global Lipschitz property of harmonic function This section is devoted to construct a harmonic function which is not Lipschitz. In the graph (see Fig. 1), there are vertices x, y, {xi 1 i 2 ···in : n  1 and i 1 i 2 · · · in ∈ {1, 2}n } and { y i 1 i 2 ···in : n  1 and i 1 i 2 · · · in ∈ {1, 2}n }, where each vertex has three neighbors. The vertex x has 3 neighbors y, x1 and x2 . By induction, for n  1, xi 1 ···in−1 in has 3 neighbors xi 1 ···in−1 and xi 1 ···in 1 and xi 1 ···in 2 , and y i 1 ···in−1 in has 3 neighbors y i 1 ···in−1 and y i 1 ···in 1 and y i 1 ···in 2 . Let a = (3 +

√

5 )/2 > 1, then

(a + 1/a)/3 = 1. On this tree, we can define a harmonic function f which is not global Lipschitz (see Fig. 2).

(3.1)

Y. Lin, L. Xi / J. Math. Anal. Appl. 366 (2010) 673–678

677

Fig. 1. A tree of degree 3.

Fig. 2. Harmonic function on the tree.

Let f (x) = 1, f (x1 ) = a, f (x2 ) = 0, f ( y ) = a−1 , f ( y 1 ) = f ( y 2 ) = (3a−1 − 1)/2. For any n  1, we write [1]n = 1 · · 1 and  · n

let

f (x[1]n ) = an

and

f (x[1]n 2 ) = 0.

(3.2)

On the other vertices of tree, we obtain values of f by induction, for example, assume that f (xi 1 ···ik ) and f (xi 1 ···ik−1 ) have been defined, we let

f (xi 1 ···ik 1 ) = f (xi 1 ···ik 2 ) =

3 f (xi 1 ···ik ) − f (xi 1 ···ik−1 ) 2

;

(3.3)

similarly, assume that f ( y j 1 ··· jk ), f ( y j 1 ··· jk−1 ) have been defined, we let

f ( y j 1 ··· jk 1 ) = f ( y j 1 ··· jk 2 ) =

3 f ( y j 1 ··· jk ) − f ( y j 1 ··· jk−1 ) 2

.

(3.4)

Here dx = 3 for any vertex x. By (3.1)–(3.4), we have

3 f (x) =



f ( y ),

(3.5)

y ∼x

which implies that f is harmonic on the graph. We notice that

   f (x[1]n ) − f (x n−1 ) = (a − 1)an−1 → ∞. [1 ]

(3.6)

That means f is not a global Lipschitz function. Acknowledgment The authors thank Xiao Zhong for the discussion of the paper.

References [1] [2] [3] [4] [5]

S.Y. Cheng, S.T. Yau, Differential equations and Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (3) (1975) 333–354. F. Chung, Spectral Graph Theory, CBAS Reg. Conf. Ser., vol. 72, Amer. Math. Soc., 1997. I. Holopainen, P.M. Soardi, A strong Liouville theorem for p-harmonic functions on graphs, Ann. Acad. Sci. Fenn. Math. 22 (1997) 205–226. Y. Lin, S.T. Yau, Ricci curvature and eigenvalue estimate on locally finite graphs, preprint. P. Koskela, K. Rajala, N. Shanmugalingam, Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces, J. Funct. Anal. 202 (2003) 147–173.

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[6] A. Petrunin, Harmonic functions on Alexandrov spaces and their applications, Electron. Res. Announc. Amer. Math. Soc. 9 (2003) 135–141. [7] M. Rigoli, M. Salvatori, M. Vignati, Subharmonic functions on graphs, Israel J. Math. 99 (1997) 1–27. [8] P.M. Soardi, Potential Theory on Infinite Networks, Lecture Notes in Math., vol. 1590, Springer, Berlin, 1994.