Gradient Estimates and Entropy Formulae for Weighted p-Heat Equations on Smooth Metric Measure Spaces

Acta Mathematica Scientia 2013,33B(4):963–974 http://actams.wipm.ac.cn

GRADIENT ESTIMATES AND ENTROPY FORMULAE FOR WEIGHTED p-HEAT EQUATIONS ON SMOOTH METRIC MEASURE SPACES∗

)1

Yuzhao WANG (


)2

Jie YANG (

Wenyi CHEN (

)1

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430071, China 2. College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China E-mail: [email protected]; [email protected]; [email protected]

Abstract Let (M, g, e−f dv) be a smooth metric measure space. In this paper, we consider two nonlinear weighted p-heat equations. Firstly, we derive a Li-Yau type gradient estimates for the positive solutions to the following nonlinear weighted p-heat equation ∂u = ef div(e−f |∇u|p−2 ∇u) ∂t on M × [0, ∞), where 1 < p < ∞ and f is a smooth function on M under the assumption ´ that the m-dimensional nonnegative Bakry-Emery Ricci curvature. Secondly, we show an ´ entropy monotonicity formula with nonnegative m-dimensional Bakry-Emery Ricci curvature which is a generalization to the results of Kotschwar and Ni [9], Li [7]. Key words gradient estimates; weighted p-heat equation; entropy monotonicity formula; ´ m-Bakry-Emery Ricci curvature 2010 MR Subject Classification

1

58J35; 35K55

Introduction

A smooth metric measure space is a triple (M n , g, e−f dv), where M is a complete Riemannian manifold of dimension n, f a smooth real-valued function on M , and dv the Riemannian volume form. Smooth metric measure spaces carry a natural analogs of the Ricci tensor, that is m´ dimensional Bakry-Emery Ricci curvature tensor (m ≥ n) Ricm f + Ric + ∇∇f −

df ⊗ df . m−n

´ When m = ∞, Ric∞ f + Ricf = Ric + ∇∇f is the original Bakry-Emery Ricci curvature tensor, which was studied firstly in diffusion processes[1], then extensively investigated in the theory of the Ricci flow [3] as Ricf = λg is precisely the gradient Ricci soliton equation. ∗ Received

June 11, 2012; revised October 26, 2012.

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Smooth metric measure spaces also carry a natural analog of the Laplace-Beltrami operator, f -Laplacian (also called the drifting Laplacian or Witten Laplacian), is defined by ∆f = ∆ − ∇f · ∇, which is self-adjoint operator in the sense of dµ + e−f dv, namely it satisfies the following integration by parts Z Z Z h∇u, ∇vidµ = − ∆f uvdµ = − ∆f vudµ ∀ u, v ∈ C0∞ (M ). M

M

M

In a pioneer paper [5], Li and Yau gave a sharp gradient estimate ut |∇u|2 n − ≥− , u u2 2t

(1.1)

for positive solutions to the heat equation on Riemannian manifolds via the maximum principle. Later on, the similar technique was employed by Hamilton and Perelman in the study of Ricci flow as well as other geometric flow, see a survey [12] and the references therein. There are many works concerning in the generalization of Li-Yau’s estimate to the case of f -Laplacian, ∂t u = ∆f u. See [2, 4, 6, 10, 11]. One of our goals here is to consider the gradient estimate of the weighted p-heat equations ∂u = ∆p,f u + ef div(e−f |∇u|p−2 ∇u) ∂t

(1.2)

for p > 1 and f ∈ C ∞ (M ). This weighted p-heat equation is a natural extension of heat equation from a variational point of view and was studied by Kotschwar and Ni [9] for the case f = constant. They obtained some interesting results about gradient estimates and entropy formulae. Combining the usual maximum principle approach with some geometric techniques including the use of a nonlinear Bochner formula and curvature-dimension inequality [6, 15], we obtain the following sharp estimate. Theorem 1.1 Assume (M n , g, e−f dv) is a compact smooth metric measure space with ´ nonnegative m-Barky-Emery Ricci curvature and u is a smooth nonnegative solution to (1.2) with p > 2n/(n + 1) in the region |∇u| > 0. Then for all t ∈ (0, T ), we have ut |∇u|p mβ − ≥− , 2 u u t

(1.3)

where β = 1/(p + m(p − 2)). Another goal of this paper is to consider the parabolic equation associated to the operator Λp,f , i.e., if v is a solution to the equation ∂v = Λp,f v + ef div(e−f |∇v|p−2 ∇v) − |∇v|p . ∂t

(1.4)

v Let u = exp(− p−1 ), then the corresponding equation for u is

∂up−1 = (p − 1)p−1 ef div(e−f |∇u|p−2 ∇u). ∂t We have a similar sharp gradient estimate.

(1.5)

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Theorem 1.2 Assume (M n , g, e−f dv) is a compact smooth metric measure space with ´ nonnegative m-Barky-Emery Ricci curvature and u is a smooth nonnegative solution to (4.1). Then for any p > 1, |∇v|p + vt = (p − 1)p−1

|∇u|p ut m − (p − 1) ≤ . up u pt

(1.6)

In the last part of this paper, we are concerned with the monotonicity formula for entropy, which paly an important role in Ricci flow [14], then Ni [13] showed that a monotonicity entropy formula for heat equation in static manifold with nonnegative Ricci curvature, and he and his cooperator extend this result to the case of nonlinear p-heat equation. Recently, Li [7] proved an analogue of Perelman’s and Ni’s monotonicity formula for the W-entropy of the heat kernel of the f -Laplacian on complete Riemannian manifolds with non-negative m-dimensional Bakry´ Emery Ricci curvature. For the case of Fokker-Planck equation see [8]. Inspired by their works, we obtain an entropy monotonicity formula for weighted p-heat equation on smooth metric measure space. More precisely, we have Theorem 1.3 Let (M, g, e−f dv) be a compact smooth metric measure space. For any R p > 1, let u be a positive solution to (4.1) satisfying M up−1 dµ = 1. Then ! 2 Z p −1 ∂ 1 m p−2 2 ∇i ∇j v − aij + w Wp,f (u, t) = −pt Ricf (∇v, ∇v) up−1 dµ w ∂t pt M A 2 Z  m−n pt wp/2−1 ∇f · ∇v − up−1 dµ. (1.7) − m−n M pt u u

p−2 i j 2 ik jl Here w = |∇v|2 , aij = gij − p−1 |∇u|2 , and can be viewed as a metric tensor, |T |A = A A Tij Tkl ij for any 2-tensor T where (A ) is the inverse of (aij ). The entropy Z Wp,f (ϕ, t) = (t|∇ϕ|p + ϕ − m)up−1 dµ (1.8) M

Γ(m/2+1) e−ϕ p 1 ∗ m m , where p = m ∗ p−1 assumed π 2 (p∗p−1 p) p Γ(m/p +1) t p ∂ Ricm f ≥ 0, then for all t > 0, we have ∂t Wp,f (u, t) ≤ 0.

is defined with up−1 =

to be finite.

In particular, if Remark 1.4 When p = 2, (1.7) is precisely the entropy formula in [7]. The structure of this paper is as follows. In Section 2, we will derive a nonlinear weighted Bochner type formula needed in the later proof. Section 3 is devoted to the proof of Theorem 1.1. In Section 4, we will prove Theorem 1.2 and then deduce a nonlinear entropy formula in Theorem 1.3.

2

A Nonlinear Weighted Bochner Type Formula Let u be a weighted p-harmonic function, that is u satisfies the equation ∆p,f u = ef div(e−f |∇u|p−2 ∇u) = 0.

(2.1)

Let v = −(p − 1) log u. It is easy to see that v satisfies Λp,f (v) = ef div(e−f |∇v|p−2 ∇v) − |∇v|p = 0.

(2.2)

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If

|∇u|2 , u2 then, expressed in terms of w, (2.2) has the equivalent form p Λp,f (v) = ( − 1)wp/2−2 h∇w, ∇vi + wp/2−1 ∆f v − wp/2 , 2 where ∆f = ∆ − ∇f · ∇ is the f -Laplacian. Assume that w > 0 over some region of M , defined w = |∇v|2 = (p − 1)2

Lf (ψ) + ef div(e−f wp/2−1 A(∇ψ)) − pwp/2−1 h∇v, ∇ψi, where

(2.3)

(2.4)

∇v ⊗ ∇v , w which can be checked easily to be nonnegative definite in general and positive definite for p > 1. Note that the operator Lf is the linearized operator for Λp,f . The first result is a computational lemma, which is a nonlinear Bochner type formula relating the nonlinear operator with its linearization. Lemma 2.1 Let w = |∇v|2 , operator Lf is defined in (2.4), then   p  Lf (w) = 2wp/2−1 |∇∇v|2 + Ricf (∇v, ∇v) + 2h∇Λp,f (v), ∇vi + − 1 wp/2−2 |∇w|2 . (2.5) 2 ´ Here ∇∇v is the Hessian of v and Ricf is the Barky-Emery Ricci tensor of M . A = id + (p − 2)

Proof Direct calculation shows that    ∇v ⊗ ∇v  Lf (w) = ef div e−f wp/2−1 id + (p − 2) (∇w) − pwp/2−1 h∇v, ∇wi w   = ∆p,f w + (p − 2)ef div e−f wp/2−2 h∇v, ∇wi∇v − pwp/2−1 h∇v, ∇wi p  = wp/2−1 ∆f w + − 1 wp/2−2 |∇w|2 + (p − 2)wp/2−2 h∇v, ∇wi∆f v 2  p/2−2 +(p − 2)w h∇∇v, ∇wi + h∇v, ∇∇wi · ∇v p  +(p − 2) − 2 wp/2−3 h∇v, ∇wi2 − pwp/2−1 h∇v, ∇wi   2  p p/2−1 =w − 1 wp/2−2 |∇w|2 2|∇∇v|2 + 2h∇∆f v, ∇vi + 2Ricf (∇v, ∇v) + 2 p  p/2−2 p/2−3 +(p − 2)w h∇v, ∇wi∆f v + (p − 2) −2 w h∇v, ∇wi2 2   +(p − 2)wp/2−2 h∇∇v, ∇wi + h∇v, ∇∇wi · ∇v − pwp/2−1 h∇v, ∇wi, where we use the generalized Bochner formula ∆f w = 2|∇∇v|2 + 2h∇∆f v, ∇vi + 2Ricf (∇v, ∇v). Taking the gradient of both sides of (2.3) and computing its product with ∇v, we have p p/2−1 w h∇w, ∇vi + h∇Λp,f (v), ∇vi 2  p   = ∇v · ∇ − 1 wp/2−2 h∇w, ∇vi + wp/2−1 ∆f v  p2 p  p = − 1 ( − 2)wp/2−3 h∇w, ∇vi2 + − 1 wp/2−2 h∇w, ∇vi∆f v 2 2 2   p p/2−2 + −1 w h∇∇v, ∇wi + h∇v, ∇∇wi · ∇v + wp/2−1 h∇∆f v, ∇vi. 2

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Combining the above two calculations, we prove the claimed identity (2.5). Now, we consider the linearization of the operator ∆p,f at u, given by Lf (ψ) + ef div(e−f hp/2−1 A(∇ψ)).

967 2

(2.6)

Here h = |∇u|2 and A is the tensor, namely, Aij = gij + (p − 2)

ui uj , h

then, expressed in terms of h, (2.1) has the equivalent form p  ∆p,f (u) = − 1 hp/2−2 h∇h, ∇ui + hp/2−1 ∆f u. 2

(2.7)

By using of a calculation similar to Lemma 2.1, we obtain   p  Lf (h) = 2hp/2−1 |∇∇u|2 + Ricf (∇u, ∇u) + 2h∇∆p,f (u), ∇ui + − 1 hp/2−2 |∇h|2 . (2.8) 2

3

Gradient Estimate for Weighted p-Heat Equation In this section, we consider the weighted p-heat equation (1.2), ∂u = ∆p,f u = ef div(e−f |∇u|p−2 ∇u), ∂t

which is the gradient flow for the weighted p-energy functional Z Ep,f (u) + |∇u|p e−f dv.

(3.1)

M

The following lemma is a straightforward calculation and very similar to (2.5) and (2.8). Lemma 3.1 Let h = |∇u|2 . Then   ∂ Lf − u = (p − 2)ut (3.2) ∂t and

     p ∂ Lf − h = 2hp/2−1 |∇∇u|2 + Ricf (∇u, ∇u) + − 1 hp/2−2 |∇h|2 . ∂t 2

(3.3)

The following calculation is essential for the main result in this section. Lemma 3.2 Suppose u : M × [0, T ) → R is a smooth, positive solution to (1.2) with p > 1. For any α > 0, defined |∇u|p ut Ffα + −α . u2 u Then, on the region |∇u| > 0, the following estimate holds ! 2   ∂ 1 ∇u ⊗ ∇u Ricf (∇u, ∇u) α p−2 ∇∇u Lf − Ff = ph u − p−1 + ∂t u2 u2 A +(p − 2)(Ff1 )2 + (α − 1)(p − 2)

u2t − 2(p − 1)hp/2−1 h∇Ffα , ∇ log ui. (3.4) u2

where |T |2A = Aik Ajl Tij Tkl for any 2-tensor T , (Aij ) is the inverse of (Aij ), namely, Aij = g ij + (p − 2)

ui uj . h

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In particular, if Ricm f ≥ −Kg for some K ≥ 0, we have     ∂ u2 p + m(p − 2) α Lf − Ff ≥ (Ff1 )2 + (α − 1)(p − 2) t2 ∂t m u hp−1 −pK 2 − 2(p − 1)hp/2−1 h∇Ffα , ∇ log ui. u Proof By the definition of Lf , we have that   ∂ Lf − ut = 0, ∂t and Lemma 3.1 implies that     ∂ Lf − hp/2 = php−2 |∇∇u|2A + Ricf (∇u, ∇u) . ∂t

Vol.33 Ser.B

(3.5)

(3.6)

(3.7)

In fact,   ∂ p Lf − hp/2 = ef div(e−f hp/2−1 A(∇hp/2 )) − hp/2−1 ht ∂t 2 p   p  = ef div e−f hp/2−1 hp/2−1 A(∇h) − hp/2−1 ht 2  2  p p/2−1 p p p/2−2 = h (Lf − ∂t )h + −1 h ∇h · (hp/2−1 A(∇h)) 2 2 2      p p − 1 hp/2−2 |∇h|2 = hp/2−1 2hp/2−1 |∇∇u|2 + Ricf (∇u, ∇u) + 2 2    pp 1 + − 1 hp−3 |∇h|2 + (p − 2) h∇u, ∇hi2 2 2 h  (p − 2)2 h∇u, ∇hi2  p − 2 |∇h|2 + = php−2 Ricf (∇u, ∇u) + php−2 |∇∇u|2 + 2 h 4 h2   = php−2 |∇∇u|2A + Ricf (∇u, ∇u) , where

|∇∇u|2A = Aik Ajl uij ukl p − 2 ik j l (p − 2)2 i j (u u uij )(uk ul ukl ) + (g u u uij ukl + g jl ui uk uij ukl ) 2 h h (p − 2)2 p−2 = |∇∇u|2 + h∇u, ∇hi2 + |∇h|2 . 4h2 2h Using the general formula Q L Q Q 2β(p − 1) p/2−1 f Lf β = β − β β+1 Lf (u) − h h∇Q, ∇ui u u u uβ+1 Q +β(β + 1)(p − 1)hp/2 β+2 , (3.8) u together with (3.2) (3.6) and (3.7), we then obtain       ∂ hp/2 1 ∂ hp/2 ∂ p/2 Lf − = 2 Lf − h − 2 3 Lf − u ∂t u2 u ∂t u ∂t = g ik g jl uij ukl +

p/2 4(p − 1) p/2−1 p/2 p/2 h h h∇h , ∇ui + 6(p − 1)h u3 u4  p−2  h hp/2 = p 2 |∇∇u|2A + Ricf (∇u, ∇u) − 2(p − 2) 3 ut u u hp−2 hp −2p(p − 1) 3 h∇h, ∇ui + 6(p − 1) 4 u u

−

(3.9)

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and   ∂ ut u2 hp/2−1 hp/2 Lf − = −(p − 2) 2t − 2(p − 1) h∇ut , ∇ui + 2(p − 1) 3 ut . 2 ∂t u u u u

(3.10)

Note that h∇Ffα , ∇ui =

php/2−1 2hp/2+1 α ut h∇h, ∇ui − − h∇ut , ∇ui + αh 2 . 2 3 2u u u u

So, after multiplying both sides by α, (3.10) becomes   ∂  ut  u2 hp−1 Lf − α = −α(p − 2) t2 − p(p − 1) 3 h∇h, ∇ui ∂t u u u +4(p − 1)

hp hp/2−1 + 2(p − 1) h∇Ffα , ∇ui. 4 u u

Combining this with (3.9), we have   
∂ hp−2  hp/2 Lf − Ffα = p 2 |∇∇u|2A + Ricf (∇u, ∇u) − 2(p − 2) 3 ut ∂t u u hp−2 hp −p(p − 1) 3 h∇h, ∇ui + 2(p − 1) 4 u u u2t hp/2−1 +α(p − 2) 2 − 2(p − 1) h∇Ffα , ∇ui. u u

(3.11)

(3.12)

Equation (3.4) then follows from (3.12) and the identity 2 hp−2 1 ∇u ⊗ ∇u hp hp−2 2 p−2 ∇∇u = p − |∇∇u| h∇h, ∇ui + p . ph − p(p − 1) A u p−1 u2 u2 u3 u4 A For (3.5), we observe from (1.2) and (2.7) that

 hp/2−2 hp/2  p hp/2−1 hp/2 ut = − 1 h∇h, ∇ui − ∆f u − − u2 u u2 2 u u   1 ∇u ⊗ ∇u ∇∇u hp/2−1 = hp/2−1 trA − + ∇f · ∇u. 2 p−1 u u u

Ff1 =

´ Applying the properties of m-Bakry-Emery Ricci tensor Ricm f ≥ −Kg and the standard in2 2 equality n|T |A ≥ (trA T ) for a two-tensor T , we have ! 2 1 ∇u ⊗ ∇u Ricf (∇u, ∇u) p−2 ∇∇u ph u − p−1 + u2 u2 A   2 php−2 p  1 hp/2−1 |∇u · ∇f |2 m ≥ F − ∇f · ∇u + Ricf (∇u, ∇u) + n f u u2 m−n hp−1 p 1 2 ≥ (Ff ) − pK 2 , m u where we use the inequality (a − b)2 ≥

a2 b2 − , 1+δ δ

δ=

m−n . n

With the help of above lemma, we can prove Theorem 1.1.

2

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p−1 . The direct computation Proof For α = 1, define the pressure quantity φ + p−1 p−2 u implies   p−2 1 ∇u ⊗ ∇u ∇∇u p−1 ∇∇φ = u − u p−1 u2

and

hp/2−1 ∇f · ∇u. u ≥ 0, inequality (3.5) has the equivalent form

∆p,f φ = −Ff1 = |∇φ|p−2 trA (∇∇φ) − So that in the case α = 1 and Ricm f ∂

 1 − L − 2(p − 1)|∇φ|p−2 ∇φ · ∇ ∆p,f φ ≥ (∆p,f φ)2 . ∂t mβ

Thus by the maximum principle, we get the global estimate ∆p,f φ =

ut |∇u|p mβ − ≥− , u u2 t

where β = 1/(p + m(p − 2)), p > 2m/(m + 1). 2 Remark 3.3 When p = 2, m = n and f = constant, estimate (1.3) is exactly the classic Li-Yau estimate (1.1). Moreover, this estimate (1.3) indeed is sharp in view of the explicit source-type solutions (see [9]) Hp,m (x, t) +

1 tmβ

β p−1 (2 − p)  |x|  p−1 1+ p tβ 1

p

p−1 ! p−2

(3.13)

+

to equation (1.2) on Rm for which (1.3) is an equality. Remark 3.4 Due to the degeneration of equation (1.2) in the region |∇u| = 0, so we cannot directly obtain above conclusion. While we may get an estimate of above form by a limiting procedure following the argument in [9], which was obtained for p-heat equation in the usual complete Riemannian manifold.

4

A Nonlinear Entropy Formula for Weighted p-Heat Equation In this section, we consider another weighted p-heat equation (4.1), ∂up−1 = (p − 1)p−1 ef div(e−f |∇u|p−2 ∇u). ∂t

(4.1)

Recall the operator Lf defined as Lf (ψ) = ef div(e−f wp/2−1 A(∇ψ)) − pwp/2−1 h∇v, ∇ψi.

(4.2)

For any v, letting w = |∇v|2 , the proof of Lemma 2.1 yields the nonlinear Bochner formula   p  Lf (w) = 2wp/2−1 |∇∇v|2 + Ricf (∇v, ∇v) + 2h∇v, ∇(Λp,f v)i + − 1 wp/2−2 |∇w|2 . (4.3) 2 A bit of computation together with this formula yields the following lemma. Lemma 4.1 Let w = |∇v|2 , then   ∂ Lf − vt = 0, ∂t

(4.4)

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    p  ∂ Lf − w = 2wp/2−1 |∇∇v|2 + Ricf (∇v, ∇v) + − 1 wp/2−2 |∇w|2 . ∂t 2

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(4.5)

Lemma 4.2 For any α > 0, defined Ffα + |∇v|p + αvt = wp/2 + αvt . Then     ∂ Ffα = pwp−2 |∇∇v|2A + Ricf (∇v, ∇v) . Lf − ∂t

(4.6)

Proof By using of formulae (4.4) and (4.5) in above lemma, computing like in (3.7), we have

      ∂ ∂ α Lf − Ff = Lf − wp/2 = pwp−2 |∇∇v|2A + Ricf (∇v, ∇v) . ∂t ∂t

2

Proof of Theorem 1.2 Notice that (1.4) implies that   p Ff1 = wp/2 + vt = wp/2 + wp/2−1 ∆f v + ( − 1)wp/2−2 h∇w, ∇vi − wp/2 2 p = wp/2−1 ∆f v + ( − 1)wp/2−2 h∇w, ∇vi 2 p/2−1 =w trA (∇∇v) − wp/2−1 ∇f · ∇v. Hence, combining Cauchy-Schwartz inequality wp−2 |∇∇v|2A ≥

1 p/2−1 (w trA (∇∇v))2 n

with Ricm f (∇v, ∇v) ≥ −Kw, we get   pwp−2 |∇∇v|2A + Ricf (∇v, ∇v)

  |∇f · ∇v|2 p 1 (Ff + wp/2−1 ∇f · ∇v)2 + pwp−2 Ricm (∇v, ∇v) + f n m−n p 1 2 p−1 ≥ (Ff ) − pKw . m

≥

´ In the case that M is compact with nonnegative m-dimension Barky-Emery Ricci curvature, we obtain at once the desired global estimate |∇v|p + vt = (p − 1)p−1

ut m |∇u|p − (p − 1) ≤ . up u pt

2

Ramark 4.3 We can check that for p > 1, the following function u0,m +

π

−m/2

(p

∗p−1

−m/p

pt)

1 ! p−1 ∗  Γ(m/2 + 1) |x − x0 |p  exp − 1 Γ(m/p∗ + 1) (tp∗p−1 p) p−1

(4.7)

is a fundamental solution of (4.1) on Rm , and the equality holds for this function in (1.6). So p the estimate is sharp. Here p∗ = p−1 . Motivated by the Ni’s entropy formula for the heat equation [13], we have the following conservation law. Proposition 4.4 For any smooth ψ, Z Z    d ∂ ψup−1 dµ = − Lf ψ up−1 dµ, (4.8) dt M ∂t M

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where dµ = e−f dv, f ∈ C ∞ (M ). In particular, if ψ satisfies ∂  − Lf ψ = 0, ∂t R p−1 then M ψu dµ is a constant along equations (4.1) and (4.9). u Proof Note that ∇u = − p−1 ∇v, then

(4.9)

(p − 1)p−1 |∇u|p−2 ∇u = −up−1 |∇v|p−2 ∇v. Integrating by parts, we obtain Z Z
d p−1 ψu dµ = ψt up−1 + ψ(up−1 )t dµ dt M ZM  Z   ∂ = − Lf ψ up−1 dµ + Lf ψ up−1 dµ ∂t M M Z f −f p−1 p−2 − e div(e u |∇v| ∇v)dµ M Z Z    ∂ − Lf ψ up−1 dµ − |∇v|p−2 hA(∇ψ), ∇up−1 idµ = ∂t M M Z −(p − 1) |∇v|p−2 h∇ψ, ∇viup−1 dµ. M

The result follows from the observations that ∇up−1 = −up−1 ∇v and Z − |∇v|p−2 hA(∇ψ), ∇up−1 idµ ZM Z =− |∇v|p−2 h∇ψ, ∇up−1 idµ − (p − 2) |∇v|p−4 h∇ψ, ∇vih∇v, ∇up−1 idµ M M Z = (p − 1) |∇v|p−2 h∇ψ, ∇viup−1 dµ. M

2 Now we define Nash’s entropy Np,f (u, t) + Np,f (u, t) −

m log t = p

Following [9] and [7], we also define Fp,f (u, t) + Wp,f (u, t) +

Z

up−1 vdµ −

M

d dt Np,f (u, t)

m log t. p

and

d (tNp,f (u, t)) = Np,f (u, t) + tFp,f (u, t). dt

For the proof we need the following result. Proposition 4.5 Let w = |∇v|2 , we have Z Z d 1 p−1 Np,f (u, t) = Ff u dµ = wp/2−1 (trA (∇∇v) − ∇f · ∇v) up−1 dµ, dt M M Z Z
d 1 p−1 F u dµ = −p wp−2 |∇∇v|2A + Ricf (∇v, ∇v) up−1 dµ. dt M f M Proof Direct calculation shows that ∂  − Lf v = (p − 1)|∇v|p − (p − 2)ef div(e−f |∇v|p−2 ∇v). ∂t

(4.10) (4.11)

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Hence by using of identity (4.8) and the observation Z Z Z 1 p−1 f −f p−2 p−1 Ff u dµ = e div(e |∇v| ∇v)u dµ = M

we get d Np,f (u, t) = dt

M

Z

M

|∇v|p up−1 dµ,

M


(p − 1)|∇v|p − (p − 2)ef div(e−f |∇v|p−2 ∇v) up−1 dµ =

Z

Ff1 up−1 dµ.

M

The second identity follows from (4.6) and (4.8). 2 ´ Proof of Theorem 1.3 Note that when the Bakry-Emery Ricci curvature of M is nonnegative, (4.11) yields the monotonicity of “energy” Z F (u, t) + |∇v|p up−1 dµ. M

Proposition 4.5 implies d Fp,f (u, t) = Np,f (u, t) = dt and

d Fp,f (u, t) = −p dt By observing that

  m 1 Ff − up−1 dµ pt M

Z

Z


m wp−2 |∇∇v|2A + Ricf (∇v, ∇v) up−1 dµ + 2 . pt M

d d Wp,f (u, t) = 2Fp,f (u, t) + t Fp,f (u, t) dt dt Z = −pt

ZM


wp−2 |∇∇v|2A + Ricf (∇v, ∇v) up−1 dµ + 2 p−2

|∇∇v|2A

Z

Ff1 up−1 dµ −

M

m pt


+ Ricf (∇v, ∇v) up−1 dµ

= −pt w Z M m + wp/2−1 (trA (∇∇v) − ∇f · ∇v) up−1 dµ − . pt M Note that

2 p −1 w 2 ∇i ∇j v − 1 aij = wp−2 |∇∇v|2A − 2 wp/2−1 trA (∇∇v) + n . pt A pt (pt)2

By completing the square, we can prove 2 Z p −1 d n−m 1 2 Wp,f (u, t) = − pt w ∇i ∇j v − aij up−1 dµ dt pt pt M Z ZA p/2−1 p−1 +2 w h∇f, ∇viu dµ − pt wp−2 (Ricf (∇v, ∇v)) up−1 dµ M

M

2 p −1 1 2 ∇i ∇j v − aij up−1 dµ = −pt w pt A ZM
p−1 −pt wp−2 Ricm dµ f (∇v, ∇v) u Z

M

pt − m−n

2 p/2−1 m − n p−1 w ∇f · ∇v − u dµ. pt M

Z

Thus, we obtain the desired result. It is obvious that the entropy is monotone nonincreasing if Ricm 2 f ≥ 0.

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Acknowledgements The first author would like to thank Professor Xiang-dong Li for his interest and useful discussion. This work is partially supported by the Fundamental Research Fund for the Central Universities. References ´ [1] Bakry D, Emery M. Diffusion hypercontractivitives//S´ eminaire de Probabilit´ es XIX, 1983/1984. Lecture Notes in Math, Vol 1123. Berlin: Springer-Verlag, 1985: 177–206 [2] Chen Li, Chen Wenyi. Gradient estimates for positive smooth f -harmonic functions. Acta Math Sci, 2010, 30B(5): 1614–1618 [3] Chow B, Lu P, Ni L. Hamilton’s Ricci Flow. Lectures in Contemporary Mathematics 3. Science Press and American Mathematical Society, 2006 [4] Huang Guangyue, Ma Bingqing. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch Math, 2010, 94(3): 265–275 [5] Li P, Yau S T. On the parabolic kernel of the Schr¨ odinger operator. Acta Math, 1986, 156: 153–201 [6] Li Xiangdong. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J Math Pures Appl, 2005, 84(9): 1295–1361 [7] Li Xiangdong. Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via BakryEmery Ricci curvature. Math Ann, 2012, 353(2): 403–437 [8] Li Xiangdong. Perelman’s W-entropy for the Forkker-Plank equation over complete Riemannian manifolds. Bull Sci Math, 2001, 125: 871–882 [9] Kotschwar B, Ni L. Gradient estimate for p-harmonic functions, 1/H flow and an entropy formula. Ann ´ Norm Sup´ Sci Ec er, 2009, 42(4): 1–36 [10] Munteanu O, Wang Jiaping. Smooth metric measure spaces with non-negative curvature. Comm Anal Geom, 2011, 19(3): 451–486 [11] Munteanu O, Wang Jiaping. Analysis of weighted Laplacian and applications to Ricci solitons. Comm Anal Geom, 2012, 20(1): 55–94 [12] Ni Lei. Monotonicity and Li-Yau-Hamilton inequalities//Surv Differ Geom 12: Geometric flows. International Press, 2008: 251–301 [13] Ni Lei. The entropy formula for linear heat equation. J Geom Anal, 2004, 14: 87–100 [14] Perelman G. The entropy formula for the Ricci flow and its geometric applications. http://arXiv.org/abs /maths0211159 ´ [15] Wei Guofang, Wylie Will. Comparison geometry for the Bakry-Emery Ricci tensor. J Differ Geom, 2009, 83(2): 377–405