WANG Yu-Zhao, ZHANG Hui-Ting

(School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China)

Abstract: In this paper, we study the concavity of the entropy power on Riemannian manifolds.By using the nonlinear Bochner formula and Bakry-mery method,we prove p-Rnyi entropy power is concave for positive solutions to the weighted doubly nonlinear diffusion equations on the weighted closed Riemannian manifolds with CD(−K,m) condition for some K ≥0 and m ≥n,which generalizes the cases of porous medium equation and nonnegative Ricci curvature.

Keywords: concavity; p-Rnyi entropy power; weighted doubly nonlinear diffusion equations; m-Bakry-mery Ricci curvature

1 Introduction

Let (M,g,dµ) be a weighted Riemannian manifold equipped with a reference measuredµ=e−fvol, there is a canonical differential operator associated to the triple given by the weighted Laplacian

A weighted Riemannian manifold (M,g,dµ) is said to satisfy the curvature dimensional conditionCD(−K,m) if for every functionu,

In fact, by the enhanced Bochner formula (see P.383 in Villani’s book [2])

the curvature dimensional conditionCD(−K,m) form∈(−∞,0)∪[n,∞) is equivalent tom-dimensional Bakry-mery Ricci curvatureis bounded below by −K, i.e.,

In this paper, we consider the weighted doubly nonlinear diffusion equation

where ∆p,f·efdiv(e−f|∇· |p−2∇·) is the weightedp-Laplacian, which appears in non-Newtonian fluids, turbulent flows in porous media, glaciology and other contexts.We are mainly concerned on the concavity of the Rnyi type entropy power with respecting to the equation (1.2) on the weighted Riemannian manifolds.

In the classic paper [3], Shannon defined the entropy powerN(X) for random vectorXon Rn,

whereuis the probability density ofXandH(X) is the information entropy.Moreover, he proved the entropy power inequality (EPI) for independent random vectorsXandY,

Later, Costa [4]proved EPI when one of a random vector is Gaussian and established an equivalence between EPI and the concavity of Shannon entropy powerN(u)whenusatisfies heat equation∂tu=∆u, that is,

Moreover, Villani [2]gave a short proof by using of the Bakry-mery identities.In a recent paper, Savarand Toscani [5]proved the concavity of the Rnyi entropy powerNγ(u) along the porous medium equation∂tu=∆uγon Rn, whereNγ(u) is given by

In [6], the first author and coauthors studied the the concavities ofp-Shannon entropy power for positive solutions top-heat equations on Riemannian manifolds with nonnegative Ricci curvature.

On the other hand,Li and Li[11]proved the concavity of Shannon entropy power for the heat equation∂tu=∆fuand the concavity of Rnyi entropy power for the porous medium equation∂tu=∆fuγwithγ> 1 on weighted Riemannian manifolds withCD(0,m) orCD(−K,m) condition, and also on (0,m) or (−K,m) super Ricci flows.

Inspired by works mentioned above, we studyp-Rnyi type entropy power for the weighted doubly nonlinear diffusion equation (1.2) on the weighted Riemannian manifolds and prove its concavity under the curvature dimensional conditionCD(0,m)andCD(−K,m).

Let us define the weightedp-Rnyi entropyRp(u) andp-Rnyi entropy powerNp(u) on the weighted Riemannian manifolds,

wherebandσare constants

andEp(u) is given by

Whenp=2 andf=const., (1.6) reduces to (1.4).

Theorem 1.1Let (M,g,dµ) be a weighted closed Riemannian manifold.Ifuis a positive solution to (1.2) on (M,g,dµ), then we have

whereaijis the inverse ofandIp(u) is the weighted Fisher information in (3.3).

By modified the definition ofp-Rnyi entropy power, we can also obtain the concavity under the curvature dimensional conditionCD(−K,m).

Theorem 1.2Ifuis a positive solution to (1.2) on the weighted closed Riemannian manifolds with curvature dimensional conditionCD(−K,m) forK≥0, definep-K-Rnyi entropy powerNp,Ksuch that

where

we obtain

This paper is organized as follows.In section 2,we will prove some identities as lemmas,and they are important tools for proving theorems.In section 3, we will finish the proofs of the main results.

2 Some Useful Lemmas

According to [7], the pressure function is given by

thenvsatisfies the following equation

Let us recall the weighted entropy functional

and the weightedp-Rnyi entropy

wheredµ=e−fdVol.Hence, the weightedp-Rnyi entropy power is given by

In order to obtain our results, we need to prove following lemmas.Motivated by the works of [8], we apply analogous methods in this paper.

Lemma 2.1For any two functionshandg, let us define the linearized operator of the weightedp-Laplacian at pointv,

ProofThe proof is a direct result by the definition ofLp,fand integration by parts.

Lemma 2.2We have the modified weightedp-Bochner formula

where

ProofAccording to [9], we have

Using the definition ofm-Bakry-Emery Ricci curvature in (1.1), we find

which completes the proof of (2.8).In particular, whenm>norm<0, we have

Lemma 2.3Ifu,vsatisfies equations (1.2) and (2.2), we obtain that

ProofBy the definitions ofAandLp,f, then

By the definition ofvin (2.1), we haveu∇v=bv∇u.Hence

and

A direct calculation shows that

Combining (2.14) with (2.15), we can show (2.13).

Lemma 2.4Letube a positive solution to (1.2) andvsatisfies (2.2), then

and

where Γ2,Ais defined in (2.10).

ProofBy integrating by parts, we can obtain

where

Using identities (2.2), (2.8), (2.12) and (2.13), we find that

According to the properties of the linearized operatorLp,f(2.6) and (2.7), we have

where we use the fact

Hence, we get

which is (2.17).

3 Proofs of Theorems

Proof of Theorem 1.1Letσbe a constant,Np(u)=(Ep(u))σ,a direct computation implies

By using of identities (2.16), (2.10) and (2.17), we get that

We can choose a proper constantσsuch that

In fact, we can obtain a precise form of the second order derivative of the weightedp-Rnyi entropyNp(u).LetIp(u)be the weighted Fisher information with respect toRp(u),

Applying identities (3.1) and (3.2), we get

Combining this with (3.3), one has

Thus, we obtain an explicit expression aboutMoreover, ifandb>0,σ<0, then

Proof of Theorem 1.2Motivated by [10], formula (1.8) can be rewritten as

where we use the definition ofκin (1.10) and variaitonal formula in (2.16), that is

Defining a functionalNp,Ksuch that [11]

A direct computation yields

Hence,when(M,g,dµ)satisfies the curvature dimensional conditionCD(−K,m),i.e.,−KgwithK> 0,m≥n, by formula (3.5), we obtain the concavity ofNp,K, that isFurthermore, we obtain an explicit variational formula (1.11), which finishes the proof of Theorem 1.2.

AcknowledgementThe first author would like to thank Professor Xiang-Dong Li for his interest and illuminating discussion.