Research On The Rigidity And Concavity Of Entropy Power For The Nonlinear P-Laplace Type Equations

This paper studies the rigidity and concavity of the entropy power for the nonlinear p-Laplace type equations on compact Riemannian manifolds,including the rigidity for n-Laplace type equations with exponential nonlinearity and the concavity of the entropy power for the double nonlinear diffusion equations in different measure spaces.The most used tools are integration by parts and Bochner's formula.More precisely are as follows:Chapter 1,introduction is divided into three sections.The first section introduces the research background and main results of rigidity;the second section introduces the research background and main results of the concavity for the entropy power;the third section is outline of the paper.Including research motivation?purpose?methods?results etc.Chapter 2 and Chapter 3 are the main parts of this paper.Chapter 2 describes the rigidity for n-Laplace type equations with exponential nonlinearity,that is-1/p?pu+?=eu,when p=n,by using some lemmas,combining with p-Bochner formula,determining the range of parameters,proving a rigidity result for this equations on the compact Riemannian manifold.Chapter 3 proves the concavity of p-Renyi entropy power for the double nonlinear diffusion equations.This chapter is composed of two sections.In the first section,by building two trace-free tensors,selecting the appropriate pressure term f,combining Bakry-Emery method,we gives a new proof that the concavity of p-Renyi entropy power for the following double nonlinear diffusion equations on compact Riemannian manifolds(?)tu=div(u1/p|?f|p-2?f).In the second section,we consider the weighted double nonlinear diffusion equations(?)tu=?p,?u?.Proving the concavity of p-Renyi type entropy power for this equation on the weighted closed Riemannian manifolds with the curvature dimension CD(0,m)and CD(-K,m).