A Research Of The Proof Of The Minkowski Inequality By P-capacitary Potentials

In this paper,we discuss the proof of the Minkowski inequality by using pcapacitary potentials.In the first part,we talk about the proof of the Minkowski inequality in Euclidean space.We calculate the conformal transformation of the p-Laplace equation,we also prove the Kato-type identity and the monotonicity of some function,then the Minkowski inequality is proved by using the monotonicity.In the second part,we talk about the proof of the Minkowski inequality in Riemannian manifolds,with rionnegative Ricci curvature and Euclidean Volume Growth.We list the conditions that Riemannian manifolds should be satisfied when using p-capacitary potentials and also introduce the Li-Yau-type estimation.Then we talk about the proof of two Monotonicity-Rigidity Theorems and the Minkowski inequality.This paper is divided into four chapters.In chapter 1,we introduce some backgrounds,the main results and the structure of this paper.In chapter 2,we talk about the proof of the Minkowski inequality in Euclidean space.In section 2.1,we introduce p-capacitary potentials and conformal transformation.In section 2.2,we discuss how to prove the monotonicity of some function by two monotonicity formulas.In section 2.3,we use the monotonicity to prove the Minkowski inequality.In chapter 3,we introduce the proof of the Minkowski inequality in Riemannian manifolds.In section 3.1,we collect some preliminaries on p-capacitary potentials in Riemannian manifolds.In section 3.2,we talk about the proof of two Monotonicity-Rigidity Theorems.In section 3.3,we use the Monotonicity-Rigidity Theorems to discuss the proof of the Minkowski inequality.In chapter 4,we summarize the contents of this paper and give some future prospects of the Minkowski inequality.