﻿ 带扰动的Einstein-Yang/Mills方程局部解的存在性 Existence of Local Solutions to Perturbed Einstein-Yang/Mills Equations

Vol.06 No.05(2017), Article ID:21686,7 pages
10.12677/AAM.2017.65081

Existence of Local Solutions to Perturbed Einstein-Yang/Mills Equations

Xu Wang

Department of Mathematics, Yunnan Minzu University, Kunming Yunnan

Received: Jul. 24th, 2017; accepted: Aug. 9th, 2017; published: Aug. 14th, 2017

ABSTRACT

In this paper, we will give a rigorous proof of existence of local solutions to perturbed Einstein- Yang/Mills equations with gauge group SU(2), here we require the existence of zero point for A, and we consider the area in Holder spaces.

Keywords:Einstein-Yang/Mills Equations, Perturbed Term, Holder Spaces, Existence of Solutions

1. 引言

Yang/Mills理论，是现代规范场理论的基础。由杨振宁和米尔斯在1954年首先提出来，通过后来许多学者于1960年到1970年代引入对称性自发破缺与渐进自由的观念，发展成今天的标准模型。杨–米尔斯理论作为克雷数学研究所提出的新前年七大问题之一，在当今物理界和数学界都是热门的问题，本文讨论静态球对称Einstein-Yang/Mills方程在Holder空间中局部解的存在性，这里Einstein度量 [1] ：

(1.1)

SU(2)Yang/Mills曲率-2形式 [2] ：

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

2. 准备知识

(2.1)

。则由(1.3)，(1.6)得

(2.2)

(2.3)

(2.4)

(2.5)

，由泰勒展开并引入算子B有，解得，得

(2.6)

3. 局部解的存在唯一性

，由于，则：

，于是为压缩映射得证 [5] 。

(3.1)

(3.2)

(3.3)

(3.4)

Existence of Local Solutions to Perturbed Einstein-Yang/Mills Equations[J]. 应用数学进展, 2017, 06(05): 685-691. http://dx.doi.org/10.12677/AAM.2017.65081

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