几类偏微分方程的显式径向对称解
几类偏微分方程的显式径向对称解(外文翻译,论文5000字)
摘要：本文研究了在全空间下几类偏微分方程的径向对称解。将主要围绕几类重要的非线性椭圆方程开展研究，例如无穷拉普拉斯方程、p-Laplace方程、极小曲面方程、Monge-Ampere方程，k-容许，k-Hessian方程等。给出了关于这些方程的一些定义及其推理，并运用该定义和推理解决了一些实际问题。当方程结构比较复杂时，相应的常微分方程的求解也具有一定的难度。
本文的第一章解释了径向对称解应用的范围及在学术上的意义。第二章列举了本文要用到的调和函数的定义即径向对称解，并计算了一类特殊p-Laplace方程的径向对称解。第三章则列举了Monge-Ampere方程，k-容许，和k-Hessian方程的一些定义，并求出了特殊非线性椭圆型方程Monge-Ampere方程，k-Hessian方程的径向对称解。但这些只是一些特殊例子，径向对称解在实际的学术研究中还有更加普遍且便捷的应用。
关键词：径向对称解，调和函数，Monge-Ampere方程，k-容许，k-Hessian方程，椭圆型方程
Explicit radial symmetry of several partial differential equations
Abstract:In this paper, we study the radial symmetric solutions of several partial differential equations in full space. This paper mainly focuses on several important nonlinear elliptic equations. For example, the infinite Laplace equation, p-Laplace equation, minimum surface equation, Monge-Ampere equation,k-admissible and k-Hessian equation. Some definitions and meanings of these equations are given, and some practical problems are solved by using these definitions and meanings. The first chapter of this article explains the extent of use radial symmetry and its implications in the academic world. The second chapter enumerates the definition of harmonic functions used in this paper, namely radial symmetry, and calculates the radial symmetry solution of a special type of p-Laplace equation. The third chapter lists the definitions of the Monge-Ampere equation, the k-admissibility, the k-Hessian equation, and the special non-linear elliptic equation. We solve some radial symmetric solutions of the Monge-Ampere equation, and the k-Hessian equation. But these are just a few examples, and radial symmetry is a more common and convenient application in actual academic research.
Keywords:radial symmetry solution,, harmonic function, Monge-Ampere equation, k-admissible k-Hessian equation, elliptic equations.
目 录
摘要.........................................................3
第一章 概述..................................................5
第二章 调和函数及p-Laplace方程
1.1径向对称解...............................................7
1.2调和函数..................................................7
1.3 半线性椭圆方程...........................................9
1.4 本章小结................................................12
第三章 Monge-Ampere方程和k-Hessian方程......................13
2.1 Monge-Ampere方程.....................................13
2.2 k-Hessian方程..........................................15 [来源：http://Doc163.com]
2.3 k-容许.................................................16
3.4 本章小结...............................................20
参考文献.....................................................22
致谢.........................................................23