## 调和函数的性质与应用+文献综述

Title    Properties and Application of Harmonic Functions
Abstract
In this paper,we firstly discussed the maximum principle of harmonic functions and the Perron family deduced by subharmonic functions.Then we transformed the judgment of the Dirichlet problem to the existence of barrier function by using Perron family,thus we got the sufficient condition of the solution of the Dirichlet problem.Then we utilized the Green's function method to obtain the general solution of the Dirichlet problem,and we also discussed some other boundary value problems.In the end,we discussed the Harmonic Measure and discussed the Phragmén-Lindelöf theorem.
The paper mainly consists of the following parts:
Chapter one introduced some background knowledge of the harmonic function;
Chapter two discusses some properties of harmonic functions,and deduced Perron family by the nature of subharmonic functions,which we could use to solve the Dirichlet problem.

Chapter three mainly used the Green's function method to discuss the Dirichlet problem,and also discussed some other boundary value problems.
Chapter four mainly discussed the Harmonic Measure and discussed the Phragmén-Lindelöf theorem.
Keywords  harmonic function; boundary value problem; Green's function; Maximum principle

1  引言 1
2  调和函数的性质 1
2.1  调和函数 2
2.2  调和函数极值原理 3
2.3  调和函数序列的收敛性质 4
2.4  次调和函数的性质 5
3  调和函数的边值问题 8
3.1  Dirichlet问题有解的充分条件  8
3.2  Dirichlet问题的Green函数法 10
3.3  调和函数的Riemann边值问题 13
4  调和测度和Phragmén-Lindelöf定理   17
4.1  调和测度    17
4.2  Phragmén-Lindelöf定理   22

1  引言

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