# 狄利克雷问题

## 一般解

${\displaystyle u(x)=\int _{\partial D}\nu (s){\frac {\partial G(x,s)}{\partial n}}ds\,}$

${\displaystyle {\frac {\partial G(x,s)}{\partial n}}={\widehat {n}}\cdot \nabla _{s}G(x,s)=\sum _{i}n_{i}{\frac {\partial G(x,s)}{\partial s_{i}}}\,}$

${\displaystyle f(x)=-{\frac {\nu (x)}{2}}+\int _{\partial D}\nu (s){\frac {\partial G(x,s)}{\partial n}}ds.\,}$

${\displaystyle G(x,s)=0}$ ${\displaystyle s\in \partial D}$ ${\displaystyle x\in D}$

### 存在性

${\displaystyle \partial D\in C^{(1,\alpha )},\,}$ ${\displaystyle 0<\alpha ,\,}$

## 例子：二维单位圆盘

 ${\displaystyle u(z)={\begin{cases}{\frac {1}{2\pi }}\int _{0}^{2\pi }f(e^{i\psi }){\frac {1-\vert z\vert ^{2}}{\vert z-e^{i\psi }\vert ^{2}}}d\psi ,\\f(z),\end{cases}}}$ 如果${\displaystyle z\in D,\,}$ 如果${\displaystyle z\in \partial D.\,}$

${\displaystyle u}$ 在闭单位圆盘${\displaystyle {\bar {D}}}$ 上连续在${\displaystyle D}$ 内调和。

${\displaystyle G(z,x)=-{\frac {1}{2\pi }}\log \vert z-x\vert +\gamma (z,x)}$

${\displaystyle \Delta _{x}\gamma (z,x)=0,\,}$